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University of Southern California Dissertations and Theses
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Essays on service systems with matching
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Essays on service systems with matching
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ESSAYS ON SERVICE SYSTEMS WITH MATCHING by Erhun ¨ Ozkan A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) August 2018 Copyright 2018 Erhun ¨ Ozkan Dedicated to my family. ii Acknowledgments I could not write this dissertation without the help that I got from various faculty in USC. First and foremost, I would like to thank to my primary advisor Amy R. Ward for her amazing guidance during my entire Ph.D. studies. I learned a lot from her great knowledge, wisdom, and intelligence and I really appreciate her patience for my progress during my Ph.D. studies. Secondly, I would like to thank to my co-advidor Ramandeep Randhawa for his valuable advice and sharing his great knowledge, wisdom, and intelligence with me. I would like to thank to my dissertation committee members Kimon Drakopoulos, who gave me valuable advice and feedback, and Remigijus Mikulevicius, who taught me difficult mathematical concepts in an understandable way. I would like to thank to Paat Rusmevichientong and Song-Hee Kim for their help and guidance during my Ph.D. studies. I would like to thank to Vishal Gupta, Milan Miric, Greys Sosic, Sampath Rajagopalan, and Peng Shi for their valuable advice and feedback. I would like to thank to Dongyuan Zhan and Jeunghyun Kim for their help and guidance as fellow and senior Ph.D. students. Lastly, I would like to express my deepest gratitude to the Data Sciences and Operations Department of the Marshall School of Business for giving me the opportunity to make a Ph.D. studies in an amazing research environment. iii Contents Acknowledgments iii List of Tables viii List of Figures ix Abstract xi 1 On the Control of Fork-Join Networks 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Stochastic Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Scheduling Control and Network Dynamics . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 The Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Asymptotic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 A Sequence of Fork-Join Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Fluid and Diffusion Scaled Processes . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 The Approximating Diffusion Control Problem . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Proposed Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Asymptotic Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7 Weak Convergence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7.1 Case I: a 3 < A (Slow Departure Pacing Policy) . . . . . . . . . . . . . . . 26 1.7.2 Case II: 3 A (Static Priority Policy) . . . . . . . . . . . . . . . . . . . . . . . 28 1.8 Proof of Proposition 1.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.8.1 Proof of Convergence of (1.55) (Up Intervals) . . . . . . . . . . . . . . . . . . . . . 32 1.8.2 Proof of Convergence of (1.56) (Down Intervals) . . . . . . . . . . . . . . . . . . . 36 1.9 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.9.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.9.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.10 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.10.1 Task Dependent Holding Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.10.2 Networks with Arbitrary Number of Forks . . . . . . . . . . . . . . . . . . . . . . 54 1.10.3 Networks with More Than Two Job Types . . . . . . . . . . . . . . . . . . . . . . . 57 1.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 iv 2 Dynamic Matching for Real-time Ridesharing 68 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.3 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 A Ridesharing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2.1 Admissible Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.3 An Asymptotically Optimal CLP-Based Matching Policy . . . . . . . . . . . . . . . . . . . 78 2.3.1 A Large Matching Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.2 An Asymptotic CLP Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.3.3 A CLP-Based Randomized Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4 An Asymptotically Optimal LP-Based Randomized Policy . . . . . . . . . . . . . . . . . . 83 2.4.1 An LP-Based Randomized Matching Policy . . . . . . . . . . . . . . . . . . . . . . 83 2.4.2 Jointly Optimizing Pricing and Matching . . . . . . . . . . . . . . . . . . . . . . . 85 2.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.5.1 Additional Proposed Matching Policies . . . . . . . . . . . . . . . . . . . . . . . . 88 2.5.2 Simulation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3 Joint Pricing and Matching in Ridesharing Systems 96 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3 Futility of Optimization in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3.1 Constant and Origin Based Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3.2 Origin and Destination Based Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.4 Constant Pricing and No Cross Matching Equilibriums . . . . . . . . . . . . . . . . . . . . 109 3.4.1 The best equilibriums among the CP equilibriums . . . . . . . . . . . . . . . . . . . 109 3.4.2 Homogeneous Customer Valuation Distributions . . . . . . . . . . . . . . . . . . . 111 3.5 Cross Matching Equilibriums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5.1 Patient Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5.2 Impatient Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.6 Simulation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.1 Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A Technical Appendix to Chapter 1 126 A.1 Proof of Proposition 1.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.1.1 Proof of Lemma A.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.2 Proofs of the Results with Standard Methodology . . . . . . . . . . . . . . . . . . . . . . . 139 A.2.1 Proof of Proposition 1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.2.2 Proof of Proposition 1.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.2.3 Proof of Theorem 1.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.2.4 Proof of Proposition 1.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A.2.5 Proof of Lemma 1.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.2.6 Proof of Lemma 1.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.2.7 Proof of Lemma 1.8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 v A.2.8 Proof of Lemma 1.10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.2.9 Proof of Lemma 1.10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.2.10 Proof of Lemma 1.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A.3 Detailed Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 B Technical Appendix to Chapter 2 162 B.1 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 B.2 Proof of Proposition B.1.1 (Properties of Fluid Limits) . . . . . . . . . . . . . . . . . . . . 166 B.2.1 Proof of (B.21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 B.3 A Regulator Mapping Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 B.4 Proofs of Theorems 2.3.2, 2.4.2, and 2.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 B.4.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 B.4.2 Proof of Theorem 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 B.4.3 Proof of Theorem 2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 B.4.4 Proof of Theorem 2.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.5 Lemma Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.5.1 Proof of Lemma 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.5.2 Proofs of Lemmas 2.3.2 and 2.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.5.3 Proof of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.5.4 Proof of Lemma 2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.5.5 Proof of Lemma 2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.6 Relative Compactness in spaceD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 C Technical Appendix to Chapter 3 201 C.1 Proofs of Lemmas 3.2.1, 3.2.2, and 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 C.1.1 Proof of Lemma 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 C.1.2 Proof of Lemma 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 C.1.3 Proof of Lemma 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 C.2 Proofs of Theorem 3.3.2 and Proposition 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 204 C.2.1 An Upper Bound Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 204 C.2.2 Proof of Theorem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 C.2.3 Proof of Proposition 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 C.3 Proofs of Theorems 3.3.1 and 3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 C.3.1 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 C.3.2 Proof of Theorem 3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 C.4 Relationship between OP and ODP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 C.4.1 Construction of the Hypothetical Ridesharing Network . . . . . . . . . . . . . . . . 212 C.4.2 Proof of Theorem C.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 C.5 Proof of Proposition 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 C.5.1 An optimal equilibrium when prices are given and customers are patient . . . . . . . 219 C.5.2 An Optimization Problem Characterizing Optimal Equilibriums . . . . . . . . . . . 224 C.5.3 Extension of the Results with Origin and Destination Based Pricing . . . . . . . . . 226 C.6 Proofs of Theorems 3.4.1 and 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 C.6.1 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 C.6.2 Proof of Theorem 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 C.7 Proof of Theorem 3.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 C.8 Solution of Example 3.4.1 and Proof of Proposition 3.4.2 . . . . . . . . . . . . . . . . . . . 232 C.8.1 Solution of Example 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 vi C.8.2 Proof of Proposition 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 C.9 Information about Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 vii List of Tables 1.1 Parameter choices at each instance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.2 Average and maximum deviations of the cost of the policies from the lowest realized average cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.3 An optimal solution set for the optimization problem (1.113). . . . . . . . . . . . . . . . . . 53 2.1 The percentage of all customers matched in the simulation experiment corresponding to Figure 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.1 The 9 different settings used in the simulation experiments. . . . . . . . . . . . . . . . . . . 121 A.1 Detailed results of the simulation experiments: Average queue lengths with their 95% con- fidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C.1 Illustration of the constructedy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 viii List of Figures 1.1 (Color online) An example of a fork-join processing network with arbitrary number of job types, arbitrary number of forks associated with each job type, and a single shared server. . . 1 1.2 (Color online) A fork-join processing network with two job types and a single shared server. 2 1.3 Illustration of the down 1, down 2, and up intervals when a 3 < A . There are three possible cases associated with the down intervals: only a down 1 interval exists, only a down 2 interval exists, or both exist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 (Color online) Percentage deviations of the costs of the policies from the lowest realized average cost (L(i) fori2f1; 2;:::; 18g) in the first 18 instances whenh a = 2 andh b = 1. . 50 1.5 (Color online) Percentage deviations of the costs of the static, randomized, and the randomized- 2=3 policies from the lowest realized average cost (L(i) fori2f1; 2;:::; 18g) in the first 18 instances whenh a = 2 andh b = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.6 A fork-join processing network with two job types and arbitrary number of forks. . . . . . . 55 1.7 (Color online) A fork-join processing network with arbitrary job types and a single shared server. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.8 A fork-join processing network with three job types and two shared servers. . . . . . . . . . 61 1.9 A fork-join processing network with seven job types and six shared servers. . . . . . . . . . 66 2.1 An intuitive explanation of why the CD policy may not assign the right driver to a customer. 69 2.2 A queueing-inspired visualization of our model formulation. . . . . . . . . . . . . . . . . . 75 2.3 The parameters of the first simulation experiment are: Q i (0) = 0, i = 0:1, F ii = 1, w ij = 1 for alli;j2f1; 2; 3g, andn2f1; 10; 100g. . . . . . . . . . . . . . . . . . . . . . 90 2.4 The percentage of all customers matched in the simulation experiment corresponding to Figure 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.5 The parameters of the second simulation experiment are:Q i (0) = 0, F ii = 1,w ij = 1 for all i;j2f1; 2; 3g,n2f0:1; 1; 10; 100g, i = is constant ini and2f10 2 ; 10 3 ; 10 4 ; 10 5 g, T = 1800, 1 (t) = 0 for allt2 [0;T=2]; and 1 (t) = 2n for allt2 [T=2;T ]. . . . . . . . . 92 3.1 An example illustrating the effect of matching decisions on the number of matchings. . . . . 97 3.2 The pricing and the matching decisions of the firm when only CP and OP are considered. . . 107 3.3 (Color online) The pricing and the matching decisions of the firm when ODP is included. . . 109 3.4 (Color online) G(p) andp G(p) when = 0:25 in Example 3.4.1. . . . . . . . . . . . . . . . 112 3.5 (Color online) Results of Example 3.5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.6 (Color online) Results of Example 3.5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.7 (Color online) Results of the simulation experiments associated with the subset2f0:1; 0:2;:::; 1g of the numerical experiments in Figure 3.5b. . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.8 (Color online) Results of the simulation experiments associated with the subseta2f1; 3; 5;:::; 19g of the numerical experiments in Figure 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . 123 ix C.1 The original network (on the left) and the hypothetical network (on the right) in Example C.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 x Abstract We study operational problems in service systems in which matching plays an important role. Specifically, we focus on operational problems in fork-join networks and ridesharing platforms. In the first chapter of the dissertation, we study scheduling decisions in fork-join networks. In the second and third chapters, we study matching and pricing decisions in ridesharing platforms. A fork-join network is a special type of queueing network in which processing of jobs occurs both sequen- tially and in parallel. Fork-join networks are prevalent in many application domains, such as computer systems, healthcare, manufacturing, and project management. The parallel processing of jobs gives rise to synchronization constraints that can be a main reason for job delay. In comparison with feedforward queueing networks that have only sequential processing of jobs, the approximation and control of fork-join networks is less understood. Therefore, in the first chapter of the dissertation, we study a specific fork-join processing network with multiple job types in which there is first a fork operation, then there are activities that can be performed in parallel, and then a join operation. The difficulty is that some of the activities that can be performed in parallel require a shared resource. We solve the scheduling problem for that shared server (that is, which type of job to prioritize at any given time) when that server is in heavy traffic and propose a scheduling policy that is asymptotically optimal in diffusion scale. Then, we extend our proposed scheduling policy to more general fork-join processing networks. Ridesharing platforms are online mobile platforms which match paying customers who need a ride with drivers who provide transportation. Some examples of these platforms are Uber and Lyft in the USA, Didi Chuxing in China, Ola in India, and Grab in Southeast Asia. When a customer requests a ride, the ridesharing firm should charge a price and offer a driver to the customer. The matching decisions affect the overall number of customers matched because they impact whether or not future available drivers will be close to the locations of arriving customers. The pricing decisions are important because they have opposite effect on the customer demand and driver supply. As the price in an area increases, customer demand decreases but the driver supply (roughly speaking) increases in that area. xi Since customer demand and driver supply change dramatically over time, an ideal ridesharing model should have time dependent parameters and the customer and driver arrival rates should depend on the pricing and matching decisions. However, such a model is very difficult to analyze. Therefore, in the second chapter of the dissertation, we present a model in which the prices are given, customer and driver arrival rates are time dependent but exogenous, and we optimize the matching decisions. We propose to base the matching decisions on the solution to a continuous linear program (CLP) that accounts for (i) the differing arrival rates of customers and drivers in different areas of the city, (ii) how long customers are willing to wait for driver pick-up, and (iii) the time-varying nature of all the aforementioned parameters. We prove asymptotic optimality of a forward-looking CLP-based policy in a large market regime. We leverage that result to also prove the asymptotic optimality of a myopic LP-based matching policy when drivers are fully utilized. In the third chapter of the dissertation, we present a ridesharing model in which the customer and driver behaviors are endogenous but the parameters are time homogeneous. We jointly optimize the pricing and matching decisions. We prove that neither optimizing the pricing decisions while keeping the matching decisions simple nor optimizing the matching decisions while keeping the pricing decisions simple provides better performance than the simple pricing and matching decisions do. (Simple pricing decisions mean that the firm charges the same price in all areas of the city and simple matching decisions mean that a customer can be offered a driver only from the same area.) In other words, the firm cannot achieve an optimal perfor- mance by focusing on only the pricing (matching) decisions and ignoring the matching (pricing) decisions. Therefore, the pricing and the matching decisions should be optimized simultaneously. Moreover, we derive conditions under which simple pricing and matching decisions are optimal and we derive conditions under which sophisticated pricing and matching decisions are necessary for optimal performance. xii Chapter 1 On the Control of Fork-Join Networks 1.1 Introduction Networks in which processing of jobs occurs both sequentially and in parallel are prevalent in many appli- cation domains, such as computer systems (Xia et al. (2007)), healthcare (Armony et al. (2015)), manu- facturing (Dallery and Gershwin (1992)), project management (Adler et al. (1995)), and the justice system (Larson et al. (1993)). The parallel processing of jobs gives rise to synchronization constraints that can be a main reason for job delay. Although delays in fork-join networks can be approximated under the common first-come-first-served (FCFS) scheduling discipline (Nguyen (1993, 1994)), there is no reason to believe FCFS scheduling minimizes delay. !"#$%! &'() !"#$%" &'() !"#$%# &'() Figure 1.1: (Color online) An example of a fork-join processing network with arbitrary number of job types, arbitrary number of forks associated with each job type, and a single shared server. Our objective in this chapter is to devise control policies that minimize delay (or, more generally, holding costs) in fork-join networks with multiple customer classes that share processing resources. For a con- crete motivating example, consider the patient-flow process associated with the emergency department at Saintemarie University Hospital (cf. Hublet et al. (2011)). An arriving patient is first triaged to determine condition severity, and then (after some potential waiting) moves to the patient management phase before 1 being discharged. The patient management phase begins with the vital signs being taken and a first eval- uation. Then, depending on the condition, there may be laboratory tests and radiology exams. Simple laboratory tests on the patient’s blood and urine can be performed in parallel with the patient receiving a radiology exam, such as a CT scan. The discharge decision - whether the patient can return home or should be admitted to the hospital - cannot be made until all test results are received. Roughly speaking, we can imagine a process flow diagram such as that in Figure 1.1, where the patient type corresponds to the con- dition severity determined at triage, the isolated operations correspond to the laboratory tests (which are necessarily associated with each individual patient), and the shared operation corresponds to the use of the CT scanner. The CT scanner is an expensive machine, and, as can be seen from the case teaching note (cf. Hublet et al. (2011)), has a large impact on patient wait time. This motivates us to study the problem of how to schedule a shared resource that is used in parallel with other resources. ! " # $ % & ' " ! $ & ' # % ( ) !* +,-./!0123/4" # 5 +,-./$ 0123/4" % 5 & ' & ( Figure 1.2: (Color online) A fork-join processing network with two job types and a single shared server. The simpler fork-join network shown in Figure 1.2 serves to illustrate why fork-join network control is difficult. In that network, there are two arriving job types (a andb), seven servers (numbered 1 to 7), and ten buffers (numbered 1 to 10). We assumeh a is the cost per unit time to hold a typea job, andh b to hold a typeb job. Typea (b) jobs are first processed at server 1 (2), then “fork” into two jobs, one that must be processed at server 3 (5) and the other at server 4, and finally “join” together to complete their processing at server 6 (7). There is synchronization because the processing at server 6 (7) cannot begin until there is at least one job in both buffers 7 and 8 (9 and 10). Server 4 processes both job types, but can only serve one job 2 at a time. The control decision is to decide which job type server 4 should prioritize. This decision could be essentially ignored by serving the jobs in the order of their arrival regardless of type (that is, implementing FCFS policy). Another option is to always prioritizing the more expensive job type, in accordance with the well-knownc-rule. Then, ifh a A h b B where A ( B ) is the rate at which server 4 processes type a (b) jobs, server 4 always prefers to work on a type a job over a type b job. However, when there are multiple jobs waiting at buffers 8 and 10, and no jobs waiting at buffer 9, it may be preferable to have server 4 work on a typeb job instead of a typea job (and especially if also no jobs are waiting at buffer 7). This is because server 4 can prevent the “join” server 7 from idling without being the cause of server 6’s forced idling. (Server 3 is the cause.) The fork-join network control problem is too difficult to solve exactly, and so we search for an asymptotic solution. We do this under the assumption that server 4 is in heavy traffic. Otherwise, the scheduling control in server 4 has negligible impact on the delay of typea and typeb jobs. We further assume the servers 6 and 7 are in light traffic, which focuses attention on when the required simultaneous processing of jobs at those servers forces idling. The servers 1, 2, 3, and 5 can all be in either light or heavy traffic. In the aforementioned heavy traffic regime, we formulate and solve an approximating diffusion control problem (DCP). The DCP solution matches the number of jobs in buffer 4 to that in buffer 3, except when the total number of jobs waiting for processing by server 4 is too small for that to be possible. The implication is that when server 3 is in light traffic, so that buffer 3 is empty, buffer 4 is empty and all jobs waiting to be processed by server 4 are typeb jobs. Otherwise, when server 3 is in heavy traffic, the control at server 4 must carefully pace its processing of typea jobs to prevent “getting ahead” of server 3. Our proposed policy is motivated by the observation from the DCP solution that there is no reason to have fewer typea jobs in buffer 4 than in buffer 3. If server 3 can process jobs at least as quickly as server 4 can process typea jobs, then the control under which server 4 gives static priority to typea jobs performs well. Otherwise, we introduce a slow departure pacing (SDP) control in which server 4 slows its processing of typea jobs to match the departure process of typea jobs from buffer 4 to the one from buffer 3. SDP is a robust idea that is applicable to the more general network topology shown in Figure 1.1. To see this, we formulate and solve approximating DCPs for the fork-join network in Figure 1.2 but with task dependent holding costs (cf. Section 1.10.1), a fork-join network with an arbitrary number of “forks” (cf. Figure 1.6 in Section 1.10.2), and fork-join networks with more than two job types (cf. Figure 1.7 in Section 1.10.3, Figure 1.1 and Remark 1.10.5 in Section 1.10.3, and Figure 1.8 in Section 1.10.3). In many cases, the DCP solutions suggest that, depending on the processing capacities of the servers and the network state, the servers in the network that process more than one job type should either give static priority to the 3 more expensive job types or slow the departure process of these more expensive jobs in order to sometimes prioritize the less expensive jobs. This prevents unnecessary forced idling of the downstream “join” servers that process the less expensive job types, without sacrificing the speed at which the more expensive job types depart the network. We prove that our proposed policy is asymptotically optimal in heavy traffic for the fork-join network in Figure 1.2. To do this, we prove that under our proposed policy there is weak convergence to the DCP solu- tion (cf. Theorem 1.5.1), and that the DCP solution provides a stochastic lower bound on the holding cost under any policy at every time instant (cf. Theorem 1.6.1). This is a strong form of asymptotic optimality, which also implies asymptotic optimality with respect to both the expected infinite horizon discounted cost and the expected total cost over a finite horizon (cf. Theorem 1.6.2). That rigorous analysis suggests that our translations of the DCP solutions relevant to the more general topologies shown in Figures 1.1 and 1.8 should also perform well. The weak convergence result when the network operates under the SDP control is a major technical challenge. This is because the SDP control is a dynamic control that depends on the network state. In order to obtain the weak convergence, we must carefully construct random intervals on which we know the job type server 4 is prioritizing. Although this idea is similar in spirit to the random interval construction in Bell and Williams (2001) and Ghamami and Ward (2013), the proof to show convergence on the intervals is much different, due to the desired matching of the typea job departure processes from the servers 3 and 4. More specifically, the interval construction is determined by tracking and comparing the job counts in buffers 3 and 4, because the SDP policy prioritizes typea jobs when the number of jobs in buffer 4 exceeds that in buffer 3, and prioritizes typeb jobs otherwise. Then, the keys to obtaining the desired weak convergence result are as follows. On the intervals on which type a jobs are prioritized, we must bound the difference between two renewal processes having different rates. On the intervals on which typeb jobs are prioritized, we require a rate of convergence result for a pair of single server queues in tandem in which the service rate of the downstream server is strictly greater than the external arrival rate. Finally, when we piece those intervals together, we see the DCP solution arise. The remainder of this chapter is organized as follows. We conclude this section with a literature review and a summary of our mathematical notation. Section 1.2 specifies our model and Section 1.3 provides our asymptotic framework. We construct and solve an approximating DCP in Section 1.4. We introduce the SDP 4 control in Section 1.5, and specify when the proposed policy is SDP and when it is static priority. Section 1.6 proves the asymptotic optimality of the proposed policy, and Sections 1.7 and 1.8 prove weak convergence under the proposed policy. We separate out our rate of convergence result for a pair of single server queues in tandem, as that is a result of interest in its own right. Section 1.9 provides extensive simulation results. In Section 1.10, we construct and solve approximating DCPs for a broader class of fork-join networks. Section 1.11 makes concluding remarks and proposes a future research direction. We also provide the proofs of the results that use more standard methodology as well as more detailed simulation results in Appendix A. 1.1.1 Literature Review The inspiration for this work came from the papers Nguyen (1993, 1994). Nguyen (1993) establishes that a feedforward FCFS fork-join network with one job type and single-server stations in heavy traffic can be approximated by a reflected Brownian motion (RBM), and Nguyen (1994) extends this result to include multiple job types. The dimension of the RBM equals the number of stations, and its state space is a poly- hedral region. In contrast to the RBM approximation for feedforward queueing networks (Harrison (1996), Harrison and Van Mieghem (1997), Harrison (1998, 2006)), the effect of the synchronization constraints in fork-join networks is to increase the number of faces defining the state space. Also in contrast to feedfor- ward queueing networks, that number is increased further when moving from the single job type to multiple job type scenario. Although delay estimates for fork-join networks follow from the results of Nguyen (1993, 1994), they leave open the question of whether and how much delays can be shortened by scheduling jobs in a non-FCFS order. To solve the scheduling problem, we follow the “standard Brownian machinery” proposed in Harrison (1996). This is typically done by first introducing a heavy-traffic asymptotic regime in which resources are almost fully utilized and the buffer content processes can be approximated by a function of a Brownian motion, and second formulating an approximating Brownian control problem. Often, the dimension of the approximating Brownian control problem can be reduced by showing its equivalence with a so-called workload formulation (Harrison and Van Mieghem (1997), Harrison (2006)). The intriguing difference when the underlying network is a fork-join network is that the join servers must be in light traffic to arrive at an equivalent workload formulation. The issue is that otherwise the approximating problem is non- linear. This light traffic assumption is asymptotically equivalent to the assumption that processing times are instantaneous. Our simulation results suggest that our proposed control that is asymptotically optimal when the join servers are in light traffic also performs very well when the join servers are in heavy traffic. 5 The papers Pedarsani et al. (2014a,b) are some of the few studies we find that consider the control of fork-join processing networks. In both papers, there are multiple job classes, but in Pedarsani et al. (2014a) the servers can cooperate on the processing of jobs and in Pedarsani et al. (2014b) they cannot. Their focus is on finding robust policies in the discrete-time setting that do not depend on system parameters and are rate stable. They do not determine whether or not their proposed policies minimize delay, which is our focus. The paper Gurvich and Ward (2014) seeks to minimize delay, but in the context of a matching queue network that has only “joins” and no “forks”. An essential question to answer when thinking about controls for multiclass fork-join networks, as can be seen from the papers Lu and Pang (2016a,b, 2017), Atar et al. (2012), is whether or not the jobs being joined are exchangeable; that is, whether or not a task forked from one job can be later joined with a task forked from a different job. Exchangeability is generally true in the manufacturing setting, because there is no difference between parts with the same specifications, and generally false in healthcare settings, because all paperwork and test results associated with one patient cannot be joined with another patient. The papers Lu and Pang (2016a,b, 2017) develop heavy traffic approximations for a non-exchangeable fork-join network with one arrival stream that forks into arrival streams to multiple many-server queues, and then must be joined together to produce one departure stream. The heavy-traffic approximation for the non-exchangeable network is different than for the exchangeable network, and the non-exchangeability assumption increases the problem difficulty. Their focus, different than ours, is on the effect of correlation in the service times, and there is no control. The paper Atar et al. (2012) looks at a fork-join network in which there is no control decision if jobs are exchangeable, and shows that the performance of the exchangeable network lower bounds the performance of the non-exchangeable network. Then, they propose a control for the non- exchangeable network that achieves performance very close to the exchangeable network. In comparison to the aforementioned papers, the exchangeability assumption is irrelevant in our case. This is because we assume head-of-line processing for each job type, so that the exact same typea (b) jobs forked from server 1 (2) are the ones joined at server 6 (7). 1.1.2 Notation and Terminology The set of nonnegative integers is denoted byN and the set of strictly positive integers are denoted byN + . Thek dimensional (k2 N + ) Euclidean space is denoted byR k ,R + denotes [0; +1), and 0 k is the zero vector inR k . For x;y 2 R, x_y := maxfx;yg, x^y := minfx;yg, and (x) + := x_ 0. For any x2R,bxc (dxe) denotes the greatest (smallest) integer which is smaller (greater) than or equal tox. The superscript 0 denotes the transpose of a matrix or vector. 6 For eachk2N + ,D k denotes the the space of all! :R + !R k which are right continuous with left limits. Let 02D be such that 0(t) = 0 for allt2R + . For!2D andT2R + , we letk!k T := sup 0tT j!(t)j. We considerD k endowed with the usual SkorokhodJ 1 topology (cf. Chapter 3 of Billingsley (1999)). Let B(D k ) denote the Borel-algebra onD k associated with SkorokhodJ 1 topology. By Theorem 11.5.2 of Whitt (2002),B(D k ) coincides with the Kolmogorov-algebra generated by the coordinate projections. For stochastic processesW r ,r2 R + , andW whose sample paths are inD k for somek2 N + , “W r ) W ” asr!1 means that the probability measures induced byW r on (D k ;B(D k )) weakly converge to the one induced byW on (D k ;B(D k )) asr!1. Forx;y2D,x_y,x^y, and (x) + are processes inD such that (x_y)(t) :=x(t)_y(t), (x^y)(t) :=x(t)^y(t), and (x) + (t) := (x(t)) + for allt2R + . Forx2D, we define the mappings ; :D!D such that for allt2R + , (x)(t) := sup 0st (x(s)) + ; (x)(t) :=x(t) + (x)(t); where is the one-sided, one-dimensional reflection map, which is introduced by Skorokhod (1961). LetZ =f1; 2;:::;mg andX i be a process inD for eachi2Z. Then (X i ;i2Z) denotes the process (X 1 ;X 2 ;:::;X m ) inD m . We denotee as the deterministic identity process inD such thate(t) =t for all t2R + and “” denotes the composition map. We abbreviate the phrase “uniformly on compact intervals” by “u.o.c.”, “almost surely” by “a.s.”. We let a:s: ! denote almost sure convergence and d = denote “equal in distribution”. We repeatedly use the fact that convergence in theJ 1 metric is equivalent to u.o.c. convergence when the limit process is continuos (cf. page 124 in Billingsley (1999)). LetfX n ;n2Ng be a sequence in D andX2D. ThenX n !X u.o.c. asn!1, ifkX n Xk T ! 0 asn!1 for allT2R + .I denotes the indicator function andBM q (; ) denotes a Brownian motion with drift vector and covariance matrix which starts at pointq. The big-O notation is denoted byO(); i.e., ifx : R + ! R andy : R + ! R are two functions, thenx(t) = O(y(t)) ast!1 if and only if there exist constantsC andt 0 such that jx(t)j Cjy(t)j for allt t 0 . Lastly,o p () is the little-o in probability notation; i.e., iffX n ;n2Ng and fY n ;n2 Ng are sequences of random variables, thenX n = o p (Y n ) if and only ifjX n j=jY n j converges in probability to 0. 1.2 Model Description We consider the control of the fork-join processing network depicted in Figure 1.2. In this network, there are 2 job types, 7 servers, 10 buffers, and 8 activities. The set of job types is denoted byJ , whereJ =fa;bg and a and b denote the type a and type b jobs, respectively. The set of servers is denoted byS, where 7 S =f1; 2;:::; 7g. The set of buffers is denoted byK, whereK =f1; 2;:::; 10g, and the set of activities is denoted byA whereA =f1; 2; 3;A;B; 5; 6; 7g. Except for server 4, each server is associated with a single activity. Server 4 is associated with two activities, denoted byA andB, which are processing type a jobs from buffer 4 and typeb jobs from buffer 5, respectively. Both server 6 and server 7 deplete jobs from 2 different buffers. Note that these two servers are join servers and process jobs whenever both of the corresponding buffers are nonempty. Hence, both server 6 and server 7 are associated with a single activity, namely activities 6 and 7, respectively. Lets :A!S be a function such thats(j) denotes the server in which activityj,j2A takes place. Letf :Knf1; 2g!A be a function such thatf(k) denotes the activity which feeds bufferk,k2Knf1; 2g. Lastly, letd :K!A be a function such thatd(k) denotes the activity which depletes bufferk,k2K. For example,s(A) =s(B) = 4,f(4) = 1 andd(4) =A. 1.2.1 Stochastic Primitives We assume that all the random variables and stochastic processes are defined in the same complete proba- bility space ( ;F;P),E denotes the expectation underP, andP(A;B) :=P(A\B). We associate the external arrival time of each job and the process time of each job in the corresponding activities with the sequence of random variablesf v j (i);j2J[Ag 1 i=1 and the strictly positive constants f j ;j2Jg andf j ;j2Ag. We assume that for eachj2J[A,f v j (i)g 1 i=1 is a strictly positive, indepen- dent and identically distributed (i.i.d.) sequence of random variables mutually independent off v k (i)g 1 i=1 for all k 2 (J[A)nfjg, E[ v j (1)] = 1, and the variance of v j (1) is denoted by 2 j . For j 2J , let v j (i) := v j (i)= j be the inter-arrival time between the (i 1)st andith typej job. Then, j and j are the arrival rate and the coefficient of variation of the inter-arrival times of the typej jobs, wherej2J . For j2f1; 3;A; 6g (j2f2;B; 5; 7g), letv j (i) := v j (i)= j be the service time of theith incoming typea (b) job associated with the activityj. Then, j and j are the service rate and the coefficient of variation of the service times related to activityj,j2A. For eachj2J[A, letV j (0) := 0 and V j (n) := n X i=1 v j (i) 8n2N + ; S j (t) := supfn2N :V j (n)tg: (1.1) Then, S j is a renewal process for eachj2J[A. Ifj2J , S j (t) counts the number of external type j arrivals until timet; ifj2A,S j (t) counts the number of service completions associated with activityj until timet given that the corresponding server works continuously on this activity during [0;t]. 8 1.2.2 Scheduling Control and Network Dynamics LetT j (t),j2A, denote the cumulative amount of time servers(j) devotes to activityj during [0;t]. Then, a scheduling control is defined by the two dimensional service time allocation process (T A ;T B ). Although a scheduling control indirectly affects (T 6 ;T 7 ), since we do not have any direct control on servers 6 and 7, we exclude (T 6 ;T 7 ) from the definition of the scheduling control. Let, I s(j) (t) :=tT j (t); j2AnfA;Bg; (1.2a) I 4 (t) :=tT A (t)T B (t); (1.2b) denote the cumulative idle time of the servers during the interval [0;t]. For anyj2A,S j (T j (t)) denotes the total number of service completions related to activityj in servers(j) up to timet. For anyk2K, let Q k (t) be the number of jobs waiting in bufferk at timet,t2R + , including the jobs that are being served. Then, for allt2R + , Q 1 (t) :=S a (t)S 1 (T 1 (t)) 0; Q 2 (t) :=S b (t)S 2 (T 2 (t)) 0; (1.3a) Q k (t) :=S f(k) (T f(k) (t))S d(k) (T d(k) (t)) 0; k2Knf1; 2g: (1.3b) For simplicity, we assume that initially all buffers are empty; i.e.,Q k (0) = 0 for allk2K. Later, we relax this assumption in Remark 1.8.2. We have V j (S j (T j (t)))T j (t)<V j (S j (T j (t)) + 1); for allj2A andt2R + ; (1.4) which implies that we consider only head-of-the-line (HL) policies, where jobs are processed in FCFS order within each buffer. In this network, a task associated with a specific job cannot join a task originating in another job under the HL policies. It is straightforward to see that work-conserving policies are more efficient than the others in this network. Hence, we ensure that all of the servers work in a work-conserving fashion by the following constraints: For allt2R + , I j () is nondecreasing andI j (0) = 0 for allj2S; (1.5a) I s(d(k)) (t) increases if and only ifQ k (t) = 0; for allk2f1; 2; 3; 6g; (1.5b) 9 I 4 (t) increases if and only ifQ 4 (t)_Q 5 (t) = 0; (1.5c) I 6 (t) increases if and only ifQ 7 (t)^Q 8 (t) = 0; (1.5d) I 7 (t) increases if and only ifQ 9 (t)^Q 10 (t) = 0: (1.5e) A scheduling policy := (T A ;T B ) is admissible if it satisfies the following conditions: For any T i , i2A,I j ,j2S, andQ k ,k2K satisfying (1.2), (1.3), (1.4), and (1.5), T j (t) is measurable (i.e.,T j (t)2F) for allt2R + andj2fA;Bg; (1.6a) T j () is continuous and nondecreasing withT j (0) = 0 for allj2fA;Bg; (1.6b) I 4 () is continuous and nondecreasing withI 4 (0) = 0: (1.6c) Conditions (1.6a)–(1.6c) imply that we consider a wide range of scheduling policies including the ones which can anticipate the future. The class of admissible policies includes some conditions imposed without loss of optimality and some that are more restrictive. The HL assumption implies that the ordering of jobs is preserved at the “join” servers, which means tasks forked from one job are the exact ones later joined together. Without the HL assumption, we would need to specify whether or not jobs are exchangeable (discussed in the last paragraph of the literature review). The HL assumption is not restrictive when the server does not know the exact job size ahead of time and job size distributions are the same. However, the HL assumption is restrictive in our setting because an admissible scheduling policy can anticipate the future. For example, we expect that using the shortest-remaining processing time (SRPT) rule at servers 1, 2, 3, and 5, would improve performance, because SRPT is known to minimize the combined time jobs spend waiting and in service, which would ensure the fastest possible movement of jobs from buffers 1, 2, 3, and 6 to buffers 7 and 10 (Schrage (1968)). Finally, the work-conserving assumption is not restrictive when holding costs are type dependent, but is restrictive when they are task dependent; see our extension to task-dependent holding costs in Section 1.10.1 and Remark 1.10.2. 10 1.2.3 The Objective A natural objective is to minimize the discounted expected total holding cost. Let h a and h b denote the holding cost rate per job per unit time for a type a and b job, respectively; and > 0 be the discount parameter. Moreover, let Z(t) :=h a (Q 3 (t) +Q 4 (t) +Q 7 (t) +Q 8 (t)) +h b (Q 5 (t) +Q 6 (t) +Q 9 (t) +Q 10 (t)): ThenZ(t) denote the total cost rate in the buffers 3; 4;:::; 10 at timet, t2 R + . SinceQ 1 (t) andQ 2 (t) are independent of the scheduling policy, we exclude these processes from the definition ofZ(t). Then, the expected total discounted cost under admissible policy is J =E Z 1 0 e t Z(t)dt ; (1.7) and our objective is to find an admissible policy which minimizes (1.7). Another natural objective is to minimize the expected total cost up to timet,t2R + , which is J =E Z t 0 Z(s)ds : (1.8) Yet another possible objective is to minimize the long-run average cost per unit time, J = lim sup t!1 1 t E Z t 0 Z(s)ds : (1.9) We focus on a more challenging objective which is minimizing P (Z(t)>x); for allt2R + andx> 0: (1.10) It is possible to see that any policy that minimizes (1.10) also minimizes the objectives (1.7), (1.8), and (1.9). In this specific network, for allt2R + , Q 3 (t) +Q 7 (t) =Q 4 (t) +Q 8 (t); Q 5 (t) +Q 9 (t) =Q 6 (t) +Q 10 (t): (1.11) 11 By (1.11), a policy is optimal under the objective (1.10) if and only if it is optimal under the objective of minimizing P (h a (Q 3 (t) +Q 7 (t)) +h b (Q 6 (t) +Q 10 (t))>x); for allt2R + andx> 0: (1.12) 1.3 Asymptotic Framework It is very difficult to analyze the system described in Section 1.2 exactly. Even if we can accomplish this very challenging task, it is even less likely that the optimal control policy will be simple enough to be expressed by a few parameters. Therefore, we focus on finding an asymptotically optimal control policy under diffusion scaling and the assumption that server 4 is in heavy traffic. We first introduce a sequence of fork-join systems and present the main assumptions done for this chapter in Section 1.3.1. Then we formally define the fluid and diffusion scaled processes and present convergence results for the diffusion scaled workload facing server 4 and the diffusion scaled queue length processes associated with servers 1, 2, 3, and 5 in Section 1.3.2. The question left open is to determine which control results in the cost-minimizing queue length processes in buffers 4 and 5, that determine the proportion of typea andb jobs in the workload facing server 4. 1.3.1 A Sequence of Fork-Join Systems We consider a sequence of fork-join systems indexed by r where r!1 through a sequence of values in R + . Each queueing system has the same structure defined in Section 1.2 except that the arrival and service rates depend on r. To be more precise, in the rth system, we associate the external arrival time of each job and the process time of each job in the corresponding activities with the sequence of random variablesf v j (i);j2J[Ag 1 i=1 , which we have defined in Section 1.2.1, and the strictly positive constants f r j ;j2Jg andf r j ;j2Ag such that v r j (i) := v j (i)= r j , j2J is the inter-arrival time between the (i 1)st andith typej job andv r j (i) := v j (i)= r j ,j2f1; 3;A; 6g (j2f2;B; 5; 7g) is the service time of theith incoming typea (b) job associated with the activityj in therth system. Therefore, r j ,j2J and r j , j2A are the arrival rates and service rates in therth system whereas the coefficient of variations are the same with the original system defined in Section 1.2. From this point forward, we will use the superscriptr to show the dependence of the stochastic processes to therth queueing system. Next, we present our assumptions related to the system parameters. The first one is related to cost param- eters. Assumption 1.3.1. Without loss of generality, we assume thath a A h b B . 12 This assumption implies that it is more expensive to keep typea jobs than typeb jobs in server 4. Second, we make the following assumptions related to the stochastic primitives of the network. Assumption 1.3.2. There exists a non-empty open neighborhood,O, around 0 such that for all2O, E[e v j (1) ]<1; for allj2J[A: Assumption 1.3.2 is the exponential moment assumption for the inter-arrival and service time processes. This assumption is common in the queueing literature, cf. Harrison (1998), Bell and Williams (2001), Maglaras (2003), Meyn (2003). Our final assumption concerns the convergence of the arrival and service rates, and sets up the heavy traffic asymptotic regime. Assumption 1.3.3. 1. r j ! j > 0 for allj2J asr!1. 2. r j ! j > 0 for allj2A asr!1. 3. a = A + b = B = 1. 4. r ( r a = r A + r b = r B 1)! 4 = A asr!1, where 4 is a constant inR. 5. Asr!1, r( r a r 1 )! 1 2R[f1g; r( r b r 2 )! 2 2R[f1g; r( r a r 3 )! 3 2R[f1g; r( r b r 5 )! 5 2R[f1g: 6. Asr!1,r( r a r 6 )!1 andr( r b r 7 )!1. 7. Ifh a A =h b B , there exists anr 0 2R + such thath a r A h b r B for allrr 0 . Parts 3 and 4 of Assumption 1.3.3 are the heavy traffic assumptions for server 4. Note that, if server 4 was in light traffic, its processing capacity would be high, thus any work-conserving control policy would perform well. By Parts 1, 2, and 3, we have A > a and B > b . Part 5 states that each of the servers 1, 2, 3, and 5 can be either in light or heavy traffic. On the one hand, if i =1 for somei2f1; 2; 3; 5g, then serveri is in light traffic 1 . On the other hand, if i 2R, then serveri is in heavy traffic. Part 6 states 1 Note thata <1 implies that server 1 is in light traffic. Moreover, it is possible that server 1 is in light traffic anda =1, e.g., consider the case r a = 1r 0:5 and r 1 = 1 for allr. 13 that the two join servers are in light traffic. Note that, Atar et al. (2012), Gurvich and Ward (2014), Lu and Pang (2016a,b, 2017) assume that the service processes in the join servers are instantaneous. Hence, Part 6 of Assumption 1.3.3 extends the assumptions on the join servers made in the literature. Lastly, Part 7 of Assumption 1.3.3 is done for technical reasons that occur only whenh a A =h b B (cf. Section 1.6). For simplicity, we assume that Q r k (0) = 0; for allk2K andr2R + : Later, we relax this assumption in Remark 1.8.2. Assumptions 1.3.1, 1.3.2, and 1.3.3 are assumed throughout the chapter. 1.3.2 Fluid and Diffusion Scaled Processes In this section, we present the fluid and diffusion scaled processes. For allt2R + , the fluid scaled processes are defined as S r j (t) :=r 2 S r j (r 2 t); j2J[A; T r j (t) :=r 2 T r j (r 2 t); j2A; (1.13a) Q r j (t) :=r 2 Q r j (r 2 t); j2K; I r j (t) :=r 2 I r j (r 2 t); j2S; (1.13b) and the diffusion scaled processes are defined as ^ S r j (t) :=r 1 S r j (r 2 t) r j r 2 t ; j2J; ^ S r j (t) :=r 1 S r j (r 2 t) r j r 2 t ; j2A; (1.14a) ^ T r j (t) :=r 1 T r j (r 2 t); j2A; ^ I r j (t) :=r 1 I r j (r 2 t); j2S; (1.14b) ^ Q r j (t) :=r 1 Q r j (r 2 t); j2K: (1.14c) By the functional central limit theorem (FCLT), cf. Theorems 4.3.2 and 7.3.2 of Whitt (2002), we have the following weak convergence result. ^ S r j ;j2J[A ) ~ S j ;j2J[A asr!1, (1.15) where ~ S j is a one-dimensional Brownian Motion for eachj2J[A such that ~ S j d = BM 0 (0; j 2 j ) for j2J , ~ S j d =BM 0 (0; j 2 j ) forj2A and each ~ S j is mutually independent of ~ S i ,i2 (J[A)nfjg. 14 Fort2R + , let us define the workload process (up to a constant scale factor) W r 4 (t) :=Q r 4 (t) + r A r B Q r 5 (t): (1.16) Then, W r 4 (t)= r A is the expected time to deplete buffers 4 and 5 at timet, if no more jobs arrive at these buffers after timet in therth system. Let W r 4 (t) := r 2 W r 4 (r 2 t) and ^ W r 4 (t) := r 1 W r 4 (r 2 t) denote the fluid and diffusion scaled workload processes, respectively. Next, we present the convergence of the fluid scaled processes under any work-conserving policy. Proposition 1.3.1. Under any work-conserving policy (cf. (1.5)), asr!1 Q r k ;k2K; W r 4 ; T r j ;j2A a:s: ! Q k ;k2K; W 4 ; T j ;j2A u.o.c.; where Q k = 0 for allk2K, W 4 = 0, and T j (t) = ( a = j )t forj2f1; 3;A; 6g and T j (t) = ( b = j )t for j2f2;B; 5; 7g for allt2R + . The proof of Proposition 1.3.1 is presented in Appendix A.2.1. We will use Proposition 1.3.1 to prove convergence results for the diffusion scaled processes. Considering Assumption 1.3.3 Part 5, letH :=fi2f1; 2; 3; 5g : i 2Rg; i.e.,H is the set of servers which are in heavy traffic among the servers 1, 2, 3, and 5. LetjHj be the cardinality of the setH, and for eachi2f1; 2; 3; 5g, let i := 8 > < > : 1; if serveri is in heavy traffic; i.e.,i2H, 0; if serveri is in light traffic; i.e.,i = 2H. For allt2R + , let S r 1 (t) := 1 ^ S r 1 T r 1 (t) r( r a r 1 )t + (1 1 ) ^ S r a (t) ^ Q r 1 (t) ; (1.17a) S r 2 (t) := 2 ^ S r 2 T r 2 (t) r( r b r 2 )t + (1 2 ) ^ S r b (t) ^ Q r 2 (t) : (1.17b) Then, by using (1.17), we define the so-called “netput process” for each buffer as ^ X r k (t) := ^ S r l (t) ^ S r d(k) T r d(k) (t) +r( r l r d(k) )t; for (k;l)2f(1;a); (2;b)g; (1.18a) ^ X r k (t) := S r l (t) ^ S r d(k) T r d(k) (t) +r( r i r d(k) )t; for (k;l;i)2f(3; 1;a); (6; 2;b)g; (1.18b) 15 ^ X r k (t) := S r l (t) ^ S r d(k) T r d(k) (t) +r r d(k) r i r d(k) i d(k) ! t; for (k;l;i)2f(4; 1;a); (5; 2;b)g; (1.18c) ^ X r k (t) := ^ S r l (t) ^ Q r i (t) ^ Q r j (t) ^ S r d(k) T r d(k) (t) +r( r l r d(k) )t; for (k;l;i;j)2f(7;a; 1; 3); (8;a; 1; 4); (9;b; 2; 5); (10;b; 2; 6)g: (1.18d) Let ^ Q r := 2 6 6 6 6 6 6 6 6 6 4 ^ Q r 1 ^ Q r 2 ^ Q r 3 ^ Q r 6 ^ W r 4 3 7 7 7 7 7 7 7 7 7 5 ; ^ X r := 2 6 6 6 6 6 6 6 6 6 4 ^ X r 1 ^ X r 2 ^ X r 3 ^ X r 6 ^ X r 4 + r A r B ^ X r 5 3 7 7 7 7 7 7 7 7 7 5 ; ^ I r := 2 6 6 6 6 6 6 6 6 6 4 ^ I r 1 ^ I r 2 ^ I r 3 ^ I r 5 ^ I r 4 3 7 7 7 7 7 7 7 7 7 5 ; (1.19) := 2 6 6 6 6 6 6 6 6 6 4 1 2 3 1 1 5 2 2 4 1 1 2 A B 2 3 7 7 7 7 7 7 7 7 7 5 ; R r := 2 6 6 6 6 6 6 6 6 6 4 r 1 0 0 0 0 0 r 2 0 0 0 1 r 1 0 r 3 0 0 0 2 r 2 0 r 5 0 1 r 1 2 r A r B r 2 0 0 r A 3 7 7 7 7 7 7 7 7 7 5 ; (1.20) and letR be a 5 5 matrix which is the component-wise limit ofR r . Then, we have ^ Q r = ^ X r +R r ^ I r ; (1.21) ^ Q r k = ^ X r k + r d(k) ^ I r s(d(k)) ; k2f7; 8; 9; 10g: (1.22) Let us define := 0 B B B B B B B B B @ 1 2 3 6 4 1 a ( 2 a + 2 1 ) 0 a 2 1 0 a 2 1 2 0 b ( 2 b + 2 2 ) 0 b 2 2 A B b 2 2 3 a 2 1 0 Cov(3; 3) 0 Cov(3; 4) 6 0 b 2 2 0 Cov(6; 6) Cov(4; 6) 4 a 2 1 A B b 2 2 Cov(3; 4) Cov(4; 6) Cov(4; 4) 1 C C C C C C C C C A (1.23) 16 where Cov(3; 3) := a 1 2 1 + (1 1 ) 2 a + 2 3 ; Cov(3; 4) := a 1 2 1 + (1 1 ) 2 a ; Cov(4; 6) := A B b 2 2 2 + (1 2 ) 2 b ; Cov(6; 6) := b 2 2 2 + (1 2 ) 2 b + 2 5 ; Cov(4; 4) := a 1 2 1 + (1 1 ) 2 a + 2 A + A B 2 b 2 2 2 + (1 2 ) 2 b + 2 B : (1.24) Then, we have the following weak convergence result. Proposition 1.3.2. Under any work-conserving policy (cf. (1.5)), ^ Q r 1 ; ^ Q r 2 ; ^ Q r 3 ; ^ Q r 6 ; ^ W r 4 ) ~ Q 1 ; ~ Q 2 ; ~ Q 3 ; ~ Q 6 ; ~ W 4 asr!1, (1.25) where ~ Q i = 0 for each i = 2 H and ~ Q i ;i 2 H; ~ W 4 is a semimartingale reflected Brownian motion (SRBM) associated with the data P jHj ; H ; H ;R H ; 0 jHj . P jHj is the nonnegative orthant inR jHj ; H is anjHj-dimensional vector derived from the vector (cf. (1.20)) by deleting the rows corresponding to each i,i = 2H; H andR H arejHjjHj-dimensional matrices derived from (cf. (1.23)) andR (cf. (1.20)) by deleting the rows and columns corresponding to eachi,i = 2H, respectively; and 0 jHj is the origin inP jHj . The state space of the SRBM isP jHj ; H and H are the drift vector and the covariance matrix of the underlying Brownian motion of the SRBM, respectively;R H is the reflection matrix; and 0 jHj is the starting point of the SRBM. The formal definition of an SRBM can be found in Definition 3.1 of Williams (1998b). The proof of Proposition 1.3.2 is presented in Appendix A.2.2. 1.4 The Approximating Diffusion Control Problem In this section, we construct an approximating diffusion control problem (DCP) with non-rigorous mathe- matical arguments. However, the solution of the DCP will help us to guess a heuristic control policy. Then, we will prove the asymptotic optimality of this heuristic control policy rigorously in Sections 1.5, 1.6, 1.7, and 1.8. Parallel with the objective (1.12), consider the objective of minimizing P h a ^ Q r 3 (t) + ^ Q r 7 (t) +h b ^ Q r 6 (t) + ^ Q r 10 (t) >x ; 8t2R + ; x> 0; (1.26) 17 for some r. Motivated by Assumption 1.3.3 Part 6, let us pretend that the service processes at servers 6 and 7 happen instantaneously. Since the diffusion scaled queue length process weakly converges to 0 in a light traffic queue, we believe that considering instantaneous service rates in the downstream servers will not change the behavior of the system in the limit. In this case, jobs can accumulate in buffer 7 (8) only at the times buffer 8 (7) is empty. Similarly, jobs can accumulate in buffer 10 (9) only at the times buffer 9 (10) is empty. By this fact and (1.11), Q r 7 = (Q r 4 Q r 3 ) + ; Q r 8 = (Q r 3 Q r 4 ) + ; Q r 9 = (Q r 6 Q r 5 ) + ; Q r 10 = (Q r 5 Q r 6 ) + : (1.27) By (1.27), objective (1.26) is equivalent to minimizing P h a ^ Q r 3 (t) + ^ Q r 4 (t) ^ Q r 3 (t) + +h b ^ Q r 6 (t) + ^ Q r 5 (t) ^ Q r 6 (t) + >x ;8t2R + ; x> 0: (1.28) From the objective (1.28), we need approximations for ^ Q r 3 , ^ Q r 4 , ^ Q r 5 , and ^ Q r 6 and we will achieve this goal by lettingr!1. By (1.14c), (1.16), and Proposition 1.3.2, we know that ^ Q r 3 ; ^ Q r 6 ; weakly converges to ~ Q 3 ; ~ Q 6 and ^ Q r 4 + ( r A = r B ) ^ Q r 5 weakly converges to ~ W 4 . At this point, let us assume that ^ Q r 4 ; ^ Q r 5 ) ~ Q 4 ; ~ Q 5 asr!1: Then we construct the following DCP: For eachx> 0 andt2R + , min P h a ~ Q 3 (t) + ~ Q 4 (t) ~ Q 3 (t) + +h b ~ Q 6 (t) + ~ Q 5 (t) ~ Q 6 (t) + >x ; s.t. ~ Q 4 (t) + A B ~ Q 5 (t) = ~ W 4 (t); (1.29) ~ Q k (t) 0; for allk2f4; 5g: Intuitively, we want to minimize the total cost by splitting the total workload in server 4 to buffers 4 and 5 in the DCP (1.29). Now, we will consider DCP (1.29) pathwise. Let! (!2 ) denote a sample path of the processes in DCP (1.29) and for anyF : !D,F (! t ) denote the value of the processF at timet in the sample path!. Then, consider the following optimization problem for each!2 andt2R + . min h a ~ Q 3 (! t )_ ~ Q 4 (! t ) +h b ~ Q 5 (! t ) ~ Q 6 (! t ) + ; (1.30a) 18 s.t. ~ Q 4 (! t ) + A B ~ Q 5 (! t ) = ~ W 4 (! t ); (1.30b) ~ Q k (! t ) 0; for allk2f4; 5g: (1.30c) Note that, we exclude the termh b ~ Q 6 (! t ) from the objective function (1.30a) because ~ Q 6 (! t ) is independent of the decision variables ~ Q 4 (! t ) and ~ Q 5 (! t ). Although the objective function (1.30a) is nonlinear, the optimization problem (1.30) is easy to solve because it has linear constraints. Moreover, Lemma 1.4.1, provides a closed-form solution for (1.30). Lemma 1.4.1. Consider the optimization problem min h a (q 3 _q 4 ) +h b (q 5 q 6 ) + ; s.t. q 4 + A B q 5 =w 4 ; q 4 0; q 5 0; whereq 4 andq 5 are the decision variables,h a ;h b 0, A ; B > 0, andh a A h b B . Thenq 4 =q 3 ^w 4 andq 5 = ( B = A )(w 4 q 3 ) + is an optimal solution of this problem. Proof: Replacingq 5 with ( B = A )(w 4 q 4 ) gives us the following equivalent optimization problem which has only one decision variable. min h a (q 3 _q 4 ) +h b ( B = A ) (w 4 q 4 ( A = B )q 6 ) + ; (1.31a) s.t. 0q 4 w 4 : (1.31b) The objective function (1.31a) is a continuous function ofq 4 . There are two cases to consider. First, suppose that w 4 q 3 . Then, the objective function (1.31a) is a nonincreasing function of q 4 , so q 4 = w 4 is an optimal solution. Second, suppose that w 4 > q 3 . When q 4 q 3 , the objective function (1.31a) is a nonincreasing function ofq 4 . Whenq 4 > q 3 , the objective function (1.31a) is a nondecreasing function of q 4 becauseh a A h b B . Hence,q 4 =q 3 is an optimal solution whenw 4 >q 3 . Therefore,q 4 =q 3 ^w 4 andq 5 = ( B = A )(w 4 q 4 ) = ( B = A )(w 4 q 3 ) + is an optimal solution of the optimization problem (1.31). Therefore, by Lemma 1.4.1, we see that an optimal solution of the optimization problem (1.30) is ~ Q 4 (! t ) = ~ W 4 (! t )^ ~ Q 3 (! t ) and ~ Q 5 (! t ) = ( B = A ) ~ W 4 (! t ) ~ Q 3 (! t ) + for all!2 andt2 R + . This result and (1.27) imply the following proposition. 19 Proposition 1.4.1. ~ Q 4 ; ~ Q 5 ; ~ Q 7 ; ~ Q 8 ; ~ Q 9 ; ~ Q 10 = ~ Q 3 ^ ~ W 4 ; B A ~ W 4 ~ Q 3 + ; 0; ~ Q 3 ~ W 4 + ; ~ Q 6 B A ~ W 4 ~ Q 3 + + ; B A ~ W 4 ~ Q 3 + ~ Q 6 + ! (1.32) is an optimal solution of the DCP (1.29). Note that the right hand side of (1.32) is independent of the scheduling policies by Proposition 1.3.2. Therefore, a control policy in which the corresponding processes weakly converge to the right hand side of (1.32) is a good candidate for an asymptotically optimal policy. In the next section, we formally introduce the proposed policy. 1.5 Proposed Policy Our objective is to propose a policy under which the diffusion scaled queue-length processes track the DCP solution given in Proposition 1.4.1. This is because the DCP solution provides a lower bound on the asymptotic performance of any admissible policy, as we will prove in Section 1.6 (see Theorem 1.6.1). The DCP solution in Proposition 1.4.1 matches the content level of buffer 4 to that of buffer 3, except when the buffer 3 content level exceeds the total work facing server 4 (that is, the combined contents of buffers 4 and 5). This ensures that server 4 never causes server 6 to idle because of the join operation, as is evidenced by the fact that buffer 7 is always empty whereas buffer 8 sometimes has a positive content level in (1.32). At the same time, server 4 prevents any unnecessary idling of server 7 by devoting its remaining effort to process the contents of buffer 5. Therefore, the key observation from the DCP solution is that there is no reason for the departure process of the more expensive type a jobs from server 4 to exceed that of server 3. Instead of ever letting server 4 “get ahead”, it is preferable to have server 4 work on typeb jobs, so as to prevent as much forced idling at server 7 due to the join operation as possible. The only time there should have been more cumulative typea job departures from server 4 than from server 3 is when the total number of jobs facing server 4 is less than that facing server 3. In that case, server 4 can outpace server 3 without forcing additional idling at server 7. The intuition in the preceding paragraph motivates the following departure pacing policy, in which server 4 gives priority to typea jobs when the number of typea jobs in buffer 4 exceeds that in buffer 3 and gives priority to the typeb jobs in buffer 5 otherwise. 20 Definition 1.5.1. Slow Departure Pacing (SDP) Policy. The allocation process (T A ;T B ) satisfies Z 1 0 I (Q 3 (t)<Q 4 (t)) d (tT A (t)) = 0; (1.33a) Z 1 0 I (Q 3 (t)Q 4 (t);Q 5 (t)> 0) d (tT B (t)) = 0; (1.33b) Z 1 0 I (Q 3 (t)Q 4 (t);Q 4 (t) +Q 5 (t)> 0) dI 4 (t) = 0; (1.33c) together with (1.2), (1.3), (1.4), and (1.5). It is possible to see that (T A ;T B ) that satisfies (1.33) also satisfies (1.6), and so is admissible. IfQ 3 (t) < Q 4 (t), (1.33a) ensures that server 4 gives priority to buffer 4. If Q 3 (t) Q 4 (t) andQ 5 (t) > 0, (1.33b) ensures that server 4 gives priority to buffer 5. (1.33c) ensures a work-conserving control policy in server 4 whenQ 3 (t)Q 4 (t). When 3 < A , so that server 3 is the slower server, we use the slow departure pacing policy to determine when server 4 can allocate effort to processing typeb jobs without increasing typea job delay. Otherwise, when 3 A , there is almost never extra processing power to allocate to typeb jobs, and so a static priority policy will perform similarly to the slow departure pacing policy (see Remark 1.9.1 regarding our numerical results). Definition 1.5.2. Static Priority Policy. The allocation process (T A ;T B ) satisfies Z 1 0 I (Q 4 (t)> 0) d (tT A (t)) = 0; (1.34a) Z 1 0 I (Q 4 (t) +Q 5 (t)> 0) dI 4 (t) = 0; (1.34b) together with (1.2), (1.3), (1.4), and (1.5). It is possible to see that (T A ;T B ) that satisfies (1.34) also satisfies (1.6), and so is admissible. (1.34a) ensures that server 4 gives static priority to buffer 4 and (1.34b) ensures a work-conserving control policy in server 4. The proposed policy is the SDP policy in Definition 1.5.1 when 3 < A and is the static priority policy in Definition 1.5.2 when 3 A . We have the following weak convergence result associated with the proposed policy. Theorem 1.5.1. Under the proposed policy, ^ Q r k ;k2K; ^ W r 4 ) ~ Q k ;k2K; ~ W 4 asr!1, 21 where ~ Q 1 ; ~ Q 2 ; ~ Q 3 ; ~ Q 6 ; ~ W 4 is defined in Proposition 1.3.2 and ~ Q 4 ; ~ Q 5 ; ~ Q 7 ; ~ Q 8 ; ~ Q 9 ; ~ Q 10 is defined in Proposition 1.4.1. The proof of Theorem 1.5.1 is presented in Section 1.7. Theorem 1.5.1, the continuous-mapping theo- rem (see, for example Theorem 3.4.3 of Whitt (2002)), and Theorem 11.6.6 of Whitt (2002) establish the asymptotic behavior of the objective function (1.12) under the proposed policy. Corollary 1.5.2. Under the proposed policy, for allt2R + andx> 0, we have lim r!1 P h a ^ Q r 3 (t) + ^ Q r 7 (t) +h b ^ Q r 6 (t) + ^ Q r 10 (t) >x =P h a ~ Q 3 (t) +h b ~ Q 6 (t) + B A ~ W 4 (t) ~ Q 3 (t) + ~ Q 6 (t) + ! >x ! : (1.35) Remark 1.5.1. In the classical open processing networks, if a server is in light traffic, then the corre- sponding diffusion scaled buffer length process converges to 0 (see Theorem 6.1 of Chen and Mandelbaum (1991)). However, we see in Theorem 1.5.1 that although servers 6 and 7 are in light traffic, ~ Q 9 and ~ Q 10 are non-zero processes (moreover ~ Q 8 is a non-zero process when server 3 is in heavy traffic). Therefore, even though a join server has more than enough processing capacity, significant amount of jobs can accumulate in the corresponding buffers due to the synchronization requirements between the jobs in different buffers. This makes the control of fork-join networks more challenging than the one of classical open-processing networks. Remark 1.5.2. We will prove that the SDP policy is asymptotically optimal when a 3 < A and the static priority policy is asymptotically optimal when server 3 is in light traffic (cf. Sections 1.7.1 and 1.7.2 and Remark 1.7.1). Hence, both the SDP and the static priority policies are asymptotically optimal when server 3 is in light traffic and 3 < A . However, the simulation experiments show that the static priority policy performs poorly when 3 is close to a but the SDP policy performs very well even when 3 > A (cf. Section 1.9.2). This is why our proposed policy is the SDP policy when a 3 < A and is the static priority policy when 3 > A . When the comparison between the parameters is unknown, we recommend using the SDP policy. This is because the aforementioned simulation experiments show the SDP policy performs well over a much larger range of parameters than where it is proven to be asymptotically optimal. Remark 1.5.3. The proposed policy is a preemptive policy. However, it is often preferred to use a non- preemptive policy. To specify a non-preemptive policy, we must specify which type of job server 4 chooses to process each time server 4 becomes free and there are both typea and typeb jobs waiting in buffers 4 22 and 5. The non-preemptive version of the SDP policy has server 4 choose to serve a typea job when the number of jobs in buffer 4 exceeds that in buffer 3 or buffer 5 is empty, and to serve a typeb job otherwise. The non-preemptive version of the static priority policy has server 4 always choose a typea job. We expect the performance of the non-preemptive version of our proposed policy to be indistinguishable from our proposed policy in our asymptotic regime, and we verify that the former policy performs very well by our numeric results in Section 1.9. Remark 1.5.4. The non-preemptive version of the proposed policy (cf. Remark 1.5.3) is relatively easy to implement in practice. Implementation of the non-preemptive static priority policy is trivial, because that requires only local state information (whether or not buffer 4 is empty). For the SDP policy, the information required to decide which buffer server 4 will next serve is the number of jobs in buffers 3 and 4 - in other words, no information on the state of upstream or downstream buffers is needed. 1.6 Asymptotic Optimality In this section, we prove that the proposed policy is asymptotically optimal with respect to the objective function (1.12), and then show that the former result implies asymptotic optimality with respect to both the expected total discounted cost (1.7) and the expected total cost in a finite time horizon (1.8). Theorem 1.6.1. Let =f r ;r 1g be an arbitrary sequence of admissible policies. Let the diffusion scaled queue length process in bufferk under in therth system be denoted by ^ Q ;r k for allk2K. Then for allt2R + andx> 0, we have lim inf r!1 P h a ^ Q ;r 3 (t) + ^ Q ;r 7 (t) +h b ^ Q ;r 6 (t) + ^ Q ;r 10 (t) >x P h a ~ Q 3 (t) +h b ~ Q 6 (t) + B A ~ W 4 (t) ~ Q 3 (t) + ~ Q 6 (t) + ! >x ! : (1.36) Theorem 1.6.1 together with Corollary 1.5.2 state that the proposed policy is asymptotically optimal with respect to the objective (1.12). However, although showing that optimality under the objective (1.12) implies optimality under the objectives (1.7), (1.8), and (1.9) is straightforward, showing that asymptotic optimality under the objective (1.12) implies asymptotic optimality under the objectives (1.7), (1.8), and (1.9) is not. This is done in our next result for the objectives (1.7) and (1.8). This is not done for the objective (1.9) because that would require changing the order of the limits with respect to r and t, which is outside the scope of this chapter. However, the simulation experiments in Section 1.9.2 show that the proposed policy performs very well under the objective (1.9). 23 In preparation of the statement of the next result, we define the following notation. For an arbitrary sequence of admissible policies =f r ;r 1g, let the diffusion scaled total cost rate in the buffers 3; 6; 7; 10 at timet under in therth system be denoted by ^ Z ;r (t); i.e., for allt2R + , ^ Z ;r (t) :=h a ^ Q ;r 3 (t) + ^ Q ;r 7 (t) +h b ^ Q ;r 6 (t) + ^ Q ;r 10 (t) : For all t 2 R + , let ^ Z r (t) denote the diffusion scaled total cost rate in the buffers 3; 6; 7; 10 under the proposed policy at timet and ~ Z(t) :=h a ~ Q 3 (t) +h b ~ Q 6 (t) + B A ~ W 4 (t) ~ Q 3 (t) + ~ Q 6 (t) + ! : Then, we have the following result. Theorem 1.6.2. Let =f r ;r 1g be an arbitrary sequence of admissible policies. Then, lim r!1 E Z 1 0 e t ^ Z r (t)dt =E Z 1 0 e t ~ Z(t)dt lim inf r!1 E Z 1 0 e t ^ Z ;r (t)dt ; (1.37) and for allT2R + , lim r!1 E Z T 0 ^ Z r (t)dt =E Z T 0 ~ Z(t)dt lim inf r!1 E Z T 0 ^ Z ;r (t)dt : (1.38) The proof of Theorem 1.6.2 follows by Theorems 1.5.1 and 1.6.1 and showing uniform integrability. For that reason, we prove Theorem 1.6.1 below, and prove Theorem 1.6.2 in Appendix A.2.3. In comparison to the literature, both Ata and Kumar (2005) and Dai and Lin (2008) prove asymptotic optimality with respect to an objective parallel to (1.12), but neither of them establish asymptotic optimality with respect to objectives parallel to (1.7) and (1.8). Proof of Theorem 1.6.1: For notational convenience, we will drop the superscript in this proof. Let us consider the term in the left hand side of (1.36). By (1.11) and the fact thatQ r k 0 for allk2K, we have Q r 7 (t) (Q r 4 (t)Q r 3 (t)) + ; Q r 10 (t) (Q r 5 (t)Q r 6 (t)) + ; 8t2R + : Therefore, it is enough to prove for allt2R + andx> 0, lim inf r!1 P h a ^ Q r 3 (t) + ^ Q r 4 (t) ^ Q r 3 (t) + +h b ^ Q r 6 (t) + ^ Q r 5 (t) ^ Q r 6 (t) + >x 24 P h a ~ Q 3 (t) +h b ~ Q 6 (t) + B A ~ W 4 (t) ~ Q 3 (t) + ~ Q 6 (t) + ! >x ! : (1.39) By (1.15), Proposition 1.3.1, and Theorem 11.4.5 of Whitt (2002) (joint convergence when one limit is deterministic), we have ^ S r j ;j2J[A; T r i ;i2A ) ~ S j ;j2J[A; T i ;i2A asr!1: (1.40) Now, we use Skorokhod’s representation theorem (cf. Theorem 3.2.2 of Whitt (2002)) to obtain the equiv- alent distributional representations of the processes in (1.40) (for which we use the same symbols and call “Skorokhod represented versions”) such that all Skorokhod represented versions of the processes are defined in the same probability space and the weak convergence in (1.40) is replaced by almost sure convergence u.o.c. Then we have ^ S r j ;j2J[A; T r i ;i2A a:s: ! ~ S j ;j2J[A; T i ;i2A ; u.o.c. asr!1: (1.41) We will consider the Skorokhod represented versions of these processes from this point forward and prove (1.39) with respect to these processes. By (1.3), (1.14), (1.16), (1.18), (1.21), (1.41), and Proposition 1.3.2, we have ( ^ Q r 3 ; ^ W r 4 ; ^ Q r 6 ) a:s: ! ( ~ Q 3 ; ~ W 4 ; ~ Q 6 ); u.o.c.; (1.42) where ^ W r 4 = ^ Q r 4 + r A r B ^ Q r 5 ; a.s.; (1.43) and all of the processes in (1.42) and (1.43) have the same distribution with the original ones. Then by Fatou’s lemma, the term in the left hand side of (1.39) is greater than or equal to P lim inf r!1 h a ^ Q r 3 (t) + ^ Q r 4 (t) ^ Q r 3 (t) + +h b ^ Q r 6 (t) + ^ Q r 5 (t) ^ Q r 6 (t) + >x : (1.44) For eacht2R + and sufficiently larger such thath a r A h b r B (note that such anr exists by Assumption 1.3.1 and Parts 2 and 7 of Assumption 1.3.3), we will find a pathwise lower bound on the term h a ^ Q r 3 (t) + ^ Q r 4 (t) ^ Q r 3 (t) + +h b ^ Q r 6 (t) + ^ Q r 5 (t) ^ Q r 6 (t) + 25 =h a ^ Q r 3 (t)_ ^ Q r 4 (t) +h b ^ Q r 5 (t) ^ Q r 6 (t) + +h b ^ Q r 6 (t): (1.45) From this point forward, we will consider the sample paths in which ^ Q r k (t),k2f3; 4; 5; 6g are finite for all r and t. By (1.42) and (1.43), these sample paths occur with probability one when r is sufficiently large. Let ! be a sample path and ! t be defined as in Section 1.4. By (1.43), (1.45), and the fact that ^ Q r 6 (! t ) is independent of the control policy, we can construct the optimization problem in Lemma 1.4.1 withq k = ^ Q r k (! t ) fork2f3; 4; 5; 6g andw 4 = ^ W r 4 (! t ) for which ^ Q r 4 (! t ) = ^ Q r 3 (! t )^ ^ W r 4 (! t ); ^ Q r 5 (! t ) = r B r A ^ W r 4 (! t ) ^ Q r 3 (! t ) + (1.46) is an optimal solution. Therefore, by (1.46), a pathwise lower bound on (1.45) under the admissible policy r is h a ^ Q r 3 (t) +h b ^ Q r 6 (t) + r B r A ^ W r 4 (t) ^ Q r 3 (t) + ^ Q r 6 (t) + ! : (1.47) When we take the lim inf r!1 of the term in (1.47), by (1.42) and the continuous-mapping theorem (specif- ically we use the continuity of the mapping () + and Theorem 4.1 of Whitt (1980), which shows the conti- nuity of addition), (1.44) is greater than or equal to P h a ~ Q 3 (t) +h b ~ Q 6 (t) + B A ~ W 4 (t) ~ Q 3 (t) + ~ Q 6 (t) + ! >x ! : (1.48) Note that the lower bound in (1.48) is independent of control by Proposition 1.3.2 and (1.42). Therefore, (1.48) proves (1.36) for the Skorokhod represented versions of the processes. Since these processes have the same joint distribution with the original ones, (1.48) also proves Theorem 1.6.1. 1.7 Weak Convergence Proof In this section, we prove Theorem 1.5.1. We consider the cases a 3 < A and 3 A separately. Note that the proposed policy is the SDP policy in (1.33) in the first case and the static priority policy in (1.34) in the second case. The proof of the second case is straightforward, but the proof of the first case is complicated because the SDP policy is a continuous-review and state-dependent policy. 1.7.1 Case I: a 3 < A (Slow Departure Pacing Policy) The following result plays a crucial role in the weak convergence of ^ Q r 4 and ^ Q r 5 under the proposed policy. 26 Proposition 1.7.1. Under the proposed policy, for all> 0 andT2R + , lim r!1 P ^ Q r 4 ^ Q r 3 ^ ^ W r 4 T > = 0: (1.49) The proof of Proposition 1.7.1 is presented in Section 1.8. By (1.25), (1.49), and Theorems 3.4.3 and 11.4.7 of Whitt (2002) (continuous-mapping and convergence-together theorems, respectively), we have the joint convergence result ^ Q r 1 ; ^ Q r 2 ; ^ Q r 3 ; ^ Q r 4 ; ^ Q r 6 ; ^ W r 4 ) ~ Q 1 ; ~ Q 2 ; ~ Q 3 ; ~ Q 3 ^ ~ W 4 ; ~ Q 6 ; ~ W 4 asr!1: (1.50) By (1.16), (1.50), and continuous-mapping theorem, we have ^ Q r 5 ) B A ( ~ W 4 ~ Q 3 ) + asr!1: (1.51) At this point, we invoke the Skorokhod representation theorem again for all the processes in (1.40) and we will use the same symbols again. Then, we can replace the weak convergence in (1.40), (1.50), and (1.51) with almost sure convergence u.o.c. for the Skorokhod represented versions of the processes. Next, we will consider server 6. Let ^ Z r 6 := ^ Q r 7 ^ ^ Q r 8 . Then, by (1.18d) and (1.22), ^ Z r 6 (t) = ^ U r 6 (t) + r 6 ^ I r 6 (t); ^ U r 6 (t) := ^ S r a (t) ^ Q r 1 (t) ^ Q r 3 (t)_ ^ Q r 4 (t) ^ S r 6 ( T r 6 (t)) +r( r a r 6 )t; (1.52) where ^ U r 6 is defined in (1.52). Since ^ I r 6 (t) increases only if ^ Z r 6 (t) = 0, we have a Skorokhod problem with respect to ^ U r 6 . By the uniqueness of the solution of the Skorokhod problem (cf. Theorem 6.1 of Chen and Yao (2001)) r 6 ^ I r 6 = ( ^ U r 6 ) and ^ Z r 6 = ( ^ U r 6 ) for eachr. By (1.41) and the fact that ^ Q r 1 ; ^ Q r 3 ; ^ Q r 4 a:s: ! ~ Q 1 ; ~ Q 3 ; ~ Q 3 ^ ~ W 4 u.o.c. asr!1, r 6 ^ I r 6 +r( r a r 6 )e; ^ Z r 6 a:s: ! ~ S a + ~ Q 1 + ~ Q 3 + ~ S 6 T 6 ; 0 ; u.o.c.; (1.53) asr!1 by Lemma 6.4 (ii) of Chen and Yao (2001). By (1.18d), (1.22), and (1.53), asr!1, ^ Q r 7 = ^ S r a ^ Q r 1 ^ Q r 3 ^ S r 6 T r 6 +r( r a r 6 )e + r 6 ^ I r 6 a:s: ! ~ S a ~ Q 1 ~ Q 3 ~ S 6 T 6 ~ S a + ~ Q 1 + ~ Q 3 + ~ S 6 T 6 =0 u.o.c.; 27 ^ Q r 8 = ^ S r a ^ Q r 1 ^ Q r 4 ^ S r 6 T r 6 +r( r a r 6 )e + r 6 ^ I r 6 a:s: ! ~ S a ~ Q 1 ~ Q 3 ^ ~ W 4 ~ S 6 T 6 ~ S a + ~ Q 1 + ~ Q 3 + ~ S 6 T 6 = ~ Q 3 ~ W 4 + u.o.c. By the same way, we can derive the following result for server 7: ^ Q r 9 ; ^ Q r 10 a:s: ! ~ Q 6 B A ~ W 4 ~ Q 3 + + ; B A ~ W 4 ~ Q 3 + ~ Q 6 + ! u.o.c.; asr!1: Since the Skorokhod represented versions of the processes have the same joint distribution with the orig- inal ones, when the Skorokhod represented versions of the processes converge almost surely u.o.c., then the original processes weakly converge and we get the desired result. 1.7.2 Case II: 3 A (Static Priority Policy) When 3 A , server 3 is in light traffic because Assumption 1.3.3 Parts 1, 2, and 3 imply A > a . Then ^ Q r 3 ) ~ Q 3 =0 asr!1 by Proposition 1.3.2. Since server 4 gives static priority to buffer 4 over buffer 5 when 3 A in the proposed policy, then buffer 4 acts like a light traffic queue and ^ Q r 4 )0 = ~ Q 3 ^ ~ W 4 asr!1. This implies that all of the workload in server 4 accumulates in buffer 5 and ^ Q r 5 ) ( B = A ) ~ W 4 asr!1 by (1.16) and Proposition 1.3.2. The convergence proof of all other processes is very similar to the one presented in Section 1.7.1. Remark 1.7.1. It is straightforward to see that the proof presented above holds when server 3 is in light traffic. Hence, Theorem 1.5.1 and the Corollary 1.5.2 holds under the static priority policy whenever server 3 is in light traffic. Therefore, as stated in Remark 1.5.2, the static priority policy is asymptotically optimal whenever server 3 is in light traffic. From this point forward, we will only consider the case a 3 < A and we prove Proposition 1.7.1 under this case in the following section. 1.8 Proof of Proposition 1.7.1 The SDP policy is a dynamic policy that changes the relative priorities of typea andb jobs depending on the system state. The analysis of such a policy requires different arguments to showQ r 4 is close enough to Q r 3 ^W r 4 to satisfy (1.49), depending on which class has priority. This motivates us to partition the interval [0;r 2 T ] according to the aforementioned priority rules, so that we can break the proof of (1.49) into two different parts. 28 We begin with the observation that type a jobs are given priority at all times t2 [0;r 2 T ] for which Q r 4 (t) > Q r 3 (t), and typeb jobs are given priority otherwise. Then, we define “up” intervals during which Q r 4 (t) > Q r 3 (t) and “down” intervals during which Q r 4 (t) Q r 3 (t) as follows. In the rth system, let r n : !R + [f+1g be such that r 0 := 0 and r 2n1 := infft> r 2n2 :Q r 3 (t) =Q r 4 (t) 1g; 8n2N + ; (1.54a) r 2n := infft> r 2n1 :Q r 3 (t) =Q r 4 (t)g; 8n2N + : (1.54b) For completeness, if r n 0 = +1 for somen 0 2N + , we define r n := +1 for allnn 0 . Finally, we bound (1.49) using the “up” and “down” intervals so that P ^ Q r 4 ^ Q r 3 ^ ^ W r 4 T > =P sup 0tr 2 T jQ r 4 (t)Q r 3 (t)^W r 4 (t)j>r ! P sup t2[0;r 2 T ]\ S 1 n=1 [ r 2n1 ; r 2n ) jQ r 4 (t)Q r 3 (t)^W r 4 (t)j>r ! (1.55) +P sup t2[0;r 2 T ]\ S 1 n=0 [ r 2n ; r 2n+1 ) jQ r 4 (t)Q r 3 (t)^W r 4 (t)j>r ! ; (1.56) where> 0 andT2R + are arbitrary. For the proof, it is sufficient to prove that both of the probabilities in (1.55) and (1.56) converge to 0 asr!1. The reason why the probability in (1.55) converges to 0 asr!1 relies on the up interval construction. During an up interval,Q r 4 > Q r 3 , soQ r 3 ^W r 4 = Q r 3 by (1.16), and server 4 gives priority to typea jobs. Since server 4 is faster than server 3 at processing typea jobs,Q r 4 never becomes much larger thanQ r 3 . We make this argument rigorous in Section 1.8.1 below. The argument to see the probability in (1.56) converges to 0 as r ! 1 requires the splitting of the down intervals, as shown in Figure 1.3. To see this, first observe that at the beginning of a down interval, Q r 3 ( r 2n ) =Q r 4 ( r 2n ). Define the first time buffer 5 empties in a down interval as r; 2n := 8 > > > > < > > > > : inf t2 [ r 2n ; r 2n+1 ) :Q r 5 (t) = 0 ; if Q r 5 (t) = 0 for some t 2 [ r 2n ; r 2n+1 ) where r 2n <1, +1; otherwise: (1.57) Note that it is possible that the next up interval starts before buffer 5 empties, in which case r; 2n :=1. For [ r 2n ; r; 2n ^ r 2n+1 ), since server 4 works only on type b jobs and server 3 does not complete a job 29 !"#$%&'()*+ ,-.%#/#$%&'()*+ ,-.%#0#$%&'()*+ ! "#$" %& ! "# %& ! "#$' %& ! "# %&() * + %, - ./ %, 0* 1 %, - ./ %, * + %, - ./$2 %, 0* 1 %, - ./$2 %, 34 * 5 %, - ./ %,() 0%6 * + %, 7 8 * 1 %, 7 (97:;- ./$2 %, (- ./$. %, < * + %, - ./$. %, 0 * 1 %, - ./$. %, * + %, 7 0* 1 %, 7 (%%%* 5 %, 7 = 6%%% 97: ;- ./ %, (- ./ %,() < * 1 %, 7 >* + %, 7 (97: ;- ./ %,() (- ./$2 %, < 1($-($&2#&-#3455'(#6 1($-($&2#&-#3455'(#7 Figure 1.3: Illustration of the down 1, down 2, and up intervals when a 3 < A . There are three possible cases associated with the down intervals: only a down 1 interval exists, only a down 2 interval exists, or both exist. (otherwise an up interval would start), Q r 3 (t) = Q r 4 (t); and since Q r 3 (t)^W r 4 (t) = Q r 3 (t) by (1.16), Q r 4 (t)Q r 3 (t)^W r 4 (t) = 0 trivially. During the interval [ r; 2n ; r 2n+1 ) (when it exists),Q r 5 has arrival rate strictly less than service rate. Hence,Q r 5 stays close to 0, so thatQ r 4 W r 4 . Since alsoQ r 4 Q r 3 by the down interval construction,Q r 3 ^W r 4 Q r 4 . IfQ r 5 ( r 2n ) = 0, then r 2n = r; 2n , so that the second half of the above argument works for this case as well. The rigorous argument (given in Section 1.8.2 below) to show that Q r 5 stays close to 0 during down 2 intervals requires understanding the behavior of a pair of single server queues in tandem, when the service rate of the downstream (second) queue strictly exceeds the arrival rate to the upstream (first) queue. That argument uses two results that are of interest in their own right, which are presented below as standalone results (i.e., using separate notation). A convergence rate result for a queue (either standalone or in tandem) with arrival rate strictly less than the service rate Letfu k ;k 2 N + g,fv k ;k 2 N + g, andfw k ;k 2 N + g be three independent sequences of strictly positive and i.i.d. random variables such thatE[u 1 ] =E[v 1 ] =E[w 1 ] = 1. We assume that there exists an open interval centered at zero denoted by ( ; ) where > 0 such thatE[e u 1 ]<1, E[e v 1 ]<1, andE[e w 1 ]<1 for all2 ( ; ). This implies that all moments ofu 1 ,v 1 , andw 1 are finite. Consider sequences of queueing systems indexed byr wherer!1 through a sequence of values in R + and each queueing system is a pair of two single server queues in tandem (servers 1 and 2, respectively). The buffer capacity is infinite and the service discipline is work-conserving and FCFS in each server in each queueing system. Letu r k :=u k = r ,v r k :=v k = r 1 , andw r k :=w k = r 2 for allk2N + andr, where r , r 1 , r 2 are strictly positive constants for allr. In therth queuing system, the inter-arrival time between the (k1)th 30 andkth job arriving in the system after time 0 isu r k , and the service time of thekth job in servers 1 and 2 arev r k andw r k , respectively, for allk2N + andr. LetQ r k (t) denote the total number of jobs in the bufferk and in the serverk at timet2R + in therth system for allk2f1; 2g. We assume thatQ r 1 (0) andQ r 2 (0) are mutually independent random variables which take values inN and are independent of all other stochastic primitives for allr. Proposition 1.8.1. Suppose that r ! , r 1 ! 1 , and r 2 ! 2 asr!1, where, 1 , and 2 are strictly positive constants such that< 1 ^ 2 . Fix arbitraryn2N + and> 0, and suppose that there exists anr 0 1 such that ifrr 0 , P Q r 1 (0)_Q r 2 (0)> (( r 1 ^ r 2 r )^)r 7 C 0 r 2n1 e C 1 r ; (1.58) for some constantsC 0 > 0 andC 1 > 0 which are independent ofr. Then, there exists anr 1 1 such that ifrr 1 , P sup 0tr n Q r 1 (t)>r +P sup 0tr n Q r 2 (t)>r C 2 r 2n1 e C 3 r ; (1.59) whereC 2 > 0 andC 3 > 0 are constants which are independent ofr. The proof of Proposition 1.8.1 is presented in Appendix A.1. Proposition 1.8.1 also provides an exponential convergence rate result (specifically, (1.59)) for a stan- dalone GI=GI=1 queue, which will be useful in the later analysis. The way to use Proposition 1.8.1 in reference to aGI=GI=1 queue is to artificially construct a second (downstream) server having service rate that exceeds the arrival rate to theGI=GI=1 queue of interest. A pathwise comparison between two single server queues Consider two single server queues both with work-conserving and FCFS service discipline and infinite buffer capacity. LetQ k (t) denote the total number of jobs in the buffer and in the server in thekth single server queue at timet for allk2f1; 2g andt2R + . Letfa 1 n ;a 2 n ;b n ;n2N + g be a sequence of real numbers such thata 1 n a 2 n 0 andb n > 0 for alln2N + . In the kth queue, the nth inter-arrival and service times are a k n and b n , respectively, for all n2 N + and k2f1; 2g. Hence, the two queues have the same service times but the inter-arrival times in the second queue are shorter than the ones in the first queue. IfQ k (0) > 0 for somek2f1; 2g, thena k n = 0 for all n2f1; 2;:::;Q k (0)g; i.e., the inter-arrival time sequencefa k n ;n2 N + g includes the jobs that are in the system at time 0 too. Therefore, sincea 1 n a 2 n 0 for alln2N + , we haveQ 2 (0)Q 1 (0). Moreover, if a 2 n = 0 for alln2N + , thenQ 2 (0) =1 and the server never idles. LetD k n denote the service completion time of thenth job in thekth queue for alln2N + andk2f1; 2g. Then, we have the following result. 31 Proposition 1.8.2. Suppose that sup 0tT Q 1 (t) < 1 for all T 2 R + . Then, sup 0tT Q 1 (t) sup 0tT Q 2 (t) for allT2R + . Moreover,D 1 1 D 2 1 andD 1 n D 1 n1 D 2 n D 2 n1 for alln2f2; 3;:::g; i.e., the inter-departure times in the first queue are longer than the ones in the second queue. The proof of Proposition 1.8.2 is presented in Appendix A.2.4. 1.8.1 Proof of Convergence of (1.55) (Up Intervals) Throughout Section 1.8.1, let 2 (0; 1) be an arbitrary constant, = ( ) > 0 be a constant such that 4=(1 +)< . LetN r :=d( r 3 + 1 )r 2 Te where 1 > 0 is an arbitrary constant. Since there is a service completion in server 3 at r 2n1 for eachn2N + , then P r 2N r 1 r 2 T P S r 3 (r 2 T )N r P S r 3 (r 2 T ) ( r 3 + 1 )r 2 T ! 0 asr!1; (1.60) where (1.60) is by functional strong law of large numbers (FSLLN) for renewal processes (cf. Theorem 5.10 of Chen and Yao (2001)). Since, by construction,Q r 4 (t)>Q r 3 (t) for allt2 [ r 2n1 ; r 2n ) andn2N + , thenQ r 3 (t)^W r 4 (t) =Q r 3 (t) for allt2 [ r 2n1 ; r 2n ) andn2N + by (1.16), and the probability in (1.55) is equal to P sup t2[0;r 2 T ]\ S 1 n=1 [ r 2n1 ; r 2n ) Q r 4 (t)Q r 3 (t)>r ! P r 2N r 1 r 2 T +P sup t2[0;r 2 T ]\ S 1 n=1 [ r 2n1 ; r 2n ) Q r 4 (t)Q r 3 (t)>r; r 2N r 1 >r 2 T ! : (1.61) Note that the first probability in the right hand side of inequality (1.61) converges to 0 asr!1 by (1.60). Hence, it is enough to consider the second probability in the right hand side of (1.61) which is less than or equal to N r X n=1 P sup r 2n1 t< r 2n Q r 4 (t)Q r 3 (t)>r; r 2n1 r 2 T ! = N r X n=1 P sup r 2n1 t< r 2n S r 3 (T r 3 (t))S r A (T r A (t))>r; r 2n1 r 2 T ! (1.62) = N r X n=1 P sup 0t< r 2n r 2n1 S r 3 (T r 3 ( r 2n1 +t))S r A (T r A ( r 2n1 +t))>r; r 2n1 r 2 T ! N r X n=1 P sup 0t< r 2n r 2n1 S r 3 (T r 3 ( r 2n1 ) +t)S r A (T r A ( r 2n1 ) +t)>r; r 2n1 r 2 T ! (1.63) 32 where (1.62) is by (1.3b). We obtain (1.63) in the following way. Server 4 works on buffer 4 during the whole up interval by construction. However, server 3 can be idle during an up interval. Hence, we have for allt2 [0; r 2n r 2n1 ) T r A ( r 2n1 +t) =T r A ( r 2n1 ) +t; T r 3 ( r 2n1 +t)T r 3 ( r 2n1 ) +t; (1.64) which gives (1.63). We expect the sum in (1.63) to converge to zero because each term is the difference between two renewal processes with different rates, and the faster renewal process is the one being subtracted. To formalize this intuition, it is helpful to bound those differences by using the processes E r;n 3 (t) := sup ( k2N : k X l=1 v r 3 (l +B r n )t ) ; E r;n A (t) := sup ( k2N : k X l=1 v r A (l +A r n )t ) ; where P 0 l=1 x l := 0 for any sequencefx l ;l2 N + g,A r n := S r A (T r A ( r 2n1 )), andB r n := S r 3 (T r 3 ( r 2n1 )). Then by (1.3b) and (1.54a),A r n =B r n 1. We have the following result. Lemma 1.8.1. For alln2N + andt2R + , we haveS r 3 (T r 3 ( r 2n1 ) +t)S r A (T r A ( r 2n1 ) +t)E r;n 3 (t) E r;n A (t) + 1. The proof of Lemma 1.8.1 is presented in Appendix A.2.5. By Lemma 1.8.1, (1.63) is less than or equal to N r X n=1 P sup 0t< r 2n r 2n1 E r;n 3 (t)E r;n A (t)>r 1; r 2n1 r 2 T ! (1.65) N r X n=1 P sup 0t<( r 2n r 2n1 )_(r T ) E r;n 3 (t)E r;n A (t)>r 1; r 2n1 r 2 T ! = N r X n=1 P sup 0t< r 2n r 2n1 E r;n 3 (t)E r;n A (t)>r 1; r 2n r 2n1 >r T; r 2n1 r 2 T ! + N r X n=1 P sup 0t<r T E r;n 3 (t)E r;n A (t)>r 1; r 2n r 2n1 r T; r 2n1 r 2 T ! N r X n=1 P r 2n r 2n1 >r T; r 2n1 r 2 T (1.66) + N r X n=1 P sup 0t<r T E r;n 3 (t)E r;n A (t)>r 1; r 2n1 r 2 T ! : (1.67) We will show that both of the terms in (1.66) and (1.67) converge to 0 asr!1. 33 Proof of Convergence of (1.66) (Length of Up Intervals) In this section, we show that the sum in (1.66) converges to 0 asr!1. This implies that the length of the up intervals within the time interval [0;r 2 T ] iso p (r ) for any > 0. LetE r n denote the event inside the probability in (1.66); i.e., E r n := r 2n r 2n1 >r T; r 2n1 r 2 T : (1.68) By (1.3b) and the fact thatQ r 4 (t)>Q r 3 (t) for allt2 [ r 2n1 ; r 2n ), (1.66) is equal to N r X n=1 P sup r 2n1 t< r 2n1 +r T S r A (T r A (t))S r 3 (T r 3 (t))< 0;E r n ! = N r X n=1 P sup 0t<r T S r A (T r A ( r 2n1 +t))S r 3 (T r 3 ( r 2n1 +t))< 0;E r n ! N r X n=1 P sup 0t<r T S r A (T r A ( r 2n1 ) +t)S r 3 (T r 3 ( r 2n1 ) +t)< 0;E r n ! (1.69) N r X n=1 P sup 0t<r T E r;n A (t)E r;n 3 (t)< 1; r 2n1 r 2 T ! ; (1.70) where (1.69) is by (1.64), and (1.70) is by Lemma 1.8.1. Similar to (1.63), we consider the difference of two renewal processes in (1.70). We want to show that the probability that the number of renewals associated with the renewal process with higher renewal rate is always less than the number of renewals of the one with smaller renewal rate within a time interval of length r T converges to zero asr!1. Let E r n;1 := ( sup 0t<r T jE r;n 3 (t) r 3 tj 2 r T; sup 0t<r T E r;n A (t) r A t 2 r T ) ; (1.71) where 2 is an arbitrary constant such that 0< 2 < 3 ^ (( A 3 )=2). Then, the sum in (1.70) is equal to N r X n=1 P sup 0t<r T E r;n A (t)E r;n 3 (t)< 1; r 2n1 r 2 T;E r n;1 ! (1.72) + N r X n=1 P sup 0t<r T E r;n A (t)E r;n 3 (t)< 1; r 2n1 r 2 T; (E r n;1 ) c ! ; (1.73) where superscriptc denote the complement of a set. Note that, on the setE r n;1 , we have for allt2 [0;r T ), r 3 t 2 r TE r;n 3 (t) r 3 t + 2 r T; r A t 2 r TE r;n A (t) r A t + 2 r T: (1.74) 34 This implies that the sum in (1.72) is less than or equal to N r X n=1 P sup 0t<r T r A t 2 r T ( r 3 t + 2 r T )< 1 ! =N r P (( r A r 3 )r T 2 2 r T < 1) (1.75) =N r I r A r 3 2 1 2r T < 2 ! 0 asr!1; (1.76) where (1.75) is by Assumption 1.3.3 Part 2 and the fact that A > 3 , and (1.76) is by the fact that 2 < ( A 3 )=2. Now, let us look at the sum in (1.73), which is less than or equal to N r X n=1 P r 2n1 r 2 T; (E r n;1 ) c (1.77) N r X n=1 P sup 0t<r T jE r;n 3 (t) r 3 tj> 2 r T; r 2n1 r 2 T ! (1.78) + N r X n=1 P sup 0t<r T E r;n A (t) r A t > 2 r T; r 2n1 r 2 T ! : (1.79) It is straightforward to see that the sums in (1.78) and (1.79) converges to 0 asr!1 by the following result, whose proof is presented in Appendix A.2.6. Lemma 1.8.2. For all > 0, 2 > 0 such that 2 < 3 ^ (( A 3 )=2), andj2f3;Ag, N r X n=1 P sup 0t<r T E r;n j (t) r j t > 2 r T; r 2n1 r 2 T ! ! 0; asr!1: (1.80) Note that Lemma 1.8.2 extends Lemma 9 of Ata and Kumar (2005) to a renewal process that starts from a random time. Consequently, the sum in (1.66) converges to 0 asr!1. Remark 1.8.1. The sum in (1.66) converges to 0 for all > 0 asr!1. We need < 1 in the next section (see (1.84)). Proof of Convergence of (1.67) In this section, we will show that the sum in (1.67) converges to 0 asr!1. The sum in (1.67) is less than or equal to N r X n=1 P r 2n1 r 2 T; (E r n;1 ) c + N r X n=1 P sup 0t<r T E r;n 3 (t)E r;n A (t)>r 1; r 2n1 r 2 T;E r n;1 ! ; (1.81) 35 whereE r n;1 is defined in (1.71). Note that the first sum in (1.81) converges to zero with the same way (1.77) does. The second sum in (1.81) is less than or equal to N r X n=1 P sup 0t<r T E r;n 3 (t)E r;n A (t)>r 1;E r n;1 ! N r X n=1 P sup 0t<r T r 3 t + 2 r T ( r A t 2 r T )>r 1 ! (1.82) =N r P (2 2 r T >r 1) (1.83) =N r I (2 2 r T >r 1)! 0 asr!1; (1.84) where (1.82) is by (1.74), (1.83) is by the fact that r A > r 3 whenr is sufficiently large, and (1.84) is by the fact that < 1. Therefore, we prove that (1.55) converges to 0 asr!1. In the next section, we will prove that (1.56) converges to 0 asr!1. 1.8.2 Proof of Convergence of (1.56) (Down Intervals) In this section, we consider the down intervals and prove the convergence of (1.56). LetN r 2 :=d( r A + 3 )r 2 Te where 3 > 0 is an arbitrary constant. Since there is a service completion of a typea job in server 4 at r 2n for alln2N + , then P r 2N r 2 r 2 T P S r A (r 2 T )N r 2 P S r A (r 2 T ) ( r A + 3 )r 2 T ! 0 asr!1; (1.85) where (1.85) is by FSLLN. Let r n (t) :=jQ r 4 (t)Q r 3 (t)^W r 4 (t)j: (1.86) Then, the probability in (1.56) is less than or equal to P r 2N r 2 r 2 T +P sup t2[0;r 2 T ]\ S 1 n=0 [ r 2n ; r 2n+1 ) r n (t)>r; r 2N r 2 >r 2 T ! : (1.87) 36 Note that the first probability in (1.87) converges to 0 asr!1 by (1.85) and the second probability in (1.87) is less than or equal to N r 2 X n=0 P sup r 2n t< r 2n+1 ^r 2 (T + 4 ) r n (t)>r; r 2n r 2 T ! = N r 2 X n=0 P " sup r 2n t r; 2n ^r 2 (T + 4 ) r n (t) ! _ sup r; 2n ^r 2 (T + 4 )<t< r 2n+1 ^r 2 (T + 4 ) r n (t) !# I( r; 2n <1) + " sup r 2n t< r 2n+1 ^r 2 (T + 4 ) r n (t) # I( r; 2n =1)>r; r 2n r 2 T ! ; (1.88) where 4 > 0 is an arbitrary constant introduced to cover the time instantr 2 T . For completeness, we define sup t2; X(t) = 0 for anyX2 D. By definition (cf. (1.57)), r; 2n =1 implies that down 2 interval does not exist within [ r 2n ; r 2n+1 ), thus buffer 5 never becomes empty during the corresponding down interval. Hence, server 4 does not work on buffer 4 during the same interval and the down interval ends with the first service completion in server 3. Therefore, when r; 2n =1, Q r 4 (t) = Q r 3 (t) and r n (t) = 0 for all t2 [ r 2n ; r 2n+1 ) by (1.86) and the fact thatQ r 4 (t) W r 4 (t) for allt2 R + (cf. (1.16)). Moreover, when r; 2n <1, by the same logic, we haveQ r 4 (t) =Q r 3 (t) and r n (t) = 0 for allt2 [ r 2n ; r; 2n ]. Therefore, the only nonzero term in (1.88) is the one associated with the down 2 interval ( r; 2n ; r 2n+1 ), and (1.88) is equal to N r 2 X n=0 P " sup r; 2n ^r 2 (T + 4 )<t< r 2n+1 ^r 2 (T + 4 ) r n (t) # I( r; 2n <1)>r; r 2n r 2 T ! : (1.89) From the preceding argument, it is enough to show that the term in (1.89) converges to 0 asr!1. The term in (1.89) is equal to N r 2 X n=0 P " sup r; 2n <t< r 2n+1 ^r 2 (T + 4 ) r n (t) # I( r; 2n <1)>r; r 2n r 2 T; r; 2n r 2 (T + 4 ) ! N r 2 X n=0 P sup r; 2n <t< r 2n+1 ^r 2 (T + 4 ) r n (t)>r; r; 2n r 2 (T + 4 ) ! = N r 2 X n=0 P sup r; 2n <t< r 2n+1 ^r 2 (T + 4 ) Q r 3 (t)^W r 4 (t)Q r 4 (t)>r; r; 2n r 2 (T + 4 ) ! (1.90) N r 2 X n=0 P sup r; 2n <t< r 2n+1 ^r 2 (T + 4 ) r A r B Q r 5 (t)>r; r; 2n r 2 (T + 4 ) ! ; (1.91) 37 where (1.90) is by (1.16), (1.86), and the fact thatQ r 4 (t)Q r 3 (t) for allt2 ( r; 2n ; r 2n+1 ) by construction; and (1.91) is by (1.16). Let 5 := inf rr 1 ( r B = r A ) for somer 1 1. By Assumption 1.3.3 Part 2, 5 > 0 whenr 1 is sufficiently large. Then, whenrr 1 , the sum in (1.91) is less than or equal to N r 2 X n=0 P sup r; 2n <t< r 2n+1 ^r 2 (T + 4 ) Q r 5 (t)>r 5 ; r; 2n r 2 (T + 4 ) ! : (1.92) Note that, when r; 2n <1,Q r 5 ( r; 2n ) = 0 for alln2N by construction. Each probability within the sum (1.92) resembles the probability that the supremum of the buffer length of the downstream queue in a pair of single server queues in tandem is at leastO(r) within a down 2 interval. Since server 4 gives priority to buffer 5 during a down 2 interval, and by Assumption 1.3.3 Parts 1, 2, and 3, whenr is sufficiently large, the arrival rate of typeb jobs, r b , is strictly less than the rate at which server 4 processes type b jobs, r B , we expect this probability to be exponentially small, by Proposition 1.8.1. Then, the sum in (1.92), which hasO(r 2 ) terms, should converge to 0. The issue is that we cannot directly apply Proposition 1.8.1, because of (i) the random variables r; 2n and r 2n+1 and (ii) the verification of the initial condition (1.58). Instead, we use Proposition 1.8.2 to construct an upper bound queueing system that satisfies the conditions of Proposition 1.8.1. Construction of an upper bound queueing system The approach to constructing the upper bound queueing system is different depending on if b < 2 (in which case the upper bound system will be a pair of tandem queues) or b = 2 (in which case the upper bound system will be a standalone GI=GI=1 queue). This is because if b < 2 , server 2 experiences non-negligible idletime, which affects its departure process. Then, the key is to construct a “good set” on which we can control the size of buffer 2. Otherwise, we think of server 2 as being always busy, which is simpler. In both cases, Proposition 1.8.1 will apply to the constructed upper bound queueing system because b < B , as observed in the preceding paragraph. Let,N r 3 :=d( r b + r 2 + r B )r 2 (T + 4 )e and for alln2N, ~ r; 2n := 8 > < > : inf t r; 2n :S r 2 (T r 2 (t))S r 2 (T r 2 ( r; 2n )) = 1 ; if r; 2n <1, +1; otherwise: (1.93) Then, ~ r; 2n is the first time server 2 makes a service completion after r; 2n ,Q r 5 (t) = 0 for allt2 [ r; 2n ; ~ r; 2n ), andQ r 5 (~ r; 2n ) = 1. On the set r; 2n <1, if ~ r; 2n > r 2 (T + 4 ), then the next job arrival to buffer 5 will be 38 afterr 2 (T + 4 ) by (1.93), which implies that sup r; 2n <t<r 2 (T + 4 ) Q r 5 (t) = 0. Hence, the sum in (1.92) is equal to N r 2 X n=0 P sup ~ r; 2n t< r 2n+1 ^r 2 (T + 4 ) Q r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 ) ! : (1.94) Case 1: b < 2 In this section, we prove that the sum in (1.94) converges to 0 asr!1 under the assumption that b < 2 . Let for alln2N, E r n;2 := n S r b (~ r; 2n )_S r 2 (T r 2 (~ r; 2n ))_S r B (T r B (~ r; 2n ))N r 3 ; sup 0tr 2 (T + 4 ) Q r 2 (t)L r 1 o ; (1.95) whereL r := (( r 2 ^ r B r b )^ 5 )r=7. Then the sum in (1.94) is less than or equal to N r 2 X n=0 P sup ~ r; 2n t< r 2n+1 ^r 2 (T + 4 ) Q r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 );E r n;2 ! (1.96) + N r 2 X n=0 P ~ r; 2n r 2 (T + 4 ); (E r n;2 ) c : (1.97) The sum in (1.97) converges to 0 becauseQ r 2 is aGI=GI=1 queue with arrival rate strictly less than service rate when r is sufficiently large, and the number of external type b job arrivals and service completions associated with the activities 2 andB in an interval of lengthO(r 2 ) can be bounded with high probability using the rate of the renewal processes. We have the following result whose proof is presented in Appendix A.2.7. Lemma 1.8.3. Asr!1, P N r 2 n=0 P ~ r; 2n r 2 (T + 4 ); (E r n;2 ) c ! 0. To complete the proof of Proposition 1.7.1 when b < 2 , it remains only to show the sum in (1.96) converges to 0 asr!1. We do this in the next several paragraphs by using sample path arguments to construct an upper bound (depending onr) for each term in the sum in (1.96), and then showing this upper bound converges to 0 asr!1. First, we coupleQ r 2 andQ r 5 with an hypothetical pair of two single server queues in tandem in which typeb jobs are given priority even beyond time r 2n+1 in server 4, so that the random time at the end of the interval (~ r; 2n ; r 2n+1 ^r 2 (T + 4 )) can be ignored. Then, we create the second hypothetical queueing system in which the first external typeb job arrival occurring after time ~ r; 2n occurs at ~ r; 2n , in order to derive a pathwise upper bound on the largest queue seen over a time interval in buffer 5 by Proposition 1.8.2. Next, we create the third hypothetical queueing system which is a pair of two single server queues in tandem with i.i.d. inter-arrival and service times. Finally, we invoke Proposition 1.8.1 on the third hypothetical queueing system to obtain the desired convergence result. 39 For each n2f0; 1;:::;N r 2 g, let us define a new hypothetical queueing system in which we use the same control until r 2n+1 but after this time epoch, server 4 always gives preemptive priority to buffer 5 over buffer 4. We call this hypothetical system as “system 1-n”. Let the queue length process for bufferk, k2K beQ (1);n;r k in system 1-n. Then, for alln2f0; 1;:::;N r 2 g, we haveQ (1);n;r 5 (~ r; 2n ) =Q r 5 (~ r; 2n ) = 1, Q (1);n;r 5 (t) = Q r 5 (t) for allt < r 2n+1 , andQ (1);n;r 2 (t) = Q r 2 (t) for allt2 R + by construction. Then, the sum in (1.96) is less than or equal to N r 2 X n=0 P sup ~ r; 2n t<r 2 (T + 4 ) Q (1);n;r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 );E r n;2 ! N r 2 X n=0 N r 3 X i=0 N r 3 X j=0 N r 3 X k=0 L r 1 X l=0 P sup ~ r; 2n t<r 2 (T + 4 ) Q (1);n;r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 ); S r b (~ r; 2n ) =i; S r 2 (T r 2 (~ r; 2n )) =j; S r B (T r B (~ r; 2n )) =k; Q (1);n;r 2 (~ r; 2n ) =l ! ; (1.98) where (1.98) is by (1.95). Next, for each n2f0; 1;:::;N r 2 g, let us construct another hypothetical queueing system, which we call “system 2-n”, by modifying system 1-n in the following way. Let the queue length process for buffer k, k2K be Q (2);n;r k in system 2-n. When ~ r; 2n r 2 (T + 4 ), the next external type b job arrival after time ~ r; 2n , which occurs at time V r b (S r b (~ r; 2n ) + 1), occurs instead at time ~ r; 2n . The subsequent external inter-arrival times are againfv r b (S r b (~ r; 2n ) +m);m 2g. Hence, in the system 2-n, the first external inter- arrival time after ~ r; 2n is shorter than the one in the system 1-n, and all other external inter-arrival times are the same with the ones in the system 1-n. Service times associated with the activity 2 after ~ r; 2n are fv r 2 (S r 2 (T r 2 (~ r; 2n )) +m);m2 N + g. SinceQ r 5 (t) = Q (1);n;r 5 (t) = Q (2);n;r 5 (t) = 0 for allt2 [ r; 2n ; ~ r; 2n ) and soS r B (T r B ( r; 2n )) = S r B (T r B (~ r; 2n )), then the service times associated with the activityB after ~ r; 2n are fv r B (S r B (T r B (~ r; 2n )) +m);m2 N + g. Hence, the service times are the same with the ones in system 1-n. Therefore, inter-departure times from server 2 in the system 2-n are shorter than the ones in the system 1-n by Proposition 1.8.2. Since inter-departure times from server 2 are the inter-arrival times to buffer 5, a second application of Proposition 1.8.2 shows that sup ~ r; 2n t<r 2 (T + 4 ) Q (1);n;r 5 (t) sup ~ r; 2n t<r 2 (T + 4 ) Q (2);n;r 5 (t): 40 Moreover, Q (2);n;r 2 (~ r; 2n ) = Q (1);n;r 2 (~ r; 2n ) + 1 andQ (2);n;r 5 (~ r; 2n ) = Q (1);n;r 5 (~ r; 2n ) = 1. Then, the sum in (1.98) is less than or equal to N r 2 X n=0 N r 3 X i=0 N r 3 X j=0 N r 3 X k=0 L r X l=1 P sup ~ r; 2n t<r 2 (T + 4 ) Q (2);n;r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 ); S r b (~ r; 2n ) =i; S r 2 (T r 2 (~ r; 2n )) =j; S r B (T r B (~ r; 2n )) =k; Q (2);n;r 2 (~ r; 2n ) =l ! ; (1.99) and the external inter-arrival times and the service times in system 2-n after ~ r; 2n are fv r b (S r b (~ r; 2n ) +m);m 2g;fv r 2 (S r 2 (T r 2 (~ r; 2n )) +m);m2N + g;fv r B (S r B (T r B (~ r; 2n )) +m);m2N + g; (1.100) respectively. At this point we use the i.i.d. property of the external inter-arrival and service times. For all n2f0; 1;:::;N r 2 g and i;j;k2f0; 1;:::;N r 3 g, the external inter-arrival times to buffer 2 and the ser- vice times associated with the activities 2 and B are equal tofv r b (m); m2N + g,fv r 2 (m); m2N + g, fv r B (m); m2N + g in distribution, respectively, by (1.100) in (1.99). Then, for eachl2f1; 2;:::;L r g, let us construct a hypothetical queueing system, which is called “system 3-l”, where the buffer length process is denoted byQ (3);l;r k , k2K, the external inter-arrival times to buffer 2 and the service times associated with the activities 2 andB arefv r b (m); m2N + g,fv r 2 (m); m2N + g,fv r B (m); m2N + g, respectively, Q (3);l;r 2 (0) =l andQ (3);l;r 5 (0) = 1. Then, the sum in (1.99) is less than or equal to N r 2 (N r 3 ) 3 L r X l=1 P sup 0tr 2 (T + 4 ) Q (3);l;r 5 (t)>r 5 ! : (1.101) Let us fix an arbitraryl2f1; 2;:::;L r g. Consider a hypothetical queueing system which is the same of the system 3-l except thatv r b (m) = 0 for allm2f1; 2;:::;L r lg; i.e., the initial queue length in buffer 2 isL r instead ofl. In the hypothetical queueing system, since the external inter-arrival times to buffer 2 are shorter than the ones in system 3-l, inter-departure times from server 2 and so the inter-arrival times to buffer 5 are also shorter and the supremum of the queue length seen in buffer 5 is larger than the one in system 3-l by Proposition 1.8.2. Therefore, the sum in (1.101) is bounded above by N r 2 (N r 3 ) 3 L r P sup 0tr 2 (T + 4 ) Q (3);L r ;r 5 (t)>r 5 ! ! 0; asr!1; (1.102) 41 where the convergence is by Proposition 1.8.1 and the fact that b < 2 ^ B , Q (3);L r ;r 2 (0) = L r , and Q (3);L r ;r 5 (0) = 1. Case 2: b = 2 In this section, we prove that the sum in (1.94) converges to 0 asr!1 under the assumption that b = 2 . Now, Lemma 1.8.3 does not hold because we cannot show that the largest queue seen in buffer 2 over a time interval with lengthO(r 2 ) is bounded above byL r 1 with high probability when b = 2 . However, we can solve this problem by modifying the part of the proofs starting from (1.95) in the following way. First, we modify (1.95) by E r n;2 := S r B (T r B (~ r; 2n ))N r 3 : (1.103) Then, again, the sum in (1.94) is less than or equal to the sum of the terms in (1.96) and (1.97), and the sum in (1.97) converges to 0 asr!1 by the convergence of the sum in (A.97) to 0 (cf. the proof of Lemma 1.8.3 in Appendix A.2.7, specifically see (A.99)). We again use sample-path arguments to construct an upper bound (depending onr) to the sum in (1.96). First, let us construct the hypothetical system 1-n as before for alln2f0; 1;:::;N r 2 g. By (1.3b) and the fact thatQ (1);n;r 5 (~ r; 2n ) = 1 on the setf~ r; 2n r 2 (T + 4 )g, S r 2 (T r 2 (~ r; 2n )) =S r B (T r B (~ r; 2n )) + 1; (1.104) on the setf~ r; 2n r 2 (T + 4 )g. By (1.103) and (1.104), the sum in (1.96) is less than or equal to N r 2 X n=0 N r 3 X i=0 P sup ~ r; 2n t<r 2 (T + 4 ) Q (1);n;r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 ); S r 2 (T r 2 (~ r; 2n ))1 =S r B (T r B (~ r; 2n )) =i ! : (1.105) Next, let us construct the hypothetical system 2-n in the following way for all n2f0; 1;:::;N r 2 g. We assume that at time ~ r; 2n , all remaining external type b jobs arrive immediately to buffer 2; i.e., all of the external inter-arrival times are 0 after ~ r; 2n . Thus,Q (2);n;r 2 (~ r; 2n ) =1 and so server 2 never becomes idle after time ~ r; 2n . Service times associated with the activities 2 andB after ~ r; 2n arefv r 2 (S r 2 (T r 2 (~ r; 2n )) +m);m2 N + g andfv r B (S r B (T r B (~ r; 2n )) +m);m2N + g; i.e., the service times are the same with the ones in system 1-n. Hence, inter-departure times from server 2 in the system 2-n are shorter than the ones in the system 42 1-n by Proposition 1.8.2. Since inter-departure times from server 2 are the inter-arrival times to buffer 5, a second application of Proposition 1.8.2 shows that sup ~ r; 2n t<r 2 (T + 4 ) Q (1);n;r 5 (t) sup ~ r; 2n t<r 2 (T + 4 ) Q (2);n;r 5 (t): Moreover,Q (2);n;r 5 (~ r; 2n ) =Q (1);n;r 5 (~ r; 2n ) = 1. Then, the sum in (1.105) is less than or equal to N r 2 X n=0 N r 3 X i=0 P sup ~ r; 2n t<r 2 (T + 4 ) Q (2);n;r 5 (t)>r 5 ; ~ r; 2n r 2 (T + 4 ); S r 2 (T r 2 (~ r; 2n ))1 =S r B (T r B (~ r; 2n )) =i ! ; (1.106) and the inter-arrival times to buffer 5 and service times associated with the activityB in system 2-n after ~ r; 2n are fv r 2 (S r 2 (T r 2 (~ r; 2n )) +m);m2N + g; fv r B (S r B (T r B (~ r; 2n )) +m);m2N + g; (1.107) respectively. At this point we use the i.i.d. property of the service times. For all n 2 f0; 1;:::;N r 2 g and i 2 f0; 1;:::;N r 3 g, the inter-arrival times to buffer 5 and the service times associated with the activ- ity B are equal tofv r 2 (m); m2N + g andfv r B (m); m2N + g in distribution, respectively, by (1.107) in (1.106). Let us construct a hypothetical GI=GI=1 queue, which is called “system 3”, where the buffer length process is denoted byQ (3);r 5 , the inter-arrival and service time sequences arefv r 2 (m); m2N + g and fv r B (m); m2N + g, respectively, andQ (3);r 5 (0) = 1. Then, the sum in (1.106) is less than or equal to N r 2 N r 3 P sup 0tr 2 (T + 4 ) Q (3);r 5 (t)>r 5 ! ! 0; asr!1; (1.108) where the convergence is by Proposition 1.8.1 applied to a standalone GI=GI=1 queue and the fact that 2 < B andQ (3);r 5 (0) = 1. Remark 1.8.2. We assume thatQ r k (0) = 0 for allk2K up to now. However, it is straightforward to see that the results presented so far hold under the following weaker assumption. Assumption 1.8.1. For eachr, (Q r k (0);k2K) is a nonnegative random vector which takes values inN 10 and is independent of all other stochastic primitives such that 1. Q r 3 (0) =Q r 4 (0),Q r 7 (0) =Q r 8 (0),Q r 5 (0) =Q r 6 (0), andQ r 9 (0) =Q r 10 (0) for eachr. 2. r 1 Q r k (0)) 0 andr 2 Q r k (0) a:s: ! 0 for allk2K asr!1. 43 3. If b < 2 , then for all> 0, there exists anr 0 () 1 such that ifrr 0 (), P (Q r 2 (0)r)C 0 r 3 e C 1 r ; for some constantsC 0 > 0 andC 1 > 0 which are independent ofr. 4. There exists anr 1 1 such that sup rr 1 E ^ Q r k (0) 2 <1; 8k2K: Assumption 1.8.1 Part 1 guarantees that the initial buffer lengths do not violate (1.11) at time t = 0. We need Assumption 1.8.1 Part 2 because of two main reasons. First, since we consider cases in which some of the servers are in light traffic, in order to obtain the weak convergence of the diffusion scaled queue length processes corresponding to these servers to 0, we need the first convergence result in Assumption 1.8.1 Part 2. Second, in order to obtain the a.s. convergence of the fluid scaled queue length processes in each buffer to 0 u.o.c., we need the second convergence result in Assumption 1.8.1 Part 2. We need Part 3 of this assumption in order to invoke Proposition 1.8.1 in the proof of Lemma 1.8.3, specifically in (A.100). Lastly, we need Assumption 1.8.1 Part 4 in order to satisfy the required uniform integrability condition in the proof of Theorem 1.6.2 (cf. condition (A.54)). Remark 1.8.3. It is possible to weaken the exponential moment assumption (cf. Assumption 1.3.2) for some of the inter-arrival and service time processes in order to prove the asymptotic optimality of the proposed policy with respect to the objectives (1.7), (1.8), and (1.12) (cf. Corollary 1.5.2, Theorem 1.6.1, and Theorem 1.6.2). First, in Section 1.8.1, we use the exponential moment assumption only for the service time processes associated with activities 3 and A in the proof of Lemma 1.8.2. However, we can relax this assumption in the following way. We need 4=(1 +) < < 1 (cf. proof of Lemma 1.8.2). Since 2 (0; 1) can be chosen arbitrarily close to 1, any strictly greater than 3 satisfies 4=(1 +) < < 1. Since we needE[(v r j (1)) 2+2 ] <1 forj2f3;Ag in the proof of Lemma 1.8.2 (cf. (A.92) and (A.94)), E[(v r j (1)) 8+ ]<1,j2f3;Ag for some > 0 is sufficient to get the convergence result associated with (1.55). Second, in Section 1.8.2, we have used the exponential moment assumption for the inter-arrival times of typeb jobs (cf. (1.102) and (A.100), which is in the proof of Lemma 1.8.3), service times associated with activity 2 (cf. (1.102), (1.108), and (A.100)), and service times associated with activityB (cf. (1.102) and (1.108)). For the inter-arrival times of typea jobs and the service times associated with activities 1, 5, 44 6, and 7, we only need finite second moment assumption to use the FCLT in Sections 1.7.1 and 1.7.2 and invoke Lemma A.2.1 in the proof of Theorem 1.6.2. 1.9 Simulation In this Section, we use discrete-event simulation to test the performance of the non-preemptive version of the proposed policy which is described in Remark 1.5.3. We consider 36 different test instances and at each instance, we compare the performance of the non-preemptive version of the proposed policy with the performances of 4 other non-preemptive control policies. The instances are designed to consider various cases related to the processing capacities of the servers and variability in the service times. We describe the simulation setup in Section 1.9.1, then we present the results of the experiments in Section 1.9.2. 1.9.1 Simulation Setup At each instance, the arrival processes of type a and b jobs are independent Poisson processes with rate one, thus a = b = 1 2 . At each instance, servers 5, 6, and 7 are in either heavy or light traffic, where heavy and light traffic mean 95% and 70% long-run utilization rates of the corresponding server, respectively. For example, when server 6 is in heavy (light) traffic at an instance, then 6 = 1=0:95 ( 6 = 1=0:7), which gives the desired long-run utilization rate. At each instance, servers 1 and 2 are in light traffic and server 4 is in heavy traffic such that A = B = 2=0:95. At each instance, we assume that 3 2 f1=0:95; 1=0:7125; 1=0:35g to test the performance of the proposed policy when a 3 , a < 3 < A , and 3 > A , respectively. We use three different distribution types for the service time processes, namely Erlang-3, Exponential, and Gamma distributions. When the service time process associated with an activity is Gamma distributed, we choose the distribution parameters such that the squared coefficient of variation of the corresponding service time process is 3. Note that the squared coefficient of variations in Erlang-3 and Exponential distributions are 1=3 and 1, respectively. Therefore, Erlang-3, Exponential, and Gamma distributions correspond to the low, moderate, and high level variability in the service time processes, respectively. At each instance, we use the same distribution type for all service time processes. Table 1.1 shows the parameter choices related to the service time processes at each instance. For example, at instance 7, 3 = 1=0:7125, server 5 is in heavy traffic, and servers 6 and 7 are in light traffic. Moreover, the service time processes associated with each activity are Erlang-3 distributed at this instance. In the 2 For convenience in notation, we have dropped the superscriptr on the parameters. The reader should understand that each set of parameters (a; b ;1;2;:::;7) is associated with a particularr, and is not the limit parameter that appear in Assumption 1.3.3. 45 first 18 instances, the downstream servers, i.e. servers 6 and 7, are in light traffic. Note that we prove the asymptotic optimality of the proposed policy under this assumption (cf. Assumption 1.3.3 Part 6). However, in order to test the robustness of the proposed policy, we consider the cases in which server 6 or 7 is in heavy traffic in the last 18 instances. Table 1.1: Parameter choices at each instance. 3 5 6 7 Service Time Variability Instance 1=0:95 1=0:7125 1=0:35 1=0:95 1=0:7 1=0:95 1=0:7 1=0:95 1=0:7 Low Moderate High 1 X X X X X 2 X X X X X 3 X X X X X 4 X X X X X 5 X X X X X 6 X X X X X 7 X X X X X 8 X X X X X 9 X X X X X 10 X X X X X 11 X X X X X 12 X X X X X 13 X X X X X 14 X X X X X 15 X X X X X 16 X X X X X 17 X X X X X 18 X X X X X 19 X X X X X 20 X X X X X 21 X X X X X 22 X X X X X 23 X X X X X 24 X X X X X 25 X X X X X 26 X X X X X 27 X X X X X 28 X X X X X 29 X X X X X 30 X X X X X 31 X X X X X 32 X X X X X 33 X X X X X 34 X X X X X 35 X X X X X 36 X X X X X Even though ^ W r 4 weakly converges to the same limit independent of the control (cf. Proposition 1.3.2), ^ W r 4 is policy dependent in the pre-limit. Since our class of admissible scheduling policies is very large, and includes the ones that can anticipate the future, we cannot construct a pre-limit lower bound based on the solution of DCP (1.29) given in Proposition 1.4.1. Hence, there is no pre-limit lower bound on performance. Furthermore, we cannot use the DCP solution to develop an approximate pre-limit lower bound on the average holding cost, because that requires knowing the stationary distribution of the relevant SRBM and that is only straightforward to find under very specific conditions (cf. Harrison and Williams (1987), Dai and Harrison (1992), Dieker (2011), Dai et al. (2014)). The relevant SRBM does not have a product form stationary distribution in our case (cf. Harrison and Williams (1987)), when it is at least two-dimensional. 46 Therefore, we compare the performances of 5 different non-preemptive control policies. The first one is the non-preemptive version of the proposed policy (see Remark 1.5.3 for the definition). Since the proposed policy is the SDP policy and the static priority policy when a 3 < A and 3 A , respectively, the second and the third control policies that we consider are the non-preemptive versions of the SDP and the static priority policies, respectively. The fourth policy that we consider is the FCFS policy, in which whenever server 4 is ready to process a new job, it chooses the job which has arrived the earliest to the buffers 4 or 5. The fifth policy that we consider is the randomized policy, in which whenever server 4 is ready to process a job and if both of the buffers 4 and 5 are non-empty, server 4 chooses a typea job with half probability. If only one of the buffers 4 and 5 is non-empty, then server 4 chooses the job from the non-empty one. We have used Omnet++ discrete-event simulation freeware in our experiments. At each instance asso- ciated with each control policy that we considered, we have done 30 replications. At each replication, we have created approximately 1 million typea jobs and 1 million typeb jobs and we have considered the time interval in which the first 50; 000 typea jobs and the first 50; 000 typeb jobs arrive as the warm-up period. 1.9.2 Simulation Results In this section, we present the results of the simulation experiments. We assume thath b = 1 in all experi- ments and consider varioush a values such thath a h b . In each experiment, we use the sameh a value at all instances. LetP denote the set of the five control policies that we consider in the simulation experiments andQ p;j k (i) denote the average length of bufferk,k2K in replicationj,j2f1; 2;:::; 30g with respect to policyp, p2P at instance i, i2f1; 2;:::; 36g. Then, Q p k (i) := (1=30) P 30 j=1 Q p;j k (i) is the average length of bufferk corresponding to policyp at instancei. Note that, we have proved the asymptotic optimality of the proposed policy with respect to the objectives (1.7), (1.8), and (1.12) but not with respect to the average cost objective (1.9) (cf. Section 1.6). Still, our results suggest that our proposed policy performs well with respect to (1.9), which is a natural objective to consider in simulation experiments. Hence, we use the objective (1.9). Recalling the equality (1.11), we only need to computeJ p (i) :=h a Q p 3 (i) + Q p 7 (i) +h b Q p 6 (i) + Q p 10 (i) , which is the average total holding cost in buffers 3, 6, 7, and 10 per unit time corresponding to policyp at instancei. LetL(i) := min p2P J p (i) denote the lowest average cost among the policies inP at instancei. We present the detailed results of the simulation experiments in Table A.1, which is in Appendix A.3. This table contains Q p 3 (i) + Q p 7 (i) and Q p 6 (i) + Q p 10 (i) for eachi andp with their 95% confidence intervals. 47 Table 1.2 shows the average and maximum deviations of the cost of the policies from the lowest realized average costs among the first and last 18 instances for differenth a values. For example, for givenh a , the “Avg.” and “Max.” columns corresponding to policyp and the first 18 instances are 1 18 18 X i=1 100 J p (i)L(i) L(i) ; max i2f1;2;:::;18g 100 J p (i)L(i) L(i) ; respectively. In the “All” row, “Avg.” (“Max.”) column corresponding to policy p denotes the average (maximum) of the values in the “Avg.” (“Max.”) column among allh a values. Table 1.2: Average and maximum deviations of the cost of the policies from the lowest realized average cost. First 18 instances Proposed SDP Static FCFS Randomized h a Avg. Max. Avg. Max. Avg. Max. Avg. Max. Avg. Max. 1 1:6% 6:4% 1:8% 6:5% 9:1% 30:7% 3:7% 9:5% 4:1% 8:7% 1:25 0:0% 0:3% 0:3% 1:4% 6:2% 25:8% 6:2% 15:3% 6:7% 14:6% 1:5 0:1% 0:8% 0:5% 1:7% 5:3% 22:2% 10:0% 20:7% 10:6% 20:2% 1:75 0:2% 1:4% 0:7% 2:0% 4:6% 19:3% 13:6% 25:9% 14:2% 25:4% 2 0:3% 2:0% 0:8% 2:3% 4:0% 17:1% 16:9% 30:8% 17:5% 30:4% 3 0:8% 4:1% 1:6% 4:1% 2:5% 11:3% 28:5% 48:0% 29:2% 48:0% 4 1:3% 5:8% 2:5% 5:8% 1:8% 8:4% 37:8% 65:3% 38:6% 65:5% 10 3:1% 11:6% 5:2% 11:6% 0:4% 2:8% 68:9% 125:1% 70:1% 123:3% All 0:9% 11:6% 1:7% 11:6% 4:2% 30:7% 23:2% 125:1% 23:9% 123:3% Last 18 instances 1 4:1% 17:0% 4:1% 16:1% 9:5% 25:6% 0:8% 5:2% 1:6% 6:5% 1:25 2:3% 11:4% 2:4% 11:4% 6:9% 22:1% 1:6% 8:4% 2:4% 10:1% 1:5 1:4% 7:4% 1:5% 7:4% 5:3% 19:3% 2:9% 11:1% 3:8% 13:0% 1:75 0:7% 3:8% 0:8% 3:8% 4:0% 17:3% 4:1% 13:3% 5:0% 15:5% 2 0:2% 1:4% 0:4% 1:4% 3:2% 15:8% 5:3% 15:3% 6:2% 17:7% 3 0:4% 2:0% 0:7% 2:0% 2:1% 11:5% 10:9% 24:8% 11:8% 26:1% 4 0:6% 2:5% 1:1% 2:5% 1:6% 8:8% 15:2% 35:9% 16:2% 37:3% 10 1:4% 7:0% 2:3% 7:0% 0:4% 2:5% 29:2% 81:9% 30:3% 84:0% All 1:4% 17:0% 1:7% 16:1% 4:1% 25:6% 8:7% 81:9% 9:7% 84:0% According to the results, the proposed policy in general performs the best with respect to both the average and maximum deviations from the lowest realized average cost. The SDP policy performs very close to the proposed policy, which shows that it performs well even when 3 > A . As expected, the performance of the static priority policy becomes better ash a increases, and it performs the best whenh a is much larger thanh b . Performances of the FCFS and the randomized policies become worse ash a increases, which is expected because these policies do not give more priority to typea jobs than typeb jobs. On average, these two policies perform the worst with respect to both the average and maximum deviations from the lowest realized average cost. At the first 18 instances, whenh a = 10, the static priority and the proposed policies perform the best and the second best, respectively. This is not surprising given that in all of the instances in which server 3 is in light traffic, both the static priority and the proposed policies are asymptotically optimal (cf. Remarks 1.5.2 48 and 1.7.1). The superior performance of the static priority policy in the pre-limit can be attributed to the high holding cost of typea jobs. At the last 18 instances, the proposed policy still performs the best in average, which suggests that the performance of the proposed policy is robust with respect to the processing capacities of the downstream servers. Another interesting result that we see from Table 1.2 is that the percentage deviation of the worst per- forming policy from the lowest realized average cost at the first 18 instances is much higher than the same deviation at the last 18 instances. This result is due to the fact that at least one of the downstream servers is in heavy traffic at the last 18 instances and this decreases the waiting time of the jobs for the ones that they are going to be matched in the join servers. In other words, synchronization requirements between the jobs in different buffers become less important in this case. Hence, the effect of the control policy on the system performance is less important when at least one of the downstream servers is in heavy traffic, which justifies Assumption 1.3.3 Part 6. Moreover, FCFS policy performs the best whenh a = 1 at the last 18 instances because all jobs are equally expensive and this policy approximately matches the arrival time of the same type of jobs to the downstream buffers. Figure 1.4 shows 100 (J p (i)L(i))=L(i) for each policyp,p2P at eachi,i2f1; 2;:::; 18g when h a = 2 andh b = 1. Since A = B andh a = 2h b , it makes more sense to give more priority to typea jobs. This is why in addition to the policies inP, we also consider a randomized-2=3 policy, in which whenever server 4 becomes available and if both of the buffers 4 and 5 are non-empty, server 4 chooses a typea job with 2=3 probability. If only one of the buffers 4 and 5 is non-empty, then server 4 chooses the job from the non-empty one. In Figure 1.4a, we see that the proposed and the SDP policies perform well at each instance. SDP policy performs well even at the instancesf13; 14;:::; 18g, where 3 > A . However, the static priority policy does not perform well in the first 6 instances, where server 3 is in heavy traffic. This result is expected because when server 4 gives static priority to buffer 4 under the condition that server 3 is in heavy traffic, jobs in buffer 8 will wait for the jobs in buffer 7. In this case, it is more efficient to give priority to the type b jobs in server 4 and this is exactly what the SDP policy does. The static priority policy performs the best at the instancesf13; 14;:::; 18g, where 3 > A . Moreover, its performance is close to the one of the SDP policy at the instancesf7; 8;:::; 12g, where a < 3 < A . This result is not surprising because static priority policy is asymptotically optimal when server 3 is in light traffic (cf. Remark 1.7.1). On the one hand, at the instancesf7; 8; 9g, the static priority policy performs better than the SDP policy because server 5 is in heavy traffic at these instances, hence server 4 should not give priority to the typeb jobs at all. On the 49 other hand, at the instancesf10; 11; 12g, SDP policy performs better than the static priority policy, because server 5 is in light traffic at these instances; hence giving priority to typeb jobs in server 4 decreases the waiting time of the jobs in buffer 10. 1 6 12 18 Instance number 0 5 10 15 20 % deviation Proposed SDP Static (a) Proposed, SDP, and static policies. 1 6 12 18 Instance number 0 5 10 15 20 25 30 35 % deviation Proposed Rand.-2/3 FCFS Rand. (b) Proposed, FCFS, rand., and rand.-2=3 policies. Figure 1.4: (Color online) Percentage deviations of the costs of the policies from the lowest realized average cost (L(i) fori2f1; 2;:::; 18g) in the first 18 instances whenh a = 2 andh b = 1. Figure 1.4b shows that the FCFS and the randomized policies perform worse than the proposed policy at each instance. The randomized-2=3 policy performs well only at the instancesf7; 8;:::; 12g. Figure 1.5, where we compare the static priority, randomized, and randomized-2=3 policies, explains this result in the following way. On the one hand, at the first 6 instances, we see that the randomized and randomized-2=3 policies perform the best and the second best among these three policies, respectively. This implies that when server 3 is in heavy traffic, giving more priority to type a jobs decreases the performance. On the other hand, at the instancesf7; 8;:::; 18g, the static priority and randomized-2=3 policies perform the best and the second best among these three policies, respectively. This implies that when server 3 is in light traffic, giving more priority to typea jobs increases the performance. In summary, the processing capacity of server 3 affects the performance of the control policy significantly. This is consistent with our theoretic results showing that an asymptotically optimal policy should not give static priority to type a jobs when server 3 is in heavy traffic. Remark 1.9.1. Figure 1.4a shows that the SDP policy performs very well at the instancesf13; 14;:::; 18g, where 3 > A . Moreover, Table 1.2 shows that the SDP policy performs close to the proposed policy at all cases. Hence, we conjecture that the SDP policy is asymptotically optimal even when 3 A . When 3 A , the proofs associated with the up intervals (cf. Section 1.8.1) do not work. Note that we consider 50 1 6 12 18 Instance number 0 5 10 15 20 25 30 35 % deviation Static Rand.-2/3 Rand. Figure 1.5: (Color online) Percentage deviations of the costs of the static, randomized, and the randomized- 2=3 policies from the lowest realized average cost (L(i) fori2f1; 2;:::; 18g) in the first 18 instances when h a = 2 andh b = 1. the difference of two renewal processes with rates A and 3 , respectively in the proofs associated with the up intervals (cf. (1.63)). We show that the renewal process with rate A stays sufficiently close to the one with rate 3 during the up intervals when 3 < A . However, this argument does not hold when 3 A . 1.10 Extensions The ideas in this chapter apply to more complex networks. We first consider a fork-join network in which there are task dependent holding costs in Section 1.10.1. Next, we consider a fork-join network in which jobs fork into an arbitrary number of jobs in Section 1.10.2. Lastly, we consider fork-join networks with more than two job types in Section 1.10.3. In each of these extensions, we first construct and solve an approximating DCP, then interpret a control policy from the solution. Since we do not prove asymptotic optimality for any of the extensions, the suggested control policies are all heuristics. 1.10.1 Task Dependent Holding Costs So far, we assume that the holding cost rate of a typea (b) job in buffers 1, 3, 4, 7, and 8 (2, 5, 6, 9, and 10) are the same and denoted by h a (h b ). In this section, we extend this assumption by considering task dependent holding cost rates. We denote the holding cost rate of a job in bufferk ash k for allk2K and we assumeh k 0 for allk2K. In this case, our objective in the DCP is to minimize P 10 X k=1 h k ~ Q k (t)>x ! ; 8t2R + ; x> 0: (1.109) 51 Since the control policy in server 4 has no effect on the costs in buffers 1, 2, 3, and 6, the objective (1.109) can be simplified to minimizing P h 4 ~ Q 4 (t) +h 5 ~ Q 5 (t) +h 7 ~ Q 7 (t) +h 8 ~ Q 8 (t) +h 9 ~ Q 9 (t) +h 10 ~ Q 10 (t)>x ; 8t2R + ; x> 0: (1.110) By first simplifying objective (1.110) by (1.27), which is the instantaneous service process assumption in the downstream servers, we can modify the DCP (1.29) in the following way: For eachx> 0 andt2R + , min P h 4 ~ Q 4 (t) +h 5 ~ Q 5 (t) +h 7 ~ Q 4 (t) ~ Q 3 (t) + +h 8 ~ Q 3 (t) ~ Q 4 (t) + +h 9 ~ Q 6 (t) ~ Q 5 (t) + +h 10 ~ Q 5 (t) ~ Q 6 (t) + >x ; (1.111) s.t. ~ Q 4 (t) + A B ~ Q 5 (t) = ~ W 4 (t); ~ Q k (t) 0; for allk2f4; 5g: When we consider the DCP (1.111) pathwise, we have the following optimization problem for allt2R + and! in except a null set: min h 4 ~ Q 4 (! t ) +h 5 ~ Q 5 (! t ) +h 7 ~ Q 4 (! t ) ~ Q 3 (! t ) + +h 8 ~ Q 3 (! t ) ~ Q 4 (! t ) + +h 9 ~ Q 6 (! t ) ~ Q 5 (! t ) + +h 10 ~ Q 5 (! t ) ~ Q 6 (! t ) + ; (1.112) s.t. ~ Q 4 (! t ) + A B ~ Q 5 (! t ) = ~ W 4 (! t ); ~ Q k (! t ) 0; for allk2f4; 5g: We can solve the optimization problem (1.112) by the following lemma whose proof is presented in Appendix A.2.8. Lemma 1.10.1. Consider the optimization problem min h 4 q 4 +h 5 q 5 +h 7 (q 4 q 3 ) + +h 8 (q 3 q 4 ) + +h 9 (q 6 q 5 ) + +h 10 (q 5 q 6 ) + ; (1.113a) s.t. q 4 + A B q 5 =w 4 ; (1.113b) q 4 0; q 5 0; (1.113c) 52 whereq 4 andq 5 are the decision variables, all of the parameters are nonnegative, and A ; B > 0. Then, there exists an optimal solution among the four solutions given in Table 1.3 below. Table 1.3: An optimal solution set for the optimization problem (1.113). Solution # q 4 q 5 1 0 ( B = A )w 4 2 q 3 ^w 4 ( B = A )(w 4 q 3 ) + 3 (w 4 ( A = B )q 6 ) + q 6 ^ ( B = A )w 4 4 w 4 0 Remark 1.10.1. The optimal solution of Lemma 1.10.1 is strictly dependent on the parameters (q 3 ;q 6 ;w 4 ); i.e., the optimal solution can change as q 3 , or q 6 , or w 4 changes. For example, consider an example in which A = B = 1,h 4 = h 5 = 1,h 8 = h 10 = 2,h 7 = h 9 = 3, and (q 3 ;q 6 ;w 4 ) = (0:01; 0; 1). Then, the objective function values corresponding to the four solutions in Table 1.3 are (3:02; 2:98; 3:97; 3:97), respectively. This implies that an optimal solution is the second solution in Table 1.3, which is (q 4 ;q 5 ) = (0:01; 0:99). Next, we consider the same example but this time (q 3 ;q 6 ;w 4 ) = (1; 0:01; 1); i.e., q 3 andq 6 change. In this case, the objective function values corresponding to the four solutions in Table 1.3 are (4:98; 1:03; 1:02; 1:03), respectively. This implies that an optimal solution is the third solution in Table 1.3, which is (q 4 ;q 5 ) = (0:99; 0:01). By Lemma 1.10.1, for eacht2R + and! in except a null set, an optimal solution of DCP (1.112) is among the four solutions presented below, which then motivates a control policy. 1. If ~ Q 4 (! t ); ~ Q 5 (! t ) = 0; B A ~ W 4 (! t ) ; (1.114) then server 4 should give static priority to typea jobs. 2. If ~ Q 4 (! t ); ~ Q 5 (! t ) = ~ Q 3 (! t )^ ~ W 4 (! t ); B A ~ W 4 (! t ) ~ Q 3 (! t ) + ; (1.115) then server 4 should use the proposed policy. Note that the solution in (1.115) is the same as the one in Proposition 1.4.1. 3. If ~ Q 4 (! t ); ~ Q 5 (! t ) = ~ W 4 (! t ) A B ~ Q 6 (! t ) + ; ~ Q 6 (! t )^ B A ~ W 4 (! t ) ! ; (1.116) then server 4 should use the proposed policy but this time it should prioritize typeb jobs over type a jobs. In other words, if B 5 , server 4 should give static priority to typeb jobs; otherwise, it 53 should use the SDP policy to pace the departure process of typeb jobs from buffer 5 with the one from buffer 6. 4. If ~ Q 4 (! t ); ~ Q 5 (! t ) = ~ W 4 (! t ); 0 ; (1.117) then server 4 should give static priority to typeb jobs. Since the optimal solution can change as the process ~ Q 3 (! t ); ~ Q 6 (! t ); ~ W 4 (! t ) changes with time (cf. Remark 1.10.1), we interpret the following dynamic control policy: Whenever server 4 makes a service completion, the system controller computes the objective function values of DCP (1.112) corresponding to the four solutions in (1.114) - (1.117) and implements the control policy corresponding to the solution with the minimum objective function value until the next service completion epoch in server 4. Remark 1.10.2. Recall that an admissible policy is work-conserving (cf. (1.5)). However, work-conserving service disciplines are not necessarily optimal in this extension. For example, ifh 3 <h 7 , then delaying the process of jobs intentionally in server 3 at some specific times can be more cost beneficial than processing jobs in a work-conserving fashion in the same server. Therefore, depending on the holding cost rates, we may achieve a better system performance by allowing control (i.e., intentional delays) in servers 1, 2, 3, 4 and 5. 1.10.2 Networks with Arbitrary Number of Forks In this section, we consider the network presented in Figure 1.6, in which typea andb jobs fork intog 1 + 1 andg 2 + 1 number of jobs, respectively, whereg 1 ;g 2 2N + ; and server 4 is the only server which processes both job types. Parallel with objective (1.12), we consider the objective of minimizing P h a (g 1 + 1) Q r U L 1 (t) +Q r D L 1 (t) +h b (g 2 + 1) Q r U R 1 (t) +Q r D R 1 (t) >x ; (1.118) for allt2R + andx> 0. We assume that servers 6 and 7 are in light traffic, hence we can assume that the service processes in these servers are instantaneous as in Section 1.4. Then, similar to (1.27), we have Q r D L 1 = Q r 4 _ max i2f2;:::;g 1 g Q r U L i Q r U L 1 + ; Q r D R 1 = Q r 5 _ max i2f2;:::;g 2 g Q r U R i Q r U R 1 + : (1.119) 54 1 ܮ ଵ 4 2 ܴ ଵ 6 7 1 2 4 5 8 9 ܮ ଶ ܴ ଶ ܴ మ ܮ భ ܷ ͳ ܦ ͳ ܷ ଶ ܦ ଶ ܷ భ ܦ భ ܷ ͳ ோ ܷ ଶ ோ ܷ మ ோ ܦ ͳ ோ ܦ ଶ ோ ܦ మ ோ Type ܾ jobs ( ݄ ) ߤ ߤ Type ܽ jobs ( ݄ ) Figure 1.6: A fork-join processing network with two job types and arbitrary number of forks. From (1.119), defining ~ h a :=h a (g 1 + 1) and ~ h b :=h b (g 2 + 1), and the fact thatx + (yx) + =x_y for allx;y2R, the objective (1.118) is equivalent to minimizing P ~ h a Q r 4 (t)_ max i2f1;:::;g 1 g Q r U L i (t) + ~ h b Q r 5 (t)_ max i2f1;:::;g 2 g Q r U R i (t) >x ; (1.120) for allt2R + andx> 0. Parallel with Proposition 1.3.2, we can prove the following weak convergence result under any work- conserving control policy: ^ Q r 1 ; ^ Q r 2 ; ^ Q r U L i ;i2f1; 2;:::;g 1 g; ^ Q r U R j ;j2f1; 2;:::;g 2 g; ^ W r 4 ) ~ Q 1 ; ~ Q 2 ; ~ Q U L i ;i2f1; 2;:::;g 1 g; ~ Q U R j ;j2f1; 2;:::;g 2 g; ~ W 4 asr!1; (1.121) where the limiting process is the zero process for the buffers whose corresponding dedicated server is in light traffic and an SRBM for the buffers whose corresponding dedicated server is in heavy traffic and the workload process in server 4. Then, by (1.120), (1.121), and using the technique that we use to derive the DCP (1.29), we construct the following DCP for this network. For eachx> 0 andt2R + , min P ~ h a ~ Q 4 (t)_ max i2f1;:::;g 1 g ~ Q U L i (t) + ~ h b ~ Q 5 (t)_ max i2f1;:::;g 2 g ~ Q U R i (t) >x ; 55 s.t. ~ Q 4 (t) + A B ~ Q 5 (t) = ~ W 4 (t); (1.122) ~ Q k (t) 0; for allk2f4; 5g: When we consider the DCP (1.122) pathwise, we have the following optimization problem for allt2R + and! in except a null set: min ~ h a ~ Q 4 (! t )_ max i2f1;:::;g 1 g ~ Q U L i (! t ) + ~ h b ~ Q 5 (! t )_ max i2f1;:::;g 2 g ~ Q U R i (! t ) ; (1.123a) s.t. ~ Q 4 (! t ) + A B ~ Q 5 (! t ) = ~ W 4 (! t ); (1.123b) ~ Q k (! t ) 0; for allk2f4; 5g; (1.123c) where ~ Q k (! t ),k2f4; 5g are the decision variables. Note that the objective function (1.123a) is equal to ~ h a ~ Q 4 (! t )_ max i2f1;:::;g 1 g ~ Q U L i (! t ) + ~ h b max i2f1;:::;g 2 g ~ Q U R i (! t ) + ~ h b ~ Q 5 (! t ) max i2f1;:::;g 2 g ~ Q U R i (! t ) + : (1.124) Since all of the buffers U R i , i2f1;:::;g 2 g are independent of the control, minimizing the objective in (1.124) is equivalent to minimizing the objective (1.125a) below, thus the optimization problem (1.123) is equivalent to the following one: min ~ h a ~ Q 4 (! t )_ max i2f1;:::;g 1 g ~ Q U L i (! t ) + ~ h b ~ Q 5 (! t ) max i2f1;:::;g 2 g ~ Q U R i (! t ) + ; (1.125a) s.t. ~ Q 4 (! t ) + A B ~ Q 5 (! t ) = ~ W 4 (! t ); (1.125b) ~ Q k (! t ) 0; for allk2f4; 5g: (1.125c) Without loss of generality, let us assume that ~ h a A ~ h b B . Then by Lemma 1.4.1, an optimal solution of the optimization problem (1.125) is ~ Q 4 ; ~ Q 5 = ~ W 4 ^ max i2f1;:::;g 1 g ~ Q U L i ; B A ~ W 4 max i2f1;:::;g 1 g ~ Q U L i + ! : (1.126) We can interpret the solution (1.126) in the following way: If the processing capacity of each of the servers L i ,i2f1; 2;:::;g 1 g is greater than or equal to A , then server 4 should give static priority to typea jobs all the time. Otherwise, server 4 should give priority to typea jobs whenever the number of jobs in buffer 4 56 is strictly greater than the maximum of the number of jobs in buffersU L i ,i2f1; 2;:::;g 1 g. In other words, server 4 should pace the departure process of typea jobs from buffer 4 with the minimum of the ones from the buffersU L i ,i2f1; 2;:::;g 1 g. Hence, we see a slightly different version of the SDP policy. 1.10.3 Networks with More Than Two Job Types In this section we consider fork-join networks with more than two job types. We first consider a network with an arbitrary number of job types and a single shared server. Then we consider a network with three job types and two shared servers. Networks with Arbitrary Job Types and a Single Shared Sever In this section, we consider the network presented in Figure 1.7, where there aren job types such thatn is arbitrary andn 2. Server 1 is a shared server which processes all job types, whereas all other servers process only a single job type. Upon arriving to the system, each job is first processed in a server, then it is forked into two jobs which are processed in the shared server and in a dedicated server, respectively, lastly the two forked jobs are joined in the corresponding downstream server and leave the system. ! !"# !"$ ! # !"% &'()*! +,-.*/" # 0 &'()*$ +,-.*/" % 0 &'()*& +,-.*/" ' 0 #"# #"$ 1 1"# 1"$ #"% 1"% Figure 1.7: (Color online) A fork-join processing network with arbitrary job types and a single shared server. A typej job can wait in buffersj, (j; 2), (j; 3), and (j; 4), where bufferj feeds the shared server, buffer (j; 2) feeds the dedicated server, buffer (j; 3) is fed by the dedicated server, and buffer (j; 4) is fed by the shared server (cf. Figure 1.7). Server 1, the shared server, processes type j jobs with rate j for all j2f1; 2;:::;ng. Leth j denote the cost per unit time to hold a typej job in the system. Without loss of 57 generality, we assume thath 1 1 h 2 2 :::h n n . We assume that server 1 is in heavy traffic and all of the downstream servers are in light traffic. Parallel with (1.16), let ^ W r := n X j=1 ^ Q r j r j denote the diffusion scaled workload process that server 1 sees. Then, parallel with Proposition 1.3.2, we can prove the following weak convergence result under any work-conserving control policy: ^ Q r j;2 ;j2f1; 2;:::;ng; ^ W r ) ~ Q j;2 ;j2f1; 2;:::;ng; ~ W asr!1; (1.127) where the limiting process is the zero process for the buffers whose corresponding dedicated server is in light traffic and an SRBM for the buffers whose corresponding dedicated server is in heavy traffic and the workload process in server 1. Since we assume that the downstream servers are in light traffic, we can assume that the service processes in these servers are instantaneous. Then, similar to (1.27), we have Q r j;3 = Q r j Q r j;2 + ; Q r j;4 = Q r j;2 Q r j + ; 8j2f1; 2;:::;ng: By using the technique that we use to derive DCP (1.29), we construct the following DCP for this network. For eachx> 0 andt2R + , min P 0 @ n X j=1 h j ~ Q j;2 (t) + ~ Q j (t) ~ Q j;2 (t) + >x 1 A ; s.t. n X j=1 ~ Q j (t)= j = ~ W (t); (1.128) ~ Q j (t) 0; for allj2f1; 2;:::;ng: When we consider the DCP (1.128) pathwise, we have the following optimization problem: min n X j=1 h j ~ Q j (! t ) ~ Q j;2 (! t ) + ; s.t. n X j=1 ~ Q j (! t )= j = ~ W (! t ); (1.129) ~ Q j (! t ) 0; for allj2f1; 2;:::;ng; 58 where ~ Q j (! t ),j2f1; 2;:::;ng are the decision variables. Note that, since ~ Q j;2 is an exogenous process for allj2f1; 2;:::;ng (cf. (1.127)), we neglect the term P n j=1 h j ~ Q j;2 (! t ) in the objective function of the optimization problem (1.129). We can solve the optimization problem (1.129) by the following lemma whose proof is presented in Appendix A.2.9. Lemma 1.10.2. Consider the optimization problem min n X j=1 h j (q j q j;2 ) + ; (1.130a) s.t. n X j=1 q j = j =w; (1.130b) q j 0; for allj2f1; 2;:::;ng; (1.130c) whereq j ,j2f1; 2;:::;ng are the decision variables, all of the parameters are nonnegative, j > 0 for all j2f1; 2;:::;ng, andh 1 1 h 2 2 :::h n n . Then, an optimal solution of the optimization problem (1.130) is q 1 =q 1;2 ^ ( 1 w); (1.131a) q j =q j;2 ^ 2 4 j w q 1;2 1 + q 2;2 2 ! + q j1;2 j1 ! + 3 5 ; 8j2f2; 3;:::;n 1g; (1.131b) q n = n w q 1;2 1 + q 2;2 2 ! + q n1;2 n1 ! + : (1.131c) Remark 1.10.3. When n = 2, the solution of Lemma 1.10.2 is the same as the solution of Lemma 1.4.1 by the fact that the notation ( 1 ; 2 ;q 1;2 ;q 2;2 ;w) in Lemma 1.10.2 corresponds to the notation ( A ; B ;q 3 ;q 6 ;w 4 = A ) in Lemma 1.4.1. Remark 1.10.4. In the optimal solution (1.131), we see thath j (q j q j;2 ) + = 0 for allj2f1; 2;:::;n1g buth n (q n q n;2 ) + can be strictly positive depending on the parameters. This implies that the objective function (1.130a) can become positive only because of type n jobs. In other words, cost can occur only because of the cheapest job type. Lemma 1.10.2 implies that an optimal solution of the optimization problem (1.129) is ~ Q 1 = ~ Q 1;2 ^ ( 1 ~ W ); (1.132a) 59 ~ Q j = ~ Q j;2 ^ 2 4 j 0 @ ~ W ~ Q 1;2 1 ! + ~ Q 2;2 2 ! + ~ Q j1;2 j1 1 A + 3 5 ;8j2f2; 3;:::;n 1g; (1.132b) ~ Q n = n 0 @ ~ W ~ Q 1;2 1 ! + ~ Q 2;2 2 ! + ~ Q n1;2 n1 1 A + : (1.132c) Then, we can interpret the following control policy from the solution (1.132): The priority ranking of the job types is 1 2:::n. Type 1 jobs: If 1 is less than or equal to the processing rate of the dedicated server for type 1 jobs, server 1 should give static priority to type 1 jobs all the time. Otherwise, server 1 should give priority to buffer 1 only when the number of jobs in buffer (1; 2) becomes less than the one in buffer 1. During the remaining time, server 1 should process the job typesj2f2;:::;ng. Typej jobs,j2f2;:::;n1g: Server 1 gives priority to typej jobs only when the higher priority job types (f1;:::;j1g) do not require any processing. Consider the time intervals in which server 1 gives priority to typej jobs. If j is less than or equal to the processing rate of the dedicated server for typej jobs, server 1 should give static priority to typej jobs all the time. Otherwise, server 1 should give priority to bufferj only when the number of jobs in buffer (j; 2) becomes less than the one in bufferj. During the remaining time, server 1 should process the job types in the setfj + 1;:::;ng. Typen jobs are processed during the remaining time. Therefore, we see a chained implementation of the proposed policy. For allj2f1;:::;n1g, by keeping bufferj less than or equal to buffer (j; 2) with minimum effort, buffer (j; 4) never causes the corresponding downstream server to idle because of the join operation and server 4 gives as much as priority to typen jobs (cf. (1.132a), (1.132b), and Remark 1.10.4). Remark 1.10.5. It is possible to extend the network in Figure 1.7 to the case where each job type forks into arbitrary number of jobs (cf. Figure 1.1). Suppose that typej jobs fork intog j + 1 jobs and ~ Q j;2;i , i2f1; 2;:::;g j g denote the buffers in front of the dedicated servers corresponding to the typej jobs. Then by the DCP (1.122) and the optimization problem (1.125) in Section 1.10.2, we need to replace ~ Q j;2 with max i2f1;:::;g j g ~ Q j;2;i andh j withh j (g j + 1) in both the DCP (1.128) and the optimization problem (1.129). We can still solve the modified version of the optimization problem (1.129) with Lemma 1.10.2. The control 60 policy that we can interpret is a modification of the control policy that we interpret from the solution (1.132) with the one from the solution (1.126). Networks with Three Job Types and Two Shared Servers In this section, we consider the network presented in Figure 1.8, where there are three job types, namely typea,b, andc jobs. There are two shared servers, which are servers 5 and 6. We formulate and solve the approximating DCP in order to derive heuristic control policies. In contrast with the networks in Figures 1.6 and 1.7, the DCP solution is not a straightforward extension of the DCP solution presented in Section 1.4. 1 4 5 2 8 9 4 1 2 7 10 5 6 13 6 3 7 3 9 8 15 10 11 12 14 ߤ ߤ ଶ ߤ ଵ ߤ Type ܽ jobs ( ݄ ) Type ܾ jobs ( ݄ ) Type ܿ jobs ( ݄ ) Figure 1.8: A fork-join processing network with three job types and two shared servers. Server 5 processes both typea andb jobs with rates A and B1 , respectively; and server 6 processes both typeb andc jobs with rates B2 and C , respectively. Leth a ,h b , andh c denote the holding cost rate per job per unit time for typea,b, andc jobs, respectively. Suppose that both servers 5 and 6 are in heavy traffic, and servers 8, 9, and 10 are in light traffic. Let ^ W r 5 := ^ Q r 5 + r A r B1 ^ Q r 6 ; ^ W r 6 := ^ Q r 7 + r B2 r C ^ Q r 8 denote the diffusion scaled workload processes (up to a constant scale factor) in servers 5 and 6, respectively. Then, parallel with Proposition 1.3.2, we can prove the following weak convergence result under any work- conserving control policy: ^ Q r 1 ; ^ Q r 2 ; ^ Q r 3 ; ^ Q r 4 ; ^ Q r 9 ; ^ W r 5 ; ^ W r 6 ) ~ Q 1 ; ~ Q 2 ; ~ Q 3 ; ~ Q 4 ; ~ Q 9 ; ~ W 5 ; ~ W 6 asr!1; (1.133) 61 where the limiting process is the zero process for the buffers whose corresponding dedicated server is in light traffic and an SRBM for the buffers whose corresponding dedicated server is in heavy traffic and the workload processes in servers 5 and 6. Since servers 8, 9, and 10 are in light traffic, we can assume that the service processes in these servers are instantaneous; and similar to (1.27), we have Q r 10 = (Q r 5 Q r 4 ) + ; Q r 15 = (Q r 8 Q r 9 ) + ; Q r 6 +Q r 12 =Q r 6 + (Q r 7 Q r 6 ) + =Q r 6 _Q r 7 : Then, by using the technique that we use to derive DCP (1.29), we construct the following DCP for this network. For anyx> 0 andt2R + , min P h a ~ Q 4 (t) + ~ Q 5 (t) ~ Q 4 (t) + +h b ~ Q 6 (t)_ ~ Q 7 (t) +h c ~ Q 9 (t) + ~ Q 8 (t) ~ Q 9 (t) + >x ! ; s.t. ~ Q 5 (t) + A B1 ~ Q 6 (t) = ~ W 5 (t); (1.134) ~ Q 7 (t) + B2 C ~ Q 8 (t) = ~ W 6 (t); ~ Q k (t) 0; for allk2f5; 6; 7; 8g: When we consider the DCP (1.134) pathwise, we have the following optimization problem: min h a ~ Q 5 (! t ) ~ Q 4 (! t ) + +h b ~ Q 6 (! t )_ ~ Q 7 (! t ) +h c ~ Q 8 (! t ) ~ Q 9 (! t ) + ; s.t. ~ Q 5 (! t ) + A B1 ~ Q 6 (! t ) = ~ W 5 (! t ); (1.135) ~ Q 7 (! t ) + B2 C ~ Q 8 (! t ) = ~ W 6 (! t ); ~ Q k (! t ) 0; for allk2f5; 6; 7; 8g; where ~ Q k (! t ),k2f5; 6; 7; 8g are the decision variables. Note that, since buffers 4 and 9 are independent of control, we neglect the termh a ~ Q 4 (! t ) +h c ~ Q 9 (! t ) in the objective function of the optimization problem (1.135). Next, we will solve the optimization problem (1.135) case by case. First, let us consider the caseh b B1 h a A andh b B2 h c C . We present the optimal solution in the following result. Lemma 1.10.3. Whenh b B1 h a A andh b B2 h c C , an optimal solution of the optimization problem (1.135) is the following: 62 1. Ifh b h a A = B1 +h c C = B2 , an optimal solution is ~ Q 5 ; ~ Q 6 ; ~ Q 7 ; ~ Q 8 = ~ W 5 ;0;0; C B2 ~ W 6 : (1.136) 2. Ifh b <h a A = B1 +h c C = B2 , an optimal solution is ~ Q 5 = max ( ~ Q 4 ^ ~ W 5 ; ~ W 5 A B1 ~ W 6 B2 C ~ Q 9 + ) ; (1.137a) ~ Q 6 = ~ Q 7 = min ( B1 A ~ W 5 ~ Q 4 + ; ~ W 6 B2 C ~ Q 9 + ) ; (1.137b) ~ Q 8 = max ~ Q 9 ^ C B2 ~ W 6 ; C B2 ~ W 6 B1 A ~ W 5 ~ Q 4 + : (1.137c) The proof of Lemma 1.10.3 is provided in Appendix A.2.10. Note that the optimal solution has different structures depending on the cost parameters. On the one hand, h b h a A = B1 +h c C = B2 intuitively implies that the holding cost of a typeb job is greater than the sum of the holding costs of a typea and type c job. Hence, ( ~ Q 6 ; ~ Q 7 ) = (0;0) in (1.136) and this implies that we should give static preemptive priority to typeb jobs in servers 5 and 6 all the time. On the other hand, maxfh a A = B1 ;h c C = B2 g h b < h a A = B1 +h c C = B2 intuitively implies that the holding cost of a typeb job is greater than the one of a typea or typec job but less than the sum of the holding costs of a typea and typec job. In this case we interpret the solution (1.137) in the following way. If we decrease both ~ Q 6 and ~ Q 7 in (1.137b), which is equivalent to giving more priority to typeb jobs, then more jobs will accumulate in buffers 10 and 15 waiting for the corresponding jobs to arrive at buffers 11 and 14, respectively. Hence, the average number of typeb jobs in the system will decrease but the average number of both typea andc jobs will increase, which is not desired becauseh b <h a A = B1 +h c C = B2 . If we decrease only ~ Q 6 in (1.137b), then the average number of typea, b, andc jobs increases, stays the same, and stays the same, respectively; and this is not desired. The case in which we decrease only ~ Q 7 in (1.137b) follows similarly. Next, suppose that we increase ~ Q 6 (t) or ~ Q 7 (t) in (1.137b), which is equivalent to giving less priority to typeb jobs. Suppose that ~ Q 6 (t) = ~ Q 7 (t) = ( B1 = A ) ~ W 5 (t) ~ Q 4 (t) + in (1.137b) for somet2R + . Since ~ Q 10 (t) = ~ Q 5 (t) ~ Q 4 (t) + , by substitution ~ Q 10 (t) = 0 in this case; i.e., server 8 does not idle due to buffer 11. Then, when we increase ~ Q 6 (t) or ~ Q 7 (t), the average number of typea,b, and c jobs stays the same, increases, and (in the best case) decreases, respectively. Sinceh c C = B2 h b , this result is not desired. The case ~ Q 6 (t) = ~ Q 7 (t) = ~ W 6 (t) ( B2 = C ) ~ Q 9 (t) + in (1.137b) follows similarly (for more intuition see the proof of Lemma 1.10.3 and Remark A.2.1 in Appendix A.2.10). 63 Therefore, (1.137) is the optimal solution and we interpret the following policy from it: Whenever (1.137b) does not hold, if ~ Q 6 ( ~ Q 7 ) is strictly greater than the right hand side of (1.137b), then server 5 (6) should give preemptive priority to typeb jobs; if ~ Q 6 ( ~ Q 7 ) is less than or equal to the right hand side of (1.137b), then server 5 (6) should give preemptive priority to typea (c) jobs. Second, let us consider the caseh a A h b B1 andh b B2 h c C . We can see that ~ Q 5 ; ~ Q 6 ; ~ Q 7 ; ~ Q 8 = ~ Q 4 ^ ~ W 5 ; B1 A ~ W 5 ~ Q 4 + ; B1 A ~ W 5 ~ Q 4 + ^ ~ W 6 ; C B2 ~ W 6 B1 A ~ W 5 ~ Q 4 + + ! (1.138) is an optimal solution of the optimization problem (1.135) for allt2R + and! in except a null set. Since the derivation of (1.138) is very similar to the proof of Lemma 1.10.3, we skip it. We can interpret a control policy from (1.138) in the following way. If server 4 processes typea jobs with a faster rate than server 5 does, then server 5 should give static priority to typea jobs. Otherwise, server 5 should use the SDP policy to pace the departure process of the typea jobs from buffer 5 with the one from buffer 4. Similarly, server 6 should use the SDP policy to pace the departure process of typeb jobs from buffer 7 with the one from buffer 6. Third, let us consider the caseh a A h b B1 andh c C h b B2 . We can see that ~ Q 5 ; ~ Q 6 ; ~ Q 7 ; ~ Q 8 = ~ Q 4 ^ ~ W 5 ; B1 A ~ W 5 ~ Q 4 + ; ~ W 6 B2 C ~ Q 9 + ; C B2 ~ W 6 ^ ~ Q 9 ! (1.139) is an optimal solution of the optimization problem (1.135). Since the derivation of (1.139) is very similar to the proof of Lemma 1.10.3, we skip it. We can interpret a control policy from (1.139) in the following way. If server 4 (7) processes typea (c) jobs with a faster rate than server 5 (6) does, then server 5 (6) should give static priority to typea (c) jobs. Otherwise, server 5 (6) should use the SDP policy to pace the departure process of type a (c) jobs from buffer 5 (8) with the one from buffer 4 (9). Lastly, the caseh b B1 h a A andh c C h b B2 is equivalent to the second case that we consider, hence we skip it. It is possible to construct DCPs for networks with more than three job types and more than two shared servers by the same methodology that we use to construct DCP (1.134). However, finding a closed-form optimal solution and interpreting a heuristic control policy is not trivial. As the number of shared servers 64 increases in the network, the dimension of the corresponding DCP, which is the number of workload con- straints, increases and finding a closed-form optimal solution becomes quite challenging. For example, there is only a single shared server in the networks considered in Figures 1.2 and 1.7 and in Sections 1.10.1 and 1.10.2. As a result, all of the corresponding DCPs (cf. (1.29), (1.111), (1.122), and (1.128)) are single dimensional; i.e., each DCP has a single workload constraint. However, when there are two shared servers as in the network in Figure 1.8, the corresponding DCP (1.134) has two workload constraints and is more difficult to solve than the single dimensional ones. In the queueing control literature, there are many studies (cf. Harrison (1998), Harrison and L´ opez (1999), Bell and Williams (2001), Mandelbaum and Stolyar (2004), Ata and Kumar (2005), Dai and Lin (2008)) in which the corresponding DCP is first converted to a single dimensional equivalent DCP and control policies are obtained by solving the latter DCP. However, in general, how to control stochastic networks whose equivalent lower dimensional DCP has more than one dimension is an open problem. In order to overcome the curse of dimensionality in the networks with more than three job types and more than two shared servers, one possible solution is to decompose the network into the ones with at most three job types and use the control policies mentioned in this chapter to control each sub-network. For example, in Figure 1.9, there are seven different job types arriving to the system, and the network is divided into three sub-networks. Then the important questions are: 1) How can a large network be decomposed into smaller ones? and 2) Which control policies should be used in the servers which belong to two different sub-networks? If there exists a server which processes more than two different job types and is in light traffic, the network can clearly be decomposed from this server and any work-conserving control policy can be used in this server, because its processing capacity is high. Otherwise, the answers are not trivial and requires further research. Remark 1.10.6. To extend our proof of asymptotic optimality for the network in Figure 1.2 to those in this section (i.e., the networks in Figures 1.1, 1.6, 1.7, and 1.8 and the network in Figure 1.2 with task dependent holding costs), we must both establish weak convergence (that is, prove the equivalent of Theorem 1.5.1) and show that the solution to the approximating DCPs are asymptotic lower bounds (that is, prove the equivalent of Theorem 1.6.1). Similar arguments as those used to prove Theorem 1.6.1 should hold for all the networks considered in this section, because the solution to each DCP is given pathwise and for allt2 R + . However, proving weak convergence is more difficult, because the shared server(s) use more complicated priority schemes. For the networks in Figures 1.1, 1.6, 1.7, and the network in Figure 1.2 with task dependent holding costs, which have one shared server, we conjecture that if the “up” and “down” intervals can be re-defined to reflect the more complicated priority scheme, so that which job type is given 65 !"#$%&'()*+,- !"#$%&'()*+,. !"#$%&'()*+,/ Figure 1.9: A fork-join processing network with seven job types and six shared servers. priority in any given interval is known, then a similar strategy to show weak convergence will work. The network in Figure 1.8 requires further thought, because there are two shared servers, and, depending on the holding costs, their workloads interact with each other through the state-dependent prioritization, as can be seen from (1.137) and (1.138). 1.11 Concluding Remarks The synchronization constraints in fork-join networks complicate their analysis. In this chapter, we have formulated and solved approximating diffusion control problems (DCPs) for a variety of fork-join networks (depicted in Figures 1.1, 1.2, 1.6, 1.7, and 1.8). The DCP solutions have mostly motivated either a slow departure pacing control or a static priority control, depending on the network parameters. We have rig- orously proved the asymptotic optimality of our proposed control for the network in Figure 1.2, and we conjecture that some of the ideas in those arguments will be helpful to establish the asymptotic optimality of the controls proposed for the networks in Figures 1.1, 1.6, 1.7, and 1.8 and the network in Figure 1.2 with task dependent holding costs. Our proposed controls will not change when light traffic queues 3 are added to the fork-join networks that we have studied. Furthermore, minor modifications to our proposed policies will accommodate the addition of heavy traffic queues before the shared or dedicated servers. For example, if we add a heavy traffic queue between server 1 and buffer 3 in Figure 1.2, even though our proofs do not cover this case, we can see that the proposed policy (or a slightly modified version of it) is still asymptotically optimal. In this modified system, if server 3 is in light traffic, then the modified network is equivalent to the one where server 3 is replaced by 3 What we mean by a queue is a server and its corresponding buffer, e.g., server 3 and buffer 3 is a queue in Figure 1.2. 66 the newly added heavy traffic queue and all the other servers have the same processing capacities with the ones in the original network. If server 3 is in heavy traffic in this modified system, then the proposed policy behaves in the following way: When the total number of jobs in buffer 3 and the buffer corresponding to the newly added queue is less than the one in buffer 4, then server 4 gives priority to buffer 4; otherwise server 4 gives priority to buffer 5. The complicated case is when there are heavy-traffic queues after the shared servers. Then, it is not clear either what the proposed policy should be or how to prove an asymptotic optimality result. An excellent topic for future research is to develop control policies for the broader class of fork-join networks with heterogeneous customer populations described in Nguyen (1994). More specifically, that paper assumes FCFS scheduling, but we believe other control policies can lead to better performance. 67 Chapter 2 Dynamic Matching for Real-time Ridesharing 2.1 Introduction Ridesharing platforms are online mobile platforms which match paying customers who need a ride with drivers who provide transportation. Some examples of these platforms are Uber and Lyft in America, Ola in India, Didi Chuxing in China, and Grab in Southeast Asia. Ridesharing companies emerged in the last ten years, as the market penetration of smartphones exploded, and have grown exponentially fast. The ridesharing companies all make use of the relatively recent ability to track the locations of the customers and drivers continuously (unlike the taxicab companies) which enables responsive and efficient market control (cf. Azevedo and Weyl (2016)). Ridesharing platforms are two-sided matching markets (cf. Rochet and Tirole (2006)) that pair customers and drivers. Typically, when a customer needs a ride, she requests the ride via an application on a smart- phone. The system controller, who is the corresponding ridesharing company, matches the customer with a driver, if both sides approve the matching. Then, the driver picks up the customer and takes her to her destination. The ridesharing company would like to ensure there is always a nearby driver to offer an arriving customer. This is because if a customer must wait too long for pick-up, then the customer may refuse the ride and use another transportation option. However, maintaining adequate driver supply is difficult. Not only do customers choose when to request a ride, but also drivers choose when to begin work, how long to work, and where to go to search for customers. The result can be dramatic changes in customer demand and driver supply across different locations and over the course of a day, which sometimes results in significant driver shortages (cf. Figures 1-3 in Hall et al. (2016)). One common operational strategy is to match customers with the closest driver (CD), and to use pricing to incentivize drivers to move to undersupplied locations. However, surges in price can lead to negative publicity (cf. White (2016), The Economist (2016), and Michallon (2016)). This leads to the question of whether better matching decisions could reduce the need for price surges. An ideal is to solve a joint pricing 68 and matching problem that accounts for the impact of differing customer and driver location. We do this in a static environment, but that joint problem is very difficult in a time-varying environment. As a first step to making progress in a time-varying environment, we assume customer and driver arrival rates are given, and focus on optimizing the matching decisions. The aforementioned time-varying arrival rates could be thought of as the result of a pricing policy as well as other underlying incentives that may be given to drivers or customers, such as demand information sharing or discount coupons – but we do not explicitly model that. The following example motivates why the matching decisions made by the CD policy can be improved. Figure 2.1 represents part of a city, partitioned into ten disjoint hexagonal areas, as is done by Uber (cf. Figure 3 in Chen and Sheldon (2015)). Suppose a customer arrives at area 1 and requests a ride. There are three drivers idle in area 4 and there is a single idle driver in area 3. Moreover, a concert recently ended in area 8, which implies a high potential customer arrival rate in that area. If the destination of the current area 1 customer is far away, the driver assigned to that customer will not return to area 1 for a long time. Then, the system controller has two options: He can either offer one of the drivers in area 4 or the driver in area 3 to the customer in area 1. Under the CD policy, the system controller offers an area 4 driver. However, offering the driver in area 3 saves all drivers in area 4 to match with the potential customers departing the concert in area 8. This prevents the potential future need to offer an area 8 customer a far-away area 3 driver, whom the customer will likely refuse, ending in no match being made. We conclude there is a nontrivial tradeoff between offering the closest driver in accordance with the CD policy and offering a farther driver in order to maximize the future number of customers matched. ! " # $ % & ' ( ) !* Figure 2.1: An intuitive explanation of why the CD policy may not assign the right driver to a customer. 69 2.1.1 Contributions We propose matching policies for ridesharing and prove their optimality in a large market asymptotic regime. We consider the objective of maximizing the number of customers matched with drivers. Our main contri- butions are as follows. Formulating an analytically tractable ridesharing system model that captures geospatial and time-varying features of the problem: In ridesharing platforms, customer demand and driver supply can change signifi- cantly over time and across locations (cf. Figures 1-3 in Hall et al. (2016)). Therefore, both the geospatial distributions of the customer demand and the driver supply and the time varying nature of them affect the potential number of matchings. In order to capture these features, we partition the city into disjoint areas as shown in Figure 2.1 and allow the customer demand and the driver supply to change over time in each area. To the best of our knowledge, there is no study in the ridesharing literature that consider time varying system parameters other than Hu and Zhou (2015). We show that ignoring the time-varying aspect of the problem can result in suboptimal system performance (see the simulation experiments in Section 2.5). Proposing asymptotically optimal matching policies based on a continuous linear program (CLP) and a linear program (LP): We consider a large matching market, in which the arrival rates of the customers and drivers grow without bound, so as to approximate the case of a large city. For any matching policy that does not know the future with certainty, we establish that the solution to a CLP is an asymptotic upper bound on the cumulative number of matchings done in a finite time horizon under fluid scaling (cf. Theorem 2.3.1). That upper bound is very strong in the sense that it is valid on almost every sample path. Then, we propose a matching policy based on an optimal solution of the CLP which is asymptotically optimal (cf. Corollary 2.3.3). The CLP leads to an LP when drivers are fully utilized or the CLP parameters are time homogeneous, which motivates an asymptotically optimal LP-based matching policy in each case (cf. Theorems 2.4.2 and 2.4.3, respectively). When pricing affects customer and driver arrival rates and parameters are time homogeneous, we provide an asymptotically optimal pricing and matching policy (cf. Corollary 2.4.4), and show drivers are fully utilized under that policy under very mild conditions (cf. Example 2.4.1). Providing simulation experiments illustrating the superior performance of the proposed policies against the CD policy: Our simulation experiments in Section 2.5 show that both the CLP-based and the LP-based proposed matching policies can significantly outperform the CD policy. Consistent with intuition, we see that demand spikes coupled with low nearby driver availability, such as in Figure 2.1, lead to the poor performance of the CD policy. This is exactly when we recommend spending the extra effort of estimating 70 parameters, such as customer and driver arrival rates, in order to be able to implement a CLP- or an LP-based matching policy. (In comparison, the CD policy requires no network information to implement.) The remainder of this chapter is organized as follows. We conclude this section with a literature review (cf. Section 2.1.2) and a summary of our mathematical notation (cf. Section 2.1.3). Section 2.2 presents our model. We formalize our large matching market regime, an asymptotic upper bound on the cumulative number of matchings, and prove that a CLP-based matching policy achieves that upper bound in Section 2.3. We provide conditions under which an LP-based matching policy achieves the asymptotic upper bound and consider a joint pricing and matching problem in Section 2.4. Section 2.5 presents some simulation experiments. Finally, we make concluding remarks in Section 2.6. The proofs of all results can be found in Appendix B. 2.1.2 Literature Review In recent years, academic research related to ridesharing platforms has grown rapidly, alongside the use of these platforms. An overview of research problems on ridesharing platforms can be seen in Azevedo and Weyl (2016). While the effects of pricing have been well-studied (cf. Chen et al. (2015), Chen and Sheldon (2015), Riquelme et al. (2015), Hall et al. (2016), Bimpikis et al. (2016), Cachon et al. (2017), Guda and Subramanian (2017), and Taylor (2017)), the effects of matching decisions have been studied relatively less. Dynamic matching control has been studied in the literature in the context of kidney exchanges (cf. ¨ Unver (2010)), housing markets (cf. Leshno (2016)), online matching platforms such as Upwork or Airbnb (cf. Arnosti et al. (2016)), and assemble-to-order manufacturing systems (cf. Plambeck and Ward (2006) and Reiman and Wang (2015)). However, the ridesharing model is different enough that it is not clear any of the results of these studies carry over. A method to control dynamic matching systems is the market thickening approach, where the system controller first waits for jobs (that is, customers and drivers in the context of this chapter) to build-up, or thicken, in the market and then makes the matching decisions. In the queueing literature, Gurvich and Ward (2014) consider a matching system where jobs with multiple classes arrive in the system dynamically and randomly and prove asymptotic optimality of a discrete review matching policy as the arrival rates of the jobs increase (i.e., a large market assumption). The discrete review policy first lets the jobs thicken in the system between the review epochs and then makes the matching decisions by solving an LP at the review epochs. Akbarpour et al. (2016) consider a dynamic matching network where jobs made to wait too long to be matched may abandon, or leave the system without being matched. They prove that if the system controller can identify which jobs are about to abandon, and match only those jobs, then waiting for other 71 jobs to arrive and thicken the market is very valuable. A straightforward implementation of the market thickening approach does not work in our case because customers expect drivers to be assigned to them immediately, and the system controller does not know when drivers will decide to finish working for the day. However, in both of the aforementioned papers, a large market assumption is used to facilitate the market thickening analysis, and that provides the key methodological insight for our chapter. The large market assumption arises in other matching contexts as well, such as Plambeck and Ward (2006), Arnosti et al. (2016), Leshno (2016), and Riquelme et al. (2015). A recent study applicable to ridesharing platforms is Hu and Zhou (2015), which consider dynamic match- ing control of a two-sided, discrete-time matching system where both supply and demand can abandon the system and the objective is to maximize the expected total discounted profit. A main modeling difference is that in their paper matchings between different types occur with either probability zero or one, whereas we assume probabilistic matching. Another difference is that our model leads to situations in which customers and drivers in the same location may not be matched (see the simulation experiments in Section 2.5.2); however, their Corollary 1 roughly states the opposite. The spirit of our methodology is drawn from the queueing control literature, where (i) an asymptotic regime is defined, (ii) a control policy is derived from the solution of an optimization problem, and (iii) asymptotic optimality of the control policy is proven (cf. Harrison (2000) and Ata and Kumar (2005)). Sim- ilar methodology is used in Braverman et al. (2016) to propose an empty car routing policy in a ridesharing model with limited driver supply. However, that methodology does not in general allow for the derivation of asymptotically optimal controls when the queueing network has time-varying arrival rates. In contrast, in the context of matching, Gurvich and Ward (2014), Hu and Zhou (2015), and our study all allow for time-varying arrival rates. Moreover, Hu and Zhou (2015) and our study both consider the situation of a time-varying distribution that represents the allowed waiting time for a match. There are some recent studies related to the stability of dynamic matching networks, cf. Caldentey et al. (2009), Adan and Weiss (2012), Buˇ si´ c et al. (2013), B¨ uke and Chen (2015), and Moyal and Perry (2017). However, stability is not an issue in our case because customers do not wait for pick-up from far away drivers and drivers leave their area if not matched with a customer quickly enough. 2.1.3 Notation and Terminology The set of nonnegative integers and strictly positive integers are denoted byN andN + , respectively. The k dimensional (k 2 N + ) Euclidean space is denoted by R k and R + denotes [0; +1). For x;y 2 R, x_y := maxfx;yg,x^y := minfx;yg, and (x) + := x_ 0. We letB(R k ) denote the Borel-algebra 72 onR k for all k2 N + andL(R) denote the collection of Lebesgue measurable subsets ofR, which is a -algebra onR. For allT2R + ,B([0;T ]) andL([0;T ]) denote the Borel and Lebesgue-algebra on the interval [0;T ], respectively. A functionf :X!R defined in measure space (X;X ) is calledX -measurable (denoted byf2X ), if it is (X;B(R))-measurable. If the measure space (X;X ) is (R;B(R)) ((R;L(R))), we say thatf is Borel (Lebesgue) measurable. For eachk2N + ,D k denotes the the space of all! :R + !R k which are right continuous with left limits. Let 0;e2 D be such that 0(t) = 0 ande(t) = t for allt2 R + . We abbreviate the phrase “independent and identically distributed” by “i.i.d.” and “almost surely” by “a.s.”. The notation a:s: ! denotes almost sure convergence. We assume that all of the random variables and stochastic processes are defined in the same complete probability space ( ;F;P),E denotes the expectation underP, andP(A;B) :=P(A\B). We let fXg denote the -field generated by the random variable X,I denote the indicator function, and? denotes independence. 2.2 A Ridesharing Model We partition the city intoN2N + disjoint areas, as illustrated in Figure 2.1, and assume that in the aggregate the individual driver decisions (regarding their work schedule and location) result in Poisson process arrivals and departures to and from each area. Specifically, typei drivers arrive at areai2N :=f1; 2;:::;Ng according to a non-homogeneous Poisson process having instantaneous rate i (t) at timet2R + , and cumu- lative rate function i (t) := R t 0 i (s)ds. The instantaneous rate at which typei drivers neither transporting customers nor on their way to pick-up a customer leave areai2N is i (t). Customers arrive in the system and request to be matched. The J 2 N + different customer types are categorized by factors such as their origin and destination area, and their priority status. Typej2J := f1; 2;:::;Jg customers arrive in accordance with a non-homogeneous Poisson process having instantaneous rate j (t) at time t2 R + , and cumulative rate function j (t) := R t 0 j (s)ds. A matching policy = ( 1 ;:::; J ) determines which driver type to offer an arriving customer. Each component j tracks the sequence of driver types offered to type j customers; i.e., when j (k) = i, then the system controller attempts to match thekth arriving typej customer with a typei driver, forj2J;k2N + ;i2N[f0g. The notation j (k) = 0 implies no driver is offered to the customer, in which case the customer is lost. The customer accepts the offered driver if the waiting time required for pick-up is not too large (in a sense specified precisely below), and otherwise the customer is lost and the offered driver stays in his current area. The implication is that customers are classified as matched or unmatched (i.e., lost) at the time of their 73 arrival, even though a matched customer must still wait to be picked-up by a driver. The process D ij (t) tracks the cumulative number of typei drivers matched to typej customers under policy in [0;t]. The number of drivers in areai at timet2R + depends on the matching policy. Then, forA i andR i unit rate Poisson processes, the number of unmatched typei drivers at timet is: Q i (t) =Q i (0) +A i ( i (t))R i Z t 0 i (s)Q i (s)ds X j2J D ij (t) 0; (2.1) wherefQ i (0);i2Ng are random variables independent of all other stochastic primitives. The second term in the right-hand side of (2.1) represents the cumulative number of driver arrivals to area i in [0;t], while the third and fourth terms represent the cumulative number of driver departures from areai in [0;t] by unmatched and matched drivers, respectively. We would like to match as many customers with drivers as possible. This is straightforward if there are many matching policies under whichQ i (t) > 0 for alli2N andt2 R + , because then there is always a driver near to an arriving customer. The difficulty is that in general not every area will have an available driver – and which areas have available drivers depends on earlier matching decisions. In this case, the arriving customer is matched only if the time required for the driver to pick-up the customer is less than the amount of time that customer is willing to wait for a driver. The time a customer is willing to wait for a driver can depend on the time of day. For example, during working hours, a customer may be more time-constrained than during non-working hours. We represent this using a step function that depends on the customer arrival time. First, we partition the time horizon into countably many disjoint intervals by defining the deterministic sequencef m ;m2Ng such that m 2R + and m < m+1 for allm2N, and m !1 asm!1. Second, to allow for potentially changing traffic conditions, we define the pick up time of a typei driver for a typej customer who arrived in the system at timet2 [ m ; m+1 ) byt ij (m)2R + for alli2N ,j2J , andm2N. Third, we denote the time thekth typej customer arrival is willing to wait for pick-up given the arrival occurred at timet2 [ m ; m+1 ) by the random variablea k j (m) for allk2N + ,j2J ,m2N. The sequencefa k j (m);k2N + g is i.i.d., and independent of all other stochastic primitives for allm2N andj2J . Then, the probability that thekth typej customer accepts a typei driver, given the arrival time ist, is F ij (t) = 1 X m=0 P a k j (m)t ij (m) I (t2 [ m ; m+1 )) (2.2) for alli2N ,j2J ,k2N + , andt2R + . 74 We connect our model to the queueing literature by showing its visual relationship to a stochastic process- ing network (cf. Harrison (2000)) in Figure 2.2. The main difference is that customers that do not accept their offered driver are lost, with the probabilities specified by F ij (t), fori2N ,j2J , andt2R + , and processing times are instantaneous. ! " ! ! " # ! $ # ! % # & " # ' " # & $ # ' $ # & ( # ' ( # ) * "" # ) * $" # ) * (" # Figure 2.2: A queueing-inspired visualization of our model formulation. The closest driver (CD) policy, denoted by CD , offers a typej customer that arrives at timet2 R + a driver type from the set argmin fi2N :Q CD i (t)>0g X m2N t ij (m)I(t2 [ m ; m+1 )): (2.3) If the set in (2.3) is not a singleton, the offered driver is chosen randomly from the closest drivers. The question is: Does the CD policy match as many customers as possible? The total cumulative number of matchings under any policy depends on the number of customers willing to wait for their offered driver. Specifically, ifE j is a unit rate Poisson process and j (k) := infft2R + : E j ( j (t)) =kg is the arrival time of thekth typej2J customer,k2N + , then D ij (t) := E j ( j (t)) X k=1 X m2N I j (k)2 [ m ; m+1 ); a k j (m)t ij (m); j (k) =i ; (2.4) 75 for alli2N ,j2J ,t2R + , and under any policy. Our objective is to find a policy that maximizes the total cumulative number of matchings made in a finite time horizon over a specified class of admissible matching policies ; that is, to solve max 2 X i2N;j2J D ij (T;!); (2.5) for all!2 . In studying this problem, we obtain some results on the more general objective max 2 X i2N;j2J w ij D ij (T;!) (2.6) that also allows the system manager to provide weightsw ij ;i2N;j2J; on the possible matchings. The objectives (2.5) and (2.6) are very strong objectives because solving either requires specifying a policy that maximizes the number of matchings on every sample path. Remark 2.2.1. In our model formulation, the driver types are formed only based on the arrival locations of the drivers. However, we can extend our results to an arbitrary (but finite) number of driver typesI2N + by updatinga k j (m) toa k ij (m) for allk2 N + ,j2J ,m2 N, andi2I :=f1; 2;:::;Ig. Then,a k ij (m) denotes the patience time of thekth typej customer for a typei driver given that she arrived in the system on the time interval [ m ; m+1 ), for allk2N + ,j2J ,m2N, andi2I. All of our results hold under this extension. 2.2.1 Admissible Policies Roughly speaking, an admissible matching policy cannot anticipate the future. This is formalized math- ematically by defining the filtrationF :=fF(t);t2 R + g that represents the information known by the system controller as time moves forward as follows. F(t) := ( A i ( i (s));E j ( j (s));R i Z s 0 i (u)Q i (u)du ;D ij (s);Q i (s);a k j (m) 8s2 [0;t]; i2N; j2J; m2N; k2f1; 2;:::;E j ( j (t))g ) : (2.7) In (2.7), we see that when a customer arrives in the system at timet, the system controller does not know how long she is willing to wait for driver pick-up. This is important because otherwise the system controller could cherry-pick certain customers to offer faraway drivers without consequence. The technical implication is that the filtrationF is not right continuous. 76 The information the system controller has at the arrival epoch of thekth typej customer is F j (k) := ( A i ( i (s^ j (k)));E j 0( j 0(s^ j (k)));R i Z (s^ j (k)) 0 i (u)Q i (u)du ! ; D ij 0((s^ j (k)));Q i ((s^ j (k))); 8i2N;j 0 2J;s2R + ; a r j 0(m);r2f1;:::;E j 0( j 0( j (k)))g;8j 0 2Jnfjg;a r j (m);r2f1;:::;k 1g;8m2N ) ; (2.8) for allk2N + andj2J . Since each of the stochastic processes that generate the-field in (2.7) is either right or left continuous and j (k) is a stopping time with respect to the filtrationF for allj2J andk2N + , the-field in (2.8) is well defined. Since j (k) j (k + 1) for allk2N + ,F j := F j (k);k2N + is a filtration for allj2J . Definition 2.2.1. (Admissible Policies) For all j 2 J , let j denote the set of discrete-time stochastic processes with domain N + and rangeN[f0g, such that for all j 2 j , j is F j -adapted (i.e., j (k)2F j (k) for allk2N + ) and ifQ i ( j (k)) = 0 for somei2N , then j (k)6= i. Let be the set ofJ-dimensional discrete-time stochastic processes such that for all2 , we have = ( 1 ; 2 ;:::; J ) where j 2 j for allj2J . Then, is the set of admissible policies. Lemma 2.2.1. The CD policy is admissible; that is, CD 2 . The proof of Lemma 2.2.1 is presented in Appendix B.5.1. Remark 2.2.2. Although the system controller cannot anticipate either the arrival times of the customers, or how long they will wait for pick-up, or how long the drivers will remain in their current area, he can accurately forecast the arrival rates of the customer and driver types and the driver acceptance probabilities associated with the customer types. In other words, he knows the functions i , j , and F ij for alli2N andj2J . Remark 2.2.3. The filtrationF can be augmented to include additional stochastic processes m : R + !R with right or left continuous sample paths for allm2N, provided that the sequencef m ;m2Ng does not contain any future information related to the processes that generateF(t) for all t2 R + . For example, if the system controller randomly chooses a driver to offer an arriving customer, thenf m ;m2 Ng includes a sequence of i.i.d. random variables that reflect the outcome of anN-sided die roll. Remark 2.2.4. Since the arrival process of each customer type is a non-homogeneous Poisson process, the probability that more than one customer arrives in the system simultaneously at some time epoch is zero. 77 Therefore, the range of all j 2 j is chosen asN[f0g instead of them-fold Cartesian product ofN[f0g for somem 2 for allj2J . Remark 2.2.5. LetF( j (k)) denote the sigma algebra defined by the stopping time j (k) as in Definition 1.2.12 of Karatzas and Shreve (1988). Another way to define j is such that for all j 2 j , j (k)2 F( j (k)) for allk2N + . We do not choose this option because provinga k j (m)?F( j (k)) for allj2J , m2 N, andk2 N + is difficult 1 , buta k j (m)?F j (k) is by construction (cf. (2.8)). This result is exactly what prevents the system controller knowing how long an arriving customer is willing to wait for driver pick-up, and so is crucial to our model and analysis. We end this section with the following technical assumption. Assumption 2.2.1. (Technicalities) The functions i :R + !R + and j :R + !R + are Borel measurable and the function i is defined such that i 2 D and i 0 for all i2N and j2J . We assume that sup t2R + i (t)<1, sup t2R + i (t)<1, and sup t2R + j (t)<1 for alli2N andj2J . The unit rate Poisson processesA i ;R i ; andE j are mutually independent for alli2N andj2J . 2.3 An Asymptotically Optimal CLP-Based Matching Policy It is very difficult to solve the optimization problem (2.6) exactly. Even if we can accomplish this very challenging task, the optimal matching policy will most likely be sample path dependent and will be very complicated. Hence, we consider a large market where the arrival rates of the customers and drivers grow without bound, and solve (2.6) under fluid scaling in that limiting regime. Section 2.3.1 specifies the large market limiting regime and the fluid scaling. Section 2.3.2 establishes that the solution to a CLP serves as an asymptotic upper bound on the objective (2.6) under fluid scaling. Section 2.3.3 provides a simple policy that can asymptotically mimic the performance of any feasible matching process for the CLP, and, therefore, can be used to specify an asymptotically optimal policy when the CLP is solvable. 2.3.1 A Large Matching Market We consider a sequence of matching systems indexed byn,n2N + . Each matching system has the same primitives with the one introduced in Section 2.2 except that the arrival rates of the drivers and customers, 1 Intuitively, one can expect thatFj(k) =F(j(k)) for allj2J andk2N+. Although such a result is proved under specific measure spaces (cf. Lemma 5.4.18 of Karatzas and Shreve (1988) and Lemma 1.3.3 of Stroock and Varadhan (2006)) or under some assumptions (cf. Theorem 1.6 of Shiryaev (2008)), none of them is applicable to our case. 78 and departure rates of unmatched drivers from their current areas depend onn. In thenth matching system, for alli2N ,j2J , andt2R + , let n i (t) := Z t 0 n i (s)ds; n j (t) := Z t 0 n j (s)ds; (2.9) where n i :R + !R + and n j :R + !R + are Borel measurable rate functions for alli2N ,j2J , and n2 N + . Moreover, n i 2 D and n i 0 for alli2N andn2 N + . A policy =f n ;n2 N + g is a sequence that specifies a policy for eachn, and is admissible if n is admissible for alln2 N + . For a policy such as CD that does not change withn, in a slight abuse of notation, we specify (i.e., = CD ) and assume n = for alln2N + . Our notational convention is to denote a process (or random variable) X in thenth system under the admissible policy byX ;n . Increasing the arrival rates without a bound and keeping the departure rates of unmatched drivers from their areas bounded results in a crowded matching system where we can use law of large numbers type of results to obtain tractable approximations for the processes of interest. Assumption 2.3.1. (Large Market) We assume that n i =n! i , n j =n! j , and n i ! i uniformly on compact intervals (u.o.c.) 2 asn!1 for alli2N andj2J . Assumption 2.3.1 does not necessarily imply that the arrival rates of all customer and driver types grow to infinity for allt2R + asn!1. In particular, arrival rates can be zero in some areas during some time periods (e.g., n i = n j = i = j =0 for somei2N andj2J and alln2N + ), as may be true in parts of the city during very early morning hours. We focus on the first-order imbalances between the driver supply and customer demand by considering fluid scaling. For alli2N ,j2J ,t2R + ,n2N + , and admissible policy, define A n i (t) :=A i (nt)=n; R n i (t) :=R i (nt)=n; E n j (t) :=E j (nt)=n; (2.10a) n i := n i =n; n j := n j =n; Q ;n i :=Q ;n i =n; D ;n ij :=D ;n ij =n: (2.10b) By (2.1), (2.10a), and (2.10b), for alli2N ,t2R + ,n2N + , and admissible policy, Q ;n i (t) = Q n i (0) + A n i ( n i (t)) R n i Z t 0 n i (s) Q ;n i (s)ds X j2J D ;n ij (t) 0: (2.11) 2 Forf 2 D andT12 R+, we letkfkT 1 := sup 0tT 1 jf(t)j. LetfXn;n2 Ng be a sequence inD andX 2 D. Then Xn!X u.o.c. asn!1, ifkXnXkT 1 ! 0 asn!1 for allT12R+. 79 Our analysis requires the initial number of drivers waiting to be very small. When the horizon length represents a working day, this corresponds to assuming the number of drivers during the late night hours is very small. In general, the horizon length is arbitrary, and can be smaller (i.e., correspond to so-called “rush” hour) or longer (i.e., represent multiple days). Assumption 2.3.2. (Initial Conditions) We assume that Q n i (0) a:s: ! 0 asn!1 for alli2N . Assumptions 2.2.1, 2.3.1, and 2.3.2 are in force throughout the chapter. 2.3.2 An Asymptotic CLP Upper Bound In this section, we derive an asymptotic upper bound on the fluid scaled objective (2.6) by solving a CLP. The decision variablesx ij : [0;T ]!R + for alli2N andj2J are such thatx ij (t) denotes the fraction of type j customers who are offered type i drivers at time t, so that j (t) F ij (t)x ij (t) approximates the instantaneous matching rate between typei drivers and typej customers. The relevant CLP is: max X i2N;j2J w ij Z T 0 j (s) F ij (s)x ij (s)ds; (2.12a) subject to: X j2J Z t 0 j (s) F ij (s)x ij (s)ds i (t); for alli2N andt2 [0;T ]; (2.12b) X i2N x ij (t) 1; for allj2J andt2 [0;T ]; (2.12c) x ij (t) 0; for alli2N ,j2J , andt2 [0;T ]; (2.12d) x ij is Lebesgue measurable for alli2N andj2J: (2.12e) Constraint (2.12b) implies that the cumulative number of customers who are assigned to typei drivers up to timet cannot be greater than the cumulative number of typei drivers arrived in the system up to timet for allt2 [0;T ] andi2N . Constraint (2.12c) implies that a customer cannot be offered more than one driver. Lemma 2.3.1. There exists an optimal solution of the CLP (2.12). The proof of Lemma 2.3.1 is presented in Appendix B.5.3. We denote an optimal solution of the CLP (2.12) by the process ~ x :=f~ x ij (t);i2N;j 2J;t2 [0;T ]g. The following theorem establishes that the optimal objective function value of the CLP (2.12) is an asymptotic upper bound on the fluid scaled objective (2.6) under any admissible policy for almost all sample paths. 80 Theorem 2.3.1. Under any admissible policy, lim sup n!1 X i2N;j2J w ij D ;n ij (T ) X i2N;j2J w ij Z T 0 j (s) F ij (s)~ x ij (s)ds; a.s. The proof of Theorem 2.3.1 is presented in Appendix B.1. The key challenge is to show that there exists a feasible matching processfx ij (t);i2N;j2J;t2 [0;T ]g (not necessarily optimal) such that the derivative of the limit of the fluid scaled cumulative matching processf D ;n ij (t);i2N;j 2J;t2 [0;T ]g corresponds to the processf j (t) F ij (t)x ij (t);i2N;j2J;t2 [0;T ]g. If the aforementioned feasible matching process is ~ x, then the upper bound in Theorem 2.3.1 is attained, meaning the policy is asymptotically optimal. The CLP (2.12) assumes that drivers will wait in their current location as long as necessary to be matched with customers, because the CLP does not incorporate i ,i2N . Although that provides a clear upper bound on the matching rates, that upper bound is not always achievable. In particular, in the case of a forward- looking CLP solution, the drivers must be patient enough to wait to be matched with closer customers arriving later in time (as in the example depicted in Figure 2.1). 2.3.3 A CLP-Based Randomized Policy We would like to find an admissible matching policy that can reproduce any feasible matching process x :=fx ij (t);i2N;j2J;t2 [0;T ]g for the CLP (2.12) (including an optimal one, ~ x). This is important because finding a feasible matching process that improves on any myopic matching policy (such as CD) may be possible even when finding an optimal CLP solution is not. The question is: How do we translate between a feasible matching process and the decision of which driver to offer an arriving customer? Definition 2.3.1. (Randomized Policy) If a type j customer arrives in the system at time t, the system controller makes a random selection from the setN[f0g such that the probability that the outcome isi is x ij (t) for alli2N and the probability that the outcome is 0 is 1 P i2N x ij (t). If the outcome isi for somei2N and there is a typei driver in the system, then the system controller offers a typei driver to the customer. If the outcome isi for somei2N but there is no typei driver in the system or if the outcome is 0, then no driver is offered to the customer (and so the customer is lost). We denote the randomized policy by R and write R (x) when we want to emphasize its dependence on the particular feasible matching processx. Under a technical condition on the associated feasible matching process for the CLP (2.12), R is admissible. 81 Lemma 2.3.2. Suppose that the feasible matching process for the CLP (2.12),x, is such thatx ij is a Borel measurable simple function for alli2N andj2J . Then, R (x) is admissible. The proof of Lemma 2.3.2 is presented in Appendix B.5.2. The randomized policy is a simple policy, because no state information is needed for its implementation. Still, its performance replicates any feasible matching process for the CLP (2.12), when drivers are willing to wait in their current locations to be matched. Theorem 2.3.2. Supposex is a feasible matching process that satisfies the condition in Lemma 2.3.2. Fix an arbitraryi2N . If i =0, then sup 0tT X j2J D R (x);n ij (t) Z t 0 j (s) F ij (s)x ij (s)ds a:s: ! 0; asn!1: The proof of Theorem 2.3.2 is presented in Appendix B.4. Theorem 2.3.2 shows that when an optimal solution ~ x to the CLP (2.12) is a Borel measurable simple function, then R (~ x) attains the upper bound in Theorem 2.3.1. However, ~ x may not satisfy the aforemen- tioned condition. Then, we do not know the associated randomized policy is admissible, and so we cannot use Theorem 2.3.2 to ensure the upper bound in Theorem 2.3.1 is achieved. However, we can approximate an optimal CLP (2.12) solution with a sequence of Borel measurable simple functions to ensure the upper bound is nearly achieved. Specifically, by Theorem 2.10 and Proposition 2.12 of Folland (1999), there exists a sequence of feasible matching processes for the CLP (2.12) denoted byf~ x r ;r2 N + g such that ~ x r :=f~ x r ij (t);i2N;j2J;t2 [0;T ]g, for alli2N ,j2J , andr2 N + , ~ x r ij is a Borel measurable simple function, 0 ~ x r ij (t) ~ x r+1 ij (t) ~ x ij (t) for allt2 [0;T ], and ~ x r ij (t)! ~ x ij (t) asr!1 for all t2 [0;T ] except on a set of zero measure. Then, by Theorem 2.3.2 and the bounded convergence theorem used on the sequencef j (t) F ij (t)~ x r ij (t);t2 [0;T ];r2N + g, we have the following corollary. Corollary 2.3.3. If i = 0 for all i2N , then for any > 0, there exists an r 0 ()2 N + such that if rr 0 (), lim n!1 X i2N;j2J D R (~ x r );n ij (T ) X i2N;j2J Z T 0 j (s) F ij (s)~ x ij (s)ds; a.s. Solving the CLP (2.12) or accurately approximating an optimal solution of it is a very challenging task (cf. Perold (1981)). If j (t) F ij (t) was a constant function oft for alli2N andj2J , then the CLP (2.12) would become a Separated Continuous Linear Program (SCLP). Although an SCLP is solvable (cf. Anderson et al. (1983) and Weiss (2008)), it is an NP-hard problem (cf. Bertsimas et al. (2015)). Hence, we will derive conditions under which CLP (2.12) can be simplified into an LP in the following section. 82 Remark 2.3.1. The CLP (2.12) can be connected to pricing as follows. Suppose that the customer and driver arrival rates depend on the pricing decisions, but are independent of the matching policy. Next assume the following sequence of events. First, the system controller determines the price function for the time interval [0;T ]. Second, the customer and driver arrival rates are realized based on the given price function. Third, the matching decisions are done continuously over the time horizon. Then, the parameters in the CLP (2.12) can be considered as outcomes of pricing decisions of the system controller. The implication is that our proposed CLP-based matching policy is asymptotically optimal (i.e., satisfies Theorems 2.3.1, 2.3.2, and Corollary 2.3.3) for a simple fixed-price function model. This motivates a modification to the CLP (2.12) in which there is a joint optimization over the price function and the matching process. 2.4 An Asymptotically Optimal LP-Based Randomized Policy LP-based matching policies arise when the system controller does not need to “save” drivers for future cus- tomers; i.e., when the system controller can be myopic. Section 2.4.1 shows a myopic LP-based randomized policy is asymptotically optimal when the drivers are always busy, or fully utilized. A myopic LP-based randomized policy is also asymptotically optimal when parameters are time homogeneous, regardless of whether or not drivers are fully utilized, and that LP can be modified to include pricing. We show in Sec- tion 2.4.2 that when pricing affects driver supply and customer demand, jointly optimizing over pricing and matching decisions results in fully utilized drivers, which provides a partial justification of the aforemen- tioned “fully utilized driver” condition. 2.4.1 An LP-Based Randomized Matching Policy The following LP is relevant at each fixedt2 [0;T ]: max X i2N;j2J w ij j (t) F ij (t)x ij (t); (2.13a) subject to: X j2J j (t) F ij (t)x ij (t) i (t); 8i2N; (2.13b) X i2N x ij (t) 1; 8j2J; (2.13c) x ij (t) 0; 8i2N; j2J; (2.13d) where the decision variables arex ij (t) for alli2N andj2J . The main difference between the LP (2.13) and the CLP (2.12) is that (2.13a) and (2.13b) in the LP (2.13) are the derivatives of (2.12a) and (2.12b) in the CLP (2.12), respectively. By Assumption 2.2.1 and the fact thatx ij (t) = 0 for alli2N andj2J is 83 feasible for the LP (2.13) for allt2 [0;T ], there exists an optimal solution of the LP (2.13) for allt2 [0;T ]. We denote an optimal solution of the LP (2.13) at timet byfx ij (t);i2N;j2Jg for allt2 [0;T ] and x :=fx ij (t);i2N;j2J;t2 [0;T ]g. Assumption 2.4.1. (Measurability) There exists anx such thatfx ij (t);t2 [0;T ]g is Lebesgue measurable for alli2N andj2J . If each of i and j is a step function for alli2N andj2J , then there exists a function which satisfies Assumption 2.4.1, and that function can be chosen as a step function. Assumption 2.4.1 is valid in the rest of Section 2.4.1. The CLP (2.12) can be solved using the LP (2.13) when drivers are fully utilized. Lemma 2.4.1. Suppose X i2N;j2J Z t 0 j (s) F ij (s)x ij (s)ds = X i2N i (t); for allt2 [0;T ] (2.14) for some matching processx that is feasible for the CLP (2.12) 3 . Ifw ij = 1 for alli2N andj2J , then the processx is an optimal solution of the CLP (2.12). The proof of Lemma 2.4.1 is presented in Appendix B.5.4. The following asymptotic LP-based upper bound on the fluid scaled objective (2.5) follows from Lemma 2.4.1 combined with Theorem 2.3.1. Corollary 2.4.1. If (2.14) holds, under any admissible policy, lim sup n!1 X i2N;j2J D ;n ij (T ) X i2N;j2J Z T 0 j (s) F ij (s)x ij (s)ds; a.s. In order to obtain the upper bound in Corollary 2.4.1, the LP (2.13) must be solved infinitely many times (for all t2 [0;T ]). However, in practice, the LP (2.13) can be solved at finitely many time epochs, and the remainingx ij (t) values can be approximated by, for example, linear interpolation or assuming thatx ij is a step function. Moreover, the time-varying parameters of the LP (2.13) can be estimated in real-time, because the matching decisions do not depend on future estimated arrival rates of the customers and drivers. The randomized policy R (x ) is myopic. In particular, an LP-based policy does not prolong driver waiting in some areas in anticipation of future demand, as can happen under a CLP-based policy. As a 3 Condition (2.14) is parallel to Assumption 1 in Harrison (2000), which introduces an optimization problem to define fully utilized resources in the queueing literature when parameters do not vary with time. 84 result, in contrast to Theorem 2.3.2, R (x ) achieves the upper bound in Corollary 2.4.1 without requiring the condition i =0 for alli2N . Theorem 2.4.2. Supposex satisfies the condition in Lemma 2.3.2, so R (x ) is admissible. If (2.14) holds andw ij = 1 for alli2N andj2J , then R (x ) achieves the asymptotic upper bound given in Corollary 2.4.1, i.e., lim n!1 X i2N;j2J D R (x );n ij (T ) = X i2N;j2J Z T 0 j (s) F ij (s)x ij (s)ds; a.s. The proof of Theorem 2.4.2 is presented in Appendix B.4. 2.4.2 Jointly Optimizing Pricing and Matching One natural intuition is that the required balance in (2.14) arises naturally when the system controller can use pricing to influence customer and driver behavior. This is because a “smart” system controller will not raise prices beyond what is needed to match driver supply with customer demand. This leads to a joint pricing and matching problem (with one such example following from Remark 2.3.1). However, any time-varying problem formulation is very difficult to solve. Therefore, we focus on a static formulation to show how (2.14) can be viewed as a consequence on good pricing decisions. We begin with the observation that we do not need the condition (2.14) to show that the randomized policy based on the solution to the LP (2.13) is asymptotically optimal when parameters are time homogeneous. Theorem 2.4.3. If i , j , and F ij are constant functions oft for alli2N andj2J , we can choosex as a constant function oft. Then,x is an optimal solution of the CLP (2.12) and R (x ) is admissible. Moreover, ifw ij = 1 for alli2N andj2J , then R (x ) achieves the asymptotic upper bound given in Theorem 2.3.1; i.e., lim n!1 X i2N;j2J D R (x );n ij (T ) = X i2N;j2J Z T 0 j (s) F ij (s)~ x ij (s)ds; a.s. The proof of Theorem 2.4.3 is presented in Appendix B.4. The question we address is: When the LP (2.13) formulation does not vary with time, and is expanded to include pricing, does an optimal solution satisfy (2.14)? To do this, suppose that, for alli2N ,j2J and n2 N + , n i and n j are constant functions that depend on the prices determined by the system controller at time 0, F ij is a time homogeneous function independent of the prices, and n i can depend on both the prices and the time. Specifically, first, the system controller sets pricesp =fp ij ;i2N;j2Jg at time 0. The prices can depend on both the customer’s typej2J (and so can be based on the customer’s origin 85 and destination area) and the areai2N where the assigned driver is currently located. Second, the time homogeneous arrival rates of the customers and drivers ( n i and n j ) and time-varying departure rates of unmatched drivers from their current areas ( n i ) are realized, and matchings are performed continuously over the time horizon [0;T ]. We extend Assumptions 2.2.1 and 2.3.1 in the following way: Assumption 2.4.2. (Technicalities 2) LetP :=fp2 R NJ + : p ij 2 [0; p];8i2N;j2Jg be the set of possible price vectors where p2 R + is the maximum chargeable price. For alli2N ,j2J ,n2 N + , p2P, andT 1 2R + , we have i ; j ; n i ; n j :P!R + , the functions i ; n i :R + P!R + is defined such that i (;p)2D, n i (;p)2D, sup t2R + i (t;p)<1, and j n i (p)=n i (p)j_ n j (p)=n j (p) _ sup t2[0;T 1 ] j n i (t;p) i (t;p)j! 0; asn!1: The system controller sets prices at time 0 by solving the following optimization problem: max X i2N;j2J w ij j (p) F ij x ij ; (2.15a) subject to: X j2J j (p) F ij x ij i (p); 8i2N; (2.15b) X i2N x ij 1; 8j2J; (2.15c) x ij 0; p ij 2 [0; p]; 8i2N; j2J; (2.15d) where the decision variables arex ij andp ij for alli2N andj2J . The optimization problem (2.15) is the LP (2.13) which is time homogeneous but modified to include pricing decisions. Sincex ij = 0 and p ij 2 [0; p] for alli2N andj2J is feasible for (2.15), the feasible region is nonempty. We cannot know whether or not there exists an optimal solution without additional assumptions on the functions i and j ,i2N ,j2J . For example, if i and j are continuous functions ofp for alli2N andj2J , then the feasible region is compact and so an optimal solution of (2.15) exists. If the price(s) in an area is equal to 0, then one can expect the total customer arrival rate to be very large, but the driver arrival rate to be 0 and increasing in price. Thus, an optimal solution should have nonzero prices. We associate the set of admissible policies given in Definition 2.2.1 with a price vectorp2P, which determines the rates of the processes generating the filtrationF =fF(t);t2 R + g representing the infor- mation known to the system controller as time moves forward (cf. (2.7)). Specifically, for a givenp2P, a policy(p) =f n (p);n2N + g is a sequence that specifies a policy for eachn, and(p) is admissible 86 if n (p) is admissible for alln2 N + . Let (x;p) be feasible matching fractions and prices for (2.15) and R;p (x) denote the randomized policy associated with the matching fractionsx (cf. Definition 2.3.1) under the price vectorp, i.e., the system controller sets the price vectorp at time 0 and then makes the matching decisions using the randomized policy with matching fractionsx. Then, we have the following corollary to Theorems 2.3.1 and 2.4.3. Corollary 2.4.4. Suppose that (x ;p ) is an optimal solution of (2.15). Then, under any admissible policy (p),p2P, lim sup n!1 X i2N;j2J w ij D (p);n ij (T ) X i2N;j2J w ij T j (p ) F ij x ij ; a.s. Moreover, ifw ij = 1 for alli2N andj2J , then R;p (x ) is asymptotically optimal, i.e., lim n!1 X i2N;j2J D R;p (x );n ij (T ) = X i2N;j2J T j (p ) F ij x ij ; a.s. If the constraint (2.15b) is binding for all driver types under given feasible matching fractions and prices, then condition (2.14) holds. Hence, we would like to know whether the constraint (2.15b) binds under “good” pricing decisions, e.g., under an optimal solution of (2.15) (if exists). The intuition that a “good” pricing policy is one under which the constraint (2.15b) binds is natural when lowering prices encourages less driver arrivals but more customer arrivals. Hence, there is no need to have idle drivers in the system at any time. That intuition is consistent with a result proved in Bimpikis et al. (2016) showing that drivers never idle (see Proposition 1 and the second and third paragraphs on page 20 therein). We provide sufficient conditions in Example 2.4.1 below under which the constraint (2.15b) binds under an optimal solution of (2.15). Example 2.4.1. There exists a customer specific baseline price for riding, denoted byc j > 0, and an area specific surge multipliers s i 2 [0; s] fori2N , where s2 R + is the maximum possible surge multiplier. Suppose that c j is constant but the system controller determines the surge multipliers at time 0. Let : J!N be a function which specifies the arrival location of each customer type. Then, the price that a type j customer needs to pay for a ride provided by a typei driver isp ij = c j s (j) for alli2N andj2J . Suppose that there existsC 2 C 1 > 0 such that C 2 X j2J :(j)=i @ j @s i C 1 ; 0 @ i @s i C 2 ; for alli2N; (2.16a) X j2J :(j)=i @ j @s k = 0; for alli;k2N such thati6=k; (2.16b) 87 X j2J :(j)=i @ j @s i X k2Nnfig @ k @s i C 1 for alli2N; (2.16c) For alli2N , when s i = 0, i = 0. (2.16d) Condition (2.16a) states that as the surge multiplier in an area decreases, the total customer arrival rate in that area increases and the driver arrival rate in that area decreases. Condition (2.16b) implies that the total customer arrival rate in an area is not affected by surge multipliers in other areas. Condition (2.16c) roughly requires that the change in the surge multiplier in an area affects that area more than it affects the other areas. Condition (2.16d) enforces that drivers do not work for free. Lemma 2.4.2. Under the conditions (2.16a)-(2.16d), there exists an optimal solution of (2.15) in which the constraint (2.15b) is binding for all driver types. The proof of Lemma 2.4.2 is presented in Appendix B.5.5. 2.5 Performance Evaluation We begin by observing that the performance of the randomized policy can likely be improved by incorpo- rating state information. This is because the randomized policy can match a customer with an area in which there are no drivers, leading to that customer being lost. To correct this, in Section 2.5.1, we introduce state-dependent LP- and CLP-based policies, that require knowledge of driver locations. Then, in Section 2.5.2, we compare the performance of those policies and the LP- and CLP-based randomized policy against the benchmark CD policy. We do this first when parameters are time homogeneous and second when they vary with time. In the first case, the LP- and CLP-based policies coincide (cf. Theorem 2.4.3), whereas in the second, when the importance of considering future customer and driver arrival rates becomes important, they do not. 2.5.1 Additional Proposed Matching Policies The randomized policy is not match conserving in the sense that a customer arriving at a time when at least one driver is available is always offered a driver. The benchmark CD policy (in (2.3)) is match conserving. We propose two new policies, one deterministic and one not, that are match conserving and can be based on a feasible matching processx for the CLP (2.12) (or the LP (2.13)). 88 Deterministic Policy: If a typej2J customer arrives in the system at timet, the system controller offers a driver type from the set argmin fi2N :Q i (t)>0g D ij (t) Z t 0 j (s) F ij (s)x ij (s)ds : (2.17) If the set in (2.17) is not a singleton, then the system controller can use any tie breaking rule which does not use any future information. If there is no driver in the system, then no driver is offered. (Note that we have suppressed the dependence ofD ij ; j , and F ij onn in the notation to emphasize that the policy can be applied directly to the model in Section 2.2, even though any results obtained on its asymptotic performance would require reference to the sequence of matching systems introduced in Section 2.3 and used in Sections 2.3 and 2.4.) Hybrid Policy: Suppose that a typej customer arrives in the system, and there is a driver in the system. Then the system controller makes a random selection in the same way explained under the randomized policy (cf. Definition 2.3.1). If the outcome isi for somei2N and there is a typei driver in the system, the system controller offers a typei driver to the customer. If the outcome isi for somei2N but there is no typei driver in the system or if the outcome is 0, the customer is offered the driver type specified in (2.3) by the CD policy. If there is no driver in the system, then the customer is lost. Remark 2.5.1. Ifx ij = 0 for alli2N andj2J such thati6= (j), then the Hybrid policy is the CD policy defined in (2.3). Lemma 2.5.1. The deterministic and hybrid policies are admissible under the condition on x in Lemma 2.3.2. The proof of Lemma 2.5.1 is presented in Appendix B.5.2. The deterministic and hybrid policies can be implemented regardless of whether or not they are admis- sible. However, in order to prove asymptotic optimality results parallel to those for the randomized policy (specifically, Corollary 2.3.3 and Theorems 2.3.2, 2.4.2, and 2.4.3), admissibility is required (and can be achieved using the approximation explained prior to Corollary 2.3.3). The proofs of the aforementioned asymptotic optimality results are much more difficult due to the state dependence. 2.5.2 Simulation Experiments In this section, we present simulation experiments where we test the performances of the CD policy and the LP- and CLP-based matching policies. We associate the LP- and CLP-based policies based on an optimal 89 solution of the LP (2.13) and an optimal solution of the CLP (2.12), respectively. We present an experiment with time homogeneous parameters and an experiment with time-varying parameters. The term “offered driver” refers to the driver offered to an arriving customer, that is determined by the matching policy. The customer may or may not accept being matched with the driver offered at the arrival timet2 [0;T ], depending on the acceptance probability F ij (t). Any variation of the word “match” means that the relevant customer has both been offered a driver, and has accepted that driver, meaning that driver picks up that customer. We assume that the larger the F ij (t) at timet2 [0;T ], the smaller the pick up time of a typei driver for a typej customer. Then, under the CD policy, the offered driver set for a typej customer arriving at timet in (2.3) is exactly argmax fi2N :Q i (t)>0g F ij (t) for allj2J andt2R + (and the match occurs if also the resulting pick-up time is lower than the time that customer is willing to wait for pick-up). We chooseN = J = 3 in both simulation experiments (so the customer type represents the customer’s arrival location). This is the smallest network size possible that allows for us to illustrate the more general insight behind when and why LP-based policies outperform CD. The reason a two area network will not work is that as long as F ii (t) F ij (t) for alli;j2f1; 2; 3g andt2 [0;T ], the LP-based policies give more priority to within-area matchings than they give to between-area matchings, which coincides with the CD policy. In contrast, the reason CLP-based policies outperform CD (as well as any other myopic policy such as an LP-based one) has to do with their potential to be forward-looking (which can be seen in a two-area network). Implementation Details. We used Omnet++ discrete-event simulation freeware. In each experiment, we started with an empty system; i.e.,Q i (0) = 0 for alli2N . At each simulation instance associated with each matching policy, we have done Rep number of replications. The maximum of the margin of errors associated with the 95% confidence interval for the percentage of all customers matched (=t 0:025;Rep1 Sample Stdv.= p Rep) among all policies in all instances is less than 0:42. ! " # $ % " # & ! ' # & % ' # & ! ( # & % ( # $ ) * "' # ) * '" # $+,, ) * '( # ) * (' # $+,- !" #" $" ) * "( # ) * (" # $ Figure 2.3: The parameters of the first simulation experiment are: Q i (0) = 0, i = 0:1, F ii = 1,w ij = 1 for alli;j2f1; 2; 3g, andn2f1; 10; 100g. 90 Time Homogeneous Parameters We present a simulation experiment which shows that our LP-based policies can potentially match more cus- tomers and drivers in a finite time horizon than the CD policy does. Figure 2.3 provides all input parameters. Since all parameters are time homogeneous, the LP- and CLP-based policies coincide (cf. Theorem 2.4.3). We omit “(t)” from the notation. Moreover,n2f1; 10; 100g measures the market size, as in Assumption 2.3.1. The intuition for why we do not expect the CD policy to perform well in this example is as follows. Under the CD policy, when a type 1 or 2 customer arrives in the system, the offered driver is type 2 (if there is one in the system) because 1 = 0, F 21 > F 31 , and F 22 > F 32 . Since 1 + 2 = 2n> 2 =n, some of the type 1 or 2 customers cannot be offered type 2 drivers and must be offered type 3 drivers. The type 2 customers who are not offered type 2 drivers are offered type 3 drivers and so 98% of those customers are matched. However, the type 1 customers offered type 3 drivers are lost because F 31 = 0. In comparison, the optimal solution of the LP (2.13) hasx 21 = 1,x 22 = 0:01, andx 32 = 0:99. In words, the LP-based policies match type 1 customers with type 2 drivers and type 2 customers with type 3 drivers, to prevent losing a non-trivial percentage of the type 1 customers. CD Rand. Det. Hyb. 50 60 70 80 90 100 % Customers Matched LP Ub. (a)n = 1 CD Rand. Det. Hyb. 50 60 70 80 90 100 % Customers Matched LP Ub. (b)n = 10 CD Rand. Det. Hyb. 50 60 70 80 90 100 % Customers Matched LP Ub. (c)n = 100 Figure 2.4: The percentage of all customers matched in the simulation experiment corresponding to Figure 2.3. Figure 2.4 shows the percentage of all customers matched with drivers in the simulation experiment (i.e., the objective (2.5), sincew ij = 1 for alli;j2f1; 2; 3g) under the CD policy and the LP-based policies. “LP Ub.” shows one hundred times the optimal objective function value of the LP (2.13) divided by the total customer arrival rate to the system, which is an approximate upper bound on the percentage of all customers matched under any admissible policy by Theorems 2.3.1 and 2.4.3. The key observations are that (i) the LP-based policies outperform the CD policy in all traffic intensities; 91 (ii) the LP-based policies are very close to the approximate upper bound based on Theorem 2.3.1 as the arrival rates become large, which verifies Theorem 2.4.3 numerically. ! " # $ % " &'( ! ) # * % ) # $ ! + # $ % + # * , - ") # , - )" # *.// , - )+ # , - +) # *./0 !" #" $" , - "+ # , - +" # * Figure 2.5: The parameters of the second simulation experiment are: Q i (0) = 0, F ii = 1,w ij = 1 for all i;j2f1; 2; 3g,n2f0:1; 1; 10; 100g, i = is constant ini and2f10 2 ; 10 3 ; 10 4 ; 10 5 g,T = 1800, 1 (t) = 0 for allt2 [0;T=2]; and 1 (t) = 2n for allt2 [T=2;T ]. Time-Varying Parameters Motivated by the example in Figure 2.1 in Section 2.1, our next simulation experiment is designed to show that CLP-based policies can significantly outperform LP-based policies and the CD-policy by taking into account the future customer and driver arrival rates. Our assumption is that the customer arrival rate in one area will spike; in particular, we assume 1 (t) = 0 for allt2 [0;T=2] and 1 (t) = 2n for allt2 [T=2;T ], where the length of the time horizon isT = 1800, andn2f0:1; 1; 10; 100g denotes the traffic intensity of the system. Figure 2.5 provides the other input parameters, which are time homogeneous, and so “(t)” is omitted. Since the total customer and driver arrival rate in the first half of the simulation (t2 [0;T=2]) aren and 2n, respectively, at least half of the arriving drivers will be idle in the first half of the simulation. This implies that the condition (2.14) does not hold in this experiment. Table 2.1 shows the percentage of all customers that are matched with drivers in the simulation experiment (i.e., the objective (2.5), sincew ij = 1 for alli;j2f1; 2; 3g) under the CD policy, the LP-based policies, and the CLP-based policies. Table 2.1 confirms that the CLP-based policies outperform the LP-based policies, and the LP-based poli- cies outperform the CD policy, although condition (2.14) does not hold. Furthermore, both the CLP-based and LP-based policies achieve the expected performance derived from the optimal solution of the CLP (2.12) and the LP (2.13) as the market sizen becomes large. Most intriguing, the performance of the CLP-based policies improves as becomes small, whereas the performance of the CD and LP-based policies is inde- pendent of. This is because the CLP-based policies are forward-looking but the CD and LP-based policies are not. As a result, the CLP-based policies are the only ones that require the drivers arriving in the first half of the simulation to be willing to wait until sometime during the second half of the simulation to be 92 Table 2.1: The percentage of all customers matched in the simulation experiment corresponding to Figure 2.5. LP-Based Policies CLP-Based Policies n CD Rand. Determ. Hybrid Rand. Determ. Hybrid 10 2 0.1 63.8 57.9 67.1 67.5 62.5 68.1 67.9 1 66.1 69.1 72.6 72.9 72.3 73.9 74.1 10 66.1 72.9 73.9 74.1 75.2 75.5 75.5 100 66.1 74.0 74.2 74.3 75.9 75.9 75.9 10 3 0.1 67.0 69.4 73.5 75.4 82.0 83.4 83.7 1 66.4 72.7 74.1 74.8 85.8 86.1 86.3 10 66.3 73.9 74.2 74.5 86.8 86.8 86.9 100 66.2 74.2 74.2 74.4 86.9 86.9 86.9 10 4 0.1 68.0 70.3 73.7 77.0 92.1 92.8 93.5 1 66.7 73.0 74.2 75.2 96.0 96.0 96.1 10 66.3 74.0 74.2 74.7 96.8 96.8 96.8 100 66.2 74.2 74.2 74.5 96.9 96.9 96.9 10 5 0.1 67.8 70.4 73.9 77.0 93.6 93.7 94.3 1 66.8 73.2 74.1 75.4 97.4 97.4 97.6 10 66.4 74.0 74.2 74.7 98.5 98.5 98.6 100 66.2 74.2 74.3 74.5 98.8 98.7 98.8 offered to a customer – and that “willingness-to-wait” is determined by the parameter. This observation is exactly the reason Theorem 2.3.2 and Corollary 2.3.3 (related to the asymptotic optimality of the potentially forward-looking CLP-based policies) require = 0 but Theorems 2.4.2 and 2.4.3 (related to the asymptotic optimality of myopic LP-based policies) do not. The next three paragraphs explain the intuition for the percentages shown in Table 2.1. The CD policy: In the first half of the simulation, i.e., in the time interval [0;T=2], the CD policy offers type 1 drivers to type 2 customers and in total 0:99n(T=2) type 2 customers are matched. In the second half of the simulation, the CD policy offers type 1 drivers to both type 1 and type 2 customers. All of the type 1 customers who are offered type 1 drivers will accept those drivers and 99% of the type 2 customers who are offered type 1 drivers will accept those drivers. Then, the arrival rate of the customers who will accept a match with type 1 drivers if offered is 2n + 0:99n = 2:99n but the arrival rate of type 1 drivers is only n. Thus, only 1=2:99 of the type 1 customers are offered and matched with type 1 drivers and 1:99=2:99 of them are lost. Hence, nT=2:99 type 1 customers are matched in the second half of the simulation. Similarly, among the 99% of the type 2 customers who accept type 1 drivers if offered, 1=2:99 of them will be offered type 1 drivers. Hence, 0:99n(T=2)=2:99 type 2 customer are matched with type 1 drivers and 0:98 1:99n(T=2)=2:99 type 2 customers are matched with type 3 drivers, and the remaining type 2 customers are lost in the second half of the simulation. Dividing the total number of matchings by the total number of arriving customers (2nT ) leads to 66:1% of all customers being matched, which agrees with Table 2.1 for largern. The LP-based policies: The optimal solution of the LP (2.13) is such thatx 12 (t) = 1 for allt2 [0;T=2] andx 11 (t) = 0:5 andx 32 (t) = 1 for allt2 [T=2;T ], and all other decision variables at all other times 93 are 0. In the first half of the simulation (t2 [0;T=2]), x 12 (t) = 1 implies type 2 customers are offered type 1 drivers, resulting in 0:99n(T=2) matchings. In the second half of the simulation (t2 [T=2;T ]), x 11 (t) = 0:5 dictates type 1 customers and drivers being matched; however, the disparity between the arrival rates results in only half the customers being matched, or n(T=2) matchings. Also, x 32 (t) = 1 results in type 2 customers matching with type 3 drivers, leading to 0:98n(T=2) matchings. Dividing the total number of matchings by the total number of arriving customers (2nT ) leads to 74:3% of all customers being matched, which agrees with Table 2.1 for largern The CLP-based policies: An optimal matching process for the CLP (2.12) has: x 32 (t) = 1 for allt2 [0;T ], x 11 (t) = 0 for all t2 [0;T=2], x 11 (t) = 1 for all t2 [T=2;T ], all other decision variables are 0. (To see that process is optimal, note that as many drivers as possible are matched.) This solution is forward-looking because the type 1 drivers arriving before timeT=2 are “saved” to be offered to the type 1 customers arriving after timeT=2. Then, 2n(T=2) type 1 customers are matched in the second half of the simulation. Sincex 32 (t) = 1 for the entire simulation, there are 0:98nT matchings for type 2 customers. Dividing the total number of matchings by the total number of arriving customers (2nT ) leads to 99% of all customers being matched, which, in contrast to the CD and LP-based policies does not agree with Table 2.1 for all larger values ofn. The issue is that not all drivers arriving in the first half of the simulation will wait for customers to arrive in the second half of the simulation. The number that wait depends on the parameter. This is why we see 99% of all customers matched in Table 2.1 only when bothn is larger and is smaller. 2.6 Concluding Remarks The decisions on which driver to offer to each arriving customer in a ridesharing system impact the overall number of customers matched. This is because those decisions determine whether or not future available drivers will be close to the locations of arriving customers. We have formulated an optimization problem whose solution serves as an asymptotic upper bound on the cumulative number of matchings as the market becomes large. That optimization problem accounts for (i) the differing arrival rates of drivers and customers in different areas of the city, (ii) how long customers are willing to wait for driver pick-up, and (iii) the time- varying nature of all the aforementioned parameters. The aforementioned optimization problem is in general a CLP, which can be difficult to solve. We estab- lish that a simple randomized matching policy can asymptotically mimic the performance of any feasible matching process for the CLP, so that there is potential to develop “good” CLP-based matching policies, 94 even when an optimal CLP solution is unknown. When an optimal CLP solution is known, then a CLP- based randomized policy asymptotically achieves the aforementioned optimization problem upper bound (assuming drivers are “patient enough”). Under the assumption that drivers are fully utilized or when the CLP parameters are time homogeneous, the CLP solution can be specified through LP solutions. Then, a LP-based randomized policy asymptotically achieves the aforementioned optimization problem upper bound. In the time homogeneous setting, when customer and driver arrival rates depend on price, we establish an asymptotic upper bound on the cumulative number of matchings by solving an optimization problem that jointly optimizes over prices and matchings, and provide a joint pricing and matching policy that achieves that upper bound. Excellent questions for future research include better understanding the joint pricing and matching problem in time-varying settings, and when customer and driver utilities are explicitly modeled. Many ridesharing companies have a car pooling option in which more than one customer can share a single driver. In this case, a “matched” driver is still regarded as potentially available. The issue, as studied in Gopalakrishnan et al. (2016), is that “adding” a customer to an already matched driver causes inconvenience to the customer(s) that are already passenger(s) (i.e., already riding with that driver). More research is required to fully understand how to make matching decisions that both account for this inconvenience and ensure the highest possible overall matching rates. 95 Chapter 3 Joint Pricing and Matching in Ridesharing Systems 3.1 Introduction The relatively recent explosive growth of ridesharing firms such as Uber and Lyft in the USA, Didi Chuxing in China, and Ola in India has brought many operational questions to the forefront. Two key questions com- mon to all ridesharing firms concern pricing and matching. The pricing decisions determine both customer demand and driver supply. When a customer requests a ride, the firm should determine which available driver to offer the customer which we call a matching decision. The matching decisions determine how long a customer must wait for driver pick-up. Both the pricing and matching decisions together affect the geospatial distribution of the drivers at any given point of time. That geospatial distribution determines the set of drivers that are available to be matched to an arriving customer. The issue is that a customer offered a far away driver may not accept the ride due to the long pick-up time. An important service metric for a ridesharing firm is the number of customers that are matched, meaning the number of customers that accept the offered rides. On the one hand, the prices should be low enough to attract the customers to request rides; on the other hand, the prices should be high enough to attract the drivers to work in the ridesharing platform. However, having many customers and drivers does not necessarily imply many matchings. The firm should ensure sufficient level of proximity of the offered drivers to the customers, because customers often do not accept far away drivers. In summary, pricing and matching decisions that result in maximum number of matchings is of primary importance. In this chapter, we jointly optimize the pricing and the matching decisions. There are many papers in the literature (e.g., Riquelme et al. (2015), Bimpikis et al. (2016), Castillo et al. (2016), Guda and Subramanian (2017), Besbes et al. (2018)) that optimize the pricing decisions under an assumed matching policy. There are also several studies in the literature that optimize the matching decisions while ignoring the pricing decisions (e.g., Hu and Zhou (2015), Banerjee et al. (2018), Chapter 2 of this thesis). Our study is the first to jointly optimize the pricing and the matching decisions. 96 A natural question is: How important is this joint optimization? The matching policy that offers an arriving customer the closest available driver is simple, intuitive, and model-blind (that is, no knowledge of customer and driver behavior patterns is assumed). The natural approach is to fix the aforementioned match- ing policy and to use pricing to ensure there are enough drivers. More specifically, under the assumption that a fixed fraction of the price a customer pays goes to the driver, raising the price increases the driver supply. Since the price can vary across locations, the pricing can be used as a main lever to influence the locations of drivers waiting to be assigned a customer. Unfortunately, as the following example shows, ignoring the matching optimization can result in subpar overall performance. Example 3.1.1. In Figure 3.1, the city is partitioned into nine hexagonal disjoint areas 1 . When the areas are small enough, prioritizing same area matchings approximates the closest driver matching policy. There are three customers and three drivers in the city. Figure 3.1a shows the consequence of matching the customers with the drivers from the same areas. Figure 3.1b provides an alternative matching in which each customer is offered a driver from a different area. The likelihood of losing one customer due to a faraway match can be higher in Figure 3.1a than in Figure 3.1b. Figure 3.1: An example illustrating the effect of matching decisions on the number of matchings. The previous argument and Example 3.1.1 motivate the desire to understand how important optimizing the pricing and matching decisions jointly is. Doing so is complicated by the fact that drivers are self-scheduling in the sense that they determine when to work, where to work, and how long to work. Therefore, our study requires an analytically tractable model for driver behavior under which we can determine the equilibrium driver arrival rates and locations for any given pricing and matching policy. The firm determines a pricing and matching policy that maximizes its desired objective in the resulting equilibrium. That objective could 1 This is as shown in Figure 3 of Chen and Sheldon (2015), which studies the effect of dynamic pricing on Uber. 97 be to maximize revenue, as in Bimpikis et al. (2016) and Besbes et al. (2018) or to maximize social welfare, as in Hu and Zhou (2015) and Castillo et al. (2016). Whereas, we focus on the objective of maximizing the number of completed matchings, which is a relevant objective to understand the interplay between the pricing and matching decisions. The first main contribution of this chapter is to show that fixing one dimension (either the pricing or matching) and optimizing in the other is not in general optimal. More specifically, we show that when the matching decisions are restricted to same area matchings, origin-based pricing does not provide more matchings than the case in which pricing is restricted to be constant across all areas does; see Theorem 3.3.1. We furthermore show that allowing non-same area matchings does not increase the number of matchings when the pricing is restricted to be constant across all areas; see Theorem 3.3.2. However, optimization in one dimension can be helpful when origin-destination based pricing is considered. The second main contribution of this chapter is to provide conditions under which restricting the prices to be constant across all areas and the matchings to be in the same area is optimal. This is true when customer valuations for the rides are convex and homogeneous; see Theorem 3.4.1. Otherwise, the percentage of completed matchings can be strictly better when the matchings are not restricted to be same area matchings; see Theorem 3.5.1. The remainder of the chapter is organized as follows. We conclude this section with a literature review; see Section 3.1.1. We present our ridesharing system model in Section 3.2. We prove the redundancy of optimization only in one dimension in Section 3.3. We derive conditions under which restricting the prices to be constant across all areas and the matchings to be in the same area is optimal in Section 3.4 and we derive conditions under which those decisions are strictly suboptimal in Section 3.5. We present simulation experiments in Section 3.6. Finally, we make concluding remarks in Section 3.7. The proofs of all results can be found in Appendix C. 3.1.1 Literature Review Academic research about ridesharing platforms has increased significantly in the recent years due to the exponential growth of the ridesharing industry. Many of the papers about ridesharing focus on the pricing decisions but ignore the matching decisions (e.g., Chen et al. (2015), Chen and Sheldon (2015), Riquelme et al. (2015), Bimpikis et al. (2016), Castillo et al. (2016), Hall et al. (2016), Cachon et al. (2017), Guda and Subramanian (2017), Besbes et al. (2018)). For example, Riquelme et al. (2015), Bimpikis et al. (2016), Guda and Subramanian (2017), and Besbes et al. (2018) assume that customers can be matched only with the drivers from the same area. Castillo et al. (2016) assume that customers can be matched only with 98 the closest available drivers. Chen et al. (2015), Chen and Sheldon (2015), and Hall et al. (2016) do not consider matching decisions at all. Since there is no geospatial distinction in Cachon et al. (2017), the matching decisions are irrelevant in this paper. Chen et al. (2015) study the fairness of dynamic pricing to the customers and the drivers and state that dynamic pricing can affect the customers negatively. Chen and Sheldon (2015), Castillo et al. (2016), Hall et al. (2016), and Cachon et al. (2017) study the effect of dynamic pricing to the system performance and all of them state that both the customers and the drivers benefit from dynamic pricing. Riquelme et al. (2015) state that in each area, static pricing is optimal as long as the firm estimates the model parameters accurately. Guda and Subramanian (2017) state that increasing the prices in areas with excess driver supply can sometimes be beneficial, which contradicts the conventional wisdom. Besbes et al. (2018) state that if there is a sudden demand peak in an area, the firm should not only increase the price in that area but also decrease the prices in nearby areas to make drivers move to the area where the demand peak is realized. There are only a few studies focusing on the matching decisions in ridesharing (e.g., Hu and Zhou (2015), Banerjee et al. (2018), and Chapter 2 of this thesis). Hu and Zhou (2015) and Banerjee et al. (2018) ignore the pricing decisions completely. Although there is a joint pricing and matching optimization model in Chapter 2, drivers are not self-scheduling in there. Therefore, to the best of our knowledge, this study is the first one that jointly optimizes the pricing and matching decisions in a ridesharing model with self- scheduling drivers. An important modeling feature of Hu and Zhou (2015) and Chapter 2 is that both studies consider a model with time dependent parameters; i.e., customer and driver arrival rates to the system can change over time. Whereas, our model has time homogeneous parameters. Hu and Zhou (2015) provide conditions under which matching the customers with the closest available drivers is optimal. In Chapter 2, we propose asymptotically optimal matching policies based on a continuous linear program and a linear program in a large market regime and show that the proposed policies can significantly outperform the policy that always offers the the closest available drivers to the customers. We also prove in Chapter 2 that forward looking matching policies can be very beneficial in settings with time dependent parameters. Banerjee et al. (2018) propose a state-dependent matching policy which achieves the asymptotically optimal system performance with the fastest possible rate as the market size increases. Similar to our study, Bimpikis et al. (2016) optimize the pricing decisions in a model with self-scheduling drivers. In Bimpikis et al. (2016), the fraction of a customer payment given to the driver in each area is a decision variable; i.e., driver compensations can be area dependent. Bimpikis et al. (2016) determine the optimal pricing decisions and the driver compensation rates. They prove that area dependent (fixed) driver compensations are optimal in the so called “unbalanced” (“balanced”) networks, in which there is 99 asymmetric (symmetric) customer flow. However, there is a fixed driver compensation scheme in our study. Lyft uses a fixed driver compensation scheme such that the drivers always get 75% of the customer payments; see Lyft (2018). There are also ridesharing studies focusing on decisions other than pricing and matching. For example, Braverman et al. (2016) study idle driver routing decisions in a setting where drivers are centrally controlled (e.g., driverless vehicles). Gopalakrishnan et al. (2016) study car pooling decisions with specific focus on fairness of pooling to the customers. Nikzad (2017) studies the effect of competition between ridesharing firms on the customers and the drivers. Chaudhari et al. (2018) study optimal decisions for the drivers. 3.2 Model Formulation We formulate a fluid model that captures customer and driver behaviors. We capture the geospatial nature of the problem by partitioning the city intoN2N + disjoint areas, whereN + denotes the set of strictly positive integers. The pricing and the matching policies determine the equilibrium customer and driver arrival rates to various areas, and the resulting number of matchings. The firm would like to determine a pricing and a matching policy that maximize the number of matchings. Customers Customer types are based on the origin and destination areas, thus there are N 2 customer types. Type ij customers arrive at area i and their destination is area j for all i;j 2N , whereN := f1; 2;:::;Ng. Potential arrival rate of the typeij customers to the system is ij 2 R + for alli;j2N , whereR + := [0;1). Each customer arrives in the system with a valuation for a ride. We assume that those valuations are independent and identically distributed (i.i.d.) among each customer type and are associated with the cumulative distribution function (cdf) G ij . We let G ij (p) := 1G ij (p) for all p2 R + and i;j 2N ; i.e., G ij is the complement cdf of G ij . If the firm charges the price p for the rides, then the effective arrival rate of typeij customers is ij G ij (p) for alli;j2N andp2R + . Therefore, customers are price sensitive in the sense that if the prices increase (decrease), fewer (more) customers arrive in the system. We assume thatG ij (0) = 0 andG ij is continuous for alli;j2N . Each customer arrives in the system with a patience time for driver pick-up. If the firm offers a driver to a customer such that the resulting pick-up time is longer than the patience time of the customer, then the customer rejects the offered driver and leaves the system without being matched (potentially looks for other transportation options). We assume that the drivers always accept the offered customers, a customer does not cancel the ride after she is matched with a driver, and the firm does not know the patience time of a customer but knows the distribution of it. We formalize these arguments in the following way: LetU be a nonnegative random variable such that the patience time of each customer is i.i.d. and has the same distribution withU. 100 Lett ij denote the expected travel time it takes for a driver to move from areai to areaj for alli;j2N . When a type ij customer arrives in the system, if the firm offers an idle 2 driver from area k to her, then the customer accepts the driver with probability ik := P(U t ik ) for all i;j;k2N . Therefore, the customers are delay sensitive such that the probability that a customer accepts the offered driver decreases in the travel time between the customer and the driver. We assume that ii = 1 for alli2N ; i.e., customers do not reject the drivers from the same area due to the short pick-up time. However, customers can reject drivers from different areas; i.e., ik can be strictly less than 1 ifi6=k. We let :=f ik ;i;k2Ng 3 . The Decisions of the Firm The firm determines the prices for the rides and the matching policy. We let p ij 2 R + denote the price that the firm charges typeij customers for a ride for alli;j2N . The pricing policy is denoted byp :=fp ij ;i;j2Ng. Whenever a customer arrives in the system, the firm should decide whether to offer a driver to her. If the firm decides to offer a driver, it should decide which idle driver to offer the customer. We assume that the customers are indifferent to the idle drivers from the same area. Hence, the firm should decide from which area to offer an idle driver to the customer. Those decisions can be non-parametric, parametric, or even state dependent. We assume that the firm offers the drivers to the customers such that the fraction of type ij customers offered drivers from area k is equal to x ijk 2 [0; 1] for all i;j;k2N within any given time interval. Consequently, in our steady state fluid model, a matching policy is described by the matrixx :=fx ijk ;i;j;k2Ng. Notice that different matching decisions of the firm can result in the same driver-to-customer offering fractions and our goal is to derive the optimal fractions. We consider two types of pricing and matching decisions: simple and sophisticated decisions. Simple pricing decisions have constant pricing (CP) under which the firm charges the same price in all areas; i.e.,p ij =p for somep2R + for alli;j2N . Whereas, sophisticated pricing decisions have origin based pricing (OP) in whichp ij =p i 4 for somep i 2R + for alli;j2N and there existi;j2N such thatp i 6=p j . Yet another sophisticated pricing decision has origin and destination based pricing (ODP) in which there existi;j;k2N such thatp ij 6= p ik . Sophisticated matching decisions have cross matching (CM) under which the firm offers a driver from a different area to a customer; i.e.,x ijk > 0 for somei;j;k2N such thati6=k. Whereas, simple matching decisions have no cross matching (NCM) under which the firm offers a customer only the drivers from the same area; i.e.,x ijk = 0 ifi6=k for alli;j;k2N . 2 An idle driver is a driver who is neither taking a customer to her destination nor on his way to pick up a customer. 3 Bold symbols represent vectors or matrices throughout the chapter. 4 For notational convenience, we drop the destination index in the pricing vector under origin based pricing 101 Drivers We let e k denote the external idle driver arrival rate in areak for allk2N . Drivers choose where to look for customers and we lety kl denote the rate of idle drivers who move from areak to areal to look for customers in areal for allk;l2N . Then, the total idle driver arrival rate in areak is denoted by k and k = e k + X l2N y lk ; 8k2N: (3.1) We have X i;j2N ij G ij (p ij ) ik x ijk k ; 8k2N; (3.2) X k2N x ijk 1; 8i;j2N; (3.3) where (3.2) states that the total rate of customers matched with idle drivers from areak cannot exceed the total idle driver arrival rate to area k and (3.3) states that the firm cannot offer more than one driver to a customer. When a driver becomes idle, he leaves the system with probability 1, where2 [0; 1). If he stays in the system (with probability), he repositions himself to an area (including the option of staying in his current area) to look for customers. For example, if a driver drops a customer in the associated destination area, he becomes idle and leaves the system with probability 1. If he stays in the system, he repositions himself to areak for somek2N . After repositioning to areak, he will be assigned to a customer with probability P i;j2N ij G ij (p ij ) ik x ijk k : (3.4) If he is not assigned to a customer in areak, he again becomes idle and leaves the system with probability 1. If he stays in the system, he makes another repositioning decision and so on. Consequently, the parameter determines the lifetime of the drivers in the system, and we have the following flow balance equation: X l2N y kl = 0 @ X i;j2N ik G ik (p ik ) ij x ikj 1 A + 0 @ k X i;j2N ij G ij (p ij ) ik x ijk 1 A ; 8k2N; (3.5) where the first term in the right-hand side of (3.5) denotes the rate of busy drivers who make a repositioning decision after dropping a customer in areak and the second term in the right-hand side of (3.5) denote the 102 rate of idle drivers who arrived at areak, but are not assigned to customers, and then make a repositioning decision. For convenience, we let e := P k2N e k and := P k2N k . By summing both the left and right-hand sides of (3.5) overk gives us the equality P k;l2N y kl =. Then, by summing both the left and right-hand sides of (3.1) overk gives us the equality e = (1); (3.6) stating that a driver makes on average 1=(1) repositioning decisions until he leaves the system. Idle drivers make the repositioning decisions based on the revenue rates in the areas. The revenue rate in areak is denoted byR k and R k := 8 > < > : k P i;j2N ij G ij (p ij ) ik x ijk p ij ; if k > 0, 0; if k = 0, 8k2N; (3.7) where2 [0; 1] denotes the fraction of the money paid by a customer given to the driver by the firm; i.e., the firm takes 1 fraction of the money paid by the customers. We assume that is exogenously given implying that the firm charges a fixed 1 commission rate for all drivers at all times. Notice that if the total idle driver arrival rate in an area increases given that all other variables stay constant, then the revenue rate in that area decreases. We let R := max k2N R k denote the maximum revenue rate in the system. A driver’s objective is to maximize his expected total revenue during his lifetime in the system. We assume that drivers ignore the travel costs (e.g., fuel cost, depreciation cost, cost due to the time spent for traveling etc.) when they make a repositioning decision. In Section 3.6, we run simulations in which the objective of a driver is to maximize the profit instead of the revenue. We see that the number of matchings under those simulation experiments are very close to the ones derived from our steady state fluid model when the travel costs are sufficiently small. This implies that our assumption that drivers ignore the travel costs is not very strong as long as the travel costs are sufficiently small. The following lemma, whose proof is presented in Appendix C.1.1, characterizes the optimal reposition- ing decisions for the drivers. Lemma 3.2.1. An idle driver should always reposition himself to an area with the maximum revenue rate in order to maximize his expected total revenue during his lifetime in the system. 103 Since drivers reposition themselves to the areas with the maximum revenue rate, under any equilibrium 5 , we have for allk2N , if k > 0, thenR k = R: (3.8) Condition (3.8) implies that the revenue rates in the areas in which idle drivers arrive are the same. Oth- erwise, if there are two areas with different revenue rates, idle drivers in the low revenue rate area will reposition themselves to the high revenue rate area until the revenue rates in the two areas become equal. Potential external idle driver arrival rate to the system is denoted by the parameter e 2R + . Each driver has an outside option to work and the revenue rate that a driver can earn from that outside option is associated with the cdfV . A driver enters the system only if the maximum revenue rate in the system is greater than or equal to the revenue rate that he can obtain from the outside option. Hence, probability that a driver enters the system isV R . Then, e = e V R : (3.9) Therefore, as the maximum revenue rate in the system increases (decreases), more (fewer) drivers enter the system. We assume thatV (0) = 0 andV is continuous. Equilibrium Since there is a network game played by the firm and the drivers, we are interested in equi- libriums. The firm announces the prices and the matching policy at the beginning of the time horizon. Then, the customer demand is realized and the drivers position themselves in the city over time. We let :=f k ;k2Ng, e :=f e k ;k2Ng, andy :=fy kl ;k;l2Ng. Definition 3.2.1. (Equilibrium) For any givenfx;pg, a vectorfx;p;; e ;yg is an equilibrium if it is nonnegative and satisfies the constraints (3.1), (3.2), (3.3) (3.5), (3.8), and (3.9). The equilibrium concept in Definition 3.2.1 is similar to the Wardrop equilibrium (see Wardrop (1952)) in the sense that each driver is infinitesimally small and thus has negligible effect on the overall system performance. By (3.6) and the assumption that drivers ignore travel costs when they make repositioning decisions, we can ignore the endogenous variablesf e ;yg and the constraints (3.1) and (3.5), and we can update the constraint (3.9) in the equilibrium definition as formally stated in the following lemma. Lemma 3.2.2. Suppose thatfx;p;g is nonnegative and satisfies the constraints (3.2), (3.3), and (3.8), and the equality = e 1 V ( R): (3.10) 5 The formal definition of equilibrium will be presented later in Definition 3.2.1. 104 Then, there exists a vectorf e ;yg that can be characterized in closed form such thatfx;p;; e ;yg is an equilibrium. The proof of Lemma 3.2.2 is presented in Appendix C.1.2. Lemma 3.2.2 states that we can ignore the vectorf e ;yg in the equilibrium definition. Firm’s Optimization Problem Since we study the interplay between the pricing and matching decisions of the firm, a relevant objective is to maximize the number of matchings (see Banerjee et al. (2018) and Chapter 2 of this thesis). Hence, we consider the objective of maximizing the total matching rate in the system. Consequently, the firm’s optimization problem is the following: max x;p X i;j;k2N ij G ij (p ij ) ik x ijk (3.11a) such thatfx;p;; e ;yg is an equilibrium. (3.11b) Remark 3.2.1. Both Braverman et al. (2016) and Chapter 2 of this thesis prove convergence of a pre-limit ridesharing model in which there is randomness and there are finite numbers of customers and drivers to a fluid model as the numbers of the customers and the drivers grow without a bound. By simulation exper- iments (see Section 3.6), we show that our steady state fluid model accurately approximates ridesharing systems in which there is randomness and there are finite but many customers and drivers (i.e., a large mar- ket). Therefore, our steady state fluid model can be interpreted as a fluid limit of a ridesharing model in the large market regime. Remark 3.2.2. Recall that we partition the city intoN disjoint areas. On the one hand, the size of the areas should be large enough such that there are many customers or drivers in some of them because our steady state fluid model approximates crowded ridesharing systems well as shown in simulation experiments (see Section 3.6). On the other hand, the size of the areas should be small enough such that the associated city partitioning can capture the geospatial supply or demand differences. 3.3 Futility of Optimization in One Dimension Intuitively, when customer demand differs significantly among different areas, the firm may benefit from employing origin based pricing, origin and destination based pricing, and/or cross matching in order to decrease the geospatial supply and demand imbalances (hence, to increase the number of matchings). In this section, we study the benefit of sophisticated decisions and establish the following surprising result: the firm does not benefit from employing sophisticated decisions in only one dimension (either pricing or 105 matching). Instead, as we show later in Sections 3.4 and 3.5, in order for such decisions to lead to an increase in the number of matchings, sophisticated pricing and matching decisions should be implemented simultaneously. First, we will consider constant pricing and origin based pricing in Section 3.3.1. Then, we will extend our results by allowing origin and destination based prices in Section 3.3.2. 3.3.1 Constant and Origin Based Pricing We show that there is no need for origin based prices when there is no cross matching and there is no need for cross matching when the prices are constant. Hence, the firm should use either origin based pricing and cross matching together or neither of them. Futility of origin based pricing when there is no cross matching We prove that there is no need for origin based prices when there is no cross matching. Theorem 3.3.1. For any origin based pricing and no cross matching equilibrium, there exists a constant pricing and no cross matching equilibrium with greater or equal total matching rate. The proof of Theorem 3.3.1 is presented in Appendix C.3.1. To see why this result holds, let us consider an origin based pricing and no cross matching equilibrium with strictly positive total matching rate. By definition of origin based pricing, there exists an areak2N in which there are idle drivers and the price is higher than some other area with idle drivers. By the revenue rate equivalence condition (see (3.8)) and the fact that there is no cross matching, it must be the case that the total idle driver arrival rate in areak is strictly greater than the matching rate in that area; i.e., the constraint (3.2) is not binding in areak. If the firm decreases the price in areak, more customers will arrive at that area. Then, the firm can match these additional customers with the excess drivers in areak and can increase the matching rate in that area. The firm should decrease the price in areak until all drivers in that area are matched with the customers, which maximizes the matching rate in areak. If the firm repeats these steps in all areas, then the drivers will not idle in the resulting system. Given that the drivers do not idle, by the revenue rate equivalence condition (see (3.8)) and the fact that there is no cross matching, an equilibrium can be obtained only when the price in each area is the same. Therefore, for any origin based pricing and no cross matching equilibrium, there exists a constant pricing and and no cross matching equilibrium with greater or equal total matching rate. The proof of Theorem 3.3.1 formalizes this argument. Futility of cross matching when there is constant pricing We prove that there is no need for cross matching when the prices are constant. 106 Theorem 3.3.2. For any equilibrium with constant pricing, there exists a constant pricing and no cross matching equilibrium with greater or equal total matching rate. The proof of Theorem 3.3.2 is presented in Appendix C.2.2. To see why this result holds, let us consider an equilibrium with the constant pricep2R + . Since each driver earnsp amount of revenue by serving a customer, the ratio of the total revenue rate generated for all drivers and the total matching rate in the system is equal to the constantp for all equilibriums with the constant pricep and strictly positive total matching rate. Therefore, under equilibriums with a constant price, if the total revenue rate generated for all drivers increases (decreases), then the total matching rate increases (decreases) and vice versa. Consequently, the objective of the drivers and the objective of the firm are perfectly aligned with each other. Hence, the drivers position themselves within the city according to the customer demand such that there is no need for cross matching and the total matching rate is maximized. The proof of Theorem 3.3.2 formalizes this argument. Optimal Pricing and Matching Decisions Theorems 3.3.1 and 3.3.2 prove that origin based pricing and cross matching do not provide any value when applied separately. This implies that in order to maximize the total matching rate, we need to consider two types of decisions as illustrated in Figure 3.2: a) decisions that set constant prices and do not cross match, b) decisions that set origin based prices and perform cross matching. !" #" $!% ! !% ! !"#$%&'( )*&$&'( Figure 3.2: The pricing and the matching decisions of the firm when only CP and OP are considered. 3.3.2 Origin and Destination Based Pricing When we allow origin and destination based pricing, optimizing only in the pricing dimension can increase the number of matchings. In other words, when there is no cross matching, origin and destination based pricing can provide strictly better performance than the constant pricing can. Nevertheless, we derive suffi- cient conditions under which there is no need for origin and destination based pricing when there is no cross matching. Theorem 3.3.3. Suppose that the cdf’s associated with the customer valuations for the rides are convex and homogeneous among the customer types arriving at the same areas; i.e., there exist cdf’sfG i ;i2Ng such thatG i is convex and continuous, G i (0) = 0, andG ij = G i for alli;j2N . Then, for any origin and 107 destination based pricing and no cross matching equilibrium, there exists a constant pricing and no cross matching equilibrium with greater or equal total matching rate. The proof of Theorem 3.3.3 is presented in Appendix C.3.2. Under the conditions in Theorem 3.3.3, origin and destination based pricing do not provide any value when applied without cross matching. Under those conditions, both G i (p) andp G i (p) are concave inp2 R + for alli2N . Therefore, charging an average price to all customer types arriving at an area instead of charging different prices to them increases both the total customer arrival rate and the total revenue rate generated by the customers in that area. Therefore, charging an average price instead of charging different prices increases the number of matchings in each area. Furthermore, we know from Theorem 3.3.1 that constant pricing provides a better performance than origin based pricing does when there is no cross matching. Consequently, constant pricing provides a better performance than origin and destination based pricing does when there is no cross matching. The proof of Theorem 3.3.3 formalizes this argument. If the conditions in Theorem 3.3.3 do not hold, origin and destination based pricing and no cross matching equilibriums can be optimal as illustrated by the following example. Example 3.3.1. Suppose that there are three areas; i.e.,N = 3. Let 12 = 13 = 1, all other ij = 0, = 0:8, e = 1, and = 0:9. LetV U[0; 10],G 12 U[0; 10],G 13 U[0; 2], whereU[a;b] denotes continuous uniform distribution on the interval [a;b]R + . Let =e 6 , wheree denotes anNN matrix with all entries equal to 1. Since customers arrive only at area 1, origin based pricing is irrelevant in the sense that there is no need for charging different prices in different areas. Hence, by Theorem 3.3.2, we compare the best constant pricing and no cross matching equilibrium, the best origin and destination based pricing and no cross matching equilibrium, and the best origin and destination based pricing and cross matching equilibrium. The best constant pricing and no cross matching equilibrium has the constant price 1:429 and the total matching rate 1:143. The best origin and destination based pricing and no cross matching equilibrium is optimal and has the prices p 12 = 4:239, p 13 = 0:239 and the total matching rate 1:457. Since type 12 customers are less price sensitive than the type 13 customers, the firm charges them a higher price than it charges type 13 customers under the optimal equilibrium. The high price attracts many drivers to area 1 and the low price attracts many customers to area 1. Consequently, the firm maximizes the total matching rate. Figure 3.3 is an extension of Figure 3.2 with origin and destination based pricing. 6 If the relation=,, or is used with vectors or matrices, then the relation holds component-wise. 108 !" #" #&" $!% ! ! !% ! ! !"#$%&'( )*&$&'( Figure 3.3: (Color online) The pricing and the matching decisions of the firm when ODP is included. Henceforth, CP equilibriums will denote the ones with constant prices and no cross matching by The- orem 3.3.2, OP equilibriums will denote the equilibriums with origin based prices and cross match- ing by Theorem 3.3.1, ODP+NCM equilibriums (ODP+CM equilibriums) will denote the equilibriums with origin and destination based prices and no cross matching (cross matching), and NCM equilibriums (CM equilibriums) will denote the equilibriums with no cross matching (cross matching). We let z X () denote the supremum of the total matching rate amongX equilibriums under a given parameter, where X2fCP;OP;ODP +NCM;ODP +CM;NCM;CMg. Since the total matching rate under an NCM equilibrium is independent of the non-diagonal components of, we will drop from the notation associ- ated with the NCM equilibriums. 3.4 Constant Pricing and No Cross Matching Equilibriums So far we prove that extending the simple pricing and matching decisions only in one dimension does not increase the number of matchings under some conditions. This result brings up the question: When are simple pricing and matching decisions optimal? To answer this question, first, we derive the best CP 7 equilibriums in Section 3.4.1. If the customer valuation distributions for the rides are homogeneous among customer types, intuitively constant pricing may be enough to achieve the optimal performance. In Section 3.4.2, we prove that the aforementioned claim is not necessarily correct and we derive sufficient conditions under which the best CP equilibriums are optimal. 3.4.1 The best equilibriums among the CP equilibriums Consider a CP equilibrium in which there are excess customers in the system such that some customers cannot be matched due to lack of drivers. If the firm increases the price by a sufficiently small amount, then the total customer arrival rate to the system will decrease, but the total driver arrival rate to the system will increase because the remaining customers will pay more for each ride. Therefore, if there are excess 7 The best CP equilibriums denote the best equilibriums among the CP equilibriums. 109 customers, the firm can increase the total matching rate by increasing the price until all of the remaining customers are matched. Next, consider a CP equilibrium in which there are excess drivers in the system such that some drivers must remain idle due to lack of customers. If the firm decreases the price by a sufficiently small amount, then the total customer arrival rate to the system will increase and those additional arriving customers can be matched with the excess drivers. Therefore, if there are excess drivers, the firm can increase the total matching rate by decreasing the price until drivers never idle. Consequently, under the best constant price, drivers should never idle and all of the customers arriving in the system should be served. This intuition is illustrated by the equality X i;j2N ij G ij (p) = e 1 V (p): (3.12) The left and right-hand sides of (3.12) are the total customer arrival rate to the system and the maximum rate of customers that can be served by the drivers, respectively, under the constant pricep2R + . The best constant price is a solution of (3.12) such that the customer demand and the driver supply are balanced with each other. The following proposition formalizes this argument. Proposition 3.4.1. Letp 2R + be a solution of (3.12), which always exists and is unique ifG ij is strictly increasing for all i;j 2 N or V is strictly increasing. Under the constant price p , there exists a CP equilibrium that can be characterized in closed form such that the total matching rate under that equilibrium is greater than or equal to the total matching rate under any CP equilibrium and equal toz CP where z CP = X i;j2N ij G ij (p ) = e 1 V (p ): (3.13) Moreover, under that equilibrium, drivers never idle; i.e., constraint (3.2) is binding for allk2N , and all of the customers arriving in the system are served. The proof of Proposition 3.4.1 is presented in Appendix C.2.3. Under the best CP equilibriums specified in Proposition 3.4.1, idle drivers position themselves such that total idle driver arrival rate in an area is equal to the total customer arrival rate in that area. Therefore, revenue rate of each driver type is equal top , thus the revenue rate equivalence condition (see (3.8)) holds and we obtain an equilibrium. Next, we study the optimality of the best CP equilibriums when the customer valuation distributions for the rides are homogeneous among customer types. 110 3.4.2 Homogeneous Customer Valuation Distributions If the cdf’s associated with the customer valuations for the rides are homogeneous among customer types, intuitively homogeneous (i.e., constant) prices may be enough to maximize the number of matchings. If that is the case, then there is no need for cross matching by Theorem 3.3.2, which implies the optimality of the best CP equilibriums. However, we show that the aforementioned claim is not necessarily correct. Nevertheless, we prove that if the cdf’s associated the customer valuations for the rides are convex and homogeneous, then the best CP equilibriums are optimal. In that case, charging an average price instead of different prices increases both the total customer and the total driver arrival rates to the system which increases the total matching rate. Otherwise, if the cdf’s associated the customer valuations for the rides are homogeneous but non-convex, the firm may increase both the total customer and the total driver arrival rates to the system by charging different prices to different customer types. In that case, the system will attract many drivers due to the high prices and attract many customers due to the low prices. Then the firm can maximize the total matching rate by cross matching. The following theorem states that if the cdf’s associated with the customer valuations for the rides are homogeneous among customer types and convex, the best CP equilibriums are optimal. Theorem 3.4.1. Suppose that the cdf’s associated with the customer valuations for the rides are convex and homogeneous among customer types; i.e., there exists a cdfG such thatG is convex and continuous, G(0) = 0, andG ij =G for alli;j2N . Then, the CP equilibriums specified in Proposition 3.4.1 have the highest total matching rate among all equilibriums. The proof of Theorem 3.4.1 is presented in Appendix C.6.1. We prove that for any given equilibrium with the price vectorp, there exists a CP equilibrium with the constant price P i;j2N ij p ij = P i;j2N ij and greater or equal total matching rate. SinceG is convex and nondecreasing, both G(p) andp G(p) are concave inp2R + . Therefore, charging a weighted average price in all areas instead of charging different prices in different areas increases both the total customer arrival rate to the system and the total revenue rate generated by the customers which increases the total driver arrival rate to the system. Consequently, charging a weighted average price increases the total matching rate. However, Theorem 3.4.1 does not necessarily hold if the cdf’s associated with the customer valuations for the rides are homogeneous among customer types but non-convex as illustrated in the following example whose solution is presented in Appendix C.8.1. 111 Example 3.4.1. Suppose that there are two areas; i.e.,N = 2. Let 11 = 22 = 0, 12 = 21 = 1, thus constant and origin based pricing are sufficient in this example. LetV U[0;A] for someA2 [3;1), =e,2 (0; 0:5) be an arbitrary constant, andG 12 =G 21 =G where G 8 > < > : U[0; 1]; with probability 1; U[2; 3]; with probability: We interpretG such that at each area, 1 fraction of the customers has the cdfU[0; 1], the remaining fraction has the cdf U[2; 3], but the firm cannot distinguish the cdf of the individual customers. The complement cdf G and the associated revenue functionp G(p) is depicted in Figure 3.4 for the case = 0:25. Since2 (0; 0:5), the associated revenue functionp G(p) is bimodal andp = 1 is a local minimum. Suppose that e =(A(1)) = 2, which implies that the best constant pricep = 1 andz CP = 2 by Propostion 3.4.1. Then there exists an OP equilibrium with the total matching rate strictly greater thanz CP . Moreover, in this example, sincez NCM =z CP andz CM (e) =z OP (e) by Theorems 3.3.1 and 3.3.2, respectively, we havez CM (e) =z OP (e)>z NCM =z CP . For example, if = 0:25, thenz CM (e)=z NCM = 128%. p 1 1 p 2 2 3 0 0.2 0.4 0.6 0.8 1 p 1 1 p 2 2 3 0 0.1 0.2 0.3 0.4 0.5 Figure 3.4: (Color online) G(p) andp G(p) when = 0:25 in Example 3.4.1. In a neighborhood around p = 1, G(p) is convex and p G(p) resembles a convex function as seen in Figure 3.4. Then, there exists a price vector (p 1 ;p 2 ) such thatp 1 <p ,p 2 >p , G(p 1 ) + G(p 2 )> 2 G(p ), p 1 G(p 1 ) +p 2 G(p 2 )> 2p G(p ), and (p 1 G(p 1 ) +p 2 G(p 2 ))=( G(p 1 ) + G(p 2 ))>p . In other words, under the price vector (p 1 ;p 2 ), more customers arrive in the system and they pay more for a ride on average, which attracts more drivers to the system. The customers accumulate in area 1 due to the low price and the drivers are attracted to the system due the high price in area 2. Then the firm maximizes the total matching rate by cross matching. 112 Building on Example 3.4.1, we can in fact prove that the ratioz CM (e)=z NCM is unbounded. Proposition 3.4.2. Suppose that the cdf’s associated with the customer valuations for the rides are homo- geneous among customer types; i.e., there exists a cdfG such thatG ij =G for alli;j2N . LetD denote the set of cdfs that are continuous and equal to 0 at origin. Then, sup G;V2D z CM (e; (G;V )) z NCM (G;V ) =1: The proof of Proposition 3.4.2 is presented in Appendix C.8.2. In summary, if the cdf’s associated with the customer valuations for the rides are homogeneous among customer types but non-convex, then origin based pricing and cross matching can be beneficial. In that case, high prices attract drivers to the system, low prices attract customers to the system, and the firm can maximize the total matching rate by cross matching. 3.5 Cross Matching Equilibriums So far we have proved that the performance gap between the best CM equilibriums and the best NCM equilibriums can be arbitrarily large (see Proposition 3.4.2). In this section, we will derive a lower bound on that performance gap under some assumptions. We will prove that if the cdf’s associated with the customer valuations for the rides are heterogeneous among customer types, then the best CM equilibriums strictly outperform the best NCM equilibriums under some assumptions. In that case, the firm should charge low prices to the price sensitive customers and high prices to the less price sensitive customers. The low prices will attract a lot of customers and the high prices will attract a lot of drivers to the system. Then, the firm can maximize the number of matchings by cross matching. The following example illustrates this result. Example 3.5.1. Suppose that N = 2 and 12 = 21 = for some 2 [0; 1]. Let 11 = 22 = 0, 12 = 21 = 1, e = 1, = 0:9,V U[0; 10],G 12 U[0; 10], andG 21 U[0; 2]. Figures 3.5a and 3.5b depict maxfz CM ();z NCM g andz NCM for2f0; 0:05; 0:1; 0:15;:::; 1g when = 0:8 and = 0:4, respectively. Since origin and destination based pricing is irrelevant in this example, we havez NCM =z CP and maxfz CM ();z NCM g = maxfz OP ();z CP g by Theorems 3.3.1 and 3.3.2. In Figure 3.5, we see that the optimal total matching rate is nondecreasing and continuous in and has a threshold structure. When = 0, cross matching is impossible and so the best NCM equilibriums are trivially optimal. As customers become more patient; i.e., as increases, then cross matching becomes more valuable because it provides more flexibility to the firm for the matching decisions. When exceeds a specific threshold, then the best CM equilibriums strictly outperform the best NCM equilibriums. That 113 0 0.2 0.4 0.6 0.8 1 1.1 1.2 1.3 1.4 1.5 Total Matching Rate (a) = 0:8 0 0.2 0.4 0.6 0.8 1 0.7 0.8 0.9 1 1.1 Total Matching Rate (b) = 0:4 Figure 3.5: (Color online) Results of Example 3.5.1. threshold can be even 0 as seen in Figure 3.5b, which implies that cross matching becomes beneficial as soon as the firm can implement it. In Section 3.5.1, we focus on the case where customers are patient; i.e., = e. When the customers are patient, the joint pricing and matching problem simplifies into a pricing problem which enables us to derive a lower bound on the performance gap between the best CM and the best NCM equilibriums. Then, we consider impatient customers in Section 3.5.2 such that we will verify our observations from Figure 3.5 theoretically. 3.5.1 Patient Customers In this section, we assume that the customers are patient; i.e., =e. Under the patient customer assumption, we simplify the joint pricing and matching problem into a much simpler pricing problem, which facilitates the derivation of the optimal equilibriums. Then, we prove that under uniformly distributed customer and driver valuations, the best CM equilibriums perform strictly better than the best NCM equilibriums do and we derive a lower bound on the associated performance gap. The optimal equilibriums when the customers are patient When the customers are patient, the optimization problem (3.11) simplifies into the following pricing opti- mization problem: max p X i;j2N ij G ij (p ij ) (3.14a) such that X i;j2N ij G ij (p ij ) = e 1 V P i;j2N ij G ij (p ij )p ij P i;j2N ij G ij (p ij ) ! ; (3.14b) 114 p ij 0; 8i;j2N: (3.14c) We maximize the total customer arrival rate to the system in the objective (3.14a). Similar to (3.12), the constraint (3.14b) states that all of the customers arriving in the system are served and drivers never idle. This intuition is formalized in the following proposition which presents some properties of the optimization problem (3.14). Proposition 3.5.1. 1. Letp be a feasible point of (3.14), which exists. Then, there exist multiple equilib- riums that can be characterized in closed form with price vectorp and the total matching rate equal to the objective function value associated withp. Furthermore, under each of those equilibriums, all of the customers arriving in the system are served, drivers never idle, and each individual driver is assigned to a customer equally likely. 2. The optimal objective function value of (3.14) is an upper bound on the total matching rate under any equilibrium under any parameter. The proof of Proposition 3.5.1 is presented in Appendix C.5. Proposition 3.5.1 states that if the customers are patient, we can identify an optimal equilibrium by an optimal solution of (3.14). The performance of an equilibrium with no cross matching is independent of the non-diagonal components of. Therefore, if the optimal objective function value of (3.14) is equal to z CP , the best CP equilibriums have the highest matching rate among all equilibriums independent of the fact that whether = e or not by Proposition 3.5.1 Part 2. When the customers are patient, the firm can match a customer with a driver from any area. Hence, when the customers are patient, where the drivers position themselves does not affect the number of matchings, but how the drivers are allocated to different customer types matter. Consequently, for any given feasible price vector of (3.14), we can construct multiple equilibriums as stated in Proposition 3.5.1 Part 1. Optimality of CM equilibriums under uniformly distributed customer and driver valuations We consider the special case in which the cdf’s associated with the customer valuations for the rides and the cdf associated with the driver valuations for the maximum revenue rate are uniformly distributed. We prove that if the customer valuations are heterogeneous among customer types, then the best CM equilibriums perform strictly better than the best NCM equilibriums do. Let i := P j2N ij for all i2N ; i.e., i denotes the total potential customer arrival rate in areai. The following theorem provides a lower bound gap between the performances of the best CM equilibriums and the best NCM equilibriums. 115 Theorem 3.5.1. Suppose thatV U[0;A] andG ij U[0;a i ] for alli;j2N , whereA2 (0;1) and a i 2R + for alli2N . Without loss of generality, suppose thata 1 a 2 :::a N and if i = 0 for some i2N , thena i = 0. Consider the following three conditions: C1 We havep , the best constant price defined in Proposition 3.4.1, is such thata n > p a n+1 for somen2f2; 3;:::;Ng. Hence, under the best CP equilibrium, at least two customer types arrive in the system. C2 Customers arriving in the system under the best CP equilibrium have heterogeneous valuations for the rides; i.e., there existi;j2f1; 2;:::;ng such thata i 6=a j . C3 Aa 1 . Under the conditions C1-C3, we have z CM (e)z NCM C n X i=j+1 n1 X j=1 i j (a i a j ) 2 a i a j > 0; (3.15) whereC > 0 is a constant dependent on the model primitives andn is defined in condition C1. The proof of Theorem 3.5.1 is presented in Appendix C.6.2. Theorem 3.5.1 states that if the cdf’s asso- ciated with the customer valuations for the rides are heterogeneous and uniformly distributed and the cdf associated with the driver valuations for the maximum revenue rate is uniformly distributed, then the best CM equilibriums perform strictly better than the best NCM equilibriums do under some conditions. By Theorem 3.3.3,z CP =z NCM under the conditions stated in Theorem 3.5.1. This is why conditions C1 and C2 are about the best CP equilibriums. Sincep is the unique solution of (3.12) and the left and right-hand sides of (3.12) are nonincreasing and nondecreasing inp, condition C1 is equivalent to n X i=1 i G ij (a n )< e (1) V (a n ); n+1 X i=1 i G ij (a n+1 ) e (1) V (a n+1 ); (3.16) for somen2f2; 3;:::;Ng. Condition C2 enforces heterogeneity among the customer types. Condition C3 ensures that all of the potential drivers will not arrive in the system under the best CM or the best NCM equilibriums, thus the total matching rate will not be maximized trivially. Otherwise, for example, if A<p , then all of the drivers will arrive in the system under the best CP equilibrium, which is also one of the best NCM equilibriums. Since drivers never idle under the best CP equilibrium (see Proposition 3.4.1), it is an optimal equilibrium trivially. We relax condition C3 in the proof of Theorem 3.5.1. The following example illustrates the statement of Theorem 3.5.1. 116 Example 3.5.2. Suppose that there are two areas; i.e.,N = 2. Let 11 = 22 = 0, 12 = 21 = 1, = 0:8, e = 1, = 0:9, =e,V U[0; 10],G 21 U[0; 2], andG 12 U[0;a] for somea2R + . LetLB denote the lower bound gap defined in the right-hand side of (3.15). Figure 3.6a depicts maxfz CM (e);z NCM g, LB +z NCM , andz NCM fora2f1; 2;:::; 20g 8 . Since origin and destination based pricing is irrelevant in this example, we havez NCM = z CP and maxfz CM ();z NCM g = maxfz OP ();z CP g by Theorems 3.3.1 and 3.3.2. Letp denote the best constant price (see Proposition 3.4.1) and (p 1 ;p 2 ) denote the optimal price vector under one of the best CM equilibriums. Figure 3.6b depicts the optimal prices under the best NCM and the best CM equilibriums fora2f1; 2;:::; 20g. 1 2 5 10 15 20 0.6 0.8 1 1.2 1.4 1.6 1.8 Total Matching Rate 1 2 5 10 15 20 0 1 2 3 4 5 6 Optimal Prices Figure 3.6: (Color online) Results of Example 3.5.2. Figure 3.6a shows that the performance gap between the best CM equilibrium and the best NCM equi- librium increases as the customer valuations for the rides become more heterogeneous. When the customer valuations for the rides are homogeneous; i.e., the casea = 2, then the best NCM equilibrium is optimal as proven in Theorem 3.4.1. Otherwise, the best CM equilibrium is optimal as proven in Theorem 3.5.1. Moreover, Figure 3.6a shows thatLB, the lower bound gap defined in (3.15), is very close to the actual gap in Example 3.5.2. In Example 3.5.2,p = 2=(1:3 + 1=a) by Proposition 3.4.1. Hence, as seen in Figure 3.6b,p increases asa increases. Intuitively, under the best NCM equilibrium, asa increases, customers become less price sensitive and the firm charges a higher price in order to attract more drivers to the system to maximize the total matching rate. 8 Ifa2f13;14;:::;20g, then condition C3 of Theorem 3.5.1 is not satisfied, but the relaxed condition that we use (instead of condition C3) in the proof of Theorem 3.5.1 is satisfied. 117 Whena> 2, since the customers arriving at area 1 are less price sensitive than the ones arriving at area 2, the firm charges a high price in area 1 and a low price in the area 2 under the best CM equilibrium as shown in Figure 3.6b. The high price attracts a lot of drivers to area 1 and the low price attracts a lot of customers to area 2. Then, the firm cross matches the excess idle drivers in area 1 with the excess customers in area 2 and maximizes the total matching rate. As the customer valuations for the rides become more heterogeneous; i.e., asa increases, p 1 increases andp 2 decreases untilp 2 becomes 0. However, oncep 2 becomes 0; i.e., when the customers arriving at area 2 does not pay for the rides, then asa increases,p 1 decreases. When a2f13;:::; 20g, by the optimization problem (3.14) and Proposition 3.5.1, p 1 = 0:8a 2 0:8a 1:5 p a 10 + 4a 2 + 1:6a ; p 2 = 0; z CM (e) = 2:4a + 0:8a 0:5 p a 10 2 + 1:6a ; drivers never idle, and all of the customers arriving in the system are matched with drivers. Ifa increases, we need more customers and drivers in the system in order to increase the total matching rate. Since it is not possible to increase the arrival rate of the customers in area 2 (becausep 2 = 0), the only remaining option is to decrease p 1 so that more customers arrive at area 1. Since only the customers in area 1 pay for the rides, asa increases andp 1 decreases by a sufficiently small amount, then more customers arrive at area 1, the revenue rate in the system increases, and thus more drivers arrive in the system. Consequently, the total matching rate increases as seen in Figure 3.6a. 3.5.2 Impatient Customers We consider the case where some of the customers are impatient; i.e., there exists a non-diagonal component of strictly less than one. Recall that the optimal total matching rate is nondecreasing and continuous in the non-diagonal component of and has a threshold structure in Figure 3.5. In this section, we will verify these observations theoretically. When we use a mathematical accent or superscript with a vector or matrix (with the components of a vector or matrix), then the same mathematical accent or superscript applies to each component (to the associated vector or matrix). For example, ifa := (a 1 ;:::;a n ), then ^ a = (^ a 1 ; ^ a 2 ;:::; ^ a n ). We letkk denote the Frobenius norm. The performance of the equilibriums that do not perform cross matching is independent of the patience level of the customers. If the customers become more patient, then more customers will accept drivers from different areas, thus the performance of the equilibriums that perform cross matching will not decrease (and potentially will increase). This implies that total matching rate is nondecreasing in the parameter which is formalized by the following lemma. 118 Lemma 3.5.1. Letfx;p;; e ;yg be an equilibrium with the total matching ratez under a parameter. Let ~ be such that ~ e. Then, there exists a matching policy ~ x such thatf~ x;p;; e ;yg is an equilibrium with the total matching ratez under ~ . The proof of Lemma 3.5.1 is presented in Appendix C.1.3. Next, the following theorem, whose proof is presented in Appendix C.7, states that the total matching rate is continuous in. Theorem 3.5.2. Let > 0 be an arbitrary constant and consider any equilibrium with the total matching ratez under a given parameter. Then, there exists a(;;z) > 0 such that if ~ < (;;z) and ~ ii = 1 for alli2N , then there exists an equilibrium with the total matching rate ~ z under ~ such that jz ~ zj<. Suppose that when the customers are patient; i.e., when =e, there exists a CM equilibrium with total matching rate strictly greater than the total matching rate of the best NCM equilibriums, which is the case when the customer valuations for the rides are heterogeneous (see Theorem 3.5.1). Then, Theorem 3.5.2 implies that if is sufficiently large, the same result still holds. The following corollary to Lemma 3.5.1 and Theorem 3.5.2 explains the threshold structure that we see in Figure 3.5. Corollary 3.5.3. Suppose that ik = for alli;k2N such thati6=k for some2 [0; 1]; i.e., all of the non-diagonal components of are equal to. Suppose also thatz CM (e) > z NCM . Then, there exists a 2 [0; 1) such that if > , we havez CM ()>z NCM ; otherwisez CM ()z NCM . Corollary 3.5.3 states that there exists a threshold value in [0; 1) such that if the customer patience is strictly greater than (less than or equal to) the threshold, the best CM equilibriums perform strictly better (worse) than the best NCM equilibriums do. Corollary 3.5.3 identifies two regions associated with the customer patience. If customers are patient enough, CM equilibriums are beneficial otherwise there is no need for them. 3.6 Simulation Experiments We test the accuracy of our steady state fluid model by comparing it with a discrete event simulation model in which the customers and the drivers arrive in the system randomly, there are non-zero travel times between the areas, and the drivers make the repositioning and entrance to the system decisions at any given time based on the realized events up to that time. First, we test the accuracy of our steady state fluid model under different customer and driver arrival rate scales; i.e., under different market sizes. Second, we test 119 the sensitivity of the system performance with respect to driver patience for waiting. Recall that there is no driver waiting in our steady state fluid model. However, in the simulation experiments, idle drivers are sensitive to waiting such that if an idle driver is not matched with a customer for a long time, he can leave the system. Next, we test the sensitivity of the system performance with respect to the travel times between areas. Lastly, we test the accuracy of our steady state fluid model by introducing travel costs to the model such that the goal of each driver is to maximize his expected total profit instead of revenue. We present the simulation set-up and results in Sections 3.6.1 and 3.6.2, respectively. 3.6.1 Simulation Set-up We use the following randomized matching policy described in Definition 2.3.1 of Chapter 2 in the simula- tion experiments: Suppose that a typeij customer arrives in the system for somei;j2N . Then the firm makes a random selection from the setN[f0g such that probability that the outcome isk isx ijk for all k2N and the the probability that the outcome is 0 is 1 P k2N x ijk . If the outcome isk for somek2N and there is an idle driver in areak, then the system controller offers an idle driver from areak to the cus- tomer. If the outcome isk for somek2N but there is no idle driver in areak or if the outcome is 0, then no driver is offered to the customer (and so the customer is lost). Recall that in Chapter 2, we prove that under the aforementioned randomized matching policy, the driver offering rates to the customers converge tox as the market size increases (cf. Theorem 2.3.2). We also propose several other matching policies which are more sophisticated than the randomized one and show that the performances of those policies are very close to the one of the randomized policy via numerical experiments (cf. Section 2.5.1). We scale the customer and driver arrival rates to the system withn2f1; 5; 50g such that potential typeij customer arrival rate and potential external driver arrival rate to the system aren ij andn e , respectively, for alli;j2N . The casesn = 1,n = 5, andn = 50 represent small, medium-sized, and large markets, respectively. When a busy driver drops his customer at the destination area, he leaves the system with probability 1, and stays in the system and makes a repositioning decision with probability . In the latter case, after repositioning to an area, the idle driver joins an infinite capacity idle driver queue in that area with a patience time generated independently fromT , whereT is an exponential random variable with mean and is independent of all other stochastic primitives. The firm matches the idle drivers with the customers in each area with respect to the first-come-first-served rule. If an idle driver is not matched with a customer until his patience is over, he leaves the system with probability 1 and makes a repositioning decision with probability. Therefore, idle drivers are sensitive to waiting and too much waiting can result in driver 120 departure from the system, which can decrease the performance. However, in our fluid model, we ignore the idle driver waiting in the following way: After repositioning to areak2N , an idle driver is assigned to a customer with the probability in (3.4), otherwise he leaves the system with probability 1. If he stays in the system, he makes another repositioning decision. Letc ij denote the expected travel cost due to traveling from areai toj for alli;j2N . If a driver in areak is matched with a typeij customer, then the profit of the driver from that matching isp ij c ki c ij . When an idle driver makes a repositioning decision, he first computes for each area the average profit an idle driver has earned in that area up to that time minus the traveling cost to that area from his current location. Then, he chooses the area with the highest net profit. Notice that the idle drivers who are not matched with customers are also considered in the aforementioned average profit computation (with profit 0). When an external driver arrival to the system occurs at timet2R + , the next external driver arrival occurs after an exponentially dis- tributed time with the rate n e V (max k average profit an idle driver has earned in areak up to timet) and the external drivers arrive at the area with the highest average profit. Notice that the external idle driver arrival rate to the system can change over time due to the fluctuations in the average profit rate in the areas over time. Whereas, type ij customers arrive in the system with respect to a Poisson process with rate ij G ij (p ij ) for alli;j2N . We assume thatt ij , the travel time from areai toj, is deterministic for all i;j2N in the simulation experiments. A detailed explanation of the driver behaviors and other details about the simulation set-up are presented in Appendix. We simulate the subset2f0:1; 0:2;:::; 1g of the numerical experiments in Figure 3.5b and the subset a2f1; 3; 5;:::; 19g of the numerical experiments in Figure 3.6 under the price vector and the matching matrix associated with the best NCM and the best CM equilibriums. Therefore, we simulate (1 + 10) + (10 + 10) = 31 different numerical experiments in total. We simulate each of the 31 numerical experiments under the 9 different settings shown in Table 3.1, thus we simulate 31 9 = 279 instances in total. Table 3.1: The 9 different settings used in the simulation experiments. Setting # 1 2 3 4 5 6 7 8 9 n 1 5 50 50 50 50 50 50 50 300 300 300 50 600 300 300 300 300 (t 12 ;t 21 ) (300,300) (300,300) (300,300) (300,300) (300,300) (200,600) (700,100) (300,300) (300,300) (c 12 ;c 21 ) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0.02,0.02) (0.05,0.05) Each instance has been run 30 times. At each run, after the warm-up period, 5 million customers with valuations for the rides greater than the given prices are generated. Consequently, among the 279 instances, the maximum of the margin of errors associated with the 95% confidence interval for the performance measure (total matching rate)=n is less than 0.0014 (Margin of error =t 0:025;29 Sample stdv.= p 30). 121 3.6.2 Simulation Results Figures 3.7 and 3.8 present the simulation results. Figure 3.7 presents the results of the simulation experi- ments associated with the subset2f0:1; 0:2;:::; 1g of the numerical experiments in Figure 3.5b. Figure 3.8 presents the results of the simulation experiments associated with the subseta2f1; 3; 5;:::; 19g of the numerical experiments in Figure 3.6. In Figures 3.7 and 3.8, the solid lines (—) represent the total matching rate under our steady state fluid approximation model. Whereas, the dashed lines ( , - -, and - - -) repre- sent the (total matching rate)=n under the simulation experiments associated with various different settings specified in Table 3.1. The line S k represents the results of the simulation experiments under the setting k2f1; 2;:::; 9g specified in Table 3.1. 1 3 5 7 9 11 13 15 17 19 0.6 0.8 1 1.2 1.4 1.6 1.8 Total Matching Rate/n (a)S1,S2 - -,S3 - - - 1 3 5 7 9 11 13 15 17 19 0.6 0.8 1 1.2 1.4 1.6 1.8 Total Matching Rate/n (b)S3 - - -,S4,S5 - - 1 3 5 7 9 11 13 15 17 19 0.6 0.8 1 1.2 1.4 1.6 1.8 Total Matching Rate/n (c)S3 - - -,S6 - -,S7 1 3 5 7 9 11 13 15 17 19 0.6 0.8 1 1.2 1.4 1.6 1.8 Total Matching Rate/n (d)S3 - - -,S8 - -,S9 Figure 3.7: (Color online) Results of the simulation experiments associated with the subset 2 f0:1; 0:2;:::; 1g of the numerical experiments in Figure 3.5b. In Figures 3.7 and 3.8, we see that our steady state fluid model approximates the simulation results accurately. Since our fluid model is a deterministic model and ignores the driver patience and the travel costs, 122 0.1 0.3 0.5 0.7 0.9 1 0.8 0.9 1 1.1 Total Matching Rate/n (a)S1,S2 - -,S3 - - - 0.1 0.3 0.5 0.7 0.9 1 0.8 0.9 1 1.1 Total Matching Rate/n (b)S3 - - -,S4,S5 - - 0.1 0.3 0.5 0.7 0.9 1 0.8 0.9 1 1.1 Total Matching Rate/n (c)S3 - - -,S6 - -,S7 0.1 0.3 0.5 0.7 0.9 1 0.8 0.9 1 1.1 Total Matching Rate/n (d)S3 - - -,S8 - -,S9 Figure 3.8: (Color online) Results of the simulation experiments associated with the subset a 2 f1; 3; 5;:::; 19g of the numerical experiments in Figure 3.6. our fluid model overestimates the simulation results. However, the overestimation is not very significant as can be seen in Figures 3.7 and 3.8. Figures 3.7a and 3.8a present the simulation results associated with the settings 1, 2, and 3; i.e., we test the market size. As the market size increases, our fluid model becomes more accurate 9 . In large markets, individual customer or driver behavior does not affect the overall system performance significantly. This is parallel to our fluid model in which customers and drivers are infinitesimally small so that individual customer or driver behavior does not affect the overall performance at all. Figures 3.7b and 3.8b present the simulation results associated with the settings 3, 4, and 5; i.e., we test the driver patience. Our fluid model approximates the simulation results very accurately in all three settings. Moreover, it approximates S 5 the best, S 3 the second best, and S 4 the worst. This is expected because 9 It is well known that fluid models can approximate large matching markets accurately (cf. Remark 3.2.1). 123 our fluid model ignores the driver patience. Hence, as the drivers become more and more patient in the simulation model, the accuracy of the fluid model increases. Figures 3.7c and 3.8c present the simulation results associated with the settings 3, 6, and 7; i.e., we test the travel times. The simulation results are insensitive to the travel times between the areas and our fluid model approximates the simulation results very accurately. We consider the long run average performance of the simulation model and the rate of drivers moving between areas are the same irrespective of the travel times in the long run. Consequently, the simulation results are insensitive to the travel times. Figures 3.7d and 3.8d present the simulation results associated with the settings 3, 8, and 9; i.e., we test the travel costs. As expected, our fluid model approximates the simulation results more accurately as the travel costs decrease. Moreover, when the travel costs are sufficiently small, our fluid model approximates the simulation results very accurately. 3.7 Concluding Remarks We study the interplay between the pricing and the matching decisions of the ridesharing firms. We show that optimizing only in one dimension (either pricing or matching) while using simple decisions in the other dimension has no benefit to the firm. Hence, both the pricing and the matching decisions have first order effect on the system performance and they should be optimized simultaneously. We derive conditions under which simple pricing and matching decisions are enough to achieve the optimal performance. We also derive conditions under which the firm must use sophisticated pricing and matching decisions. Finally, we show that our ridesharing model accurately approximates more realistic ridesharing models by simulation experiments. Our ridesharing model has time homogeneous parameters. However, in reality, customer demand or driver supply can change dramatically over time. It is not obvious how to extend our model with time dependent parameters because the decisions of the firm will potentially be time dependent in that case. Moreover, we need to extend our steady state equilibrium concept to a potentially time dependent equilibrium concept, which is quite challenging. It is not even clear whether an equilibrium concept is relevant to a time dependent ridesharing system; i.e., there may not exist any equilibriums. Nevertheless, formulating a ridesharing system with time dependent parameters is a challenging but an interesting future research topic. Although we ignore the travel costs for the drivers, in the simulation experiments, we show that our model approximates ridesharing systems with travel costs accurately when the travel costs are sufficiently small. However, incorporating travel costs to our model formulation is not trivial. In our current model, there are two types of areas in an equilibrium: areas where idle drivers position themselves and areas where no idle 124 driver arrives. If there are travel costs, then there can be the third type of areas in an equilibrium: areas where idle drivers stuck after dropping customers; i.e., idle drivers may not want to leave those undesired areas due to the high travel costs. Consequently, the firm’s optimization problem becomes more complicated than the optimization problem (3.11), which makes this extension non-trivial. Nevertheless, a ridesharing model with travel costs is an interesting future research topic. We ignore the competition between ridesharing firms. There are many drivers who work both for Uber and Lyft in the USA and each ridesharing firm wants to keep the drivers in their platform in order to have adequate level of supply in the system. Hence, ridesharing firms compete not only for the customers but also for the drivers. Therefore, understanding the effect of competition between the ridesharing firms on the pricing and matching decisions is an interesting future research topic. We assume that the drivers have perfect information about the system; i.e., they know the system param- eters and the decisions of the firm and all drivers. However, in reality, the drivers can know as much as information that the firm shares with them. Therefore, what information the firm should share with the drivers is an interesting future research topic. 125 Appendix A Technical Appendix to Chapter 1 This appendix chapter is organized into three sections: A.1, A.2, and A.3. Section A.1 presents the proof of Proposition 1.8.1. Section A.2 presents the proofs of all results in the chapter with standard methodology. Section A.3 provides the detailed results of all simulation experiments mentioned in Section 1.9. A.1 Proof of Proposition 1.8.1 Without loss of generality, we will prove Proposition 1.8.1 only forr2N + . For allr2N + , letU r (0) := 0, V r 1 (0) := 0,V r 2 (0) := 0, U r (k) := k X i=1 u r i ; 8k2N + ; and A r (t) := maxfk2N :U r (k)tg; 8t2R + ; V r 1 (k) := k X i=1 v r i ; 8k2N + ; and S r 1 (t) := maxfk2N :V r 1 (k)tg; 8t2R + ; V 2 (k) := k X i=1 w r i ; 8k2N + ; and S r 2 (t) := maxfk2N :V r 2 (k)tg; 8t2R + : Without loss of generality, we assume thatA r (t;!)<1,S r 1 (t;!)<1, andS r 2 (t;!)<1 for allr2N + , t2R + , and!2 by removing aP-null set from if necessary. LetT r j (t) (I r j (t)) denote the cumulative amount of time serverj is busy (idle) up to timet, for allr2 N + ,t2 R + , andj2f1; 2g. Then, for all r2N + ,t2R + , andj2f1; 2g,T r j (t) +I r j (t) =t and Q r 1 (t) =Q r 1 (0) +A r (t)S r 1 (T r 1 (t)); Q r 2 (t) =Q r 2 (0) +S r 1 (T r 1 (t))S r 2 (T r 2 (t)): Let us define the following shifted processes. For allr2N + ,t2R + ,j2f1; 2g, andi2N, I r;i j (t) :=I r j (ir +t)I r j (ir); T r;i j (t) :=T r j (ir +t)T r j (ir); A r;i (t) :=A r (ir +t)A r (ir) r t; S r;i j (t) :=S r j (T r j (ir +t))S r j (T r j (ir)) r j T r;i j (t); 126 Q r;i j (t) :=Q r j (ir +t); X r;i 1 (t) :=Q r 1 (ir) +A r;i (t)S r;i 1 (t) + ( r r 1 )t; X r;i 2 (t) :=Q r 1 (ir) +Q r 2 (ir)Q r;i 1 (t) +A r;i (t)S r;i 2 (t) + ( r r 2 )t: Then, after some algebra, for allr2N + ,t2R + ,j2f1; 2g, andi2N, we have Q r;i j (t) =X r;i j (t) + r j I r;i j (t): For allr2N + ,j2f1; 2g, andi2N, sinceI r;i j (0) = 0,I r;i j () is nondecreasing, and R 1 0 Q r;i j (t)dI r;i j (t) = 0, we have Q r;i j ; r j I r;i j = (; ) (X r;i j ): Next, for allr2N + , let us define the following sets: G r := Q r 1 (0)_Q r 2 (0) (( r 1 ^ r 2 r )^)r 7 \ r n1 1 \ i=0 A r;i r _ S r;i 1 r _ S r;i 2 r (( r 1 ^ r 2 r )^)r 28 ; (A.1) H r 1 := r n1 1 \ i=0 Q r 1 (ir) (( r 1 ^ r 2 r )^)r 7 ; (A.2) H r 2 := r n1 1 \ i=0 Q r 2 (ir) 3(( r 1 ^ r 2 r )^)r 7 : (A.3) We have the following result related to (A.1) whose proof is presented in Appendix A.1.1. Lemma A.1.1. For all> 0, there existr 2 () 1 and strictly positive constantsC 4 ,C 5 ,C 6 , andC 7 which are independent ofr such that ifrr 2 (), P A r;i r >r C 4 r n e C 5 r ; (A.4) P S r;i 1 r >r +P S r;i 2 r >r C 6 r n e C 7 r ; (A.5) for alli2f0; 1;:::;r n1 1g. There exists an r 3 1 such that if r r 3 , we have r 1 ^ r 2 r ( 1 ^ 2 )=2 > 0. Let us fix an arbitrary r r 3 . First, we will show that G r H r 1 by induc- tion. In the setG r , we havefQ r 1 (0) (( r 1 ^ r 2 r )^)r=7g. Next, suppose that, in the setG r , 127 fQ r 1 (jr) (( r 1 ^ r 2 r )^)r=7g for allj2f1; 2;:::;ig for somei2f0; 1;:::;r n1 2g. Then, in the setG r , Q r 1 ((i + 1)r) =Q r;i 1 (r) =X r;i 1 (r) + r 1 I r;i 1 (r); =Q r 1 (ir) +A r;i (r)S r;i 1 (r) + ( r r 1 )r + sup 0tr Q r 1 (ir)A r;i (t) +S r;i 1 (t) + ( r 1 r )t + Q r 1 (ir) + (( r 1 ^ r 2 r )^)r 14 + ( r r 1 )r + sup 0tr A r;i (t) +S r;i 1 (t) + + sup 0tr (Q r 1 (ir) + ( r 1 r )t) + (( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) + ( r r 1 )r + sup 0tr (Q r 1 (ir) + ( r 1 r )t) + = (( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) + ( r r 1 )r + (Q r 1 (ir) + ( r 1 r )r) + = (( r 1 ^ r 2 r )^)r 7 + (Q r 1 (ir) + ( r r 1 )r) + = (( r 1 ^ r 2 r )^)r 7 ; where the first and second inequalities are by (A.1) and the last equality is by the induction hypothesis. Therefore,G r H r 1 . In the setG r , for alli2f0; 1;:::;r n1 1g, we have sup 0tr Q r;i 1 (t) = sup 0tr n X r;i 1 (t) + r 1 I r;i 1 (t) o ; = sup 0tr Q r 1 (ir) +A r;i (t)S r;i 1 (t) + ( r r 1 )t + sup 0st Q r 1 (ir)A r;i (s) +S r;i 1 (s) + ( r 1 r )s + (( r 1 ^ r 2 r )^)r 7 + sup 0tr Q r 1 (ir) + ( r r 1 )t + sup 0st (Q r 1 (ir) + ( r 1 r )s) + = (( r 1 ^ r 2 r )^)r 7 + sup 0tr Q r 1 (ir) + ( r r 1 )t + (Q r 1 (ir) + ( r 1 r )t) + = (( r 1 ^ r 2 r )^)r 7 + sup 0tr (Q r 1 (ir) + ( r r 1 )t) + = (( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) 2(( r 1 ^ r 2 r )^)r 7 ; (A.6) where the first inequality is by (A.1) and the last inequality is by (A.2) and the fact thatG r H r 1 . 128 Next, we will show thatG r H r 2 by induction. In the setG r ,fQ r 2 (0) (( r 1 ^ r 2 r )^)r=7g. Suppose that, in the setG r ,fQ r 2 (jr) 3(( r 1 ^ r 2 r )^)r=7g for allj2f1; 2;:::;ig for somei2 f0; 1;:::;r n1 2g. Then, in the setG r , Q r 2 ((i + 1)r) =Q r;i 2 (r) =X r;i 2 (r) + r 2 I r;i 2 (r) =Q r 1 (ir) +Q r 2 (ir)Q r;i 1 (r) +A r;i (r)S r;i 2 (r) + ( r r 2 )r + sup 0tr Q r 1 (ir)Q r 2 (ir) +Q r;i 1 (t)A r;i (t) +S r;i 2 (t) + ( r 2 r )t + Q r 1 (ir) +Q r 2 (ir) + (( r 1 ^ r 2 r )^)r 7 + ( r r 2 )r + sup 0tr Q r;i 1 (t) + sup 0tr (Q r 1 (ir)Q r 2 (ir) + ( r 2 r )t) + 3(( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )r + sup 0tr (Q r 1 (ir)Q r 2 (ir) + ( r 2 r )t) + = 3(( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )r + (Q r 1 (ir)Q r 2 (ir) + ( r 2 r )r) + = 3(( r 1 ^ r 2 r )^)r 7 + (Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )r) + = 3(( r 1 ^ r 2 r )^)r 7 ; where the first inequality is by (A.1) and the fact thatQ r;i 1 (r) 0, the second inequality is by (A.6), and the last equality is by the induction hypothesis and the fact thatG r H r 1 . Therefore,G r H r 2 . In the set G r , for alli2f0; 1;:::;r n1 1g, we have sup 0tr Q r;i 2 (t) = sup 0tr n X r;i 2 (t) + r 2 I r;i 2 (t) o ; = sup 0tr n Q r 1 (ir) +Q r 2 (ir)Q r;i 1 (t) +A r;i (t)S r;i 2 (t) + ( r r 2 )t + sup 0st Q r 1 (ir)Q r 2 (ir) +Q r;i 1 (s)A r;i (s) +S r;i 2 (s) + ( r 2 r )s + o (( r 1 ^ r 2 r )^)r 7 + sup 0tr Q r;i 1 (t) + sup 0tr n Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )t + sup 0st (Q r 1 (ir)Q r 2 (ir) + ( r 2 r )s) + o 3(( r 1 ^ r 2 r )^)r 7 + sup 0tr Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )t + (Q r 1 (ir)Q r 2 (ir) + ( r 2 r )t) + 129 3(( r 1 ^ r 2 r )^)r 7 + sup 0tr (Q r 1 (ir) +Q r 2 (ir) + ( r r 2 )t) + 3(( r 1 ^ r 2 r )^)r 7 +Q r 1 (ir) +Q r 2 (ir) (( r 1 ^ r 2 r )^)r; (A.7) where the first inequality is by (A.1) and the fact thatQ r;i 1 (t) 0 for allt2R + , the second inequality is by (A.6), and the last inequality is by the fact thatG r H r 1 \H r 2 . Next, note that P sup 0tr n Q r 1 (t)>r =P 0 @ r n1 1 [ i=0 sup 0tr Q r;i 1 (t)>r 1 A P 0 @ r n1 1 [ i=0 sup 0tr Q r;i 1 (t)>r ;G r 1 A +P ((G r ) c ); =P 0 @ r n1 1 [ i=0 sup 0tr Q r;i 1 (t)>r ;G r \H r 1 1 A +P ((G r ) c ): (A.8) Similarly, P sup 0tr n Q r 2 (t)>r P 0 @ r n1 1 [ i=0 sup 0tr Q r;i 2 (t)>r ;G r \H r 2 1 A +P ((G r ) c ): (A.9) Note that, ifrr 3 , r n1 1 [ i=0 sup 0tr Q r;i 1 (t)>r \G r \H r 1 =;; r n1 1 [ i=0 sup 0tr Q r;i 2 (t)>r \G r \H r 2 =;; (A.10) by (A.6) and (A.7), respectively. Therefore, if r r 3 , both the first probability in (A.8) and the first probability in the right hand side of (A.9) are equal to 0 and so P sup 0tr n Q r 1 (t)>r +P sup 0tr n Q r 2 (t)>r 2P ((G r ) c ): (A.11) Hence, in order to prove (1.59), we need to boundP ((G r ) c ) which is less than or equal to P Q r 1 (0)_Q r 2 (0)> (( r 1 ^ r 2 r )^)r 7 + r n1 1 X i=0 P A r;i r > (( 1 ^ 2 )^)r 56 130 + 2 X j=1 r n1 1 X i=0 P S r;i j r > (( 1 ^ 2 )^)r 56 ; (A.12) whenrr 3 by (A.1). Letr 1 :=r 0 _r 2 (( 1 ^ 2 )^)=56 _r 3 . Then, by (1.58) and Lemma A.1.1, ifrr 1 , (A.12) is less than or equal to C 0 r 2n1 e C 1 r +C 4 r 2n1 e C 5 r +C 6 r 2n1 e C 7 r 0:5C 2 r 2n1 e C 3 r ; (A.13) whereC 0 ;C 1 ;C 4 ;C 5 ;C 6 ;C 7 are strictly positive constants independent ofr,C 2 := 2(C 0 +C 4 +C 6 ), and C 3 := minfC 1 ;C 5 ;C 7 g. Therefore, (A.11), (A.12), and (A.13) prove Proposition 1.8.1. A.1.1 Proof of Lemma A.1.1 We first present some preliminary results, then prove (A.5), and lastly prove (A.4). There exists r 4 1 such that ifr r 4 , then r > 0:5 and r j > 0:5 j for allj2f1; 2g. Let us fix arbitraryr r 4 and i2f0; 1;:::;r n1 1g. Without loss of generality, we choose> 0 such that< 0:5(^ 1 ^ 2 ). Preliminary Results For allt2 [0;r] andj2f1; 2g, since 0 T r j (ir +t)T r j (ir) t, there existsf r j (t)2 [0;t] such that T r j (ir +t) =T r j (ir) +f r j (t). Then, for allt2 [0;r] andj2f1; 2g, S r;i j (t) = S r j (T r j (ir +t))S r j (T r j (ir)) r j T r j (ir +t) + r j T r j (ir) = S r j (T r j (ir) +f r j (t))S r j (T r j (ir)) r j (T r j (ir) +f r j (t)) + r j T r j (ir) sup 0st S r j (T r j (ir) +s)S r j (T r j (ir)) r j (T r j (ir) +s) + r j T r j (ir) sup 0sr S r j (T r j (ir) +s)S r j (T r j (ir)) r j (T r j (ir) +s) + r j T r j (ir) = sup 0sr S r j (T r j (ir) +s)S r j (T r j (ir)) r j s : Since the last inequality above holds uniformly for allt2 [0;r], then for allj2f1; 2g, we have S r;i j r = sup 0tr S r;i j (t) sup 0tr S r j (T r j (ir) +t)S r j (T r j (ir)) r j t : (A.14) 131 Suppose thatN 2 is partially ordered componentwise in the sense that for any (k 1 ;l 1 )2N 2 and (k 2 ;l 2 )2 N 2 , (k 1 ;l 1 ) (k 2 ;l 2 ), ifk 1 k 2 andl 1 l 2 . For all (k;l)2N 2 , let F r kl :=fQ r 1 (0); Q r 2 (0); u r m ;m2N + ; v r m ;m2f1; 2;:::;k + 1g; w r m ;m2f1; 2;:::;l + 1gg: Clearly,fF r kl ; (k;l)2 N 2 g is a multiparameter filtration (cf. page 85 in Ethier and Kurtz (1986)). Then, similar to Lemma 8.3 of Williams (1998a) and Lemmas 7.5 and 7.6 of Bell and Williams (2001) and by the fact thatS r 1 (t;!)<1 andS r 2 (t;!)<1 for allr2N + ,t2R + , and!2 , we have the following result. Lemma A.1.2. For all r 2 N + and t2 R + , T r (t) := (S r 1 (T r 1 (t));S r 2 (T r 2 (t))) is a (multiparameter) stopping time (cf. Definition 7.4 in Bell and Williams (2001)) with respect to the filtrationfF r kl ; (k;l)2N 2 g and the associated-algebra is F r T r (t) := A2F :A\fS r 1 (T r 1 (t)) =k;S r 2 (T r 2 (t) =lg2F r kl ;8(k;l)2N 2 : Then, v r S r 1 (T r 1 (t))+1 ;w r S r 2 (T r 2 (t))+1 2F r T r (t) . LetfA m ;m2 N + g be a sequence of sets such thatA m 2 B(R 2 ) for allm2N + andB r be a set such thatB r 2F r T r (t) . Then, P v r S 1 (T r 1 (t))+m ;w r S 2 (T r 2 (t))+m 2A m ;m2f2; 3;:::g;B r =P ((v r m ;w r m )2A m ;m2N + )P (B r ): Lemma A.1.2 can be proven similarly to Lemma 8.3 of Williams (1998a) and Lemma 7.6 of Bell and Williams (2001) are proven, hence we skip the proof. Next, we present some preliminary results related to large deviations theory for the renewal processS r 1 . For all2R, let `() := lnE [expf (1v 1 )g]; ` r () := lnE [expf (1= r 1 v r 1 )g]: (A.15) Then,e `() <1 for all2 ( ; ) by the exponential moment assumption onv 1 . Sincev r 1 =v 1 = r 1 , we have` r () =`(= r 1 ). Forx 0, let 1 (x) := sup >0 fx`()g; 2 (x) := sup >0 fx`()g; (A.16a) r 1 (x) := sup >0 fx` r ()g = 1 ( r 1 x); r 2 (x) := sup >0 fx` r ()g = 2 ( r 1 x): (A.16b) 132 Note that 1 and 2 are not exactly but very similar to Fenchel-Legendre transform of `() and `(), respectively (cf. Definition 2.2.2 of Dembo and Zeitouni (1998)). We have the following result. Lemma A.1.3. Both 1 and 2 are convex and nondecreasing in [0;1), 1 (0) = 2 (0) = 0, and 1 (x)> 0 and 2 (x)> 0 for allx> 0. Proof: First, let us consider 1 . Let 1 (x) := sup 2R fx`()g; 8x2R; (A.17) denote the Fenchel-Legendre transform of`() (cf. Definition 2.2.2 of Dembo and Zeitouni (1998)). Then, by Parts (a) and (b) of Lemma 2.2.5 of Dembo and Zeitouni (1998), 1 is convex, 1 (0) = 0, 1 is nondecreasing in [0;1), and for allx 0, 1 (x) = sup 0 fx`()g = 1 (x)_ 0; (A.18) where the last equality in (A.18) is by (A.16). Moreover, by Parts (a) and (c) of Lemma 2.2.5 of Dembo and Zeitouni (1998),` is convex inR,` is differentiable in ( ; ), and`(0) = ` 0 (0) = 0, where` 0 is the derivative of`. Then,` achieves the global minimum at point 0; and since it is convex,` is nondecreasing in the interval [0;1). Therefore, for any fixed x > 0, there exists an 2 (0; ) such that 1 (x) x`( ) > 0. As a result, 1 (x) = 1 (x) > 0 for allx > 0 by (A.18). Lastly, since` is convex and achieves its global minimum at point 0 in its domainR and continuous in the interval ( ; ), 1 (0) = inf >0 `() =`(0) = 0 = 1 (0); where the first equality is by (A.16). Hence, 1 (x) = 1 (x) for allx 0, which implies that 1 is convex and nondecreasing in [0;1), 1 (0) = 0, and 1 (x)> 0 for allx> 0. The proofs for 2 follows with exactly the same way. The only difference is that we consider the random variablev 1 1 instead of 1v 1 . Next, we will derive two more preliminary results. Lemma A.1.4. For all> 0, there exists anr 5 () 1 such that ifrr 5 (), we have P (S r 1 (r)> ( r 1 +)r)C 8 e C 9 r ; whereC 8 andC 9 are strictly positive constants independent ofr. 133 Proof: P (S r 1 (r)> ( r 1 +)r)P (S r 1 (r)>b( r 1 +)rc) =P (V r 1 (b( r 1 +)rc)<r) =P 0 @ b( r 1 +)rc X j=1 1 r 1 v r j > b( r 1 +)rc r 1 r 1 A P 0 @ b( r 1 +)rc X j=1 1 r 1 v r j > r 1 r 1 1 A (A.19) =P 0 @ exp 8 < : b( r 1 +)rc X j=1 1 r 1 v r j 9 = ; > exp r 1 r 1 1 A E 2 4 exp 8 < : b( r 1 +)rc X j=1 1 r 1 v r j 9 = ; 3 5 exp r 1 r 1 (A.20) = b( r 1 +)rc Y j=1 E exp 1 r 1 v r j exp r 1 r 1 = exp b( r 1 +)rc` r () r 1 r 1 (A.21) = exp b( r 1 +)rc r 1 r 1 b( r 1 +)rc ` r () ; (A.22) where> 0 is an arbitrary constant in (A.19), (A.20) is by Markov’s inequality, and the equalities in (A.21) are by the i.i.d. property of the sequencefv r j ;j2N + g and (A.15). Since (A.22) is valid for all> 0, P (S r 1 (r)> ( r 1 +)r) exp b( r 1 +)rc sup >0 r 1 r 1 b( r 1 +)rc ` r () = exp b( r 1 +)rc r 1 r 1 r 1 b( r 1 +)rc = exp b( r 1 +)rc 1 r 1 b( r 1 +)rc ; (A.23) by (A.16). Note that there exists anr 5 () 1 and 1 > 0 such that ifrr 5 (), we have r 1 b( r 1 +)rc > 1 ; and r 1 +> 1 : By Lemma A.1.3, 1 ( 1 )> 0 and 1 (x) is nondecreasing for allx 0, thus the term in (A.23) converges to 0 with exponential rate asr!1. To complete the proof, letC 8 := exp( 1 ( 1 )) andC 9 := 1 1 ( 1 ). Lemma A.1.5. For all> 0, there exists anr 6 1 independent of such that ifrr 6 , we have P v r 1 1 r 1 >r C 10 e C 11 r ; 134 whereC 10 andC 11 are strictly positive constants independent ofr. Proof: There exists anr 6 1 such that ifrr 6 , r 1 1 =2. Fix an arbitraryrr 6 . Then P v r 1 1 r 1 >r =P (v 1 1> r 1 r) =P 2 (v 1 1)> 2 r 1 r ; =P exp n 2 (v 1 1) o > exp r 1 r 2 ; E h exp n 2 (v 1 1) oi exp r 1 r 2 ; (A.24) = exp ` 2 r 1 r 2 exp n ` 2 1 r 4 o ; (A.25) where the inequality in (A.24) is by Markov’s inequality, and the equality in (A.25) is by (A.15). Since ` ( =2) < 1 because of the exponential moment assumption, (A.25) gives us the desired result. To complete the proof, letC 10 := expf`( =2)g andC 11 := 1 =4. Note that Lemmas A.1.3, A.1.4, and A.1.5 also hold for the renewal processesA r andS r 2 . Next, we will prove Lemma A.1.1 by Lemmas A.1.2, A.1.3, A.1.4, and A.1.5 in the following section. Proof of (A.5) We will first derive an exponentially decaying bound for the first probability in the left hand side of (A.5). Let r := infft2 [0;r] :jS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir)) r 1 tj>rg; where inff;g :=1 for completeness. Then, P S r;i 1 r >r P sup 0tr jS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir)) r 1 tj>r =P ( r r); (A.26) by (A.14). Let ~ V i;r (k) := S r 1 (T r 1 (ir))+k X j=S r 1 (T r 1 (ir))+2 v r j ; 8k2f2; 3;:::g; ~ V i;r 1 (k) := S r 1 (T r 1 (ir))+k X j=S r 1 (T r 1 (ir))+1 v r j ; 8k2N + ; (A.27) ~ V i;r (1) := 0 and ~ V i;r (k) = ~ V i;r 1 (k) := 0 for allk2f:::;2;1; 0g. Then, for allt2R + , fjS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir)) r 1 tj>rg =fS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir))> r 1 t +rg[fS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir))< r 1 trg fS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir))>b r 1 t +rcg[fS r 1 (T r 1 (ir) +t)S r 1 (T r 1 (ir))<d r 1 treg 135 n ~ V i;r (b r 1 t +rc)<t o [ n ~ V i;r 1 (d r 1 tre)>t o : Let us define r 1 := inf n t2 [0;r] : ~ V i;r (b r 1 t +rc)<t o ; r 2 := inf n t2 [0;r] : ~ V i;r 1 (d r 1 tre)>t o : Then, r r 1 ^ r 2 , so P ( r r)P ( r 1 ^ r 2 r)P ( r 1 r) +P ( r 2 r): (A.28) First, P ( r 1 <r) =P inf 0tr ~ V i;r (b r 1 t +rc)t < 0 P min j=brc;:::;b( r 1 +)rc ~ V i;r (j) j + 1r r 1 < 0 =P min j=brc;:::;b( r 1 +)rc ~ V i;r (j) j 1 r 1 < 2r r 1 =P min j=brc;:::;b( r 1 +)rc V r 1 (j 1) j 1 r 1 < 2r r 1 (A.29) =P min j=brc1;:::;b( r 1 +)rc1 V r 1 (j) j r 1 < 2r r 1 =P min j=brc1;:::;b( r 1 +)rc1 V r 1 (j) j r 1 > r 2 r 1 =P max j=brc1;:::;b( r 1 +)rc1 j r 1 V r 1 (j) > r 2 r 1 =P max j=brc1;:::;b( r 1 +)rc1 j r 1 V r 1 (j) > r 2 r 1 (A.30) =P exp max j=brc1;:::;b( r 1 +)rc1 j r 1 V r 1 (j) > exp r 2 r 1 =P max j=brc1;:::;b( r 1 +)rc1 exp j r 1 V r 1 (j) > exp r 2 r 1 E exp b( r 1 +)rc 1 r 1 V r 1 (b( r 1 +)rc 1) exp r 2 r 1 (A.31) =E 2 4 exp 8 < : b( r 1 +)rc1 X j=1 1 r 1 v r j 9 = ; 3 5 exp r 2 r 1 136 =E 2 4 b( r 1 +)rc1 Y j=1 exp 1 r 1 v r j 3 5 exp r 2 r 1 = b( r 1 +)rc1 Y j=1 E exp 1 r 1 v r j exp r 2 r 1 (A.32) =E exp 1 r 1 v r 1 b( r 1 +)rc1 exp r 2 r 1 (A.33) = e ` r () b( r 1 +)rc1 exp r 2 r 1 (A.34) = exp (b( r 1 +)rc 1) r 2 r 1 (b( r 1 +)rc 1) ` r () ; (A.35) where (A.29) is by Lemma A.1.2 and (A.27), > 0 is an arbitrary constant in (A.30), we use Doob’s inequality (cf. Theorem 5.4.2 of Durrett (2010)) in order to obtain the inequality in (A.31), (A.32) and (A.33) are by the i.i.d. property of the sequencefv r j ;j2N + g, (A.34) is by (A.15). Since (A.35) is valid for all> 0, by (A.16) we have P ( r 1 <r) exp (b( r 1 +)rc 1) r 1 r 2 r 1 (b( r 1 +)rc 1) ; = exp (b( r 1 +)rc 1) 1 r 2 b( r 1 +)rc 1 : (A.36) Second, P ( r 2 <r) =P sup 0tr ~ V i;r 1 (d r 1 tre)t > 0 P max j=1;:::;d( r 1 )re ~ V i;r 1 (j) j 1 +r r 1 > 0 (A.37) =P max j=1;:::;d( r 1 )re ~ V i;r 1 (j) j r 1 > r 1 r 1 =P v S r 1 (T r 1 (ir))+1 1 r 1 _ v S r 1 (T r 1 (ir))+1 1 r 1 + max j=2;:::;d( r 1 )re ~ V i;r (j) j 1 r 1 > r 1 r 1 ! (A.38) 2P v S r 1 (T r 1 (ir))+1 1 r 1 > r 1 2 r 1 +P max j=2;:::;d( r 1 )re ~ V i;r (j) j 1 r 1 > r 1 2 r 1 ; (A.39) 137 where (A.37) is by the fact that< 0:5(^ 1 ^ 2 )< r 1 , (A.38) is by (A.27). By the same way we derive (A.36), we can derive that the second probability in (A.39) is less than or equal to exp (d( r 1 )re 1) 2 r 1 2(d( r 1 )re 1) : (A.40) Next, let us consider the first probability in (A.39), which is less than or equal to 2P v S r 1 (T r 1 (ir))+1 1 r 1 > r 1 2 r 1 ; S r 1 (r n r) 2 r 1 (r n r) 1 + 2P (S r 1 (r n r)> 2 r 1 (r n r) 1) 2P max j2f1;2;:::;2 r 1 (r n r)g v r j 1 r 1 > r 1 2 r 1 + 2P (S r 1 (r n r)> 2 r 1 (r n r) 1) 4 r 1 (r n r)P v r 1 1 r 1 > r 1 2 r 1 + 2P (S r 1 (r n r)> 2 r 1 (r n r) 1): (A.41) By Lemmas A.1.4 and A.1.5, there exists anr 7 ()r 4 and strictly positive constantsC 12 andC 13 indepen- dent ofr andi such that ifr r 7 (), the sum in (A.41) is less than or equal toC 12 (r n r)e C 13 r . Next, let us consider (A.36) and (A.40). There exists anr 8 ()r 4 and 2 > 0 such that ifrr 8 (), we have r 2 b( r 1 +)rc 1 > 2 ; and r 1 2(d( r 1 )re 1) > 2 : By Lemma A.1.3, i ( 2 )> 0 and i (x) is nondecreasing for allx 0 andi2f1; 2g. Hence, ifrr 8 (), the sum of the terms in (A.36) and (A.40) is less than or equal toC 14 e C 15 r , whereC 14 := 2 expf 1 ( 2 )_ 2 ( 2 )g andC 15 := (0:5 1 )( 1 ( 2 )^ 2 ( 2 )). Letr 9 () := r 7 ()_r 8 (),C 16 := C 12 +C 14 , and C 17 :=C 13 ^C 15 . Then, ifrr 9 (), P S r;i 1 r >r C 16 r n e C 17 r ; 8i2f0; 1;:::;r n1 1g: Similarly, we can prove that there exists anr 10 () 1 and strictly positive constantsC 18 andC 19 indepen- dent ofr andi such that ifrr 10 (), P S r;i 2 r >r C 18 r n e C 19 r ; 8i2f0; 1;:::;r n1 1g: Lastly, letr 2 () :=r 9 ()_r 10 (),C 6 :=C 16 +C 18 , andC 7 :=C 17 ^C 19 . Then, we obtain (A.5). 138 Proof of (A.4) Note that proving (A.4) is equivalent to proving that there existr 2 () 0 and strictly positive constantsC 4 andC 5 which are independent ofr andi such that ifrr 2 (), P sup 0tr jS r 1 (ir +t)S r 1 (ir) r 1 tj>r C 4 r n e C 5 r ; 8i2f0; 1;:::;r n1 1g: (A.42) Hence we will explain how to prove (A.42). Let ~ F r m :=fv r k ;k2f1; 2;:::;m + 1gg 8m2N;r2N + : Then, it is easy to see that S r 1 (ir) is a stopping time with respect to the filtrationf ~ F r m ;m2 Ng for all i2f0; 1;:::;r n1 1g andr2 N + . The rest of the proof is very similar to the one of (A.5), the major differences are that 1. the definitions in (A.27) should be updated as ~ V i;r (k) := S r 1 (ir)+k X j=S r 1 (ir)+2 v r j ; 8k2f2; 3;:::g; ~ V i;r 1 (k) := S r 1 (ir)+k X j=S r 1 (ir)+1 v r j ; 8k2N + ; 2. we need to use Theorem 4.1.3 of Durrett (2010) instead of Lemma A.1.2 in order to obtain (A.29). A.2 Proofs of the Results with Standard Methodology In this section, we present the proofs of the results stated in Chapter 1 that require standard methodology. A.2.1 Proof of Proposition 1.3.1 The following beginning observations are useful. By FSLLN (cf. Theorem 5.10 of Chen and Yao (2001)), we have asr!1 S r j ;j2J[A a:s: ! S j ;j2J[A u.o.c.; (A.43) where for allt2R + , S a (t) = a t, S b (t) = b t, and S j (t) = j t for allj2A. By (A.43) and the fact that T r j (t)t for allj2A, we have for allT2R + andj2A, sup 0tT S r j ( T r j (t)) r j T r j (t) sup 0tT S r j (t) r j t a:s: ! 0; asr!1: (A.44) 139 We will derive the fluid limit results first for servers 1, 2, 3, and 5, second for server 4, lastly for servers 6 and 7. Since servers 1 and 3 and servers 2 and 5 form two independent generalized Jackson networks, by Theorem 7.23 of Chen and Yao (2001), asr!1, Q r k a:s: ! Q k := 0 u.o.c. for allk2f1; 2; 3; 6g and T r j a:s: ! T j u.o.c. for allj2f1; 2; 3; 5g where T j (t) := ( a = j )t ifj2f1; 3g and T j (t) := ( b = j )t if j2f2; 5g. Next let us consider the workload process in server 4. For allt2R + , W r 4 (t) = Q r 4 (t) + r A r B Q r 5 (t) = U r 4 (t) + r A I r 4 (t); U r 4 (t) := S r 1 ( T r 1 (t)) S r A ( T r A (t)) r A T r A (t) + r A r B S r 2 ( T r 2 (t)) S r B ( T r B (t)) r B T r B (t) r A t: Since we consider work-conserving policies (cf. (1.5c)), server 4 can be idle only if there is no workload in buffers 4 and 5. Therefore, by the uniqueness of the solution of the Skorokhod problem with respect to U r 4 (cf. Theorem 6.1 of Chen and Yao (2001)), we have r A I r 4 = ( U r 4 ) and W r 4 = ( U r 4 ) for allr. By (A.44), we have S r j T r j r j T r j a:s: ! 0 u.o.c. asr!1 for allj2fA;Bg. By (A.44), the fluid limit results corresponding to servers 1 and 2, and Theorem 5.3 of Chen and Yao (2001) (Random Time-Change theorem), we have S r j T r j a:s: ! S j T j u.o.c. as r !1 for all j 2f1; 2g. Therefore, U r 4 a:s: ! 0 u.o.c. asr!1 by Assumption 1.3.3 Part 3. Then, by the continuity of the mappings and , we have r A I r 4 a:s: ! 0 u.o.c. and W r 4 a:s: ! 0 u.o.c. asr!1. Since 0 Q r 4 W r 4 and 0 Q r 5 ( r B = r A ) W r 4 , we have Q r 4 a:s: ! 0 u.o.c. and Q r 5 a:s: ! 0 u.o.c. asr!1. Note that, Q r 4 (t) = S r 1 ( T r 1 (t)) S r A ( T r A (t)) r A T r A (t) r A T r A (t); (A.45) Q r 5 (t) = S r 2 ( T r 2 (t)) S r B ( T r B (t)) r B T r B (t) r B T r B (t): (A.46) By substituting the above results in (A.45) and (A.46), as r !1, we have T r A a:s: ! T A u.o.c. where T A (t) := ( a = A )t and T r B a:s: ! T B u.o.c. where T B (t) := ( b = B )t for allt2R + . Next, we consider server 6. Let Z r 6 := Q r 7 ^ Q r 8 . After some algebra, for allt2R + , Z r 6 (t) = X r 6 (t) + r 6 I r 6 (t); X r 6 (t) := S r 3 ( T r 3 (t))^ S r A ( T r A (t)) S r 6 ( T r 6 (t)) r 6 T r 6 (t) r 6 t: Since server 6 can be idle only if Z r 6 (t) = 0 (cf. (1.5d)), we can use the solution of the Skorokhod problem with respect to X r 6 and we have r 6 I r 6 = ( X r 6 ) and Z r 6 = ( X r 6 ). By (A.44), S r 6 T r 6 r 6 T r 6 a:s: ! 0 u.o.c. as r !1. By the fluid limit results corresponding to servers 3 and 4 and random time-change 140 theorem, we have S r 3 T r 3 a:s: ! S 3 T 3 u.o.c. and S r A T r A a:s: ! S A T A u.o.c. asr!1. Therefore, X r 6 a:s: ! X 6 u.o.c. asr!1 where X 6 (t) := ( a 6 )t for allt2R + . Then, by the fact that a 6 (cf. Assumption 1.3.3 Parts 1, 2, and 6) and the continuity of the mappings and , asr!1, we have I r 6 a:s: ! I 6 u.o.c. where I 6 (t) := (1 a = 6 )t and Z r 6 a:s: ! 0 u.o.c. By (1.2a), T r 6 a:s: ! T 6 u.o.c. asr!1 where T 6 (t) := ( a = 6 )t. Since Q r 7 (t) = S r 3 ( T r 3 (t)) S r 6 ( T r 6 (t)); Q r 8 (t) = S r A ( T r A (t)) S r 6 ( T r 6 (t)); (A.47) we have Q r 7 a:s: ! 0 u.o.c. and Q r 8 a:s: ! 0 u.o.c. asr!1 by (A.43), (A.47), and the random time-change theorem. The convergence results related to Q r k ,k2f9; 10g and T r 7 follow similarly, hence we skip them. A.2.2 Proof of Proposition 1.3.2 We provide the proof for the caseH =f2; 3; 5g, i.e servers 2, 3, and 5 are in heavy traffic together with server 4 but server 1 is in light traffic. The proofs of the other cases follow similarly. We derive the weak convergence result first for server 1, second for servers 2, 3, 4, and 5. We use the Skorokhod’s representation theorem to obtain the equivalent distributional representations of the processes in (1.40) (for which we use the same symbols and call “Skorokhod represented versions”) such that all Skorokhod represented versions of the processes are defined in the same probability space and the weak convergence in (1.40) is replaced by almost sure convergence u.o.c. Then we have (1.41) and let us consider the Skorokhod represented versions of the processes in (1.41). We first consider server 1. By (1.41), random time-change theorem, and Theorem 4.1 of Whitt (1980) (continuity of addition), we have ^ S r a ^ S r 1 T r 1 a:s: ! ~ S a ~ S 1 T 1 u.o.c. asr!1. Since server 1 works in a work-conserving fashion and is in light traffic, we have ^ Q r 1 = ^ X r 1 and r 1 ^ I r 1 = ^ X r 1 by (1.18a) and (1.21). Then by Lemma 6.4 (ii) of Chen and Yao (2001), asr!1, r 1 ^ I r 1 +r( r a r 1 )e; ^ Q r 1 a:s: ! ~ S a + ~ S 1 T 1 ; 0 ; u.o.c. (A.48) Next let us consider servers 2, 3, 4, and 5. Let ^ Q r H , ^ X r H , and ^ I r H bejHj-dimensional vectors derived from the vectors ^ Q r , ^ X r , and ^ I r (cf. (1.19)) by deleting the rows corresponding to eachi,i = 2H, respectively. 141 LetR r H denote thejHjjHj-dimensional matrix derived fromR r (cf. (1.20)) by deleting the rows and columns corresponding to eachi,i = 2H. Then by (1.19), (1.20), and (1.21), we have ^ Q r H = ^ X r H +R r H ^ I r H : By the fact that all servers work in a work-conserving fashion (cf. (1.5)) and Theorem 7.2 of Chen and Yao (2001), ^ Q r H = ^ X r H and ^ I r H = ^ X r H , where (; ) is the multidimensional reflection mapping which is Lipschitz continuous under the uniform norm. Hence let us first focus on ^ X r H . By Assumption 1.3.3 Parts 3, 4, and 5, (1.16), (1.17), (1.18), (1.41), random time-change theorem, continuity of addition, and the fact that ^ Q r 1 a:s: !0 u.o.c., asr!1, we have ^ X r H = 2 6 6 6 6 6 6 4 ^ X r 2 ^ X r 3 ^ X r 6 ^ X r 4 + r A r B ^ X r 5 3 7 7 7 7 7 7 5 a:s: ! 2 6 6 6 6 6 6 4 ~ S b ~ S 2 T 2 + 2 e ~ S a ~ S 3 T 3 + 3 e ~ S 2 T 2 ~ S 5 T 5 + ( 5 2 )e ~ S a ~ S A T A + A B ~ S 2 T 2 ~ S B T B + 4 A B 2 e 3 7 7 7 7 7 7 5 =: ~ X H u.o.c. (A.49) After some algebra, it is possible to see that ~ X H is a Brownian motion starting from 0 jHj with drift vector H and covariance matrix H . By the continuity of the multidimensional reflection mapping, we have ~ Q H = ~ X H and ~ I H = ~ X H , where ~ Q H = ~ Q 2 ; ~ Q 3 ; ~ Q 6 ; ~ W 4 0 ; ~ I H = ~ I 2 ; ~ I 3 ; ~ I 5 ; ~ I 4 0 : By Definition 3.1 of Williams (1998b), ~ Q H is an SRBM associated with the data P jHj ; H ; H ;R H ; 0 jHj . Since the Skorokhod represented version of the processes have the same joint distribution with the original ones, when the Skorokhod represented versions of the processes converge almost surely u.o.c., then the original processes weakly converge. In other words, corresponding to the original processes, we have ^ Q r 1 ; ^ Q r 2 ; ^ Q r 3 ; ^ Q r 6 ; ^ W r 4 ) 0; ~ Q 2 ; ~ Q 3 ; ~ Q 6 ; ~ W 4 asr!1: (A.50) 142 A.2.3 Proof of Theorem 1.6.2 We will only prove (1.37) because the proof of (1.38) is very similar. By Corollary 1.5.2 and Theorem 1.6.1, for allt2R + andx> 0, we have lim r!1 P ^ Z r (t)>x =P ~ Z(t)>x lim inf r!1 P ^ Z ;r (t)>x : (A.51) Moreover, lim inf r!1 E Z 1 0 e t ^ Z ;r (t)dt Z 1 0 e t lim inf r!1 E h ^ Z ;r (t) i dt (A.52a) = Z 1 0 e t lim inf r!1 Z 1 0 P ^ Z ;r (t)>x dxdt Z 1 0 e t Z 1 0 lim inf r!1 P ^ Z ;r (t)>x dxdt (A.52b) Z 1 0 e t Z 1 0 P ~ Z(t)>x dxdt = Z 1 0 e t E h ~ Z(t) i dt =E Z 1 0 e t ~ Z(t)dt ; (A.52c) where the inequality in (A.52a) is by Fubini’s theorem, Fatou’s lemma, and the fact that ^ Z ;r (t) is a non- negative random variable. The equality and inequality in (A.52b) are by equation (21.9) of Billingsley (1995) and Fatou’s lemma, respectively. The inequality and the first and second equalities in (A.52c) are by (A.51), equation (21.9) of Billingsley (1995), and Fubini’s theorem, respectively. Therefore, (A.52) proves the inequality in (1.37). Next, we will prove the equality in (1.37) by the proof technique used in Bell and Williams (2001). By Theorem 1.5.1 and the Skorokhod representation theorem, we may assume ^ Q r k ;k2K; ^ W r 4 a:s: ! ~ Q k ;k2K; ~ W 4 u.o.c. asr!1, where ~ Q 1 ; ~ Q 2 ; ~ Q 3 ; ~ Q 6 ; ~ W 4 is defined in Proposition 1.3.2 and ~ Q 4 ; ~ Q 5 ; ~ Q 7 ; ~ Q 8 ; ~ Q 9 ; ~ Q 10 is defined in Proposition 1.4.1. Then, ^ Z r ! ~ Z asr!1, (mP)a:e:; where dm := e t dt on (R + ;B(R + )). Since (R + ;B(R + )F;mP) is a probability space, the condition lim sup r!1 E Z 1 0 e t ^ Z r (t) 2 dt <1; (A.53) 143 which implies the uniform integrability of ^ Z r with respect to the expectation under mP, gives us the equality in (1.37). In order to prove (A.53), it is enough to show that there exists anr 0 1 such that, sup rr 0 Z 1 0 e t E ^ Q r k (t) 2 dt<1; 8k2f3; 6; 7; 10g: (A.54) Hence, we will prove (A.54) from this point forward. We will use the following result frequently in the rest of the proof. Lemma A.2.1. (Equation (172) of Bell and Williams (2001)) There exists anr 1 1 such that sup rr 1 E sup 0st ^ S r j (s) 2 18 1 + 4 2 j (1 +x)t + 2 j + 2 ; (A.55) where ifj2J , thenx = j ; and ifj2A, thenx = j 1 . Therefore, the term in the left hand side of (A.55), which is a function of t, is in L 1 (m) := L 1 (R + ;B(R + );m) for all j 2 J [A, where L 1 (m) is the space of integrable nonnegative functions with respect to the probability measurem. Let us first consider ^ Q r 1 . For allt2R + , we have ^ Q r 1 (t) = ^ X r 1 (t) + r 1 ^ I r 1 (t) = ^ S r a (t) ^ S r 1 T r 1 (t) +r( r a r 1 )t + r 1 ^ I r 1 (t) (A.56) = ^ S r a (t) ^ S r 1 T r 1 (t) +r( r a r 1 )t + sup 0st ^ S r a (s) + ^ S r 1 T r 1 (s) +r( r 1 r a )s + (A.57) 2 sup 0st ^ S r a (s) + 2 sup 0st ^ S r 1 (s) +r( r a r 1 )t + sup 0st (r( r 1 r a )s) + ; (A.58) where (A.56) is by (1.18a) and (1.21), (A.57) is by the fact that r 1 ^ I r 1 = ( ^ X r 1 ), and (A.58) is by the fact that T r 1 (t)t for allt2R + . If server 1 is in light traffic; i.e.,r( r a r 1 )!1 asr!1 (cf. Assumption 1.3.3 Part 5), then there exists anr 2 1 such that ifrr 2 ,r( r a r 1 )< 0 and so (A.58) is equal to 2 sup 0st ^ S r a (s) + 2 sup 0st ^ S r 1 (s) : (A.59) 1 We present a slightly different version of (172) of Bell and Williams (2001) such that instead of the terms (1 +j)t and (1+j)t in the right hand side of (A.55), there are r j t and r j t in (172) of Bell and Williams (2001), respectively. 144 If server 1 is in heavy traffic; i.e.,r( r a r 1 )! 1 2R asr!1 (cf. Assumption 1.3.3 Part 5), then there exists anr 3 1 such that ifrr 3 , (A.58) is less than or equal to 2 sup 0st ^ S r a (s) + 2 sup 0st ^ S r 1 (s) + (1 +j 1 j)t: (A.60) Therefore, by (A.58), (A.59), and (A.60), there exists anr 4 1 such that ifrr 4 , sup 0st ^ Q r 1 (s) 2 16 sup 0st ^ S r a (s) 2 + sup 0st ^ S r 1 (s) 2 + (1 +j 1 j) 2 t 2 _ 1; 8t2R + : (A.61) Then, Lemma A.2.1 and (A.61) imply that there exists anr 5 1 such that sup rr 5 E sup 0st ^ Q r 1 (s) 2 2L 1 (m); (A.62) where (A.62) holds independent of whether server 1 is in light or heavy traffic. By the same technique that we prove (A.62), we can prove that there exists anr 6 1 such that sup rr 6 E sup 0st ^ Q r 2 (s) 2 2L 1 (m): (A.63) Next, let us consider ^ W r 4 . For allt2R + , we have ^ W r 4 (t) = ^ U r 4 (t) + r A ^ I r 4 (t); (A.64) where ^ U r 4 (t) := ^ S r a (t) ^ Q r 1 (t) ^ S r A T r A (t) + r A r B ^ S r b (t) ^ Q r 2 (t) ^ S r B T r B (t) +r r A r a r A + r b r B 1 t; where (A.64) is by (1.16), (1.17), (1.18c), and (1.21), and (A.64) holds independent of whether servers 1 and 2 are in light or heavy traffic. Then, by Lemma A.2.1, (A.62), (A.63), Assumption 1.3.3 Parts 3 and 4, we can prove that there exists anr 7 1 such that sup rr 7 E sup 0st ^ U r 4 (s) 2 2L 1 (m): (A.65) 145 Since r A ^ I r 4 is nondecreasing and r A ^ I r 4 = ( ^ U r 4 ), we have E sup 0st r A ^ I r 4 (s) 2 =E r A ^ I r 4 (t) 2 =E " sup 0st ^ U r 4 (s) + 2 # E sup 0st ^ U r 4 (s) 2 : (A.66) Then, by (A.64), (A.65), and (A.66), we have sup rr 7 E sup 0st ^ W r 4 (s) 2 2L 1 (m); (A.67) which, by (1.16) and the fact that ^ Q r 4 and ^ Q r 5 are nonnegative processes, implies that sup rr 7 E sup 0st ^ Q r k (s) 2 2L 1 (m); 8k2f4; 5g: (A.68) By the same technique used to derive (A.62) and (A.67), we can prove that there exists anr 8 1 such that sup rr 8 E sup 0st ^ Q r k (s) 2 2L 1 (m); 8k2f3; 6g; (A.69) independent of whether servers 3 and 5 are in light or heavy traffic. Next, by (1.18d) and (1.22), for allt2R + , ^ Q r 7 (t)^ ^ Q r 8 (t) = ^ U r 6 (t) + r 6 ^ I r 6 (t); where ^ U r 6 (t) := ^ S r a (t) ^ Q r 1 (t) ^ Q r 3 (t)_ ^ Q r 4 (t) ^ S r 6 ( T r 6 (t)) +r( r a r 6 )t; and r 6 ^ I r 6 = ( ^ U r 6 ). Then, for allt2R + , r( r a r 6 )t + r 6 ^ I r 6 (t) = r( r a r 6 )t + sup 0st ^ S r a (s) + ^ Q r 1 (s) + ^ Q r 3 (s)_ ^ Q r 4 (s) + ^ S r 6 ( T r 6 (t))r( r a r 6 )s + sup 0st ^ S r a (s) + sup 0st ^ S r 6 (s) + X k=1;3;4 sup 0st ^ Q r k (s) + r( r a r 6 )t + sup 0st (r( r 6 r a )s) + : (A.70) 146 Since server 6 is in light traffic; i.e.,r( r a r 6 )!1 asr!1 (cf. Assumption 1.3.3 Part 6), there exists anr 9 1 such that ifrr 9 ,r( r a r 6 )< 0 and so (A.70) is equal to sup 0st ^ S r a (s) + sup 0st ^ S r 6 (s) + X k=1;3;4 sup 0st ^ Q r k (s): (A.71) Therefore, Lemma A.2.1, (A.62), (A.68), (A.69), and (A.71) imply that there exists anr 10 1 such that sup rr 10 E sup 0st r( r a r 6 )s + r 6 ^ I r 6 (s) 2 2L 1 (m): (A.72) By (1.18d) and (1.22), for allt2R + , ^ Q r 7 (t) = ^ S r a (t) ^ Q r 1 (t) ^ Q r 3 (t) ^ S r 6 ( T r 6 (t)) +r( r a r 6 )t + r 6 ^ I r 6 (t): Hence, sup 0st ^ Q r 7 (s) sup 0st ^ S r a (s) + sup 0st ^ S r 6 (s) + X k=1;3 sup 0st ^ Q r k (s) + sup 0st r( r a r 6 )s + r 6 ^ I r 6 (s) ; which, by Lemma A.2.1, (A.62), (A.69), and (A.72), implies that there exists anr 11 such that sup rr 11 E sup 0st ^ Q r 7 (s) 2 2L 1 (m): (A.73) By the same technique that we prove (A.73), we can prove that there exists anr 12 1 such that sup rr 12 E sup 0st ^ Q r 10 (s) 2 2L 1 (m): (A.74) Therefore, (A.69), (A.73), and (A.74) imply (A.54), which completes the proof. A.2.4 Proof of Proposition 1.8.2 LetU k n := P n i=1 a k i for alln2N + andk2f1; 2g, thenU k n denotes the arrival time of thenth job in the kth single server queue andU 1 n U 2 n for alln2N + andk2f1; 2g. LetA k (t) := supfn2N + :U k n tg for allt2R + andk2f1; 2g. Then,A k (t) denotes the total number of external arrivals in thekth queue up to timet,t2R + . For alln2N + andk2f1; 2g, letS k n denote the service start time of thenth job in the kth queue andW k n denote the sojourn time (total time spent in the buffer and server) of thenth job in the 147 kth queue. Then,D k n =S k n +b n andW k n =D k n U k n for alln2N + andk2f1; 2g;S k 1 =U k 1 =a k 1 and S k n =U k n _D k n1 for alln2f2; 3;:::g. Note that, D 1 1 = S 1 1 +b 1 = a 1 1 +b 1 a 2 1 +b 1 = S 2 1 +b 1 = D 2 1 . Suppose that D 1 i D 2 i for all i2f1; 2;:::;n 1g for somen 2. Then, D 1 n =S 1 n +b n =U 1 n _D 1 n1 +b n U 2 n _D 2 n1 +b n =S 2 n +b n =D 2 n : Hence,D 1 n D 2 n for alln2 N + by induction. SinceS k n = D k n b n for alln2 N + andk2f1; 2g, we haveS 1 n S 2 n for alln2N + . Moreover, for alln 2 andk2f1; 2g, W 1 1 =W 2 1 =b 1 ; (A.75) W k n =D k n U k n =b n +S k n U k n =b n +U k n _D k n1 U k n =b n + D k n1 U k n + =b n + U k n1 +W k n1 U k n + =b n + W k n1 a k n + : (A.76) Note that,W 1 1 W 2 1 by (A.75). Suppose thatW 1 i W 2 i for alli2f1; 2;:::;n 1g for somen 2. By (A.76) and the fact thata 1 n a 2 n , we haveW 1 n W 2 n . Hence,W 1 n W 2 n for alln2N + by induction. Fix an arbitraryn2N + . Then for all 0tb n , we have Q 1 (t +S 1 n ) =A 1 (t +S 1 n ) (n 1)I(t =b n ) =A 1 (t +S 1 n )A 1 (U 1 n ) +I(t<b n ) = sup ( m2N : n+m X i=n+1 a 1 i t +S 1 n U 1 n ) +I(t<b n ) = sup ( m2N : n+m X i=n+1 a 1 i tb n +W 1 n ) +I(t<b n ) sup ( m2N : n+m X i=n+1 a 2 i tb n +W 2 n ) +I(t<b n ) = sup ( m2N : n+m X i=n+1 a 2 i t +S 2 n U 2 n ) +I(t<b n ) =A 2 (t +S 2 n )A 2 (U 2 n ) +I(t<b n ) =A 2 (t +S 2 n ) (n 1)I(t =b n ) =Q 2 (t +S 2 n ); (A.77) where the inequality is by the fact thata 1 n a 2 n andW 1 n W 2 n for alln2N + . By (A.77), we have sup S 1 n tS 1 n +K Q 1 (t) sup S 2 n tS 2 n +K Q 2 (t); for allK2 [0;b n ] andn2N + : (A.78) 148 In order to prove that sup 0tT Q 1 (t) sup 0tT Q 2 (t), we will consider two different cases. First, suppose that the server is idle at timeT in the first single server queue. We consider two sub-cases. First, suppose that there is no service completion beforeT in the first single server queue. Then, it must be the case thatQ 1 (0) = 0 and there is no job arrival during [0;T ] in the first single server queue (otherwise the server would not be idle) and so sup 0tT Q 1 (t) = 0, which gives the desired result trivially. In the second sub-case, suppose that there is a service completion untilT in the first single server queue. Let the last job which departs the first queue in the interval [0;T ] be theNth job for someN2N + . Then,D 1 N T and sup 0tT Q 1 (t) = max n2f1;2;:::;Ng sup S 1 n tD 1 n Q 1 (t) max n2f1;2;:::;Ng sup S 2 n tD 2 n Q 2 (t) = sup 0tD 2 N Q 2 (t) sup 0tT Q 2 (t); (A.79) where the first inequality is by (A.78) and the last inequality is by the fact thatTD 1 N D 2 N . Second, suppose that the server is busy in the first single server queue at timeT . Without loss of generality, suppose that the server in the first queue is working on theNth job for someN2N + . Then, it must be the case thatS 1 N T <D 1 N . Hence,b N >TS 1 N and sup 0tT Q 1 (t) = max n2f1;2;:::;N1g sup S 1 n tD 1 n Q 1 (t) ! _ sup 0tTS 1 N Q 1 (t +S 1 N ) max n2f1;2;:::;N1g sup S 2 n tD 2 n Q 2 (t) ! _ sup 0tTS 1 N Q 2 (t +S 2 N ) = sup 0tTS 1 N +S 2 N Q 2 (t) sup 0tT Q 2 (t); (A.80) with the convention max n2f1;2;:::;0g x n := 0 for any real sequencefx n ;n2 N + g; the first inequality is by (A.78), and the last inequality is by the fact that S 1 N S 2 N . Therefore, (A.79) and (A.80) prove that sup 0tT Q 1 (t) sup 0tT Q 2 (t). Next we will prove thatD 1 n D 1 n1 D 2 n D 2 n1 for alln2f2; 3;:::g by induction. For allk2f1; 2g andn2f2; 3;:::g, D k n D k n1 =S k n +b n D k n1 =U k n _D k n1 +b n D k n1 = U k n D k n1 + +b n : 149 Hence, it is enough to prove that U 1 n D 1 n1 + U 2 n D 2 n1 + for alln2f2; 3;:::g. Note that, U 1 2 D 1 1 + = a 1 1 +a 1 2 (S 1 1 +b 1 ) + = a 1 2 b 1 + a 2 2 b 1 + = U 2 2 D 2 1 + : Next, suppose that U 1 i D 1 i1 + U 2 i D 2 i1 + for alli2f2;:::;n 1g for somen 3. There are two cases to consider. Case 1. Suppose that U 1 n1 D 1 n2 , meaning that the server is idle at the arrival time of the (n 1)st job in the first single server queue. Then, D 1 n1 = U 1 n1 +b n1 and so U 1 n D 1 n1 + = a 1 n b n1 + . There are two sub-cases. Case 1.1. Suppose thatU 2 n1 D 2 n2 . Then, U 2 n D 2 n1 + = a 2 n +U 2 n1 U 2 n1 b n1 + = a 2 n b n1 + a 1 n b n1 + = U 1 n D 1 n1 + . Case 1.2. Sup- pose that U 2 n1 < D 2 n2 . Then, U 2 n D 2 n1 + = a 2 n +U 2 n1 D 2 n2 b n1 + a 2 n b n1 + a 1 n b n1 + = U 1 n D 1 n1 + . Case 2. Suppose thatU 1 n1 <D 1 n2 , meaning that the server is busy at the arrival time of the (n 1)st job in the first single server queue. Without loss of generality, let the job being processed in the server of the first queue at timeU 1 n1 be thejth job wherej < n 1 and b j (0 b j b j ) denote the remaining service time for that job. Then, U 1 n1 = S 1 j +b j b j and D 1 n1 U 1 n1 = b j + P n1 i=j+1 b i . We have Q 2 (S 2 j +b j b j )Q 1 (S 1 j +b j b j ) =Q 1 (U 1 n1 ) =nj, where the inequality is by (A.77). Therefore, in the second single server queue, when the jth job is processed b j b j amount of time, (n 1)st job should have already arrived in the system, i.e.,U 2 n1 S 2 j +b j b j . This implies thatD 2 n1 U 2 n1 D 2 n1 S 2 j b j + b j = P n1 i=j b i b j + b j = b j + P n1 i=j+1 b i =D 1 n1 U 1 n1 . Then, U 1 n D 1 n1 + = a 1 n +U 1 n1 D 1 n1 + a 2 n +U 2 n1 D 2 n1 + = U 2 n D 2 n1 + , where the inequality is by the fact thatD 2 n1 U 2 n1 D 1 n1 U 1 n1 anda 1 n a 2 n . A.2.5 Proof of Lemma 1.8.1 Let us fix an arbitraryn2N + . For allt2R + , S r A (T r A ( r 2n1 ) +t) = sup ( k2N : k X l=1 v r A (l)T r A ( r 2n1 ) +t ) = sup 8 < : k +A r n :k2N; A r n X l=1 v r A (l) + k X l=1 v r A (l +A r n )T r A ( r 2n1 ) +t 9 = ; ; sup ( k +A r n :k2N; T r A ( r 2n1 ) + k X l=1 v r A (l +A r n )T r A ( r 2n1 ) +t ) (A.81) 150 =A r n + sup ( k2N : k X l=1 v r A (l +A r n )t ) =A r n +E r;n A (t); (A.82) where (A.81) is by the fact that P A r n l=1 v r A (l) T r A ( r 2n1 ) by definition ofA r n . Similar to the derivation of (A.82), we can get the following result: S r 3 (T r 3 ( r 2n1 ) +t) =B r n + sup ( k2N : k X l=1 v r 3 (l +B r n )t ) =B r n +E r;n 3 (t): (A.83) Note that, we have equality sign in (A.83) unlike the greater than or equal to sign in (A.81). This is because there is a service completion in server 3 exactly at r 2n1 by construction (cf. (1.54a)). Therefore, we have P B r n l=1 v r 3 (l) =T r 3 ( r 2n1 ), and this gives us the equality sign in (A.83). Then, for allt2R + , S r 3 (T r 3 ( r 2n1 ) +t)S r A (T r A ( r 2n1 ) +t)E r;n 3 (t)E r;n A (t) + 1; by (A.82), (A.83), and the fact thatB r n A r n = 1. A.2.6 Proof of Lemma 1.8.2 First, we present the following results from the literature which will be used later in the proof. Lemma A.2.2. (Lemma 9 of Ata and Kumar (2005)) Given> 0 andT > 2=, we have for eachj2J[A and> 0 P sup 0tT jS j (t)x j tjT ! C j (;) T 1+ ; where C j (;) := 2 + 2 1 + 2 2+2 18(2 + 2) 3=2 (1 + 2) 1=2 ! 2+2 E jv j (1) 1=x j j 2+2 2 4 4x 2 j (x j +) 2 ! 1+ + 4x 3 j 2 ! 1+ 3 5 ; (A.84) ifj2J , thenx j = j ; and ifj2A, thenx j = j . The proof of Lemma A.2.2 can be seen in Ata and Kumar (2005) (Note that we have fixed a small typo in (A.84) by replacing (2 + 2)=(1 + 2) with (2 + 2)=(1 + 2) 2+2 . This typo does not affect the results of Ata and Kumar (2005)). Since we assume exponential moment condition for the service times 151 (cf. Assumption 1.3.2), Lemma A.2.2 holds for all> 0. However, the proof presented in Ata and Kumar (2005) requires a weaker moment assumption. Lemma A.2.3. (Equation 3.67 of Hall and Heyde (1980))For any martingale with differencesZ i ; 1i n and anyp 2, we have E " n X i=1 Z i p # (18pq 1=2 ) p n p=2 max 1in E [jZ i j p ]; whereq = (1p 1 ) 1 . The proof of Lemma 1.8.2 is a modification of the proof of Lemma A.2.2 of Ata and Kumar (2005). We present the proof for the casej = 3, the other case follows similarly. Let us define E r n;3 := B r n d( r 3 + 9 )r 2 Te ; (A.85) where 9 > 0 is an arbitrary constant. Then, N r X n=1 P sup 0t<r T jE r;n 3 (t) r 3 tj> 2 r T; r 2n1 r 2 T ! N r X n=1 P r 2n1 r 2 T; E r n;3 c + N r X n=1 P sup 0t<r T jE r;n 3 (t) r 3 tj> 2 r T;E r n;3 ! : (A.86) Let us consider the first sum in (A.86). N r X n=1 P r 2n1 r 2 T; E r n;3 c N r X n=1 P r 2n1 r 2 T; S r 3 (T r 3 ( r 2n1 ))> ( r 3 + 9 )r 2 T N r X n=1 P S r 3 (r 2 T )> ( r 3 + 9 )r 2 T N r X n=1 P sup 0tr 2 T jS r 3 (t) r 3 tj> 9 r 2 T ! =N r P sup 0tr 2 T jS r 3 (t) r 3 tj> 9 r 2 T ! N r C r 3 ( 9 ; 2 ) (r 2 T ) 1+ 2 ! 0 asr!1; (A.87) where 2 > 0 is an arbitrary constant, and (A.87) is by Lemma A.2.2 and the fact that N r = O(r 2 ). Moreover, for eachr,C r 3 ( 9 ; 2 ) is the constant defined in Lemma A.2.2 associated with the renewal process S r 3 . It is straightforward to see thatC r 3 ( 9 ; 2 ) is bounded above by a constant uniformly inr, hence we get the convergence result in (A.87). 152 Next, let us consider the second sum in (A.86). Let us fixn andr. Let := infft2 [0;r T ) :jE r;n 3 (t) r 3 tj> 2 r Tg; where inff;g =1 for completeness. Then P sup 0t<r T jE r;n 3 (t) r 3 tj> 2 r T;E r n;3 ! =P <r T;E r n;3 : Let ~ V r 3 (k) := 0 for allk2N and ~ V r 3 (k) := P k i=1 v r 3 (B r n +i) for allk2N + . Then, fjE r;n 3 (t) r 3 tj> 2 r Tg =fE r;n 3 (t)> r 3 t + 2 r Tg[fE r;n 3 (t)< r 3 t 2 r Tg fE r;n 3 (t)>b r 3 t + 2 r Tcg[fE r;n 3 (t)<d r 3 t 2 r Teg =f ~ V r 3 (b r 3 t + 2 r Tc)<tg[f ~ V r 3 (d r 3 t 2 r Te)>tg: Let us define 1 := inf n t2 [0;r T ) : ~ V r 3 (b r 3 t + 2 r Tc)<t o ; 2 := inf n t2 [0;r T ) : ~ V r 3 (d r 3 t 2 r Te)>t o : Then 1 ^ 2 . Thus P <r T;E r n;3 P 1 ^ 2 <r T;E r n;3 P 1 <r T;E r n;3 +P 2 <r T;E r n;3 : (A.88) We will consider the two probabilities after the second inequality in (A.88) separately. First, 1 = inf t2 [0;r T ) : ~ V r 3 (b r 3 t + 2 r Tc) b r 3 t + 2 r Tc r 3 <t b r 3 t + 2 r Tc r 3 : Next, let us define ~ 1 := inf t2 [0;r T ) : ~ V r 3 (b r 3 t + 2 r Tc) b r 3 t + 2 r Tc r 3 < 2 r T 2 r 3 : (A.89) Whenr is sufficiently large, 2 r T > 2. This implies 2 r T 2 r 3 >t b r 3 t + 2 r Tc r 3 153 andP 1 <r T;E r n;3 P ~ 1 <r T;E r n;3 whenr is sufficiently large. Then, P ~ 1 <r T;E r n;3 P sup i=1;2;:::;b( r 3 + 2 )r Tc ~ V r 3 (i) i r 3 > 2 r T 2 r 3 ;E r n;3 ! E 2 4 sup i=1;2;:::;b( r 3 + 2 )r Tc ~ V r 3 (i) i r 3 ! 2+2 I E r n;3 3 5 2 r T 2 r 3 (2+2) =E 2 4 d( r 3 + 9 )r 2 Te X j=1 sup i=1;2;:::;b( r 3 + 2 )r Tc ~ V r 3 (i) i r 3 I (B r n =j) ! 2+2 3 5 2 r T 2 r 3 (2+2) = d( r 3 + 9 )r 2 Te X j=1 E 2 4 sup i=1;2;:::;b( r 3 + 2 )r Tc V r 3 (i +j)V r 3 (j) i r 3 I (B r n =j) ! 2+2 3 5 2 r T 2 r 3 (2+2) d( r 3 + 9 )r 2 TeE 2 4 sup i=1;2;:::;b( r 3 + 2 )r Tc V r 3 (i) i r 3 ! 2+2 3 5 2 r T 2 r 3 (2+2) ; (A.90) where> 0 is an arbitrary constant such that 4=(1 +)< ; the first inequality is by the definition of ~ 1 (cf. (A.89)); the second inequality is by Markov’s inequality; the first equality is by the definition ofE r n;3 (cf. (A.85)); the second equality is by the fact that (V r 3 (j +i)V r 3 (j))I (B r n =j) = ~ V r 3 (i)I (B r n =j) for alli;j2N + ; and the last inequality is due to the fact thatI (B r n =j) 1 andV r 3 (j +i)V r 3 (j) d =V r 3 (i) by the i.i.d. property offv r 3 (i);i2N + g. Note that V r 3 (i) i r 3 ;i2N + is a martingale. Then, by theL p maximum inequality for martingales (cf. Theorem 5.4.3 of Durrett (2010)), we see that the term in (A.90) is less than or equal to d( r 3 + 9 )r 2 Te 2 + 2 1 + 2 (2+2) E " V r 3 (b( r 3 + 2 )r Tc) b( r 3 + 2 )r Tc r 3 2+2 # 2 r T 2 r 3 (2+2) : (A.91) Then, by using Lemma A.2.3 on (A.91), the term in (A.91) is less than or equal to d( r 3 + 9 )r 2 Te 2 + 2 1 + 2 (2+2) 18(2 + 2) 3=2 (1 + 2) 1=2 ! (2+2) (b( r 3 + 2 )r Tc) 1+ E " v r 3 (1) 1 r 3 2+2 # 2 r T 2 r 3 (2+2) ; (A.92) so doesP ~ 1 <r T;E r n;3 . 154 Now, we will consider the second probability in (A.88). First, 2 = inf t2 [0;r T ) : ~ V r 3 (d r 3 t 2 r Te) d r 3 t 2 r Te r 3 >t d r 3 t 2 r Te r 3 : Next, let us define ~ 2 := inf t2 [0;r T ) : ~ V r 3 (d r 3 t 2 r Te) d r 3 t 2 r Te r 3 > 2 r T 2 r 3 : Whenr is sufficiently large, 2 r T > 2. This implies 2 r T 2 r 3 <t d r 3 t 2 r Te r 3 andP 2 <r T;E r n;3 P ~ 2 <r T;E r n;3 whenr is sufficiently large. Moreover, since r 3 > 2 when r is sufficiently large,d( r 3 2 )r Te> 0 whenr is sufficiently large. Then, P ~ 2 <r T;E r n;3 P sup i=1;2;:::;d( r 3 2 )r Te ~ V r 3 (i) i r 3 > 2 r T 2 r 3 ;E r n;3 ! P sup i=1;2;:::;b( r 3 + 2 )r Tc ~ V r 3 (i) i r 3 > 2 r T 2 r 3 ;E r n;3 ! (A.93) d( r 3 + 9 )r 2 Te 2 + 2 1 + 2 (2+2) 18(2 + 2) 3=2 (1 + 2) 1=2 ! (2+2) (b( r 3 + 2 )r Tc) 1+ E " v r 3 (1) 1 r 3 2+2 # 2 r T 2 r 3 (2+2) ; (A.94) where the inequality in (A.93) is by the fact that 2 r T 1 when r is sufficiently large (this implies d( r 3 2 )r Teb( r 3 + 2 )r Tc whenr is sufficiently large) and (A.94) is by (A.90) and (A.92). By the exponential moment assumption (cf. Assumption 1.3.2), both of the right hand sides in (A.92) and (A.94) areO(r 2 (1+) ). By (A.88) and the fact that 4=(1 +)< , we have N r X n=1 P sup 0t<r T jE r;n 3 (t) r 3 tj> 2 r T;E r n;3 ! N r X n=1 P 1 <r T;E r n;3 +P 2 <r T;E r n;3 =N r O(r 2 (1+) ) =O(r 4 (1+) )! 0; asr!1: 155 By the fact thatA r n =B r n 1 and using the same technique, we can also prove that N r X n=1 P sup 0t<r T E r;n A (t) r A t > 2 r T; r 2n1 r 2 T ! ! 0; asr!1: A.2.7 Proof of Lemma 1.8.3 Consider the sum in (1.97), which is less than or equal to N r 2 X n=1 P ~ r; 2n r 2 (T + 4 ); S r b (~ r; 2n )>N r 3 (A.95) + N r 2 X n=1 P ~ r; 2n r 2 (T + 4 ); S r 2 (T r 2 (~ r; 2n ))>N r 3 (A.96) + N r 2 X n=1 P ~ r; 2n r 2 (T + 4 ); S r B (T r B (~ r; 2n ))>N r 3 (A.97) +N r 2 P sup 0tr 2 (T + 4 ) Q r 2 (t)> (( r 2 ^ r B r b )^ 5 )r=7 1 ! : (A.98) First, consider the sum in (A.97), which is less than or equal to N r 2 X n=1 P ~ r; 2n r 2 (T + 4 ); S r B (T r B (~ r; 2n ))> ( r b + r 2 + r B )r 2 (T + 4 ) N r 2 X n=1 P S r B (r 2 (T + 4 ))> ( r b + r 2 + r B )r 2 (T + 4 ) N r 2 P sup 0tr 2 (T + 4 ) jS r B (t) r B tj> ( r b + r 2 )r 2 (T + 4 ) ! N r 2 C r B ( r b + r 2 ; 1 ) [r 2 (T + 4 )] 1+ 1 ! 0 asr!1; (A.99) where 1 > 0 is any arbitrary constant and (A.99) is by Lemma A.2.2 (cf. Appendix A.2.6). For each r,C r B ( r b + r 2 ; 1 ) is the constant defined in Lemma A.2.2 associated with the renewal processS r B . It is straightforward to see thatC r B ( r b + r 2 ; 1 ) is bounded above by a constant uniformly inr, hence we get the convergence result in (A.99). We can show that the sums in (A.95) and (A.96) converge to 0 by the same way we derive (A.99). 156 Next let us consider the sum in (A.98). There existr 11 1 and( 2 ; B ; b ; 5 )> 0 such that ifrr 11 , thenN r 3 ( b + 2 + B + 2)r 2 (T + 4 ) and (( r 2 ^ r B r b )^ 5 )r=7 1>r. Hence, ifrr 11 , the sum in (A.98) is less than or equal to ( b + 2 + B + 2)r 2 (T + 4 )P sup 0tr 2 (T + 4 ) Q r 2 (t)>r ! C 20 r 5 e C 21 r ; (A.100) whereC 20 > 0 andC 21 > 0 are constants which are independent ofr, and the inequality is by Proposition 1.8.1 and the fact that b < 2 andQ r 2 (0) = 0 for allr. Note that the right hand side of (A.100) converges to 0 asr!1, which completes the proof. A.2.8 Proof of Lemma 1.10.1 In the optimization problem (1.113), replacingq 4 withw 4 ( A = B )q 5 gives us the following equivalent optimization problem which has only one decision variable. min h 4 w 4 A B q 5 +h 5 q 5 +h 7 w 4 q 3 A B q 5 + +h 8 q 3 w 4 + A B q 5 + +h 9 (q 6 q 5 ) + +h 10 (q 5 q 6 ) + ; (A.101a) s.t. 0q 5 B A w 4 : (A.101b) The objective function (A.101a) is the sum of six different functions each of which is convex, continuous, and piecewise linear with respect to the decision variableq 5 . Since sum of finitely many convex (continuous, piecewise linear) functions is convex (continuous, piecewise linear), the objective function (A.101a) is also convex, continuous, and piecewise linear with respect to the decision variableq 5 . Then, an optimal solution should be either in the boundaries of the feasible region ofq 5 , which are 0 and ( B = A )w 4 (cf. (A.101b)), or in one of the break points which are in the interval 0; ( B = A )w 4 of the convex, continuous, and piecewise linear objective function (A.101a), which are ( B = A )(w 4 q 3 ) + andq 6 ^ ( B = A )w 4 . Therefore, the optimal solution, denoted byq 5 , is such that q 5 2 0; B A w 4 ; B A (w 4 q 3 ) + ; q 6 ^ B A w 4 : Then the optimal solution set in Table 1.3 follows by pluggingq 5 in the equality constraint (1.113b). 157 A.2.9 Proof of Lemma 1.10.2 For eachj2f1; 2;:::;ng, h j (q j q j;2 ) + is a convex and continuous function of the decision variables q i , i 2 f1; 2;:::;ng. Since sum of finitely many convex and continuous functions is also convex and continuous, then the objective function (1.130a) is also convex and continuous function of the decision variablesq j ,j2f1; 2;:::;ng. Then, a local optimum solution of the optimization problem (1.130) is also a global optimum and we will prove that the solution (1.131) is a local optimum. First, it is easy to see that the solution (1.131) satisfies the constraints (1.130b) and (1.130c), so it is feasible. We will prove that the solution (1.131) is a local optimum by showing that any deviation from this solution does not improve the objective function value. Let us fix an arbitraryj2f1; 2;:::;n 1g and considerq j . Note that the cost incurred due toq j is 0 (cf. Remark 1.10.4). Thus, decreasingq j cannot decreaseh j (q j q j;2 ) + but may increase the objective function value because at least oneq i , i2f1; 2;:::;ngnfjg will increase by (1.130b). Therefore, decreasingq j , j2f1; 2;:::;n 1g does not improve the objective function value. Next, let us increaseq j by where> 0 but sufficiently small. Increasingq j by increases the objective function value by at most h j . Since h i (q i q i;2 ) + = 0 for all i2f1; 2;:::;n 1gnfjg, decreasing q i ,i2f1; 2;:::;n 1gnfjg will not decrease the objective function value. Therefore, increasingq j but decreasingq i ,i2f1; 2;:::;n1gnfjg does not improve the objective function value. Next, let us decrease q n . There are two cases to consider. First, suppose thatq j q j;2 . Then, increasingq j by increases the objective function value exactly byh j . In this case, decreasingq n by n = j (cf. (1.130b)) can decrease the objective function value by at mosth n n = j which is less than or equal toh j becauseh j j h n n , thus the net change in the objective function value is nonnegative. Second, suppose thatq j < q j;2 . Then, increasing q j by will not increase the objective function value for sufficiently small . Note that, when q j <q j;2 , by (1.131a) and (1.131b) q j;2 j > q j j = 8 > > > < > > > : w; ifj = 1; w q 1;2 1 + q 2;2 2 + q j1;2 j1 ! + ; ifj2f2; 3;:::;n 1g: (A.102) 158 This implies that ifj = 1,q n = 0 by (1.130b) and (1.130c), and ifj2f2; 3;:::;n 1g, then by (1.131c) and (A.102), q n = n 0 @ 0 @ w q 1;2 1 + q 2;2 2 ! + q j1;2 j1 ! + q j;2 j 1 A + q n1;2 n1 1 A + ; = n q j j q j;2 j + q j+1;2 j+1 ! + q n1;2 n1 ! + ; = n 0 q j+1;2 j+1 + q n1;2 n1 ! + ; = 0: Hence, whenq j <q j;2 ,q n = 0 so it cannot be decreased by (1.130c). As a result, increasingq j by does not improve the objective function value. Therefore, increasing or decreasingq j does not improve the objective function value for allj2f1; 2;:::;n 1g. Lastly, let us consider q n . If we increase (decrease) q n , then some of the q j , j 2 f1; 2;:::;n 1g must increase (decrease) by (1.130b) and (1.130c). Since the latter change does not improve the objective function value, changing the value of q n does not improve the objective function value. Therefore, the solution (1.131) is a local optimum and also a global optimum. A.2.10 Proof of Lemma 1.10.3 By decreasing the number of decision variables from 4 to 2 by using the equality constraints and then simplifying the notation in the optimization problem (1.135), we get the following equivalent optimization problem: min h a w 5 A B1 q 6 q 4 + +h b (q 6 _q 7 ) +h c C B2 (w 6 q 7 )q 9 + ; (A.103a) s.t. 0q 6 B1 A w 5 ; 0q 7 w 6 ; (A.103b) where the decision variables are q 6 and q 7 . The objective function (A.103a) is convex and continuous function of the decision variables. Hence, a local optimum solution of the optimization problem (A.103) is also a global optimum. Note that increasingq 6 orq 7 decrease the first and third terms but increase the second term in the objective function (A.103a). Hence, the key point is to compare the total decrease in the first and third terms with the increase in the second term in (A.103a), when we increaseq 6 andq 7 from 0 to their corresponding upper bounds. We solve the optimization problem (A.103) case by case. 159 First consider the caseq 4 w 5 . Then, the first term in the objective function (A.103a) is 0 for all values of q 6 satisfying the constraints in (A.103b). Since increasingq 6 increases the second term in (A.103a),q 6 = 0 in the optimal solution. Since h b h c C = B2 by assumption, then increasing q 7 does not decrease the objective function value, thusq 7 = 0 in the optimal solution. In the caseq 9 ( C = B2 )w 6 ,q 6 =q 7 = 0 in the optimal solution by the same logic and the fact thath b h a A = B1 . Therefore, the last case to consider isw 5 >q 4 and ( C = B2 )w 6 >q 9 and we will consider it in two cases. Case 1:h b h a A = B1 +h c C = B2 By the second term in (A.103a), it is more efficient to increase q 6 andq 7 together with the same rate (if possible). Suppose thatq 6 = q 7 = 0. If we increaseq 6 andq 7 by sufficiently small> 0, then the objective function value increases by (h b h a A = B1 h c C = B2 )> 0. Therefore, we should not increaseq 6 andq 7 at all, andq 6 =q 7 = 0 in the optimal solution. Lastly, (1.136) follows by the two equality constraints of the optimization problem (1.135). Case 2: h b < h a A = B1 +h c C = B2 Suppose that q 6 = q 7 = 0. If we increase q 6 and q 7 by sufficiently small> 0, then the objective function value changes by (h b h a A = B1 h c C = B2 )< 0. Hence, it is efficient to increase q 6 and q 7 together until either q 6 = ( B1 = A )(w 5 q 4 ) or q 7 = w 6 ( B2 = C )q 9 . After this point, if we increase at least one of theq 6 orq 7 , then the objective function value increases because first or third term in (A.103a) is in its lower bound (which is 0) at this point andh b h a A = B1 andh b h c C = B2 . Therefore,q 6 = q 7 = minf( B1 = A )(w 5 q 4 ); w 6 ( B2 = C )q 9 g. Then, (1.137) follows by the two equality constraints of the optimization problem (1.135). Remark A.2.1. The solution (1.137) can be explained intuitively in the following way. Note that, we start with the solution ~ Q 6 = ~ Q 7 = 0 in the proof of (1.137), which is equivalent to giving full priority to typeb jobs in servers 5 and 6. Then we increase ~ Q 6 and ~ Q 7 , which is equivalent to giving some of the priority to typea andc jobs in servers 5 and 6, respectively. We do this until either ~ Q 10 = 0 or ~ Q 15 = 0; i.e., until either buffer 11 or buffer 14 never causes the corresponding downstream server to idle because of the join operation. At this point, we stop giving priority to typea andb jobs. A.3 Detailed Simulation Results In this section, we present the detailed results of the simulation experiments. 160 Table A.1: Detailed results of the simulation experiments: Average queue lengths with their 95% confidence intervals. Proposed Policy SDP Policy Static Priority Policy FCFS Policy Randomized Policy Randomized-2=3 Policy Ins. Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) Q p 3 (i) + Q p 7 (i) Q p 6 (i) + Q p 10 (i) 1 14.01 0.16 14.43 0.17 14.01 0.16 14.43 0.17 13.6 0.13 18.21 0.20 14.72 0.14 14.75 0.17 14.51 0.14 14.72 0.17 13.59 0.15 17.31 0.14 2 21.91 0.32 23.9 0.27 21.91 0.32 23.9 0.27 21.55 0.32 29.58 0.36 23.36 0.24 23.33 0.18 23.75 0.37 23.14 0.22 21.49 0.27 28.09 0.30 3 45.73 0.84 52.44 0.82 45.73 0.84 52.44 0.82 44.8 0.88 62.75 0.89 50.01 0.69 49.86 0.70 50.02 0.71 49.66 0.75 44.91 0.79 60.59 0.92 4 14.09 0.19 6.53 0.06 14.09 0.19 6.53 0.06 13.67 0.19 13.28 0.11 14.7 0.17 7.56 0.05 14.49 0.13 7.71 0.06 13.6 0.15 12.12 0.14 5 21.86 0.22 11.63 0.16 21.86 0.22 11.63 0.16 21.52 0.27 21.35 0.24 23.43 0.28 12.17 0.11 23.62 0.24 12.18 0.10 21.79 0.29 19.53 0.20 6 45.84 0.75 26.69 0.54 45.84 0.75 26.69 0.54 44.9 0.84 44.9 0.49 49.7 0.58 25.78 0.26 50.57 0.96 25.57 0.31 44.31 0.74 41.86 0.64 7 3.67 0.00 17.27 0.14 3.67 0.00 17.27 0.14 3.03 0.00 18.07 0.14 7.59 0.05 14.69 0.15 7.75 0.07 14.73 0.22 3.64 0.01 17.32 0.16 8 5.97 0.01 28.03 0.35 5.97 0.01 28.03 0.35 5.12 0.01 29.11 0.33 12.26 0.11 23.64 0.24 12.37 0.13 23.58 0.28 5.9 0.01 27.95 0.27 9 12.59 0.04 60.99 1.00 12.59 0.04 60.99 1.00 10.99 0.03 62.74 0.79 25.73 0.32 50.75 0.86 25.58 0.33 50.15 0.74 12.42 0.04 60.49 0.79 10 3.67 0.00 11.95 0.12 3.67 0.00 11.95 0.12 3.03 0.00 13.25 0.11 7.57 0.06 7.55 0.06 7.69 0.06 7.75 0.05 3.64 0.01 12.23 0.14 11 5.98 0.01 19.13 0.16 5.98 0.01 19.13 0.16 5.12 0.01 21.55 0.20 12.15 0.10 12.13 0.10 12.24 0.10 12.22 0.10 5.9 0.01 19.43 0.25 12 12.58 0.03 41.4 0.54 12.58 0.03 41.4 0.54 10.99 0.03 45.94 0.53 25.4 0.29 25.33 0.29 25.84 0.39 25.59 0.40 12.44 0.04 41.74 0.48 13 2.4 0.00 17.95 0.13 2.73 0.00 17.83 0.18 2.4 0.00 17.95 0.13 7.49 0.06 14.8 0.15 7.55 0.06 14.58 0.16 3.32 0.01 17.4 0.18 14 3.79 0.01 29.18 0.26 4.19 0.01 29.06 0.27 3.79 0.01 29.18 0.26 11.78 0.10 23.62 0.32 11.92 0.12 23.79 0.31 5.09 0.01 28.11 0.19 15 7.76 0.02 62.26 0.84 8.4 0.02 62.04 0.89 7.76 0.02 62.26 0.84 24.96 0.28 49.49 0.67 24.91 0.28 50.05 0.66 10.15 0.03 60.9 0.91 16 2.41 0.00 13.34 0.14 2.73 0.00 12.95 0.11 2.41 0.00 13.34 0.14 7.45 0.05 7.56 0.05 7.58 0.05 7.66 0.05 3.32 0.01 12.21 0.11 17 3.8 0.01 21.44 0.22 4.19 0.01 20.91 0.23 3.8 0.01 21.44 0.22 11.76 0.11 12.05 0.11 11.94 0.10 12.23 0.11 5.08 0.01 19.4 0.19 18 7.76 0.02 45.28 0.63 8.41 0.02 45.26 0.61 7.76 0.02 45.28 0.63 25.08 0.30 25.83 0.29 24.78 0.32 25.62 0.32 10.16 0.02 41.69 0.48 19 38.22 0.46 24.1 0.30 38.22 0.46 24.1 0.30 38.45 0.45 29.22 0.37 40.1 0.44 23.77 0.29 40.02 0.49 23.56 0.30 38.26 0.41 28.41 0.33 20 22.7 0.21 28.23 0.31 22.7 0.21 28.23 0.31 21.7 0.28 29.38 0.31 28.63 0.33 23.27 0.29 29.38 0.38 23.77 0.30 22.46 0.25 28.03 0.27 21 20.34 0.24 29.25 0.28 20.86 0.27 28.94 0.33 20.34 0.24 29.25 0.28 28.44 0.31 23.74 0.30 29.45 0.41 23.38 0.29 21.71 0.36 28.26 0.36 22 38.69 0.41 11.79 0.17 38.69 0.41 11.79 0.17 37.7 0.52 21.36 0.21 40.21 0.57 12.13 0.10 39.71 0.55 12.27 0.12 37.54 0.50 19.66 0.17 23 22.53 0.32 19.18 0.18 22.53 0.32 19.18 0.18 21.72 0.25 21.24 0.25 28.77 0.29 12.12 0.09 29.32 0.33 12.25 0.14 22.64 0.25 19.66 0.20 24 20.52 0.35 21.53 0.23 20.75 0.39 20.71 0.22 20.52 0.35 21.53 0.23 28.43 0.38 12.19 0.10 28.63 0.30 12.19 0.10 21.76 0.30 19.54 0.21 25 38.45 0.47 41.92 0.44 38.45 0.47 41.92 0.44 38.03 0.56 49.34 0.49 39.97 0.46 39.95 0.49 40.03 0.42 40.24 0.41 38.01 0.44 47.26 0.42 26 22.54 0.28 47.98 0.45 22.54 0.28 47.98 0.45 21.58 0.26 48.6 0.52 28.67 0.28 40.13 0.51 28.94 0.38 40.05 0.50 22.56 0.33 47.45 0.48 27 20.42 0.22 49.32 0.47 21.15 0.33 48.82 0.49 20.42 0.22 49.32 0.47 28.79 0.30 39.97 0.50 29.09 0.32 40.1 0.43 21.72 0.34 47.28 0.56 28 38.42 0.46 31.2 0.40 38.42 0.46 31.2 0.40 37.97 0.44 43.79 0.50 40.09 0.42 28.64 0.40 39.99 0.40 29.1 0.28 38.25 0.51 41.24 0.56 29 22.37 0.30 41.59 0.54 22.37 0.30 41.59 0.54 21.52 0.31 44.12 0.48 28.75 0.35 28.51 0.24 29.22 0.38 29.25 0.30 22.38 0.25 41.05 0.51 30 20.3 0.27 43.93 0.53 20.81 0.33 43.96 0.58 20.3 0.27 43.93 0.53 28.5 0.35 28.74 0.27 29.34 0.28 29.48 0.27 21.61 0.26 41.1 0.52 31 21.88 0.32 41.75 0.61 21.88 0.32 41.75 0.61 21.39 0.25 48.81 0.43 23.6 0.21 40.01 0.45 23.45 0.32 40.24 0.37 21.52 0.31 47.52 0.49 32 5.97 0.01 47.48 0.61 5.97 0.01 47.48 0.61 5.11 0.01 49.18 0.51 12.33 0.10 40.12 0.44 12.26 0.10 40.05 0.44 5.91 0.01 47.13 0.34 33 3.79 0.00 48.75 0.43 4.19 0.01 48.52 0.50 3.79 0.00 48.75 0.43 11.8 0.07 39.66 0.49 11.96 0.12 39.93 0.40 5.1 0.01 46.74 0.46 34 22.1 0.33 30.72 0.46 22.1 0.33 30.72 0.46 21.39 0.29 43.98 0.37 23.4 0.26 28.67 0.31 23.37 0.30 29.26 0.28 21.64 0.30 41.25 0.41 35 5.97 0.01 41.51 0.45 5.97 0.01 41.51 0.45 5.11 0.01 43.48 0.44 12.18 0.08 28.72 0.25 12.26 0.12 29.25 0.31 5.91 0.01 41.22 0.43 36 3.79 0.01 44.01 0.52 4.19 0.01 43.22 0.45 3.79 0.01 44.01 0.52 11.88 0.12 28.96 0.26 11.89 0.12 29.03 0.38 5.09 0.01 40.94 0.46 161 Appendix B Technical Appendix to Chapter 2 This appendix is organized as follows: Section B.1 provides the proof of Theorem 2.3.1, which provides an asymptotic upper bound on the fluid-scaled objective (2.6). That proof relies on the fluid equations, and the details of their derivation are given in Section B.2. Section B.3 provides a new regulator mapping result which is used in the proofs for asymptotic optimality results in Section B.4. The proofs of Lemmas 2.2.1, 2.3.1, 2.3.2, 2.4.1, 2.4.2, and 2.5.1 are found in Section B.5. Finally, we present a relative compactness result in spaceD, which is used in the derivation of fluid equations, in Section B.6. B.1 Proof of Theorem 2.3.1 The proof of Theorem 2.3.1 relies on understanding the behavior of fluid limits. In particular, the proof uses the fact that all fluid limits satisfy a set of equations that can be connected to the constraints in the CLP (2.12). The implication is that any fluid limit gives a feasible matching process for the CLP (2.12), which implies the optimal objective function value of the CLP (2.12) is an asymptotic upper bound for the objective (2.6) under fluid scale. For alli2N ,j2J ,t2R + ,n2N + , and admissible policy =f n ;n2N + g, let us define G ;n ij (t) := E j ( n j (t)) X k=1 I n j (k) =i ; (B.1) which is the cumulative number of times the system controller offers a typei driver to a typej customer up to timet and let G ;n ij := G ;n ij =n be the associated fluid scaled process. Clearly,D ;n ij G ;n ij for all i2N ,j2J ,n2N + , and admissible policy by (2.4) and (B.1). For alln2N + and admissible policy , letX ;n 2D 2NJ+3N+J be defined such that X ;n := A i n i ; E j n j ; R n i Z 0 n i (s)Q ;n i (s)ds ; Q ;n i ; D ;n ij ; G ;n ij ; 8i2N; j2J ; (B.2) where “” denotes the composition map. Let X ;n :=X ;n =n be the fluid scaled version ofX ;n . Then, we define fluid limit(s) offX ;n ;n2 N + g similar to the definition on page 299 of Dai and Tezcan (2011) in the following way. 162 Definition B.1.1. (Fluid Limit) Let us fix an arbitrary admissible policy =f n ;n2 N + g. Then, X is a fluid limit offX ;n ;n2 N + g if there exists an !2 and a subsequencefn k ;k2 N + g such that X ;n k (;!)! X u.o.c. ask!1. Proposition B.1.1. (Fluid Equations) For any admissible policy =f n ;n2N + g, there exists a full set A (i.e., P(A ) = 1) such that for any!2A ,f X ;n (;!);n2 N + g is relatively compact (i.e., every subsequence has a convergent subsequence) in the Skorokhod spaceD 2NJ+3N+J endowed with the u.o.c. topology. Thus, fluid limits exist in almost all sample paths and any fluid limit X = i ; j ; Z 0 i (s) Q i (s)ds; Q i ; D ij ; G ij ; 8i2N; j2J and satisfies the following equations for allt2R + : Q i (t) = i (t) Z t 0 i (s) Q i (s)ds X j2J D ij (t) 0; 8i2N; (B.3a) D ij (0) = G ij (0) = 0; 8i2N; j2J; (B.3b) D ij and G ij are nondecreasing and absolutely continuous for alli2N andj2J; (B.3c) X j2J D ij (t) i (t); 8i2N; (B.3d) X i2N G ij (t 2 ) G ij (t 1 ) j (t 2 ) j (t 1 ); 8j2J andt 1 ;t 2 2R + such thatt 2 t 1 , (B.3e) D ij (t) D ij ( m ) F ij ( m ) G ij (t) G ij ( m ) I (t2 [ m ; m+1 )) = 0; for allm2N,i2N , andj2J: (B.3f) The proof of Proposition B.1.1 is presented in Section B.2. Let us fix an arbitrary admissible policy and an arbitrary!2A . Then there exists a subsequencefn k ;k2N + g such that lim sup n!1 X i2N;j2J w ij D ;n ij (T;!) = lim k!1 X i2N;j2J w ij D ;n k ij (T;!): Since D ;n ij (;!) is relatively compact (cf. Proposition B.1.1) for all i2N and j 2J , there exists a subsequence offn k ;k2 N + g denoted byfn l ;l2 N + g such that D ;n l ij (;!) converges to a limit D ij () u.o.c. asl!1 for alli2N andj2J . Hence, lim sup n!1 X i2N;j2J w ij D ;n ij (T;!) = lim k!1 X i2N;j2J w ij D ;n k ij (T;!); 163 = lim l!1 X i2N;j2J w ij D ;n l ij (T;!) = X i2N;j2J w ij D ij (T ): Then, Theorem 2.3.1 follows by the following result. Proposition B.1.2. Let =f n ;n2N + g denote an arbitrary admissible policy. Under any fluid limit of fX ;n ;n2N + g, we have X i2N;j2J w ij D ij (T ) X i2N;j2J w ij Z T 0 j (s) F ij (s)~ x ij (s)ds: Next, we will prove Proposition B.1.2. Let us consider the following CLP which has the decision variables z ij : [0;T ]!R + for alli2N andj2J that replace j (t) F ij (t)x ij (t) in the CLP (2.12): max X i2N;j2J w ij Z T 0 z ij (s)ds; (B.4a) subject to: X j2J Z t 0 z ij (s)ds i (t); for alli2N andt2 [0;T ]; (B.4b) X i2N : F ij (t)>0 z ij (t) F ij (t) j (t); for allj2J andt2 [0;T ]; (B.4c) z ij (t)I F ij (t) = 0 = 0; for alli2N ,j2J , andt2 [0;T ]; (B.4d) z ij (t)I ( j (t) = 0) = 0; for alli2N ,j2J , andt2 [0;T ]; (B.4e) z ij (t) 0; for alli2N ,j2J , andt2 [0;T ]; (B.4f) z ij is Lebesgue measurable for alli2N andj2J: (B.4g) By an argument very similar to Lemma 2.3.1, there exists an optimal solution of the CLP (B.4) which is denoted byfz ij (t);i2N;j 2J;t2 [0;T ]g. We consider the CLP (B.4) instead of the CLP (2.12) because of three main reasons. First, the feasible region of the CLP (B.4) is smaller than the one of the CLP (2.12) in the sense that there exist a surjective but not bijective mapping from the feasible region of the CLP (2.12) to the one of the CLP (B.4). Second, it is easier to create connection between a fluid limit of f D ;n ij ;i2N;j2J;n2 N + g and the decision variables of the CLP (B.4) than between the same fluid limit and the decision variables of the CLP (2.12). Third, we have the following result. Lemma B.1.1. The optimal objective function values of the CLP (2.12) and the CLP (B.4) are equal. 164 Proof: First, consider the deterministic processf j (t) F ij (t)~ x ij (t);i2N;j2J;t2 [0;T ]g (remember thatf~ x ij (t);i2N;j2J;t2 [0;T ]g is an optimal solution of the CLP (2.12)). It is easy to see that this process is feasible for the CLP (B.4). Thus, X i2N;j2J w ij Z T 0 j (s) F ij (s)~ x ij (s)ds X i2N;j2J w ij Z T 0 z ij (s)ds: (B.5) Second, for alli2N ,j2J , andt2 [0;T ], let y ij (t) := 8 > < > : z ij (t) j (t) F ij (t) ; if F ij (t)> 0 and j (t)> 0, 0; otherwise: (B.6) Then, it is easy to see that the processfy ij (t);i2N;j2J;t2 [0;T ]g is feasible for the CLP (2.12), thus X i2N;j2J w ij Z T 0 j (s) F ij (s)y ij (s)ds = X i2N;j2J w ij Z T 0 z ij (s)I F ij (s)> 0; j (s)> 0 ds (B.7a) = X i2N;j2J w ij Z T 0 z ij (s)ds X i2N;j2J w ij Z T 0 j (s) F ij (s)~ x ij (s)ds; (B.7b) where the equality in (B.7b) is by (B.4d) and (B.4e). Hence, the optimal objective function values of the CLP (2.12) and the CLP (B.4) are equal to each other by (B.5) and (B.7b). Let us consider an arbitrary fluid limit of an arbitrary admissible policy. By (B.3c) and Theorem 3.35 of Folland (1999), both D ij and G ij are differentiable almost everywhere on [0;T ] for alli2N andj2J . Letd ij andg ij be the (nonnegative) derivatives of D ij and G ij , respectively, on the points where they are differentiable and without loss of generality be equal to 0 on the points where they are not differentiable for alli2N andj2J on the interval [0;T ]. By (B.3b), (B.3c), and the fundamental theorem of calculus for Lebesgue integrals (cf. Theorem 3.35 of Folland (1999)), for alli2N ,j2J , andt2 [0;T ], D ij (t) = Z t 0 d ij (s)ds; G ij (t) = Z t 0 g ij (s)ds: (B.8) Then,fd ij (t);i2N;j2J;t2 [0;T ]g satisfies (B.4b) by (B.3d) and (B.8); (B.4f) and (B.4g) by (B.3c), Theorem 3.35 of Folland (1999), and construction. Next, fix an arbitrary customer typej2J . By taking the derivatives of the both sides in (B.3e), we have X i2N g ij (t) j (t); (B.9) 165 for allt2 [0;T ]. Let us fix an arbitraryt2 [0;T ], and lett2 [ m ; m+1 ) for somem2N. By (B.3f) and (B.8), for alli2N , d ij (t) = 8 > < > : F ij ( m )g ij (t); if F ij (t)> 0, 0; otherwise; (B.10) where (B.10) holds for all t2 [ m ; m+1 ) except on a set of zero measure and we modify d ij such that d ij (t) = 0 in that set of zero measure. Then by (B.9) and (B.10) and the nonnegativity ofg ij for alli2N , X i2N : F ij (t)>0 d ij (t) F ij (t) j (t); so thatfd ij (t);i2N;j2J;t2 [0;T ]g satisfies (B.4c). Moreover, it satisfies (B.4d) by (B.10). Lastly, by (B.9), we have g ij (t)I ( j (t) = 0) = 0; 8i2N: (B.11) Thus, if F ij (t) > 0 for some i 2 N , then d ij (t)I ( j (t) = 0) = 0 by (B.10) and (B.11). Otherwise, d ij (t) = 0 by (B.10). Hence,fd ij (t);i2N;j2J;t2 [0;T ]g satisfies (B.4e) as well, so it is feasible for the CLP (B.4). Then, X i2N;j2J w ij Z T 0 z ij (s)ds X i2N;j2J w ij Z T 0 d ij (s)ds = X i2N;j2J w ij D ij (T ); (B.12) where the equality in (B.12) is by (B.8). Therefore, (B.12) together with Lemma B.1.1 proves Proposition B.1.2. B.2 Proof of Proposition B.1.1 (Properties of Fluid Limits) Let us fix an arbitrary T 1 2 R + . For all k 2 N + , let D k [0;T 1 ] denote the the space of functions with domain [0;T 1 ] and rangeR k which are right continuous with left limits. We will first prove Proposition B.1.1 except the property (B.3f) with respect to processes defined inD 2NJ+3N+J [0;T 1 ], then we will extend these results to the processes inD 2NJ+3N+J . Lastly, we will prove (B.3f). By the functional strong law of large numbers, random time-change theorem (cf. Theorems 5.10 and 5.3 of Chen and Yao (2001), respectively), and Assumption 2.3.1, A n i n i a:s: ! i ; u.o.c.; R n i a:s: !e; u.o.c.; E n j n j a:s: ! j ; u.o.c. asn!1; (B.13) 166 for alli2N andj2J . Let A 1 := n !2 : Q n i (0;!)! 0; A n i n i (;!)! i u.o.c., R n i (;!)!e u.o.c., E n j n j (;!)! j u.o.c. asn!1 for alli2N andj2J o : (B.14) Then,A 1 is a full set (i.e.,P (A 1 ) = 1) by (B.13) and Assumption 2.3.2. Let us fix an arbitrary!2A 1 and an arbitrary admissible policy =f n ;n2N + g. We omit and! from the notation up to Lemma B.2.1 below for notational convenience. Clearly, for alli2N andj2J ,f A n i n i ;n2N + g andf E n j n j ;n2N + g are relatively compact in the u.o.c. topology by (B.14). Moreover, i and j are Lipschitz continuous by Assumption 2.2.1; and for alli2N ,j2J , andt 1 ;t 2 2R + such thatt 2 t 1 , we have X i2N G n ij (t 2 ) G n ij (t 1 ) E n j n j (t 2 ) E n j n j (t 1 ); (B.15) D n ij (t 2 ) D n ij (t 1 ) G n ij (t 2 ) G n ij (t 1 ); (B.16) where (B.15) is by the fact that the system controller can offer a driver to a customer only at the customer arrival epochs, and (B.16) is by (2.4) and (B.1). Then by Lemma B.6.1 (cf. Section B.6),f G n ij ;n2 N + g andf D n ij ;n2N + g are relatively compact inD[0;T 1 ] endowed with the u.o.c. topology for alli2N and j2J such that all of their subsequential limits are Lipschitz and so absolutely continuous. Since G n ij and D n ij are nondecreasing and G n ij (0) = D n ij (0) = 0 for alli2N ,j2J ,n2N + , any fluid limits of these processes satisfy (B.3b) and (B.3c). Moreover, (B.3e) follows by (B.14) and (B.15). By Assumption 2.2.1, we define i := sup t2R + i (t)<1 and j := sup t2R + j (t)<1 for alli2N andj2J . Moreover, there exists a constant 2R + andn 0 2N + such that ifn n 0 , n i (t) for all t2R + andi2N by Assumptions 2.2.1 and 2.3.1. Let us fix an arbitraryi2N and consider the process R 0 n i (s) Q n i (s)ds. By (B.14), A n i n i (T 1 )! i (T 1 ) i T 1 <1 and Q n i (0)! 0 asn!1. Hence, there exists ann 1 2N + and a constantC <1 such that ifnn 1 , Q n i (0) + A n i n i (T 1 )C. Then Z t 2 0 n i (s) Q n i (s)ds Z t 1 0 n i (s) Q n i (s)ds = Z t 2 t 1 n i (s) Q n i (s)ds Z t 2 t 1 Q n i (s)ds (B.17a) Z t 2 t 1 Q n i (0) + A n i n i (s) ds Q n i (0) + A n i n i (T 1 ) (t 2 t 1 ) C(t 2 t 1 ) (B.17b) for alln n 0 _n 1 andt 1 ;t 2 2 [0;T 1 ] such thatt 2 t 1 , and the first inequality in (B.17b) is by (2.11). Then, R 0 n i (s) Q n i (s)ds;n2N + is relatively compact in the u.o.c. topology with Lipschitz continuous 167 subsequential limits by Lemma B.6.1 (cf. Section B.6). By the fact that R n i ! e u.o.c. (cf. (B.14)) and Lemma 11 of Ata and Kumar (2005), R n i R 0 n i (s) Q n i (s)ds ;n2N + is relatively compact in the u.o.c. topology and all of its limits are Lipschitz continuous. Hence,f Q n i ;n2N + g is also relatively compact in the u.o.c. topology with Lipschitz continuous and nonnegative limits by (2.11) and (B.14). Let us fix an arbitrary subsequencefn l ;l2N + g such that R 0 n i (s) Q n l i (s)ds;l2N + converges to a (continuous) limit under the uniform norm. Then, there exists a subsequence offn l ;l2N + g, denoted by fn k ;k2N + g, such that Q n k i Q i T 1 ! 0 ask!1 for alli2N , where Q i 2D[0;T 1 ] is nonnegative and continuous for alli2N . Then, sup t2[0;T 1 ] Z t 0 n k i (s) Q n k i (s)ds Z t 0 i (s) Q i (s)ds sup t2[0;T 1 ] Z t 0 n k i (s) Q n k i (s)ds Z t 0 i (s) Q n k i (s)ds + sup t2[0;T 1 ] Z t 0 i (s) Q n k i (s)ds Z t 0 i (s) Q i (s)ds T 1 Q n k i T 1 k n k i i k T 1 +T 1 Q n k i Q i T 1 ! 0; ask!1; by Assumption 2.3.1. Therefore, each convergent subsequence of R 0 n i (s) Q n i (s)ds;n2N + converges to R 0 i (s) Q i (s)ds u.o.c. where Q i is a fluid limit off Q n i ;n2N + g for alli2N . Then, by Lemma 11 of Ata and Kumar (2005), each convergent subsequence of R n i R 0 n i (s) Q n i (s)ds ;n2 N + converges to R 0 i (s) Q i (s)ds u.o.c. where Q i is a fluid limit off Q n i ;n2N + g for alli2N . This proves (B.3a). Next, (B.3d) follows by (B.3a) and the fact that Q i is nonnegative for alli2N . By Theorems 16.2 and 16.4 of Billingsley (1999) and the fact that T 1 2 R + is arbitrarily chosen, we can extend the results above to the processes inD 2NJ+3N+J , which proves Proposition B.1.1 except (B.3f). Lastly, in order to prove (B.3f), it is enough to prove the following result. From this point forward, we keep and! in the notation. Lemma B.2.1. For alli2N ,j2J ,m2N, and admissible policy =f n ;n2N + g, sup t2[m; m+1 ) D ;n ij (t) D ;n ij ( m ) F ij ( m ) G ;n ij (t) G ;n ij ( m ) a:s: ! 0; asn!1: (B.18) Proof: Let us fix arbitraryi2N ,j2J ,m2N, and an admissible policy =f n ;n2N + g. Then the converging number in (B.18) is equal to sup t2[m; m+1 ) 1 n E j n j (t) X k=E j n j (m)+1 I a k j (m)t ij (m) F ij ( m ) I n j (k) =i ; (B.19) 168 by (2.4), (B.1), and (2.10b). For notational completeness, for anyn;m2N and sequence of real numbers fx k ;k2Ng, ifn>m, then P m k=n x k := 0. Let Y k :=I a k j (m)t ij (m) F ij ( m ); Z n k :=I n j (k) =i ; X n k :=Y k Z n k : Then, (B.19) is equal to sup t2[m; m+1 ) 1 n E j n j (t) X k=E j n j (m)+1 X n k : (B.20) By the definition of the admissible policies (cf. Definition 2.2.1), n j (k) isF n j (k)-measurable for all k;n2N + , so doesZ n k . Moreover,a k j (m) is independent ofF n j (k) by construction (cf. (2.8)), so doesY k . Then,E h X n k jF n j (k) i = 0 for allk;n2N + becauseE [Y k ] = 0. Therefore, we expect to have a Martingale strong law of large numbers result for triangular arrays: 1 n E j n j (m)+n X k=E j n j (m)+1 X n k a:s: ! 0; asn!1: (B.21) We present the formal proof of (B.21) done by the technique introduced by de Jong (1996) in Section B.2.1. Let us first choose an arbitrary > 0 and then choose arbitrary 1 > 0 and 2 > 0 such that 1 ( 2 + j m+1 ) < . Let the set of !2 which satisfy (B.21) be denoted byA 2 . Then P(A 2 ) = 1. Let us choose an arbitrary!2A 1 \A 2 , whereA 1 is defined in (B.14). Then, there exists ann 2 ( 2 ;!)2N such thatj E n j n j ( m+1 ;!) j ( m+1 )j< 2 for allnn 2 ( 2 ;!) by (B.14). By the definition of j , (2.10a), (2.10b), and Assumption 2.2.1, E n j n j ( m+1 ;!)=n< 2 + j m+1 ; 8nn 2 ( 2 ;!): (B.22) By (B.21), there exists ann 1 ( 1 ;!)2N such that 1 n E j n j (m;!)+n X k=E j n j (m;!)+1 X n k (!) < 1 ; 8nn 1 ( 1 ;!): (B.23) 169 Let us fix an arbitraryt2 [ m ; m+1 ). On the one hand, ifE j n j (t;!)E j n j ( m ;!) n 1 ( 1 ;!), then 1 n E j n j (t;!) X k=E j n j (m;!)+1 X n k (!) < 1 E j n j (t;!)E j n j ( m ;!) n 1 E j n j ( m+1 ;!) n ; (B.24) by (B.23). On the other hand, ifE j n j (t;!)E j n j ( m ;!)<n 1 ( 1 ;!), then 1 n E j n j (t;!) X k=E j n j (m;!)+1 X n k (!) 1 n E j n j (t;!)E j n j ( m ;!) < n 1 ( 1 ;!) n : (B.25) Hence, for allt2 [ m ; m+1 ), ifnn 2 ( 2 ;!), 1 n E j n j (t;!) X k=E j n j (m;!)+1 X n k (!) < ( 1 ( 2 + j m+1 ))_ n 1 ( 1 ;!) n <_ n 1 ( 1 ;!) n ; (B.26) by (B.22), (B.24), (B.25), and the fact that 1 ( 2 + j m+1 )<. Let n 0 (;!) := maxfn 2 ( 2 ;!);n 1 ( 1 ;!)=g: (B.27) Then, for all> 0, there exists ann 0 (;!) such that ifnn 0 (;!), sup t2[m; m+1 ) 1 n E j n j (t;!) X k=E j n j (m;!)+1 X n k (!) <; (B.28) by (B.26) and (B.27). Since (B.28) is true for all!2A 1 \A 2 , the proof is complete. We complete the proof of Proposition B.1.1 by defining the full setA :=A 1 \A 2 . B.2.1 Proof of (B.21) Since the indices in the sum in (B.21) are random, we first prove thatY E j n j (m)+k ? Z n E j n j (m)+k and E[X n E j n j (m)+k ] = E[Y E j n j (m)+k ] = 0 for alln2 N + andk2 N + , so that we can use a Martingale strong law of large numbers result for triangular arrays in order to prove (B.21). 170 For notational convenience, we omit the superscript from the notation. Remember that we fix arbitrary i2N ,j2J ,m2N in Lemma B.2.1. Let ~ K n k := K n E j n j (m)+k for allK2fX;Zg andk2N + , and ~ Y k :=Y E j n j (m)+k for allk2N + . Then proving (B.21) is equivalent to proving 1 n n X k=1 ~ X n k a:s: ! 0; asn!1: (B.29) Let ~ F n j (k) :=F n j (E j n j ( m ) +k) for allk2 N + (cf. (2.8)) and ~ F n j (k) :=f;; g for allk2 ZnN + , where Z denotes the set of integers. Since n j (E j n j ( m ) +k) is a stopping time with respect to the filtrationF n for allk2N + and is increasing ink, ~ F n j (k) is well defined and ~ F n j (k);k2Z is a filtration. LetS n k :R + !R N be defined such that S n k (t;!) := A i n i (t^ n j (k;!);!);E j 0 n j 0(t^ n j (k;!);!);Q n i ((t^ n j (k;!));!); R i Z (t^ n j (k;!)) 0 n i (u)Q n i (u;!)du; ! ! ; D n ij 0((t^ n j (k;!));!); 8i2N;j 0 2J; a r^E j 0 n j 0 ( n j (k;!);!) j 0 (m;!); a r^(k1) j (m;!);8r;m2N ! ; (B.30) for allk;n2N + ,t2R + , and!2 . By (2.8), F n j (k) =fS n k (t);t2R + g: (B.31) Let (R N ) N :=R N R N :::. Then we have the following result. Lemma B.2.2. Fix arbitrary k;n2 N + and j2J . Let g be an arbitraryF n j (k)-measurable function. Then, there exist aB((R N ) N )-measurable functionf and a sequence of nonnegative real numbersft l ;l2Ng such thatg(!) =f(S n k (t l ;!);l2N) for all!2 . Proof: The proof follows by Exercise 1.5.6 of Stroock and Varadhan (2006) and the fact thatR,R N , and (R N ) N are all Polish spaces (cf. Corollary 3.39 of Aliprantis and Border (2006)). Lemma B.2.3. We have n j E j n j ( m ) +k 2 ~ F n j (k) anda E j n j (m)+k j (m)2 ~ F n j (k +l) for allj2J , k;l;n2N + , andm2N. 171 Proof: We havea E j n j (m)+k j (m)2 ~ F n j (k +l) by definition of ~ F n j (k +l) (cf. (2.8)). Let us fix arbitrary j2J ,k;n2N + , andm2N. For notational convenience, let ~ E :=E j n j ( m ). Then n j E j n j ( m ) +k = 1 X r=0 n j r +k I ~ E =r = 1 X r=0 f r S n r+k (t r l );l2N I ~ E =r (B.32) = 1 X r=0 f r S n ~ E+k (t r l );l2N I ~ E =r (B.33) where the second equality in (B.32) is by Lemma B.2.2. First, I ~ E = r 2 ~ F n j (k) for all r 2 N by definition of ~ F n j (k). Second, let y : ! (R N ) N be such that y(!) := S n ~ E(!)+k (t r l ;!);l2 N . Since B((R N ) N ) =B(R N ) B(R N ) ::: (cf. Lemma 1.2 of Kallenberg (1997)),B((R N ) N ) is generated by the sets of type B := Q l2N B l : B l 2B(R N ) (cf. Proposition 1.3 of Folland (1999)). Then, since ~ F n j (k) = n S n ~ E+k (t);t2R + o (cf. (B.31)), we have f!2 :y(!)2Bg = n !2 :S n ~ E(!)+k (t r l ;!)2B l ;l2N o = \ l2N n !2 :S n ~ E(!)+k (t r l ;!)2B l o 2 ~ F n j (k): Sincef r 2B((R N ) N ) by definition, the mapping in (B.33) is ~ F n j (k)-mesurable. Then, we have the following result. Lemma B.2.4. We havea E j n j (m)+k j (m)? ~ F n j (k) for allj2J ,k;n2N + , andm2N. Proof: For notational convenience, again let ~ E := E j n j ( m ). Let us fix arbitraryj2J ,k;n2 N + , m2N, andc2R and an arbitrary setB2 ~ F n j (k). Then P a ~ E+k j (m)<c; B = 1 X r=0 P a r+k j (m)<c; B; ~ E =r = 1 X r=0 E h I a r+k j (m)<c I(B)I ~ E =r i = 1 X r=0 E h I a r+k j (m)<c f S n ~ E+k (t l );l2N I ~ E =r i (B.34) = 1 X r=0 E h I a r+k j (m)<c f S n r+k (t l );l2N I ~ E =r i 172 = 1 X r=0 P a r+k j (m)<c E h f S n r+k (t l );l2N I ~ E =r i (B.35) =P a 1 j (m)<c 1 X r=0 E h f S n r+k (t l );l2N I ~ E =r i (B.36) =P a 1 j (m)<c 1 X r=0 P B; ~ E =r =P a 1 j (m)<c P (B) = 1 X r=0 P a 1 j (m)<c P ~ E =r P (B) = 1 X r=0 P a r+k j (m)<c P ~ E =r P (B) = 1 X r=0 P a r+k j (m)<c; ~ E =r P (B) = 1 X r=0 P a ~ E+k j (m)<c; ~ E =r P (B) (B.37) =P a ~ E+k j (m)<c P (B); (B.38) where (B.34) is by Lemma B.2.2. (B.35) is by (B.31) and the fact thata r+k j (m)?F n j (r +k) (cf. (2.8)), I ~ E = r 2F n j (r +k) for allr2 N, andf S n r+k (t l );l2N 2F n j (r +k) (see proof of Lemma B.2.3 for a similar argument explained in detail). (B.36) is by the i.i.d. property of the sequencefa r j (m);r2Ng, and the first equality in (B.37) is by the fact thata r+k j (m)?F n j (r +k) andI ~ E = r 2F n j (r +k) for allr2N. By definition of independence (cf. page 41 of Durrett (2010)), the equality in (B.38) proves the desired result. Remark B.2.1. If the filtration defined in (2.7) is generated by the sequence of processesf m ;m2 Ng defined in Remark 2.2.3, we can still prove Lemmas B.2.3 and B.2.4 by updating the definition in (B.30) such that the infinite dimensional processS n k includes the sequencef m ;m2Ng. Now we are ready to prove (B.21) by the technique introduced by de Jong (1996). For anyx2R, letbxc denote the greatest integer which is smaller than or equal tox. Then, 1 n n X k=1 ~ X n k = 1 n n X k=1 ~ X n k E h ~ X n k ~ F n j (k +bn 0:25 c 1) i + 1 n n X k=1 E h ~ X n k ~ F n j (kbn 0:25 c) i + 1 n n X k=1 E h ~ X n k ~ F n j (k +bn 0:25 c 1) i E h ~ X n k ~ F n j (kbn 0:25 c) i =:A n 1 +A n 2 +A n 3 : (B.39) We will consider each of A n 1 , A n 2 , and A n 3 separately. Let us choose an arbitrary > 0. First, let us consider 1 X n=1 P (jA n 1 j>) = 1 X n=1 P n X k=1 ~ X n k E h ~ X n k ~ F n j (k +bn 0:25 c 1) i >n ! : (B.40) 173 Note that ~ Z n k is ~ F n j (k)-measurable by Lemma B.2.3 and ~ Y k ? ~ F n j (k) by Lemma B.2.4. Moreover, E h ~ Y k i =E h I a E j n j (m)+k j (m)t ij (m) F ij ( m ) i =E " 1 X r=0 I a r+k j (m)t ij (m) I E j n j ( m ) =r # F ij ( m ) = 1 X r=0 E h I a r+k j (m)t ij (m) I E j n j ( m ) =r i F ij ( m ) = 1 X r=0 E h I a r+k j (m)t ij (m) i E I E j n j ( m ) =r F ij ( m ) (B.41) = 1 X r=0 F ij ( m )E I E j n j ( m ) =r F ij ( m ) (B.42) = F ij ( m ) 1 X r=0 E I E j n j ( m ) =r 1 ! = 0; (B.43) where (B.41) is by the fact thata r+k j (m)?F n j (r +k) (cf. (2.8)) andI E j n j ( m ) =r 2F n j (r +k) for allk2N + andr2N, and (B.42) is by definition of F ij ( m ) (cf. (2.2)). Hence, on the one hand, E h ~ X n k ~ F n j (k) i =E h ~ Y k ~ Z n k ~ F n j (k) i = ~ Z n k E h ~ Y k i = 0; for allk;n2N + ; (B.44a) E h ~ X n k ~ F n j (kl) i =E h E h ~ X n k ~ F n j (k) i ~ F n j (kl) i = 0; for allk;l;n2N + ; (B.44b) by (B.43). On the other hand, ~ X n k is ~ F n j (k +l)-measurable for allk2N + andl2N + by Lemma B.2.3. Hence, for allk;n2N + , E h ~ X n k ~ F n j (k +bn 0:25 c 1) i = 8 > < > : 0; ifn< 16, ~ X n k ; n 16: (B.45) Since ~ X n k 1 for allk2 N + andn2 N + , the sum in the the right-hand side in (B.40) is less than or equal to 15<1 by (B.45). Second, by (B.44), we have 1 X n=1 P (jA n 2 j>) = 1 X n=1 P n X k=1 E h ~ X n k ~ F n j (kbn 0:25 c) i >n ! = 0: (B.46) 174 Third, 1 X n=1 P (jA n 3 j>) = 1 X n=1 P n X k=1 E h ~ X n k ~ F n j (k +bn 0:25 c 1) i E h ~ X n k ~ F n j (kbn 0:25 c) i >n ! = 1 X n=1 P 0 @ n X k=1 bn 0:25 c1 X l=bn 0:25 c+1 E h ~ X n k ~ F n j (k +l) i E h ~ X n k ~ F n j (k +l 1) i >n 1 A 1 X n=1 P 0 @ bn 0:25 c1 X l=bn 0:25 c+1 n X k=1 E h ~ X n k ~ F n j (k +l) i E h ~ X n k ~ F n j (k +l 1) i >n 1 A 1 X n=1 bn 0:25 c1 X l=bn 0:25 c+1 P n X k=1 E h ~ X n k ~ F n j (k +l) i E h ~ X n k ~ F n j (k +l 1) i > n 2bn 0:25 c ! : (B.47) For allr2N + , let M n l (r) := r X k=1 E h ~ X n k ~ F n j (k +l) i E h ~ X n k ~ F n j (k +l 1) i : First,E M n l (r) 2r<1 andM n l (r)2 ~ F n j (l +r) (even whenl +r 0 in whichM n l (r) = 0 because ~ F n j (k) =f;; g fork2ZnN + ) for allr2N + . Second, E h M n l (r + 1) ~ F n j (l +r) i =E h E h ~ X n r+1 ~ F n j (l +r + 1) i E h ~ X n r+1 ~ F n j (l +r) i +M n l (r) ~ F n j (l +r) i = 0 +M n l (r): Therefore, for all fixedn2N + andl2Z,fM n l (r); ~ F n j (l +r);r2N + g is a martingale sequence such that E[M n l (r)] = 0. Moreover, M n l (r)M n l (r 1) 2 for allr2N + , i.e., the martingale differences are bounded by 2. Then, the sum in (B.47) is equal to 1 X n=1 bn 0:25 c1 X l=bn 0:25 c+1 P jM n l (n)j> n 2bn 0:25 c 1 X n=1 bn 0:25 c1 X l=bn 0:25 c+1 2 exp 0 @ 2 n 2bn 0:25 c 2 n X k=1 4 2 ! 1 1 A = 1 X n=1 2 2bn 0:25 c 1 exp n 2 32bnc 0:5 <1; (B.48) where the inequality is by Azuma’s inequality (cf. Theorem 6.3.3 of Ross (1996)). Finally, (B.29) follows by the fact that> 0 is arbitrary and P lim sup ( 1 n n X k=1 ~ X n k > )! =P (lim supfjA n 1 +A n 2 +A n 3 j>g) = 0; 175 where the second equality is by the fact that the sum in (B.40) is finite, (B.46), (B.48), and Borel-Cantelli Lemma (cf. Theorem 2.3.1 of Durrett (2010)). B.3 A Regulator Mapping Result We need regulator mapping results in order to prove Theorems 2.3.2, 2.4.2, and 2.4.3 in Section B.4. Since the generalized one-sided regulator mappings defined in the literature (cf. Reed and Ward (2004, 2008) and Ward and Kumar (2008)) are not applicable to our case, we introduce a new one-sided and nonlinear regulator mapping in this section. Let; :D!D be such that for allx2D andt2R + , (x)(t) := sup 0st (x(s)) + ; (x)(t) :=x(t) + (x)(t): (B.49) Then, is the conventional one-sided and one-dimensional regulator mapping (cf. Chapter 13.5 of Whitt (2002)). We define the following regulator mapping. Definition B.3.1. (A time-dependent, one-sided, and nonlinear regulator mapping): Letx;y2 D be such thatx(0) 0 and sup t2R + jy(t)j <1. The time-dependent, one-sided, and nonlinear regulator mapping M ; M :D 2 !D 2 is defined by M ; M (x;y) = (z;`) where CONDITION 1 (C1).z(t) =x(t) R t 0 y(s)z(s)ds +`(t) 0 for allt2R + . CONDITION 2 (C2).`(0) = 0,` is nondecreasing, and R 1 0 z(t)d`(t) = 0. If y = 0 in Definition B.3.1, then the time-dependent, one-sided, and nonlinear regulator mapping becomes the conventional one-sided and one-dimensional regulator mapping defined in (B.49) (cf. The- orem 6.1 of Chen and Yao (2001)); ify2D is a constant function, then it becomes the linearly generalized one-sided mapping described in Ward and Kumar (2008). In order to write the time-dependent, one-sided, and nonlinear regulator mapping in terms of the conven- tional one-sided and one-dimensional regulator mapping defined in (B.49), we make the following definition. Definition B.3.2. (Integral equation): Letx;y2D be such that sup t2R + jy(t)j<1. LetM :D 2 !D is a mapping such thatu :=M(x;y) solves the integral equation u(t) =x(t) Z t 0 y(s)(u)(s)ds; 8t2R + : (B.50) A mappingg : D! D is Lipschitz continuous with respect to the uniform norm, if for allT 1 2 R + , there exists a constant2R + which may depend onT 1 such thatkg(x)g(y)k T 1 kxyk T 1 for all x;y2D. Then, the first main result of this section is the following. 176 Proposition B.3.1. For any given x;y2 D such that x(0) 0 and sup t2R + jy(t)j <1, there exists a unique pair of functions M ; M (x;y) which satisfies conditions C1 and C2 in Definition B.3.1. More- over, 1. M (x;y) =(M(x;y)) and M (x;y) = (M(x;y)), 2. both M (;y) :D!D and M (;y) :D!D are Lipschitz continuous with respect to the uniform norm. In order to prove Proposition B.3.1, we need the following result. Lemma B.3.1. For any givenx;y2D such that sup t2R + jy(t)j <1, there exists a uniqueu2D which solves the integral equation (B.50). Moreover, the mappingM(;y) :D!D is Lipschitz continuous with respect to the uniform norm. Proof: Fix an arbitrary pairx;y2D such that y := sup t2R + jy(t)j<1. Let y :D!D be such that for allu2D andt2R + , y (u)(t) :=y(t)(u)(t). Then for allu 1 ;u 2 2D andt2R + , k y (u 1 ) y (u 2 )k t yk(u 1 )(u 2 )k t 2 yku 1 u 2 k t ; where the last inequality is by the fact that the mapping is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Thus, y is Lipschitz continuous with respect to the uniform norm. Since (B.50) is equivalent to u(t) =x(t) Z t 0 y (u)(s)ds; 8t2R + ; there exists a uniqueu2D which solves (B.50) and the mappingM(;y) :D!D is Lipschitz continuous with respect to the uniform norm by Lemma 1 of Reed and Ward (2004). Proof of Proposition B.3.1: Let us fix an arbitrary pairx;y2D such thatx(0) 0 and sup t2R + jy(t)j< 1. Letu :=M(x;y) (note that such au2 D uniquely exists by Lemma B.3.1). Moreover, let (z;`) := (; )(u). Then, by (B.49), (B.50), and the fact thatx(0) =u(0) 0, z(t) =u(t) +`(t) =x(t) Z t 0 y(s)(u)(s)ds +`(t) =x(t) Z t 0 y(s)z(s)ds +`(t) 0; 8t2R + ; 177 thus condition C1 in Definition B.3.1 is satisfied by (z;`). Sinceu(0) =x(0) 0, then`(0) = (u)(0) = 0 and`() is nondecreasing by the definition of the mapping (cf. (B.49)). Lastly, Z 1 0 z(t)d`(t) = Z 1 0 (u)(t)d (u)(t) = 0 by definition of the conventional one-sided, one-dimensional regulator mapping (cf. Theorem 6.1 of Chen and Yao (2001)). Therefore, the pair (z;`) = (; )(u) satisfies the conditions C1 and C2 in Definition B.3.1. Next, we will prove uniqueness. Let (z 1 ;` 1 ) be another pair which satisfies the conditions C1 and C2 and g2D be such that g(t) :=x(t) Z t 0 y(s)z 1 (s)ds; 8t2R + : Then,z 1 (t) =g(t) +` 1 (t) for allt2R + . By condition C2 and the uniqueness of the Skorokhod mapping (cf. Theorem 6.1 of Chen and Yao (2001)), (z 1 ;` 1 ) = (; )(g), so g(t) =x(t) Z t 0 y(s)(g)(s)ds; 8t2R + : (B.51) By Lemma B.3.1, there exists a unique solution of (B.51) which isM(x;y) = u = g. Therefore, (z 1 ;` 1 ) = (; )(g) = (; )(u) = (z;`), which proves uniqueness. Moreover, ( M ; M )(x;y) = (z;`) = (; )(u) = (; )(M(x;y)). Next, let us consider arbitraryx 1 ;x 2 2D such thatx 1 (0) 0 andx 2 (0) 0. Letu 1 :=M(x 1 ;y) and u 2 :=M(x 2 ;y). By Lemma B.3.1, the mappingM(;y) :D!D is Lipschitz continuous with respect to the uniform norm. Let y (t) be the corresponding Lipschitz constant fort2R + . Then, for allt2R + , M (x 1 ;y) M (x 2 ;y) t =k(u 1 )(u 2 )k t 2ku 1 u 2 k t 2 y (t)kx 1 x 2 k t ; where the first inequality is by the fact that the mapping is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Hence, M (;y) is Lipschitz continuous with respect to the uniform norm. Lastly, for allt2R + , M (x 1 ;y) M (x 2 ;y) t =k (u 1 ) (u 2 )k t ku 1 u 2 k t y (t)kx 1 x 2 k t ; 178 where the first inequality is by the fact that the mapping is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 1 (cf. Lemma 13.4.1 of Whitt (2002)). Hence, M (;y) is also Lipschitz continuous with respect to the uniform norm. Next, we present the following preliminary result. Lemma B.3.2. Let T 1 2 R + be an arbitrary constant, x;y2 D be such that y 0 and there exists a constantK2R + such thaty(t) K for allt2R + . Then, there exists a constantC = C(K;T 1 )2R + such thatkM(x;y)k T 1 Ckxk T 1 . Proof: Letu :=M(x;y) be the unique solution of the integral equation (B.50). Note that(u) 0 by (B.49). Then by (B.50) and the fact thaty0 and(u)0, u(t)kxk T 1 ; 8t2 [0;T 1 ]: (B.52) Letdae denote the smallest integer which is greater than or equal to a for all a 2 R. Let us parti- tion the interval [0;T 1 ] into subintervals with length 1=(4K) except the last interval which is [(d4KT 1 e 1)=(4K);T 1 ]. Then there ared4KT 1 e subintervals. Let, f n := sup ju(t)j :t2 n 1 4K ; n 4K ; 8n2f1;:::;d4KT 1 e 1g; (B.53) f d4KT 1 e := sup ju(t)j :t2 d4KT 1 e 1 4K ;T 1 : Then, kM(x;y)k T 1 =kuk T 1 = max n2f1;:::;d4KT 1 eg f n : By (B.50), u(t) =x(t) + Z t 0 y(s)(u)(s)ds; 8t2R + : (B.54) By (B.54) and the fact that the mapping is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)), u(t)kxk T 1 + 1 4K 2Kkuk 1=(4K) ; 8t2 0; 1 4K ) f 1 2kxk T 1 (B.55) where the second inequality above is by (B.52) and (B.53). By (B.54), u(t) =x(t) +x 1 4K u 1 4K + Z t 1=(4K) y(s)(u)(s)ds; 8t2 1 4K ; 1 2K 179 )u(t) 2kxk T 1 +f 1 + 1 4K 2Kkuk 1=(2K) ; 8t2 1 4K ; 1 2K )u(t) 2kxk T 1 +f 1 + 1 2 (f 1 +f 2 ); 8t2 1 4K ; 1 2K )f 2 4kxk T 1 + 3f 1 ; where the last inequality follows by (B.52) and (B.53). By induction, we can show that f n 4kxk T 1 + 2f n1 + n1 X k=1 f k ; 8n2f2;:::;d4KT 1 eg: (B.56) By (B.55) and (B.56), one can show that kuk T 1 = max n2f1;:::;d4KT 1 eg f n (d4KT 1 e + 4)!kxk T 1 : We complete the proof by definingC(K;T 1 ) := (d4KT 1 e + 4)!. The second main result of this section is the following. Lemma B.3.3. Letfx n ;n2Ng andfy n ;n2Ng be sequences inD such thatx n (0) 0 andy n 0 for alln2N, there exists a constantK2R + such thaty n (t)K for alln2N andt2R + , and there exist x2D andy2D such thatx n ! x u.o.c. andy n ! y u.o.c. asn!1. Then, M ; M (x n ;y n )! M ; M (x;y) u.o.c. asn!1. Proof: For eachn2N, letu n 2D be the unique solution of the equation u n (t) =x n (t) Z t 0 y n (s)(u n )(s)ds; 8t2R + ; (B.57) which exists by Lemma B.3.1. For eachn2N, let us define f n (t) := Z t 0 y n (s)(u n )(s)ds; 8t2R + : (B.58) Note thatf n is continuous for alln2N andu n =x n f n for alln2N by (B.57) and (B.58). LetT 1 be an arbitrary constant inR + . We will show that the sequenceff n ;n2Ng restricted to the compact domain [0;T 1 ] is relatively compact by the Arzel` a-Ascoli Theorem (cf. Theorem 7.2 of Billingsley (1999)). First, sup n2N jf n (0)j = 0<1. Second, lim !0 sup n2N sup jt 1 t 2 j jf n (t 2 )f n (t 1 )j = lim !0 sup n2N sup jt 1 t 2 j Z t 2 t 1 y n (s)(u n )(s)ds 180 lim !0 2 sup n2N ky n k T 1 sup n2N ku n k T 1 lim !0 2K sup n2N ku n k T 1 ; (B.59) lim !0 2KC(K;T 1 ) sup n2N kx n k T 1 = 0; (B.60) where the first inequality in (B.59) is by the fact that the mapping is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). The inequality in (B.60) is by Lemma B.3.2. Note that x n ! x u.o.c where x is a bounded function in [0;T 1 ], thus there exists a sufficiently large n 0 2 N such that sup nn 0 kx n k T 1 <1, and without loss of generality we assume sup n2N kx n k T 1 <1. Hence, we obtain the convergence result in (B.60). Therefore,ff n ;n2Ng restricted to the compact domain [0;T 1 ] is relatively compact. Since bothfx n ;n2 Ng andff n ;n2 Ng are relatively compact inD[0;T 1 ] endowed with the u.o.c. topology, so doesfu n ;n2Ng by (B.57) and (B.58). Let us consider an arbitrary subsequence offu n ;n2 Ng, denoted byfu n k ;k2Ng, such thatu n k !u u.o.c. ask!1 whereu2D[0;T 1 ]. Then, sup t2[0;T 1 ] u(t)x(t) + Z t 0 y(s)(u)(s)ds (B.61) = sup t2[0;T 1 ] u(t)u n k (t) +u n k (t)x(t) + Z t 0 y(s)(u)(s)ds kuu n k k T 1 +kx n k xk T 1 + sup t2[0;T 1 ] Z t 0 (y(s)(u)(s)y n k (s)(u n k )(s)) ds kuu n k k T 1 +kx n k xk T 1 + sup t2[0;T 1 ] Z t 0 (y(s)(u)(s)y n k (s)(u)(s)) ds + sup t2[0;T 1 ] Z t 0 (y n k (s)(u)(s)y n k (s)(u n k )(s)) ds kuu n k k T 1 +kx n k xk T 1 + 2T 1 kuk T 1 ky n k yk T 1 + 2T 1 Kku n k uk T 1 : (B.62) As k!1, all of the terms in (B.62) converges to 0, so (B.61) is equal to 0. Thus, u =M(x;y) and u is the unique solution of (B.50) by Lemma B.3.1. Therefore, each subsequence offu n ;n2 Ng has convergent subsequence which converges to the same limit, which implies thatu n ! u u.o.c. asn!1 where u =M(x;y). Note that the process M ; M (x n ;y n )2 D 2 is well defined for all n2 N by Proposition B.3.1. Since the mappings and are Lipschitz continuous with respect to the uniform norm, M ; M (x n ;y n ) = (; ) (M(x n ;y n )) = (; ) (u n ) ! (; ) (u) = (; ) (M(x;y)) = M ; M (x;y); u.o.c. asn!1; 181 which completes the proof. B.4 Proofs of Theorems 2.3.2, 2.4.2, and 2.4.3 We prove Theorems 2.3.2, 2.4.2, and 2.4.3 by representing the queue length process of each driver type with the time-dependent, one-sided, and nonlinear regulator mapping defined in Definition B.3.1. We first present some preliminary results that will be used in the proofs of all of the three theorems in Section B.4.1. Then, we prove Theorems 2.3.2, 2.4.2, and 2.4.3 in Sections B.4.2, B.4.3, and B.4.4, respectively. B.4.1 Preliminary Results In this section, we present some preliminary results that will be used in the proofs of Theorems 2.3.2, 2.4.2, and 2.4.3. We consider R (x) wherex =fx ij (t);i2N;j2J;t2 [0;T ]g is a feasible matching process for the CLP (2.12) such thatx ij is a Borel measurable simple function (cf. page 46 of Folland (1999)) for alli2N andj2J in this section. We start with the formal definition of the randomized policy given in Definition 2.2.1. Letf^ x ij (m);i2N;j2J;m2Ug be a sequence of real numbers inR + such thatU is a finite subset ofN. LetfB m ;m2Ug be a disjoint partition of the interval [0;T ] such thatB m is a Borel measurable set for allm2U. Then, x ij (t) := X m2U ^ x ij (m)I (t2B m ); 8t2 [0;T ];i2N;j2J: (B.63) F ij has finite range on the interval [0;T ] for alli2N andj2J by (2.2) and the fact that m !1 as m!1. Thus, without loss of generality, we choose the partitionfB m ;m2Ug such that for eachm2U, there existsl2 N such thatB m [ l ; l+1 ). By (2.12c), P i2N ^ x ij (m) 1 for allj2J andm2U. Let us define the sequence of independent random variablesfp k j (m);k2 N + ;j2J;m2Ug, which is independent of all other stochastic primitives andF(0)-measurable such that,P(p k j (m) =i) = ^ x ij (m) and P(p k j (m) = 0) = 1 P i2N ^ x ij (m) for alli2N ,j2J ,k2N + , andm2U. Note that the sequence fp k j (m);k2 N + ;j2J;m2Ug corresponds tof m ;m2Ng defined in Remark 2.2.3 associated with the randomized policy. With a slight abuse of notation, let R (x) = ( n 1 ; n 2 ;:::; n J ) in thenth system. Then, n j (k) = X i2N X m2U iI n j (k)2B m ; p k j (m) =i; Q R (x);n i ( n j (k))> 0 ; (B.64) for allj2J andk;n2N + . Then, clearly, n j (k) isF n j (k)-measurable for allj2J andk;n2N + , thus R (x) is admissible by Definition 2.2.1. 182 Let :N!N be a function such that(m) :=fl2N :B m [ l ; l+1 )g. Under R (x), for alli2N , j2J ,t2 [0;T ], andn2N + , we have D R (x);n ij (t) = E j n j (t) X k=1 X m2U I n j (k)2B m ; a k j ((m))t ij ((m)); p k j (m) =i; Q R (x);n i ( n j (k))> 0 : Let us define the following stochastic process such that for alli2N ,j2J ,n2N + , andt2 [0;T ], H R (x);n ij (t) := E j n j (t) X k=1 X m2U F ij ( (m) )^ x ij (m)I n j (k)2B m ; Q R (x);n i ( n j (k))> 0 ; = Z t 0 X m2U F ij ( (m) )^ x ij (m)I s2B m ; Q R (x);n i (s)> 0 dE j n j (s); = Z t 0 F ij (s)x ij (s)I Q R (x);n i (s)> 0 dE j n j (s); where the second equality is the Lebesgue-Stieltjes integral (cf. page 107 in Folland (1999)), and the third equality is by (2.2) and (B.63). Let H R (x);n ij := (1=n)H R (x);n ij . Then we have the following result whose proof is presented in Section B.4.1. Lemma B.4.1. For alli2N andj2J , we have D R (x);n ij H R (x);n ij T a:s: ! 0; asn!1: We omit the superscript R (x) from the notation for convenience in presentation in the rest of Section B.4.1. Let us define the following stochastic process such that for alli2N ,j2J ,n2N + , andt2 [0;T ], I n ij (t) := E j n j (t) X k=1 X m2U F ij ( (m) )^ x ij (m)I n j (k)2B m ; Q n i ( n j (k)) = 0 ; (B.65a) = Z t 0 F ij (s)x ij (s)I (Q n i (s) = 0) dE j n j (s): (B.65b) Let I n ij := (1=n)I n ij for alli2N ,j2J , andn2N + . Then, by (2.11) and some algebra, Q n i (t) = X n i (t) Z t 0 n i (s) Q n i (s)ds + X j2J I n ij (t); (B.66) 183 for alli2N ,n2N + , andt2 [0;T ], where X n i (t) := Q n i (0) + A n i n i (t) R n i Z t 0 n i (s) Q n i (s)ds Z t 0 n i (s) Q n i (s)ds X j2J D n ij (t) H n ij (t) 1 n X j2J Z t 0 F ij (s)x ij (s)dE j n j (s) for alli2N ,n2N + , andt2 [0;T ]. Without loss of generality, we define Q n i (t) := Q n i (T ); X n i (t) := X n i (T ); X j2J I n ij (t) := X j2J I n ij (T ); n i (t) := 0 (B.67) for alli2N ,n2N + , andtT for mathematical completeness, so (B.66) is well defined for allt2R + . The following result allows us to approximate X n i by a deterministic process. Lemma B.4.2. For alli2N andj2J , we have sup 0tT 1 n Z t 0 F ij (s)x ij (s)dE j n j (s) Z t 0 F ij (s)x ij (s) j (s)ds a:s: ! 0; asn!1; sup 0t<1 X n i (t) X i (t) a:s: ! 0; asn!1; where X i (t) := i (t) X j2J Z t 0 F ij (s)x ij (s) j (s)ds; for allt2 [0;T ] and X i (t) := X i (T ) for alltT . Moreover, X i is nonnegative for alli2N . The proof of Lemma B.4.2 is presented in Section B.4.1. Note that, for anyt2 [0;T ],I n ij (t) can increase only if there is a typej customer arrival at timet andQ n i (t) = 0 by (B.65). Since probability that there is also a typei driver arrival at timet is 0, thenQ n i (t) = 0 with probability 1. Hence, for alli2N and n2N + , X j2J I n ij (0) = 0; X j2J I n ij () is nondecreasing, Z 1 0 Q n i (t)d X j2J I n ij (t) = 0 a.s.; (B.68) by (B.65) and (B.67). By Assumption 2.2.1, the fact that X n i (0) 0, (B.66), (B.67), (B.68), Definition B.3.1, and Proposition B.3.1, we have Q n i ; X j2J I n ij = M ; M X n i ; n i a.s. 184 for alli2N andn2N + . There exists a constant 2R + andn 0 2N + such that n i (t) for allt2R + ,i2N , andnn 0 by Assumptions 2.2.1 and 2.3.1 and (B.67). Then, by Assumption 2.3.1, Lemma B.3.3, and Lemma B.4.2, Q n i M X i ; i T a:s: ! 0; X j2J I n ij M X i ; i T a:s: ! 0; (B.69) asn!1 for alli2N . Lastly, we have the following result. Lemma B.4.3. Fix an arbitraryi2N . If M X i ; i =0, then sup 0tT X j2J D n ij (t) Z t 0 j (s) F ij (s)x ij (s)ds a:s: ! 0; asn!1: Proof: By (B.69) and the fact that M X i ; i =0, X j2J I n ij T a:s: ! 0; asn!1. (B.70) Then, sup 0tT X j2J D n ij (t) Z t 0 j (s) F ij (s)x ij (s)ds = sup 0tT X j2J D n ij (t) H n ij (t) I n ij (t) + 1 n Z t 0 F ij (s)x ij (s)dE j n j (s) Z t 0 j (s) F ij (s)x ij (s)ds X j2J D n ij H n ij T + X j2J I n ij T + X j2J sup 0tT 1 n Z t 0 F ij (s)x ij (s)dE j n j (s) Z t 0 F ij (s)x ij (s) j (s)ds a:s: ! 0; asn!1; (B.71) where the inequality is by triangular inequality and the convergence result is by Lemma B.4.1, (B.70), and Lemma B.4.2. 185 Proof of Lemma B.4.1 The proof of Lemma B.4.1 is very similar to the one of Lemma B.2.1. For notational convenience, we omit the superscript R (x) from the notation in this section. Let us fix arbitraryi2N andj2J . Then, by definition, we have D n ij H n ij T = sup 0tT 1 n E j n j (t) X k=1 X m2U I a k j ((m))t ij ((m)); p k j (m) =i F ij ( (m) )^ x ij (m) I n j (k)2B m ; Q n i ( n j (k))> 0 : (B.72) For alln2N + andm2U, let Y k (m) :=I a k j ((m))t ij ((m)); p k j (m) =i F ij ( (m) )^ x ij (m); Z n k (m) :=I n j (k)2B m ; Q n i ( n j (k))> 0 ; W n k (m) :=Y k (m)Z n k (m); ~ W n k := X m2U W n k (m) Then, the right-hand side of (B.72) is equal to sup 0tT 1 n E j n j (t) X k=1 ~ W n k : (B.73) Note that, if we can prove that 1 n n X k=1 ~ W n k a:s: ! 0; asn!1; (B.74) then we can prove that the term in (B.73) converges to 0 as n!1 by the same way we prove Lemma B.2.1. Hence, in the remaining part of this section, we will prove (B.74), whose proof is very similar to the one of (B.21). For alln2N + , ifk2ZnN + , we letG n k :=f;; g; and ifk2N + , we let G n k := ( A i n i (s^ n j (k));E j 0 n j 0(s^ n j (k));R i Z (s^ n j (k)) 0 n i (u)Q n i (u)du ! ; D n ij 0((s^ n j (k)));Q n i ((s^ n j (k))); 8i2N;j 0 2J;s2R + ; a r j 0((m));p r j 0(m); r2f1;:::;E j 0 n j 0( n j (k))g; 8j 0 2Jnfjg;m2U 186 a r j ((m));p r j (m); r2f1;:::;k 1g; 8m2U ) : (B.75) Then,fG n k ;k2Zg is a filtration for alln2N + . Moreover,Y k (m)?G n k andZ n k (m)2G n k for allk;n2N + andm2U; andW n k (m)2G n k+l for allk;n;l2N + by construction. First, for allk;n2N + , E h ~ W n k G n k i =E " X m2U W n k (m) G n k # = X m2U E W n k (m) G n k = X m2U E Y k (m)Z n k (m) G n k ; (B.76a) = X m2U Z n k (m)E Y k (m) G n k = X m2U (Z n k (m)E [Y k (m)]) = 0: (B.76b) Second, for allk;n;l2N + , E h ~ W n k G n kl i =E h E h ~ W n k G n k i G n kl i = 0; (B.77) by (B.76b). Third, for allk;n2N + , E h ~ W n k G n k+bn 0:25 c1 i = 8 > < > : 0; ifn< 16, ~ W n k ; n 16: (B.78) Therefore, by (B.76b), (B.77), and (B.78), we can prove (B.74) by the same method that we use in Section B.2.1 (starting from (B.39)) in order to prove (B.21). Proof of Lemma B.4.2 By triangular inequality, for alli2N , X n i X i T Q n i (0) + A n i n i i T + X j2J D n ij H n ij T (B.79a) + sup 0tT R n i Z t 0 n i (s) Q n i (s)ds Z t 0 n i (s) Q n i (s)ds (B.79b) + X j2J sup 0tT Z t 0 F ij (s)x ij (s) d E n j n j (s) d j (s) : (B.79c) We will consider each term in the right-hand side (RHS) of (B.79) separately. The three terms in the RHS of (B.79a) converge to 0 a.s. by Assumption 2.3.2, (B.13), and Lemma B.4.1, respectively. Next, let us consider the term in (B.79b). Let us fix an arbitrary!2A 1 (cf. (B.14)). From the proof of Proposition B.1.1, we know that each subsequence of the term in (B.79b) has a subsequence which converges 187 to 0 on the sample path!, which implies that the term in (B.79b) itself converges to 0 asn!1 on the sample path!. SinceP(A 1 ) = 1, the term in (B.79b) converges to 0 a.s asn!1. Next let us consider the term in (B.79c) and fix an arbitrary i2N and j 2J . By (B.63), and the construction of the sequence of Borel measurable setsfB m ;m2Ug, F ij (t)x ij (t) = X m2U F ij ( (m) )^ x ij (m)I(t2B m ); 8t2 [0;T ]: Let` denote the Lebesgue measure onR. Then, for all> 0, there exists a sequence of setsfB (1) m ;m2Ug such that B (1) m is finite union of open intervals and `(B m nB (1) m ) +`(B (1) m nB m ) < for all m2U by Proposition 1.20 of Folland (1999). Hence, for all> 0, there exists a simple function such that(t) := P m2U (2)z m I(t2 B (2) m ) for allt2 [0;T ],U (2) is a finite subset ofN,fz m ;m2U (2) g is a nonnegative sequence of real numbers bounded above by 1, B (2) m is finite union of open intervals for all m2U (2) , and`(t2 [0;T ] : (t)6= F ij (t)x ij (t)) < . Moreover, for all > 0, there exists a continuous function g : R! [0; 1] such thatg(t) = 0 for allt = 2 [0;T ], g has bounded variation, and`(t2 [0;T ] : g(t)6= (t))< (see the proof of Theorem 2.26 of Folland (1999) for a specific construction and the reason whyg has bounded variation is that has finite range andB (2) m is finite union of open intervals for allm2U (2) ). LetU (3) :=ft2 [0;T ] :g(t)6= F ij (t)x ij (t)g. Then,`(U (3) )< 2. Let L n j := E n j n j j . Then, sup 0tT Z t 0 F ij (s)x ij (s) d E n j n j (s) d j (s) sup 0tT Z t 0 ( F ij (s)x ij (s)g(s))d L n j (s) + sup 0tT Z t 0 g(s)d L n j (s) : (B.80) First, let us consider the first term in (B.80), which is less than or equal to sup 0tT Z t 0 2I s2U (3) d E n j n j (s) + sup 0tT Z t 0 2I s2U (3) d j (s) = 2 Z T 0 I s2U (3) d E n j n j (s) + 2 Z T 0 I s2U (3) j (s)ds 2 n Z T 0 I s2U (3) dE n j n j (s) + 4 j ; (B.81) where (B.81) is by (2.10) and the fact that`(U (3) ) < 2 and j = sup t2R + j (t). Let us consider the first term in (B.81). By Assumption 2.3.1, there exists a sufficiently largen 0 2 N + such that ifn n 0 , then n j := sup t2[0;T ] n j (t)<1, and sup nn 0 ( n j =n)<C, whereC2R + is a constant. Let us fix an arbitrary n n 0 and consider a time homogeneous Poisson process, denoted byM n j , with rate n j and associated 188 arrival times denoted by the sequencef n j (k);k2 N + g. Since we can think that the non-homogeneous Poisson processE n j n j as being a random sample from the homogeneous Poisson processM n j (cf. page 80 of Ross (1996)), the first term in (B.81) is less than or equal to 2 n Z T 0 I s2U (3) dM n j (s) = 2 n M n j (T ) X k=1 I n j (k)2U (3) : (B.82) Note that P 0 @ 2 n M n j (T ) X k=1 I n j (k)2U (3) > p 1 A 2 n p E 2 4 M n j (T ) X k=1 I n j (k)2U (3) 3 5 = 2 n p X K2N E " K X k=1 I n j (k)2U (3) M n j (T ) =K # P M n j (T ) =K = 2 n p X K2N E " Binomial K; `(U (3) ) T !# P M n j (T ) =K = 2 n p X K2N K `(U (3) ) T P M n j (T ) =K 4 p n j T nT 4C p ; (B.83) where the first inequality is by Markov’s inequality, and the second equality is by the fact that given that M n j (T ) = K, the K arrival times n j (1);:::; n j (K) have the same distribution as order statistics corre- sponding toK independent random variables uniformly distributed on the interval [0;T ] (cf. Theorem 2.3.1 of Ross (1996)) and probability that a random variable which is uniformly distributed on [0;T ] is in the Borel measurable setU (3) is`(U (3) )=T . Therefore, (B.82) and (B.83) imply that (B.81) is bounded above by p +4 j with probability 14C p . Since> 0 is arbitrary, the first term in (B.80) is equal to 0 with probability 1 for allnn 0 . Next, let us consider the second term in (B.80) and and fix an arbitrary ! 2A 1 (cf. (B.14)). Then, L n j (;!)!0 u.o.c. asn!1. By Theorem 3.36 of Folland (1999) (integration by parts), the second term in (B.80) is equal to sup 0tT L n j (t;!)g(t) Z t 0 L n j (s;!)dg(s) sup t2[0;T ] L n j (t;!) + sup 0tT Z t 0 L n j (s;!)dg(s) : (B.84) 189 Note that the Lebesgue-Stieltjes measure induced byg can be a signed measure. By Theorem 3.3 of Fol- land (1999) (the Hahn Decomposition Theorem), there exists a positive set P and negative set N for the Lebesgue-Stieltjes measure induced byg such thatP[N = [0;T ] andP\N =;. LetV(g)<1 denote the total variation ofg. Then, the term in the RHS of (B.84) is less than or equal to sup t2[0;T ] L n j (t;!) + sup 0tT Z P\[0;t] L n j (s;!)dg(s) + sup 0tT Z N\[0;t] L n j (s;!)dg(s) sup t2[0;T ] L n j (t;!) + 2V(g) sup t2[0;T ] L n j (t;!) = sup t2[0;T ] L n j (t;!) (1 + 2V(g))! 0 asn!1: Therefore, the sum in (B.80) converges to 0 as n ! 1 a.s. so do the the term in (B.79c) and X n i X i T . Since X n i (t) = X n i (T ) for alltT (cf. (B.67)), sup 0t<1 X n i (t) X i (t) a:s: ! 0; asn!1 for alli2N: X i is nonnegative by (2.12b) and the fact thatx is a feasible matching process for the CLP (2.12). B.4.2 Proof of Theorem 2.3.2 Since i = 0 by assumption, M X i ; i = X i by Definition B.3.2. By (B.49) and the fact that M X i ; i = M X i ; i (cf. Proposition B.3.1) and X i is nonnegative (cf. Lemma B.4.2), we have M X i ; i = X i =0 for alli2N . Then, the proof follows by Lemma B.4.3. B.4.3 Proof of Theorem 2.4.2 The processx is a feasible matching process for the CLP (2.12) by Lemma 2.4.1 and it is a Borel measur- able simple function for alli2N andj2J by assumption. Then, the results of Section B.4.1 apply to this process. Moreover, we can extend the results of Lemma B.4.2 such that X i is also nondecreasing for all i2N by (2.13b), (2.12b), and the fact thatx is a feasible matching process for both the CLP (2.12) and the LP (2.13). Then, we have the following result. Lemma B.4.4.M X i ; i 0 for alli2N . Proof: Recall that X i is a nonnegative and nondecreasing function for alli2N by Lemma B.4.2 and the argument mentioned above. Let X 0 i denote the nonnegative derivative of X i for alli2N . Let us fix an arbitraryi2N . Consider the following equation: u i (t) = X i (t) Z t 0 i (s)u i (s)ds; 8t2R + : (B.85) 190 By algebra, one can see that u i (t) := exp Z t 0 i (r)dr Z t 0 X 0 i (s) exp Z s 0 i (r)dr ds; 8t2R + is a solution of the equality (B.85) such that u i 0. Then (u i ) = u i by (B.49) and u i is the unique solution of the integral equation (cf. Definition B.3.2) u i (t) = X i (t) Z t 0 i (s)(u i )(s)ds; 8t2R + ; by Lemma B.3.1. Hence,M( X i ; i ) =u i 0, which completes the proof. By (B.49), the fact that M X i ; i = M X i ; i (cf. Proposition B.3.1), and Lemma B.4.4, we have M X i ; i =0 for alli2N . Then, by Lemma B.4.3, for alli2N , sup 0tT X j2J D R (x );n ij (t) Z t 0 j (s) F ij (s)x ij (s)ds a:s: ! 0; asn!1; which gives us the desired result. B.4.4 Proof of Theorem 2.4.3 Since the parameters of LP (2.13) are time homogeneous,x can be chosen as a constant function of time and so we let x ij (t) = x ij for all i2N and j 2J . It is easy to see that the processx is a feasible matching process for the CLP (2.12). Thus, X i2N;j2J w ij j F ij Z T 0 x ij (s)ds = X i2N;j2J w ij j F ij x ij T X i2N;j2J w ij j F ij Z T 0 ~ x ij (s)ds: (B.86) Let x ij := (1=T ) R T 0 ~ x ij (s)ds for alli2N andj2J . Then it is easy to see thatf x ij ;i2N;j2Jg is feasible for the LP (2.13) when the parameters are time homogeneous, thus X i2N;j2J w ij j F ij x ij = X i2N;j2J w ij j F ij 1 T Z T 0 ~ x ij (s)ds X i2N;j2J w ij j F ij x ij : (B.87) By (B.86) and (B.87), we see thatx is an optimal solution of the CLP (2.12). Sincex is chosen as a constant function of time, R (x ) is admissible by Lemma 2.3.2. The rest of the proof of Theorem 2.4.3 is very similar to the one of Theorem 2.4.2. Since the processx is an optimal solution of the CLP (2.12) andx ij is a constant function of time and so Borel measurable simple 191 function for all i2N and j2J , the results of Section B.4.1 apply to this process. Moreover, we can extend the results of Lemma B.4.2 such that X i is also nondecreasing for alli2N by (2.13b). Therefore, Lemma B.4.4 holds and M X i ; i =0 for alli2N . Then, by Lemma B.4.3, for alli2N , sup 0tT X j2J D R (x );n ij (t) Z t 0 j (s) F ij (s)x ij (s)ds a:s: ! 0; asn!1; which gives us the desired result becausex is an optimal solution of the CLP (2.12). B.5 Lemma Proofs We first prove Lemma 2.2.1, second prove Lemmas 2.3.2 and 2.5.1 (because they are related to Lemma 2.2.1), third prove Lemma 2.3.1, then prove Lemma 2.4.1, and lastly prove Lemma 2.4.2. B.5.1 Proof of Lemma 2.2.1 Let CD = ( 1 ; 2 ;:::; J ). For notational convenience, we omit the superscript CD from the notation in this proof. Let us define the sequence of independent random variablesfq k j (l);k2 N + ;j2J;l2 N N g, which is independent of all other stochastic primitives andF(0)-measurable, such thatq k j (0; 0;:::; 0) = 0, and if P N i=1 l i > 0, thenP(q k j (l 1 ;l 2 ;:::;l N ) =i) =l i = P N i=1 l i for allk2N + ,j2J , andl2N N such thatl = (l 1 ;l 2 ;:::;l N ). For notational convenience, let us define the set S j (k) := argmin fi2N :Q i ( j (k))>0g X m2N t ij (m)I( j (k)2 [ m ; m+1 )) (B.88) for allj2J andk2N + . LetjS j (k)j denote the cardinality of the setS j (k). Then, under the CD policy, for allj2J andk2N + , let j (k) :=I (jS j (k)j = 1) X i2N iI (i2S j (k)) +I (jS j (k)j> 1)q k j Q 1 ( j (k))I(12S j (k));:::;Q N ( j (k))I(N2S j (k)) : Note that Q i ( j (k)) = lim t!1 Q i ((t^ j (k)))2F j (k) by Proposition 2.7 of Folland (1999) and definition ofF j (k) (cf. (2.8)) for all i2N , j 2J , and k2 N + . By (2.8), (B.88), and the fact that j (k)2F j (k), the random variablesI (jS j (k)j = 1), I (jS j (k)j> 1), I (i2S j (k)), andI (S j (k) =N 1 ) areF j (k)-measurable for alli2N ,j2J ,k2N + , andN 1 N . Lastly, q k j Q 1 ( j (k))I(12S j (k));:::;Q N ( j (k))I(N2S j (k)) 192 = X N 1 N I (S j (k) =N 1 ) X fl2N N :l i =0;8i= 2N 1 g q k j (l 1 ;l 2 ;:::;l N )I Q i ( j (k)) =l i ;8i2N 1 ! 2F j (k): Therefore, j 2F j for allj2J under the CD policy, so it is an admissible policy. Note that, the sequence fq k j (l);k2N + ;j2J;l2N N g corresponds tof m ;m2Ng defined in Remark 2.2.3 associated with the CD policy. B.5.2 Proofs of Lemmas 2.3.2 and 2.5.1 First, the randomized policy given in Definition 2.3.1 is admissible by the fact that (B.64) isF n j (k)- measurable for all j 2J and k;n2 N + . Second, the admissibility proof of the hybrid policy follows by the fact that it is a hybrid of the randomized policy given in Definition 2.3.1 and the CD policy defined in (2.3), and both of the latter two policies are admissible. Third, the deterministic policy is admissible because bothQ n i ( n j (k)) andD n ij ( n j (k)) areF n j (k)-measurable for alli2N ,j2J , andk;n2N + (cf. (2.8)) and the tie breaking rule does not use any future information. B.5.3 Proof of Lemma 2.3.1 We use the proof technique introduced by Levinson (1966). We first present a preliminary result from Levinson (1966), then prove Lemma 2.3.1. LetL 2 ([0;T ]) denote the space of Lebesgue measurable func- tions with domain [0;T ] and rangeR such that iff2L 2 ([0;T ]), then R T 0 f(t) 2 dt<1. We let w ! denote weak convergence inL 2 ([0;T ]) as defined in Proposition 6 in Section 8.2 of Royden and Fitzpatrick (2010). Lemma B.5.1. (Lemma 2.1 of Levinson (1966)) Letff r ;r 2 Ng be a uniformly bounded sequence of functions in L 2 ([0;T ]) such that f r w ! f for some f 2 L 2 ([0;T ]). Let f u ;f l : [0;T ]! R be defined asf u (t) := lim sup r!1 f r (t) andf l (t) := lim inf r!1 f r (t) for allt2 [0;T ]. Then,f(t) f u (t) and f(t)f l (t) for allt2 [0;T ] except on a set of zero measure. First, note that x ij = 0 for all i2N and j 2J is a feasible matching process for the CLP (2.12). Second,x ij is nonnegative and bounded for alli2N andj2J by (2.12c) and (2.12d). Third, both j and i are bounded processes on [0;T ] by Assumption 2.2.1, andw ij and F ij are bounded by definition for all i2N andj2J . Moreover, j , i , and F ij are all Borel measurable (so does the process j () F ij ()) by construction for alli2N andj2J . Letx :=fx ij (t);i2N;j2J;t2 [0;T ]g denote ta feasible matching process for the CLP (2.12),H denote the feasible region of the CLP (2.12), i.e.,H =fx :x satisfies (2.12b)-(2.12e)g, and M := sup x2H X i2N;j2J w ij Z T 0 j (s) F ij (s)x ij (s)ds<1: 193 Then, there exists a sequence of feasible matching processes for the CLP (2.12), denoted byfx r ;r2Ng, such thatx r =fx r ij (t);i2N;j2J;t2 [0;T ]g for allr2N and lim r!1 X i2N;j2J w ij Z T 0 j (s) F ij (s)x r ij (s)ds =M: By Theorem 14 in Section 8.3 of Royden and Fitzpatrick (2010), there exists a subsequence offx r ;r2Ng which is again denoted byfx r ;r2Ng for notational convenience such thatx r ij weakly converges to some x ij 2L 2 ([0;T ]) for alli2N andj2J . Let x :=f x ij (t);i2N;j2J;t2 [0;T ]g. Since X i2N;j2J w ij Z T 0 j (s) F ij (s) x ij (s)ds = X i2N;j2J w ij lim r!1 Z T 0 j (s) F ij (s)x r ij (s)ds =M; (B.89) it is enough to prove that a modification of x is inH in order to prove Lemma 2.3.1. First, let us consider (2.12b). Let us fix an arbitraryi2N andt2 [0;T ]. Then, X j2J Z t 0 j (s) F ij (s) x ij (s)ds = X j2J Z T 0 j (s) F ij (s)I(s2 [0;t]) x ij (s)ds = X j2J lim r!1 Z T 0 j (s) F ij (s)I(s2 [0;t])x r ij (s)ds i (t); (B.90) where the equality in (B.90) is by the definition of weak convergence and the inequality in (B.90) is by (2.12b). Hence, x satisfies (2.12b). Next, let us consider (2.12c) and (2.12d). Note that, sincex r ij w ! x ij for alli2N andj2J , it is easy to see that P i2N x r ij w ! P i2N x ij for allj2J . Then, by Lemma B.5.1, for allt2 [0;T ] except on a set of zero measure, X i2N x ij (t) lim sup r!1 X i2N x r ij (t) 1; x ij (t) lim inf r!1 x r ij (t) 0; 8i2N;j2J: (B.91) Hence, x satisfies (2.12c) and (2.12d) for allt2 [0;T ] except on a set of zero measure, and we denote this set of zero measure byH and its complement byH c , i.e.,H c := [0;T ]nH. Let ^ x =f^ x ij (t);i2N;j2 J;t2 [0;T ]g be such that ^ x ij (t) := x ij (t) for allt2 H c , i2N , andj2J , and ^ x ij (t) := 0 for all 194 t2 H,i2N , andj2J . Then, ^ x satisfies (2.12c) and (2.12d) by construction, and satisfies (2.12e) by Proposition 2.11.a of Folland (1999). Moreover, for alli2N andt2 [0;T ], X j2J Z t 0 j (s) F ij (s)^ x ij (s)ds = X j2J Z T 0 j (s) F ij (s)I(s2 [0;t]\H)^ x ij (s)ds + X j2J Z T 0 j (s) F ij (s)I(s2 [0;t]\H c )^ x ij (s)ds = X j2J Z T 0 j (s) F ij (s)I(s2 [0;t]\H c ) x ij (s)ds (B.92) i (t) X j2J Z T 0 j (s) F ij (s)I(s2 [0;t]\H) x ij (s)ds = i (t); (B.93) where (B.92) is by the fact thatH is a set of zero measure and ^ x ij (t) = x ij (t) for allt2H c , the inequality in (B.93) is by (B.90), and the equality in (B.93) is by the fact thatH is a set of zero measure. Therefore, ^ x satisfies (2.12b), so it is a feasible matching process for the CLP (2.12), and it is an optimal solution by (B.89) and the fact that it is equal to x almost everywhere. B.5.4 Proof of Lemma 2.4.1 Suppose thatw ij = 1 for alli2N andj2J . Then, by summing the constraint (2.12b) ini2N , we can see that an upper bound on the objective function value of the CLP (2.12) is P i2N i (T ). Any feasible matching process that satisfies condition (2.14) has the objective function value P i2N i (T ), thus it is an optimal CLP solution. Second, under any feasible matching process that satisfy condition (2.14), if any one areai is such that the constraint (2.12b) has slack, then there must be another areak such that the number of matched customers exceeds the cumulative driver arrivals (i.e., the left-hand side of the constraint strictly exceeds the right-hand side) which produces an infeasible matching process. Therefore, if w ij = 1 for alli2N andj2J , then condition (2.14) implies the CLP constraint (2.12b) binds for alli2N and t2 [0;T ] under at least one optimal CLP (2.12) solution. Suppose that ~ x is an optimal solution of the CLP (2.12) under which constraint (2.12b) is binding for all driver types at all times. Notice that ~ x satisfies (2.13c) and (2.13d) because these constraints are exactly the same of (2.12c) and (2.12d), respectively. Since constraint (2.12b) is binding for all driver types at all times under ~ x, then by taking the derivative of both right- and left-hand sides of (2.12b) with respect tot, we can 195 see thatf~ x ij (t);i2N;j2Jg (or a modification of it) satisfies (2.13b) for allt2 [0;T ], so it is feasible for the LP (2.13) for allt2 [0;T ]. Then, X i2N;j2J j (t) F ij (t)~ x ij (t) X i2N;j2J j (t) F ij (t)x ij (t); 8t2 [0;T ]; which implies X i2N;j2J Z T 0 j (s) F ij (s)~ x ij (s)ds X i2N;j2J Z T 0 j (s) F ij (s)x ij (s)ds: (B.94) Second,x satisfies (2.12e) by Assumption 2.4.1 and satisfies (2.12c) and (2.12d) by satisfying (2.13c) and (2.13d), respectively. By taking the integral of both the right- and left-hand sides in (2.13b) with respect tot, we can see thatx satisfies (2.12b) for allt2 [0;T ], so it is a feasible matching process for the CLP (2.12). Then, X i2N;j2J Z T 0 j (s) F ij (s)x ij (s)ds X i2N;j2J Z T 0 j (s) F ij (s)~ x ij (s)ds: (B.95) Therefore,x is an optimal solution of the CLP (2.12) by (B.94) and (B.95). B.5.5 Proof of Lemma 2.4.2 Since i and j are continuous functions of the surge multipliers and s i has a compact domain (so does p ij ) for alli2N andj2J , then an optimal solution of (2.15) exists. Letfp ij ;x ij ;i2N;j2Jg be feasible prices and matching fractions for the optimization problem (2.15) such that the constraint (2.15b) is not binding for some driver type(s). Letfs i ;i 2 Ng be the surge multipliers corresponding to the fp ij ;i2N;j2Jg. Letz i := P j2J j F ij x ij ,z := P i2N z i ,G i := i z i , andG := P i2N G i for all i2N . Then,z i denotes the matching rate of typei drivers,z denotes the total matching rate,G i 0 for all i2N ,G denotes the gap between the total driver arrival rate and the total matching rate, andG> 0 because the constraint (2.15b) is not binding for some driver type(s). We will show that the system controller can increasez and decreaseG to 0 by changing the surge multipliers. This implies that there exists an optimal solution of (2.15) in which the constraint (2.15b) is binding for all driver types. Consider an areai2N such thatG i > 0 (sinceG > 0, such an area exists). If we decrease s i , then P j2J :(j)=i j will increase and i may decrease by condition (2.16a). Let us decrease s i and match the additional customers arriving at areai with the excess typei drivers so thatz i increases. We should decrease s i until all typei drivers are matched with customers because decreasing s i more results in excess customers 196 which lowers z i , i.e., we should decrease s i until G i becomes equal to 0. Since we match the additional customers arriving at areai with the excess typei drivers, forj2J such that(j) =i, we should increase x ij and decreasex kj for allk2Nnfig such that the constraints (2.15c) and (2.15d) hold. Next, let us consider the change inz andG. If there existsk2Nnfig such that@ k =@s i > 0, then k decreases in areak which can decreasez k . However, total decrease in matching rates in all other areas is less than the increase inz i by condition (2.16c). Hence,z strictly increases. When k decreases,G k does not increase and it is possible that some of the customers matched with typek drivers may not find an available typek driver anymore and so leave the system without being matched. In such a case, we should decrease x kj for somej2J such that constraints (2.15c) and (2.15d) hold. If there existsk2Nnfig such that@ k =@s i 0, then k is nondecreasing. If k is strictly increasing and there are unmatched customers in the system, we prefer to not to match those customers with the additional typek drivers arriving in the system for mathematical simplicity, thusz k stays constant butG k increases. However, the increase in P k2Nnfig G k is less than the decrease in G i by condition (2.16c). Hence, G strictly decreases. In summary, we can increasez and decreaseG by decreasing s i untilG i becomes equal to 0. By condition (2.16d), such an s i exists. We propose the following algorithm which decreasesG to 0 and increasesz: LetG i = max k2N G k > 0. Decrease s i untilG i becomes equal to 0. Then, update the surge multipliers, customer and driver arrival rates, and feasible matching fractions. IfG decreases to 0, stop. Otherwise, repeat the procedure. Under this algorithm, at each step, z strictly increases, G strictly decreases. Let s i and G denotes the total change in s i andG, respectively, in a step whereG i = max k2N G k . Then,G i =(2C 2 )js i j G i =C 1 by condition (2.16a). Moreover,@G=@s i < 0 and @G @s i @ i @s i + X j2J :(j)=i @ j @s i X k2Nnfig @ k @s i C 1 by (2.16c). SinceG i G=N, jGjC 1 js i j C 1 G i 2C 2 C 1 G 2NC 2 : Therefore,G decreasesC 1 =(2NC 2 )100%> 0% at each step which implies thatG can be made arbitrarily close to 0 in finite steps and converges to 0 as the number of steps increases to infinity. 197 B.6 Relative Compactness in spaceD In this section, we present a relative compactness result in spaceD that we use in the proof of Proposition B.1.1 (cf. Section B.2). Although this result is known in folklore, we could not find a specific theorem to refer to, thus we provide one. Let T 1 2 R + be an arbitrary constant. We consider D[0;T 1 ] endowed with the usual Skorokhod J 1 topology (cf. Chapter 3 of Billingsley (1999)). For somex;y2D[0;T 1 ], letd(x;y) denote the Skorokhod J 1 distance between these processes (cf. (12.13) in Billingsley (1999)). Lemma B.6.1. LetfY n ;n2 N + g be a relatively compact sequence inD[0;T 1 ] endowed with the u.o.c. topology such that all of its subsequential limits are uniformly continuous. LetfX n ;n2N + g be a sequence inD[0;T 1 ] such that sup n2N + jX n (0)j<1; (B.96a) jX n (t 2 )X n (t 1 )jKjY n (t 2 )Y n (t 1 )j; for allt 1 ;t 2 2 [0;T 1 ] andn2N + ; (B.96b) whereK2R + is a constant. Then,fX n ;n2N + g is relatively compact inD[0;T 1 ] endowed with the u.o.c. topology and all of its subsequential limits are uniformly continuous. Moreover, if all of the subsequential limits offY n ;n2N + g are absolutely (Lipschitz) continuous, then all of the subsequential limits offX n ;n2N + g are also absolutely (Lipschitz) continuous. Proof: SincefY n ;n2 N + g is relatively compact with respect to the u.o.c. topology, it is also relatively compact with respect to the SkorokhodJ 1 topology. Then, by Theorem 12.3 of Billingsley (1999), sup n2N + kY n k T 1 <1; lim !0 sup n2N + w 0 (Y n ;) = 0; (B.97) wherew 0 is defined in equation (12.6) of Billingsley (1999). Then, sup n2N + kX n k T 1 = sup n2N + sup 0tT 1 jX n (t)j sup n2N + jX n (0)j + sup n2N + sup 0tT 1 jX n (t)X n (0)j sup n2N + jX n (0)j +K sup n2N + sup 0tT 1 jY n (t)Y n (0)j sup n2N + jX n (0)j + 2K sup n2N + kY n k T 1 <1; (B.98) where the first inequality in (B.98) is by (B.96b) and the strict inequality in (B.98) is by (B.96a) and (B.97). Next, 0 lim !0 sup n2N + w 0 (X n ;)K lim !0 sup n2N + w 0 (Y n ;) = 0; (B.99) 198 where the second inequality is by the definition ofw 0 (cf. equation (12.6) of Billingsley (1999)) and (B.96b) and the equality is by (B.97). Therefore,fX n ;n 2 N + g is a relatively compact sequence in D[0;T 1 ] endowed with the SkorokhodJ 1 topology by (B.98), (B.99), and Theorem 12.3 of Billingsley (1999). LetfX n l ;l2N + g be an arbitrary convergent subsequence offX n ;n2N + g such thatd(X n l ;X)! 0 for someX2D[0;T 1 ] asl!1. Then, there exists a subsequence offn l ;l2N + g, denoted byfn k ;k2 N + g, such thatd(Y n k ;Y )! 0 ask!1 whereY 2D[0;T 1 ] andY is uniformly continuous. Let us fix an arbitrary> 0. There exits a :=(;K)> 0 such that ifjt 2 t 1 j<,jY (t 2 )Y (t 1 )j<=(6K). Since convergence in SkorokhodJ 1 metric implies u.o.c. convergence when the limit is continuous (cf. page 124 in Billingsley (1999)), we also havekY n k Yk T 1 ! 0 ask!1. Let denote the set of continuous, strictly increasing, and bijective mappings from the domain [0;T 1 ] onto itself. Then, there exists ak 0 2N + and a sequencef k ;k2N + g in the set such that ifkk 0 , k e T 1 _ X n k k X T 1 < 8 ^; kY n k Yk T 1 < 8K ; (B.100) where the first inequality is by the definition of SkorokhodJ 1 metric (cf. (12.13) in Billingsley (1999)) and the fact thatd(X n k ;X)! 0 ask!1. Let us fix an arbitraryk k 0 . Then, for allt 1 ;t 2 2 [0;T 1 ] such thatjt 2 t 1 j, jX(t 2 )X(t 1 )j X(t 2 )X n k ( k (t 2 )) + X n k ( k (t 2 ))X n k ( k (t 1 )) + X n k ( k (t 1 ))X(t 1 ) 2 X l=1 X(t l )X n k ( k (t l )) +K Y n k ( k (t 2 ))Y n k ( k (t 1 )) 2 X l=1 X(t l )X n k ( k (t l )) +K Y n k ( k (t l ))Y ( k (t l )) +K Y ( k (t l ))Y (t l ) +KjY (t 2 )Y (t 1 )j < 2 X n k k X T 1 + 2KkY n k Yk T 1 + 3 + 6 < 4 + 4 + 2 =; where the second inequality is by (B.96b), the fourth inequality is by the definition of and (B.100), the last inequality is by (B.100). Therefore,X is uniformly continuous, thusX n l !X u.o.c. asl!1 (cf. page 124 of Billingsley (1999)). This implies thatfX n ;n2N + g is also relatively compact in the u.o.c. topology and all of its subsequential limits are uniformly continuous. 199 Next, suppose that Y is absolutely continuous. We will prove that X is also absolutely continuous. Let us fix an arbitrary 1 > 0. Letf(a l ;b l )g m l=1 be an arbitrary finite set of disjoint intervals such that (a l ;b l ) [0;T 1 ] for alll2f1; 2;:::;mg. SinceY is absolutely continuous, for any 1 > 0, there exists a 1 = 1 ( 1 ;K)> 0 such that if P m l=1 (b l a l )< 1 , then P m l=1 jY (b l )Y (a l )j< 1 =K. Then m X l=1 jX(b l )X(a l )j m X l=1 (jX(b l )X n k (b l )j +jX n k (b l )X n k (a l )j +jX n k (a l )X(a l )j) m X l=1 jX(b l )X n k (b l )j +jX n k (a l )X(a l )j +KjY n k (b l )Y (b l )j +KjY n k (a l )Y (a l )j +KjY (b l )Y (a l )j m X l=1 2kXX n k k T 1 + 2KkY n k Yk T 1 +KjY (b l )Y (a l )j = 2m kXX n k k T 1 +KkY n k Yk T 1 +K m X l=1 jY (b l )Y (a l )j; (B.101) where the second inequality is by (B.96b). By lettingk!1, the first term in (B.101) converges to 0, thus the sum of the terms in (B.101) becomes less than 1 . Hence,X is absolutely continuous. Lastly, suppose thatY is Lipschitz continuous with Lipschitz constant2R + . We will prove thatX is also Lipschitz continuous. For allt 1 ;t 2 2R + , jX(t 2 )X(t 1 )j lim sup k!1 (jX(t 2 )X n k (t 2 )j +jX n k (t 2 )X n k (t 1 )j +jX n k (t 1 )X(t 1 )j) lim sup k!1 jX(t 2 )X n k (t 2 )j +jX n k (t 1 )X(t 1 )j +KjY n k (t 2 )Y (t 2 )j +KjY n k (t 1 )Y (t 1 )j +KjY (t 2 )Y (t 1 )j lim sup k!1 2kXX n k k T 1 + 2KkY n k Yk T 1 +KjY (t 2 )Y (t 1 )j (B.102) Kjt 2 t 1 j; where the second inequality is by (B.96b). Since the first term in (B.102) converges to 0 andY is Lipschitz continuous, we obtain the last inequality, soX is Lipschitz continuous. 200 Appendix C Technical Appendix to Chapter 3 This appendix is organized as follows: The proofs of Lemmas 3.2.1, 3.2.2, and 3.5.1 are presented in Section C.1. The proofs of Theorem 3.3.2 and Proposition 3.4.1 are presented in Section C.2. The proofs of Theorems 3.3.1 and 3.3.3 are presented in Section C.3. We present an important relationship between origin based pricing and origin and destination based pricing, which will be useful later in some proofs, in Section C.4. The proof of Proposition 3.5.1 is presented in Section C.5. The proofs of Theorems 3.4.1 and 3.5.1 are presented in Section C.6. The proof of Theorem 3.5.2 is presented in Section C.7. The solution of Example 3.4.1 and the proof of Proposition 3.4.2 are presented in Section C.8. Lastly, information about the simulation set-up is presented in Section C.9. C.1 Proofs of Lemmas 3.2.1, 3.2.2, and 3.5.1 The proofs of Lemmas 3.2.1, 3.2.2, and 3.5.1 are presented in Sections C.1.1, C.1.2, and C.1.3, respectively. C.1.1 Proof of Lemma 3.2.1 Each driver makes the repositioning decisions by solving the following dynamic program (DP). The decision epochs are the times when the driver becomes idle. The state space isN[f0g where 0 represents the termination state of leaving the system. The action space isN . The transition probabilities are denoted by fq lm (k);l;m2N[f0g;k2Ng such thatq lm (k) denotes the probability that the driver who is in statel in the current stage ends up in statem after taking the actionk. In other words,q lm (k) is the probability that an idle driver in areal who repositions himself to areak becomes idle again in aream, does not leave the system, and makes another repositioning decision. Then, q 00 (k) = 1; 8k2N; q l0 (k) = 1; 8k;l2N; q lm (k) = k X i2N im G im (p im ) ik x imk ; 8k;l;m2N such thatk6=m; q lk (k) = 0 @ 1 1 k X i;j2N ij G ij (p ij ) ij x ijk + 1 k X i2N ik G ik (p ik ) ik x ikk 1 A ; 8k;l2N: 201 Notice that 0q lm (k) 1 for allk2N andl;m2N[f0g by (3.2). Lastly, the expected reward in state 0 under actionk is 0 for allk2N and the expected reward in statel under actionk isR k for allk;l2N . The aforementioned DP is a stochastic shortest path problem (see Chapter 3 of Bertsekas (2012)). Let us fix an arbitraryk 2fk2N : R k = Rg. Let denote the stationary policy in which the driver always chooses actionk ; i.e., the driver always repositions himself to an area with the highest revenue rate. Let J (l) denote the expected total revenue that the driver obtains during his lifetime in the system starting from the statel2N[f0g under the stationary policy. Let us define the operatorT such that T J(0) = 0; T J(l) = R + X m2N[f0g q lm (k )J(m); 8l2N; whereJ :N[f0g! R + is an arbitrary function. By Proposition 3.2.1 of Bertsekas (2012), J is the unique solution ofJ =T J. It is easy to see thatJ (0) = 0 andJ (l) = R=(1) for alll2N . LetJ denote the expected total revenue that the driver obtains during his lifetime in the system starting from the statel2N[f0g under an optimal policy. Let us define the operatorT such that TJ(0) = 0; TJ(l) = max k2N R k + X m2N[f0g q lm (k)J(m) ; 8l2N: By Proposition 3.2.2 of Bertsekas (2012), J is the unique solution of J = TJ. It is easy to see that J (0) = 0 andJ (l) = R=(1) for alll2N . Therefore, the stationary policy is optimal. It is easy to see that if a driver decides to reposition himself to an area with revenue rate strictly less than R, then his expected total revenue becomes strictly less than R=(1). Therefore, drivers should always reposition themselves to the areas with the maximum revenue rate. C.1.2 Proof of Lemma 3.2.2 Let e k := (1) k for allk2N . Then e = P k2N e k = (1) and e satisfies (3.9) by (3.10). Let v k := 0 @ X i;j2N ik G ik (p ik ) ij x ikj X i;j2N ij G ij (p ij ) ik x ijk 1 A ; 8k2N: 202 Then P k2N v k = 0 and v k + k 0 for all k 2N by (3.2). Without loss of generality, letN = N 1 [N 2 [N 3 whereN 1 ,N 2 , andN 3 are disjoint sets such that k = 0 for allk2N 1 , andN 1 =f1; 2;:::;ag for somea2f1;:::;Ng ifN 1 6=;; k > 0 andv k 0 for allk2N 2 , andN 2 =fa + 1;:::;bg for someb2fa + 1;:::;Ng ifN 2 6=;; k > 0 andv k < 0 for allk2N 3 , andN 3 =fb + 1;:::;Ng ifN 2 6=;: Notice thatv k 0 for allk2N 1 because of the fact thatv k + k 0 for allk2N . Let y kl := 8 > < > : 0; ifl2N 1 [N 2 , v k jv l j P i2N 3 jv i j ; ifl2N 3 , 8k2N 1 ; y kl := 8 > > > > > < > > > > > : l ; ifl =k, 0; ifl2N 1 [N 2 nfkg, v k jv l j P i2N 3 jv i j ; ifl2N 3 , 8k2N 2 ; y kl := 8 > < > : v l + l ; ifl =k, 0; ifl2Nnfkg, 8k2N 3 : LetK := ( P i2N 3 jv i j) 1 . Table C.1 shows an illustration of the constructedy. Table C.1: Illustration of the constructedy. y 1 ::: a a+1 ::: b b+1 ::: N 1 0 ::: 0 0 ::: 0 v 1 jv b+1 jK ::: v 1 jv N jK . . . . . . . . . . . . . . . . . . . . . a 0 ::: 0 0 ::: 0 v a jv b+1 jK ::: v a jv N jK a+1 0 ::: 0 a+1 0 0 v a+1 jv b+1 jK ::: v a+1 jv N jK . . . . . . . . . 0 . . . 0 . . . . . . b 0 ::: 0 0 0 b v b jv b+1 jK ::: v b jv N jK b+1 0 ::: 0 0 ::: 0 v b+1 + b+1 0 0 . . . . . . . . . . . . . . . 0 . . . 0 N 0 ::: 0 0 ::: 0 0 0 v N + N Notice that P k2N 1 [N 2 v k = P k2N 3 jv k j because of the fact that P k2N v k = 0. Then,fx;p;; e ;yg is nonnegative and satisfies (3.1) and (3.5) and so is an equilibrium. 203 C.1.3 Proof of Lemma 3.5.1 Let ~ x ijk := 8 > < > : x ijk ik = ~ ik ; if ik > 0, 0; otherwise. Then ~ x ijk x ijk and ~ ik ~ x ijk = ik x ijk for alli;j;k2N . Then, clearly,f~ x;p;; e ;yg is an equilibrium under ~ with the total matching ratez. C.2 Proofs of Theorem 3.3.2 and Proposition 3.4.1 We will present an optimization problem whose optimal objective function value is an upper bound on the total matching rate under any constant pricing equilibrium for a given constant price in Section C.2.1. Then, we will show that the aforementioned upper bound can be achieved by a constant pricing and no cross matching equilibrium in Section C.2.2. Consequently, Theorem 3.3.2 will follow. Lastly, we present the proof of Proposition 3.4.1 in Section C.2.3. C.2.1 An Upper Bound Optimization Problem We introduce an optimization problem whose optimal objective function value is an upper bound on the total matching rate under any constant pricing equilibrium for a given constant price. Let us fix an arbitrary p2R + and consider the following optimization problem. max x;; X i;j;k2N ij G ij (p) ik x ijk (C.1a) such that X i;j2N ij G ij (p) ik x ijk k ; 8k2N; (C.1b) X k2N x ijk 1; 8i;j2N; (C.1c) X k2N k =; (C.1d) = e 1 V 0 @ p X i;j;k2N ij G ij (p) ik x ijk 1 A ; if> 0; (C.1e) x ijk ; k ; 0; 8i;j;k2N; (C.1f) 204 where the decision variables arefx;;g. Notice that the objective (C.1a) is the same of the objective (3.11a) with constant prices, the constraints (C.1b) and (C.1c) are the same of the constraint (3.2) with con- stant prices and the constraint (3.3), respectively. Furthermore, the constraint (C.1e) resembles the constraint (3.9) by (3.6) and (3.8). Therefore, the optimization problem (C.1) resembles the optimization problem (3.11) in which the prices are fixed top2 R + for all customer types. The following lemma provides an optimal solution of (C.1). We letx^y := minfx;yg. Lemma C.2.1. Fix an arbitraryp2 R + . IfV (p) = 0 or ij G ij (p) = 0 for alli;j2N , thenx ijk = k = = 0 for alli;j;k2N is the optimal solution of (C.1) with the objective function value 0. Suppose thatV (p)> 0 and ij G ij (p)> 0 for somei;j2N . Let be the a solution of the equality = e 1 V 0 @ p 0 @ ^ X i;j2N ij G ij (p) 1 A 1 A ; (C.2) which exists uniquely and is strictly positive. 1. If P i;j2N ij G ij (p), let i := P j2N ij G ij (p) P i;j2N ij G ij (p) ; 8i2N; x ijk := 8 > < > : 1; ifi =k, 0; otherwise, 8i;j;k2N: (C.3) 2. If< P i;j2N ij G ij (p), let be such that P i2N i =, 0 i X j2N ij G ij (p); 8i2N; (C.4a) x ijk := 8 > < > : i = P j2N ij G ij (p); ifi =k and P j2N ij G ij (p)> 0, 0; otherwise, 8i;j;k2N: (C.4b) Then,fx;;g is an optimal solution of (C.1) with the objective function value ^ X i;j2N ij G ij (p) = e 1 V (p) ^ X i;j2N ij G ij (p): (C.5) Equation (C.2) determines the total driver supply in the system. When there are excess drivers; i.e., P i;j2N ij G ij (p), then the optimal solution of the optimization problem (C.1) described in (C.3) keeps the ratio of the customer demand and the driver supply constant among all areas and do not use cross 205 matching, thus the revenue rate in each area becomes the same. Therefore, all customers are matched and so the total matching rate is maximized. When the drivers are scarce in the system compared to the customers; i.e.,< P i;j2N ij G ij (p), then the optimal solution of the optimization problem (C.1) described in (C.4) allocates the drivers to the areas by ensuring that no driver idles and so the total matching rate is maximized. Notice that the optimal solutions in both (C.3) and (C.4) do not use cross matching. Proof: (Proof of Lemma C.2.1) We have P i;j;k2N ij G ij (p) ik x ijk by (C.1b) and (C.1d). Hence, if V (p) = 0 or ij G ij (p) = 0 for alli;j2N , there does not exist any > 0 which satisfies the equality (C.1e) becauseV (0) = 0. Then,x ijk = k = = 0 for alli;j;k2N is the only feasible point as well as the optimal solution of (C.1) with the objective function value 0. Suppose thatV (p) > 0 and ij G ij (p) > 0 for somei;j2N . Letf~ x; ~ ; ~ g be an arbitrary feasible point of (C.1). On the one hand, by (C.1d) and summing both left and right-hand sides of (C.1b) overk2N , we obtain X i;j;k2N ij G ij (p) ik ~ x ijk ~ : (C.6) On the other hand, we have X i;j;k2N ij G ij (p) ik ~ x ijk = X i;j2N ij G ij (p) X k2N ik ~ x ijk ! X i;j2N ij G ij (p); (C.7) where the inequality is by (C.1c) and the fact that ik 2 [0; 1] for alli;k2N . Moreover, if ~ > 0, we have ~ = e 1 V 0 @ p ~ X i;j;k2N ij G ij (p) ik ~ x ijk 1 A e 1 V (p); (C.8) where the equality is by (C.1e) and the inequality is by (C.6). Therefore, by (C.6), (C.7), (C.8), we have X i;j;k2N ij G ij (p) ik ~ x ijk e 1 V (p) ^ X i;j2N ij G ij (p): (C.9) Notice that the term in the right-hand side of (C.9) is an upper bound on the objective function value asso- ciated with any feasible point of the optimization problem (C.1). Consider the equality (C.2). If 0<< 0 @ X i;j2N ij G ij (p) 1 A ^ e 1 V (p); 206 then < e 1 V 0 @ p 0 @ ^ X i;j2N ij G ij (p) 1 A 1 A ; that is, the left-hand side of (C.2) is strictly less than the right-hand side of (C.2). SinceV is continuous, the left and right-hand sides of (C.2) are both continuous in, and strictly increasing and nonincreasing in, respectively. Hence, there exists a unique solution for the equality (C.2), which is denoted by and> 0. Notice that,fx;;g defined in (C.2), (C.3), and (C.4) is a feasible point of (C.1) with the objective function value^ P i;j2N ij G ij (p). There are two cases to consider. First, suppose that P i;j2N ij G ij (p). Then,fx;;g defined in (C.2), (C.3), and (C.4) has the objective function value P i;j2N ij G ij (p) which achieves the upper bound in (C.9). This implies that it is an optimal solution of (C.1). Second, suppose that < P i;j2N ij G ij (p). Then, = e V (p)=(1) by (C.2) andfx;;g defined in (C.2), (C.3), and (C.4) has the objective function value e V (p)=(1) which achieves the upper bound in (C.9). Therefore, it is an optimal solution of (C.1). The following lemma states that the optimization problem (C.1) provides an upper bound on the total matching rate under any constant pricing equilibrium for a given constant price. Lemma C.2.2. Consider a constant pricing equilibrium with constant pricep2R + . Then, total matching rate under this equilibrium is less than or equal to the optimal objective function value of the optimization problem (C.1), which is given in (C.5). Proof: Letfx;p;; e ;yg be an arbitrary equilibrium in whichp ij =p for alli;j2N for somep2R + . We will show thatfx;;g is a feasible point of (C.1). First, (C.1d) is satisfied by the definition of . Second, (C.1f) is satisfied by the definition of equilibrium (see Definition 3.2.1). Third, (C.1b) and (C.1c) are satisfied by (3.2) and (3.3), respectively. If = 0, then the desired result follows trivially. Suppose that> 0. Then, e 1 V 0 @ p X i;j;k2N ij G ij (p) ik x ijk 1 A = e 1 V p P k2N : k >0 P i;j2N ij G ij (p) ik x ijk P k2N : k >0 k ! ; (C.10) = e 1 V R = e 1 =; (C.11) where the equality in (C.10) is by (3.2), and the first, the second, and the third equalities in (C.11) are by (3.8), (3.9), and (3.6), respectively. Therefore,fx;;g satisfies the constraint (C.1e) by (C.11) and so it is 207 a feasible point of the optimization problem (C.1). Moreover, the total matching rate under the equilibrium fx;p;; e ;yg is equal to the objective function value associated with the feasible pointfx;;g by (3.11a) and (C.1a), which gives us the desired result. C.2.2 Proof of Theorem 3.3.2 By Lemma 3.2.2 and the optimal solution of the optimization problem (C.1) presented in Lemma C.2.1, the following proposition constructs a constant pricing and no cross matching equilibrium with the total matching rate equal to the optimal objective function value of the optimization problem (C.1). Therefore, the upper bound in Lemma C.2.2 is achievable by a constant pricing and no cross matching equilibrium. Proposition C.2.1. Letp ij = p for alli;j2N for somep2 R + andfx;;g be defined as in Lemma C.2.1. Then, by Lemma 3.2.2, there exists a vectorf e ;yg such thatfx;p;; e ;yg is a constant pricing and no cross matching equilibrium with the total matching rate equal to the optimal objective function value of the optimization problem (C.1), which is given in (C.5). Consequently, Theorem 3.3.2 follows by Lemma C.2.2 and Proposition C.2.1. For any given constant pricep2R + , letz CP (p) denote the total matching rate under the constant pricing and no cross matching equilibrium constructed in Proposition C.2.1. Then, by (C.5) and the definition ofz CP , z CP (p) = e 1 V (p) ^ X i;j2N ij G ij (p); z CP = sup p2R + z CP (p): (C.12) C.2.3 Proof of Proposition 3.4.1 SinceG ij andV are continuous for alli;j2N , both of the terms in the left and right-hand sides of (3.12) are continuous inp, so doesz CP (p) by (C.12). Moreover, the terms in left and right-hand sides of (3.12) are nonincreasing and nondecreasing inp, respectively. Therefore, ifp 2R + is a solution of the equality (3.12), thenz CP should satisfy (3.13) by (C.12). Whenp = 0, the sum in the left-hand side of (3.12) is equal to P i;j2N ij which is strictly positive and the term in the right-hand side of (3.12) is equal to 0. Asp increases to +1, the terms in the left and right-hand sides of (3.12) converges to 0 and e =(1), respectively. Therefore, there exists a solution of the equality (3.12). Moreover, ifG ij is strictly increasing for alli;j2N orV is strictly increasing, then the solution of the equality (3.12) is unique. Letfx ; g be the matching policy and the vector denoting the total idle driver arrival rate to each area under the constant pricing and no cross matching equilibrium defined in Proposition C.2.1 associated with the constant pricep . If P i;j2N ij G ij (p ) = e V (p )=(1) = 0, then no driver or customer arrives in 208 the system and the the constraint (3.2) is binding for allk2N trivially. Suppose that P i;j2N ij G ij (p )> 0. Then, = P i;j2N ij G ij (p ) = e V (p )=(1) is a solution of (C.2), which is unique by Lemma C.2.1. Then, for allk2N , X i;j2N ij G ij (p ) ik x ijk = X j2N kj G kj (p ) = P j2N kj G kj (p ) P i;j2N ij G ij (p ) = k ; where the equalities are by (C.3). Therefore, the constraint (3.2) is binding for allk2N . C.3 Proofs of Theorems 3.3.1 and 3.3.3 The proofs of Theorems 3.3.1 and 3.3.3 are presented in Sections C.3.1 and C.3.2, respectively. C.3.1 Proof of Theorem 3.3.1 Letfx;p;; e ;yg be an arbitrary origin based pricing and no cross matching equilibrium with the total matching ratez and the pricing vectorp =fp i ;i2Ng. LetN 1 :=fk2N : k > 0g. IfN 1 =;, then z = 0 and so the proof is trivial. Hence, suppose thatN 1 6=;. Letp := min i2N 1 p i . Then, R = p k k X j2N kj G kj (p k )x kjk = p l l X j2N lj G lj (p l )x ljl ; 8k;l2N 1 ; (C.13) = X k2N 1 k = e 1 V R ; z^ 0 @ X i2N 1 ;j2N ij G ij (p i ) 1 A ; (C.14) where (C.13) is by (3.7) and (3.8) and the definition of a no cross matching equilibrium, the second equality in (C.14) is by (3.6) and (3.9), and the inequality in (C.14) is by (3.2) and (3.3). Since Rp k for allk2N 1 by (3.2) and (C.13), then Rp. Hence, e V p =(1) by the equalities in (C.14). Consider the constant pricing and no cross matching equilibrium defined in Proposition C.2.1 associated with the constant pricep. Then, by (C.12), z CP p = e 1 V p ^ X i;j2N ij G ij (p)^ 0 @ X i2N 1 ;j2N ij G ij (p i ) 1 A z; (C.15) where the first inequality in (C.15) is by the fact that e V p =(1) andpp i for alli2N 1 , and the second inequality in (C.15) is by (C.14). Therefore, by (C.15), the total matching rate under the constant pricing and no cross matching equilibrium defined in Proposition C.2.1 associated with the constant price p is greater than or equal to the one of the origin based pricing and no cross matching equilibrium, which completes the proof. 209 C.3.2 Proof of Theorem 3.3.3 The following lemma will be useful in the proofs. Lemma C.3.1. LetS be a finite and linearly ordered set,fa i ;b i ;i 2 Sg be a set of nonnegative real numbers, andF :R + ! [0; 1] be a nonincreasing function. If P i2S a i F (b i )> 0, then P i2S a i F (b i )b i P i2S a i F (b i ) P i2S a i b i P i2S a i : (C.16) Proof: SinceF (b) 1 for allb2 R + , we have P i2S a i P i2S a i F (b i ). Thus, if P i2S a i F (b i ) > 0, both the ratio in the left-hand side of (C.16) and the ratio in the right-hand side of (C.16) are well defined. LetC := P i2S a i F (b i ) P i2S a i 1 > 0. Then, P i2S a i F (b i )b i P i2S a i F (b i ) P i2S a i b i P i2S a i = P i2S a i F (b i )b i P i2S a i P i2S a i b i P i2S a i F (b i ) P i2S a i F (b i ) P i2S a i ; =C 0 @ X i2S a 2 i F (b i )b i + X i2S:i6=j X j2S a i a j F (b i )b i X i2S a 2 i F (b i )b i X i2S:i6=j X j2S a i a j F (b j )b i 1 A ; =C 0 @ X i2S:i6=j X j2S a i a j F (b i )b i X i2S:i6=j X j2S a i a j F (b j )b i 1 A ; =C X i2S:i6=j X j2S a i a j b i (F (b i )F (b j )); =C X i2S:i>j X j2S a i a j (F (b i )F (b j )) (b i b j ); 0; where the inequality is by the fact thatF is nonincreasing. Letfx;p;; e ;yg be an arbitrary origin and destination based pricing and no cross matching equi- librium with the total matching rate z. If z = 0, the proof is trivial. Hence, suppose that z > 0. Let N 1 :=fk2N : k > 0g. Sincez> 0, we haveN 1 6=;. Then, R = k X j2N kj G k (p kj )x kjk p kj = l X j2N lj G l (p lj )x ljl p lj ; 8k;l2N 1 ; (C.17) z = X i2N 1 ;j2N ij G i (p ij )x iji X k2N 1 k = = e 1 V R ; (C.18) 210 where (C.17) is by (3.7) and (3.8) and the definition of an NCM equilibrium, the first equality in (C.18) is by (3.2) and the definition ofN 1 , the inequality in (C.18) is by (3.2), and the third equality in (C.18) is by (3.6) and (3.9). Sincez> 0, we have ifi2N 1 , then P j2N ij G i (p ij )x iji > 0. Then, let ~ p i := 8 > < > : P j2N ij x iji p ij P j2N ij x iji ; ifi2N 1 ; 0; ifi2NnN 1 ; 8i2N: Let p := min i2N 1 ~ p i . Since z > 0, we have p > 0. Consider the the constant pricing and no cross matching equilibrium defined in Proposition C.2.1 associated with the constant pricep. By (C.12), z CP p = 0 @ X i;j2N ij G i (p) 1 A ^ e 1 V p : (C.19) First, X i;j2N ij G i (p) X i2N 1 ;j2N ij G i (~ p i ) X i2N 1 0 @ X j2N ij 1 A P j2N ij x iji G i (p ij ) P j2N ij x iji ; (C.20) X i2N 1 X j2N ij x iji G i (p ij ) =z; (C.21) where the first inequality in (C.20) is by the definition ofp, the second inequality in (C.20) is by the definition of ~ p i and the concavity of G i , the inequality in (C.21) is by the fact thatx iji 2 [0; 1] for alli;j2N , and the equality in (C.21) is by (C.18). Second, let us fix an arbitraryk2N 1 . Then, R P j2N kj G k (p kj )x kjk p kj P j2N kj G k (p kj )x kjk P j2N kj x kjk p kj P j2N kj x kjk =~ p k ; (C.22) where the first and the second inequalities in (C.22) are by (3.2) and Lemma C.3.1, respectively. Since (C.22) holds for allk2N 1 , we have Rp, which gives us e 1 V p e 1 V R =z: (C.23) Therefore, (C.19), (C.21), and (C.23) imply thatz CP p z, which gives us the desired result. 211 C.4 Relationship between OP and ODP In this section, we prove that if we can characterize the optimal equilibriums in the networks where constant pricing and origin based pricing are allowed but origin and destination based pricing is excluded, we can also characterize the optimal equilibriums in the networks where origin and destination based pricing is allowed. This result is interesting in its own right and will be useful in the rest of the proofs. In order to prove this result, for any given ridesharing network with parametersfN;;; e ;V; ij ;G ij ; ik ;i;j;k2Ng, we construct a hypothetical ridesharing network by modifying the parametersfN; ij ;G ij ; ik ;i;j;k 2 Ng; and at most a single customer type arrives at each area in the hypothetical network. Hence, in the hypothetical network, ODP is irrelevant. We present the formal construction of the hypothetical ridesharing network and a simple example in Section C.4.1. Then we show a “one-to-one” relationship between the equilibriums in the original and the hypothetical networks by the following theorem. Theorem C.4.1. For any given equilibrium in an original network (including the equilibriums with origin and destination based prices), there exists an equilibrium in the associated hypothetical network with the same total matching rate and that equilibrium can be characterized in closed form. Similarly, for any given equilibrium in an hypothetical network, there exists an equilibrium in the associated original network with the same total matching rate and that equilibrium can be characterized in closed form. The proof of Theorem C.4.1 is presented in Section C.4.2. Theorem C.4.1 implies that the optimal total matching rate in the original network is equal to the one in the hypothetical network. Therefore, in order to derive the optimal equilibriums in ridesharing networks where ODP is allowed, it is enough to derive the optimal equilibriums in networks where CP and OP are allowed but ODP is not allowed. C.4.1 Construction of the Hypothetical Ridesharing Network LetN 2 :=fi2N : ij > 0; il > 0 for somej;l2N such thatj6=lg. Then,N 2 is the set of areas with multiple customer types. LetN 1 :=NnN 2 . Without loss of generality, letN 1 =f1; 2;:::;n 1g and N 2 =fn;n + 1;:::;Ng for somen2N . LetD j denote the set of the destinations of the customer types arriving at areaj for allj2N , thenD j N . Let us construct a hypothetical ridesharing network in which the set of areas in the city is denoted by ~ N :=f1; 2;:::;n 1g[f(n;k);k2 D n g[f(n + 1;k);k2 D n+1 g[[f(N;k);k 2 D N g. We will use the accent “ ˜ ” with the notation associated with the hypothetical network in this section. We let ~ ij := ij ; 8i;j2N 1 ; ~ i(j;k) := ij ; 8i2N 1 ;j2N 2 ;k2D j ; 212 ~ (i;k)j := ij ; 8i2N 2 ;j2N 1 ;k2D i ; ~ (i;k)(j;l) := ij ; 8i;j2N 2 ;k2D i ;l2D j : Let :N 1 !N[f;g denote the destination of the customers arriving in the set of areasN 1 in the original network. If there is no customer type arriving at areai2N 1 , then(i) =;. Let :N 2 !N maps an areai2N 2 to the least element of the setD i . Let us construct the customer types such that at most one customer type arrives at each area in the hypo- thetical network. If the destination of a customer isi2N 2 in the original network, then we let the destination of that customer to be (i;(i)) in the hypothetical network. Since at most a single customer type is arriving at each area in the hypothetical network, for simplicity, we will use a single index to denote the customer types in the hypothetical network instead of origin-destination index that is used in the original network. In the hypothetical network, on the one hand, if customers arrive at areai for somei2N 1 , then their destina- tion is(i) if(i)2N 1 , ((i);((i)) if(i)2N 2 . On the other hand, if customers arrive at area (i;j) for somei2N 2 andj2D i , then their destination isj ifj2N 1 , (j;(j)) otherwise. Then, ~ i := i(i) ; ~ G i :=G i(i) ; 8i2N 1 such that X j2N ij > 0; ~ (i;j) := ij ; ~ G (i;j) :=G ij ; 8i2N 2 ; j2D i ; and all other ~ i := 0. All other model primitives in the hypothetical network are the same of the ones in the original network; i.e., the parametersf;; e ;Vg are not modified. Hence, each customer type arriving in the original network also arrives in the hypothetical network with the same rate and cdf for the rides but at most one customer type arrives at each area in the latter network. Therefore, ODP is irrelevant in the hypothetical network; i.e., CP and OP cover all possible pricing schemes. We present an example of hypothetical network construction in the example below. Example C.4.1. Consider a network with two areas; i.e., N = 3, where 13 ; 31 ; 32 > 0 and all other ij = 0. Figure C.1 shows the original and the hypothetical networks. In the hypothetical network, ~ N = f1; 2; (3; 1); (3; 2)g, ~ 1 = 13 ; ~ (3;1) = 31 ; ~ (3;2) = 32 ; ~ G 1 =G 13 ; ~ G (3;1) =G 31 ; ~ G (3;2) =G 32 ; 213 ~ = 0 B B B B B B B B @ 1 2 (3; 1) (3; 2) 1 1 12 13 13 2 21 1 23 23 (3; 1) 31 32 1 1 (3; 2) 31 32 1 1 1 C C C C C C C C A ; and all other ~ ij = 0. " ! " # $ " #% #$%&' " & " #% " %# " %& $ & #$%"' ! " %'# $ " %# ! " %'& $ " %& Figure C.1: The original network (on the left) and the hypothetical network (on the right) in Example C.4.1. C.4.2 Proof of Theorem C.4.1 ! direction Letfx;p;; e ;yg be an equilibrium in the original network with total matching rate z. Without loss of generality, if ij = 0 for somei;j2N , thenp ij = 0 andx ijk = 0 for allk2N . We will construct an equilibrium in the hypothetical network with the same matching rate. Let ~ k := k ; 8k2N 1 ; ~ (k;(k)) := k ; 8k2N 2 ; ~ (k;l) := 0; 8k2N 2 ;l2D k nf(k)g; ~ p i :=p i(i) ; 8i2N 1 , if(i)6=;; ~ p (i;j) :=p ij ; 8i2N 2 ;j2D i ; ~ x ik :=x i(i)k ; 8i;k2N 1 ; ~ x i(k;(k)) :=x i(i)k ; 8i2N 1 ;k2N 2 ; ~ x (i;j)k :=x ijk ; 8i2N 2 ;j2D i ;k2N 1 ; ~ x (i;j)(k;(k)) :=x ijk ; 8i2N 2 ;j2D i ;k2N 2 ; and all other ~ p i := 0 and ~ x ik := 0 fori;k2 ~ N . We will use Lemma 3.2.2 to construct an equilibrium in the hypothetical network associated with ~ x; ~ p; ~ with the total matching ratez. First, let us consider the constraint (3.3). Fix an arbitraryi2 ~ N . We will make a case by case analysis. First, suppose thati2N 1 . Then, X k2 ~ N ~ x ik = X k2N 1 ~ x ik + X k2N 2 ~ x i(k;(k)) = X k2N 1 x i(i)k + X k2N 2 x i(i)k = X k2N x i(i)k 1 214 where the inequality is by (3.3). Second, suppose thati2 ~ NnN 1 . Then,i = (j;l) for somej2N 2 and l2D j , X k2 ~ N ~ x ik = X k2N 1 ~ x (j;l)k + X k2N 2 ~ x (j;l)(k;(k)) = X k2N 1 x jlk + X k2N 2 x jlk = X k2N x jlk 1 where the inequality is by (3.3). Hence, ~ x; ~ p; ~ satisfies (3.3). Second, let us consider the constraint (3.2). Fix an arbitraryk2 ~ N . First, suppose thatk2N 1 . Then, X i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik = X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk = X i;j2N ij G ij (p ij ) ik x ijk k = ~ k ; where the inequality is by (3.2). Second, suppose thatk2 ~ NnN 1 . By construction, it is enough to consider the areak = (l;(l)) for some arbitraryl2N 2 . Then, X i2 ~ N ~ i ~ G i (~ p i ) ~ i(l;(l)) ~ x i(l;(l)) = X i2N 1 i(i) G i(i) (p i(i) ) il x i(i)l + X i2N 2 ;j2D i ij G ij (p ij ) il x ijl = X i;j2N ij G ij (p ij ) il x ijl l = ~ (l;(l)) ; where the inequality is by (3.2). Hence, ~ x; ~ p; ~ satisfies (3.2). Third, let us consider the condition (3.8). Fix an arbitraryk;m2 ~ N such that ~ k ; ~ m > 0. It is enough to prove that ~ R k = ~ R m . First, suppose thatk2N 1 andm = (l;(l)) for somel2N 2 . Then, k ; l > 0 by construction and ~ R k = P i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik ~ p i ~ k = P i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k p i(i) + P i2N 2 ;j2D i ij G ij (p ij ) ik x ijk p ij k = P i;j2N ij G ij (p ij ) ik x ijk p ij k =R k =R l = P i;j2N ij G ij (p ij ) il x ijl p ij l (C.24) = P i2N 1 i(i) G i(i) (p i(i) ) il x i(i)l p i(i) + P i2N 2 ;j2D i ij G ij (p ij ) il x ijl p ij l = P i2 ~ N ~ i ~ G i (~ p i ) ~ i(l;(l)) ~ x i(l;(l)) ~ p i ~ l = ~ R (l;(l)) = ~ R m ; 215 where the third equality in (C.24) is by (3.8). The casesk;m2N 1 ,k = (n;(n)) for somen2N 2 and m2N 1 , andk = (n;(n)) for somen2N 2 andm = (l;(l)) for somel2N 2 follow similarly, thus we skip them. Therefore, ~ x; ~ p; ~ satisfies (3.8). Fourth, let us consider the constraint (3.10). We have ~ = by construction and ~ R := max k2 ~ N ~ R k = R by (C.24). Then, ~ = = e 1 V ( R) = e 1 V ( ~ R); where the second equality is by (3.6) and (3.9). Therefore, ~ x; ~ p; ~ satisfies (3.10). Since ~ x; ~ p; ~ is nonnegative by construction and satisfies (3.8), (3.2), (3.3), and (3.10), then by Lemma 3.2.2, there exists an equilibrium in the hypothetical network associated with ~ x; ~ p; ~ that can be charac- terized in closed form. Moreover, the associated total matching rate is equal to X i;k2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik = X i2 ~ N;k2N 1 ~ i ~ G i (~ p i ) ~ ik ~ x ik + X i2 ~ N;k2N 2 ~ i ~ G i (~ p i ) ~ i(k;(k)) ~ x i(k;(k)) ; = X k2N 1 0 @ X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk 1 A + X k2N 2 0 @ X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk 1 A = X k2N 1 X i;j2N ij G ij (p ij ) ik x ijk + X k2N 2 X i;j2N ij G ij (p ij ) ik x ijk = X i;j;k2N ij G ij (p ij ) ik x ijk =z: direction Let n ~ x; ~ p; ~ ; ~ e ; ~ y o be an equilibrium in the hypothetical network with total matching rate ~ z and ~ p =f~ p i ;i2 ~ Ng. Without loss of generality, if ~ i = 0 for somei2 ~ N , then ~ p i = 0 and ~ x ik = 0 for allk2 ~ N . We will construct an equilibrium in the original network with the same matching rate. Let k := ~ k ; 8k2N 1 ; k := X l2D k ~ (k;l) ; 8k2N 2 ; p i(i) := ~ p i ; 8i2N 1 , if(i)6=;; p ij := ~ p (i;j) ; 8i2N 2 ;j2D i ; x i(i)k := ~ x ik ; 8i;k2N 1 ; x i(i)k := X l2D k ~ x i(k;l) ; 8i2N 1 ;k2N 2 ; x ijk := ~ x (i;j)k ; 8i2N 2 ;j2D i ;k2N 1 ; x ijk := X l2D k ~ x (i;j)(k;l) ; 8i2N 2 ;j2D i ;k2N 2 ; 216 and all otherp ij := 0 andx ijk := 0 fori;j;k2N . We will use Lemma 3.2.2 to construct an equilibrium in the original network associated with x;p; with the total matching rate ~ z. First, let us consider the constraint (3.3). Fix an arbitraryi2N . We will make a case by case analysis. First, suppose thati2N 1 . Fix an arbitraryj2N . Ifj6=(i) or(i) =;, then P k2N x ijk = 0 1. Ifj =(i)6=;, then X k2N x i(i)k = X k2N 1 x i(i)k + X k2N 2 x i(i)k = X k2N 1 ~ x ik + X k2N 2 ;l2D k ~ x i(k;l) = X k2 ~ N ~ x ik 1; where the inequality is by (3.3). Second, suppose thati2N 2 . Fix an arbitraryj2N . Ifj = 2 D i , then P k2N x ijk = 0 1. Ifj2D i , then X k2N x ijk = X k2N 1 x ijk + X k2N 2 x ijk = X k2N 1 ~ x (i;j)k + X k2N 2 ;l2D k ~ x (i;j)(k;l) = X k2 ~ N ~ x (i;j)k 1; where the inequality is by (3.3). Hence, x;p; satisfies (3.3). Second, let us consider the constraint (3.2). Fix an arbitraryk2N . First, suppose thatk2N 1 . Then, X i;j2N ij G ij (p ij ) ik x ijk = X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk = X i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik ~ k = k ; where the inequality is by (3.2). Second, suppose thatk2N 2 . Then, X i;j2N ij G ij (p ij ) ik x ijk = X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk = X i2 ~ N ~ i ~ G i (~ p i ) X l2D k ~ i(k;l) ~ x i(k;l) X l2D k ~ (k;l) = k ; where the second to last equality is by the fact that ~ i(k;l) is constant inl2D k ( ~ i(k;l) = ik for alll2D k ) and the inequality is by (3.2). Hence, x;p; satisfies (3.2). Third, let us consider the condition (3.8). Fix an arbitraryk;l2N such that k ; l > 0. It is enough to prove thatR k =R l . First, suppose thatk2N 1 andl2N 2 . Then, ~ k = k > 0 and P m2D l ~ (l;m) = l > 0 by construction. LetD (1) l :=fm2D l : ~ (l;m) > 0g. Then, R k = P i;j2N ij G ij (p ij ) ik x ijk p ij k 217 = P i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k p i(i) + P i2N 2 ;j2D i ij G ij (p ij ) ik x ijk p ij k = P i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik ~ p i ~ k = ~ R k = ~ R m = P i2 ~ N ~ i ~ G i (~ p i ) ~ i(l;m) ~ x i(l;m) ~ p i ~ (l;m) ;8m2D (1) l (C.25) where the third equality in (C.25) is by (3.8). By (3.7), (3.8), (C.25), and the fact that (C.25) holds for all m2D (1) l and ~ i(l;m) is constant inm, we have R k = ~ R k = P m2D l P i2 ~ N ~ i ~ G i (~ p i ) ~ i(l;m) ~ x i(l;m) ~ p i P m2D l ~ (l;m) = P i2N 1 i(i) G i(i) (p i(i) ) il x i(i)l p i(i) + P i2N 2 ;j2D i ij G ij (p ij ) il x ijl p ij l = P i;j2N ij G ij (p ij ) il x ijl p ij l =R l : (C.26) The casesk;l2N 1 ,k2N 2 andl2N 1 , andk;l2N 2 follow similarly, thus we skip them. Therefore, x;p; satisfies (3.8). Fourth, let us consider the constraint (3.10). We have = ~ by construction and R = ~ R by (C.25) and (C.26). Then, = ~ = e 1 V ( ~ R) = e 1 V ( R); where the second equality is by (3.6) and (3.9). Therefore, x;p; satisfies (3.10). Since x;p; is nonnegative by construction and satisfies (3.8), (3.2), (3.3), and (3.10), then by Lemma 3.2.2, there exists an equilibrium in the hypothetical network associated with x;p; that can be charac- terized in closed form. Moreover, the associated total matching rate is equal to X i;j;k2N ij G ij (p ij ) ik x ijk p ij ; = X k2N 1 X i;j2N ij G ij (p ij ) ik x ijk + X k2N 2 X i;j2N ij G ij (p ij ) ik x ijk = X k2N 1 0 @ X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk 1 A + X k2N 2 0 @ X i2N 1 i(i) G i(i) (p i(i) ) ik x i(i)k + X i2N 2 ;j2D i ij G ij (p ij ) ik x ijk 1 A = X k2N 1 ;i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik + 218 + X k2N 2 0 @ X i2N 1 i(i) G i(i) (p i(i) ) ik X l2D k ~ x i(k;l) + X i2N 2 ;j2D i ij G ij (p ij ) ik X l2D k ~ x (i;j)(k;l) 1 A = X k2N 1 ;i2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik + X k2N 2 ;l2D k ;i2 ~ N ~ i ~ G i (~ p i ) ~ i(k;l) ~ x ik = X i;k2 ~ N ~ i ~ G i (~ p i ) ~ ik ~ x ik = ~ z; where in the fourth equality we use the fact that ~ i(k;l) = ik for alli2N 1 , k2N 2 , andl2 D k , and ~ (i;j)(k;l) = ik for alli;k2N 2 ,j2D i , andl2D k . C.5 Proof of Proposition 3.5.1 First, we will exclude origin and destination based pricing and consider only the constant pricing and origin based pricing. We will characterize an optimal equilibrium when the customers are patient and the prices are given in Section C.5.1. Then, we will derive an optimization problem which can be used to derive optimal equilibriums when the customers are patient and the origin and destination based pricing is excluded in Section C.5.2. Lastly, we will extend our results by including origin and destination based pricing and prove Proposition 3.5.1 in Section C.5.3. C.5.1 An optimal equilibrium when prices are given and customers are patient Suppose that the customers are patient and the price vectorp is given and destination independent; i.e., =e andp =fp i ;i2Ng. Without loss of generality, we assumep 1 p 2 :::p N . We will construct an optimal equilibrium under the given prices. The idea is to match the customers who pay the most for a ride with the drivers as much as possible. By giving priority to the most profitable customers, the firm attracts the maximum number of drivers to the system which maximizes the total matching rate. For each2 h 0; P i;j2N ij G ij (p i ) i , there existn2N and 2 [0; 1] such that = n1 X i=1 X j2N ij G ij (p i ) + X j2N nj G nj (p n ); (C.27) 219 where P 0 i=1 a i := 0 for any real sequencefa i ;i2N + g. Let R() := 8 > < > : P n1 i=1 P j2N ij G ij (p i )p i + P j2N nj G nj (p n )p n ; if 0 P i;j2N ij G ij (p i ), P i;j2N ij G ij (p i )p i ; if> P i;j2N ij G ij (p i ). (C.28) Then R() is the total revenue rate for the drivers that can be obtained from the most profitable ^ P i;j2N ij G ij (p i ) customers. It is easy to see thatR() is piecewise linear, continuous, nondecreasing, and concave in. Moreover,R()= is nonincreasing and continuous in andR()=! 0 as!1. Let, p := max i2N n p i I P j2N ij G ij (p i )> 0 o , whereI denotes the indicator function. IfV ( p) = 0; i.e., if the system is not attractive enough so that no drivers arrive in the system, then total matching rate is 0 under all equilibriums. Hence, suppose thatV ( p)> 0. Then, = e 1 V R() ; (C.29) has a unique solution which is strictly positive and denoted by. Letf k ;k2Ng be such that k 0 for allk2N and X k2N k =: (C.30) If P i;j2N ij G ij (p i ), then let x ijk := k ; 8i;j;k2N: (C.31) If< P i;j2N ij G ij (p i ), then let x ijk := 8 > > > > > < > > > > > : k =; ifi2f1; 2;::::;n 1g andj;k2N , k =; ifi =n andj;k2N , 0; otherwise, (C.32) wheren2N and 2 [0; 1] are defined in (C.27). Then, by Lemma 3.2.2, there exists a vectorf e ;yg such thatfx;p;; e ;yg is an equilibrium, wherefx;g is defined in (C.29), (C.30), (C.31), and (C.32). We denote this equilibrium and the associated total matching rate byE(p) andz(E(p)), respectively, for given p. By (C.30), (C.31) and (C.32),z(E(p)) =^ P i;j2N ij G ij (p i ). 220 The equilibriumE(p) is constructed by matching as many customers as possible starting from the most profitable ones. Since there are multiplef k ;k2Ng satisfying (C.30), there are multiple equilibriums associated withE(p) and they all have the same total matching rate, which is z(E(p)). By (C.31) and (C.32), each individual driver is assigned to a customer equally likely. Hence, where the drivers position themselves does not affect the total matching rate, but how the drivers are allocated to different customer types matter. The following result states that for any given price vectorp, the equilibriumE(p) is optimal. Lemma C.5.1. Let ~ x;p; ~ ; ~ e ; ~ y be an arbitrary equilibrium with destination independent prices and total matching rate ~ z under a given parameter which is not necessarily equal toe. Then,z(E(p)) ~ z. Proof: Letfx;p;; e ;yg denote the equilibriumE(p). The case in which ~ z = 0 is trivial. Hence, suppose that ~ z > 0. Notice that ~ ~ z by (3.2). There are two cases to consider. First, suppose that ~ z = P i;j2N ij G ij (p i ). By (3.6), (3.8), and (3.9), we have ~ = e 1 V 0 @ ~ X i;j2N ij G ij (p i )p i 1 A : Since (C.29) has a unique solution and ~ is a solution of (C.29), then = ~ and z(E(p)) = ^ P i;j2N ij G ij (p i ) = ~ z. Second, suppose that ~ z < P i;j2N ij G ij (p i ). We will use proof by contradiction technique. Suppose thatz(E(p))< ~ z. Let ~ R := P i;j;k2N ij G ij (p i ) ik ~ x ijk p i denote the total revenue rate generated in the equilibrium ~ x;p; ~ ; ~ e ; ~ y . Then, ~ = e 1 V ~ R ~ ~ z>z(E(p)) = = e 1 V R() ; (C.33) where the first equality is by (3.6), (3.8), and (3.9), and the last equality is by (C.29). We have ~ ~ z > and ~ R= ~ >R()= by (C.33) implying ~ R=~ z>R()=. Since ~ z< P i;j2N ij G ij (p i ), there existm2N and2 [0; 1] such that R(~ z) = 0 @ m1 X i=1 X j2N ij G ij (p i )p i + X j2N mj G mj (p m )p m 1 A whereR(~ z) is the revenue rate generated by the most profitable ~ z customers and we haveR(~ z) ~ R. Then, R(~ z)=~ z>R()=. SinceR()= is nonincreasing, it must be the case that ~ z, which is a contradiction. 221 Lemma C.5.1 states that the total matching rate under the equilibriumE(p) is greater than or equal to the total matching rate under any equilibrium under the price vectorp and any parameter. This is because the equilibriumE(p) matches the drivers with the most profitable customers which increases the driver supply in the system and the total matching rate is maximized. Next, we compareE(p) equilibriums with respect to the price vectorp. Lemma C.5.2. Letfx;p;; e ;yg be anE(p) equilibrium such that 6= P i;j2N ij G ij (p i ). Then, there exists anE( ~ p) equilibrium denoted by ~ x; ~ p; ~ ; ~ e ; ~ y such that z(E( ~ p)) z(E(p)) and ~ = P i;j2N ij G ij (~ p i ). Proof: Without loss of generality, let us assumep 1 p 2 :::p N . We will consider two cases. Case 1 Suppose that> P i;j2N ij G ij (p i ). Then,z(E(p)) = P i;j2N ij G ij (p i ). We will decrease the prices appropriately such that more customers and less drivers will arrive in the system and we will match the additional customers with the excess drivers. We will stop decreasing the prices at the point in which all of the customers are matched and drivers never idle. By (C.28) and (C.29), = e 1 V 0 @ X i;j2N ij G ij (p i )p i 1 A : Hence, X i;j2N ij G ij (p i )< e 1 V P i;j2N ij G ij (p i )p i P i;j2N ij G ij (p i ) ! : (C.34) SinceG ij andV are continuous for alli;j2N , the left-hand side of (C.34) is continuous and nonincreasing inp i and the right-hand side of (C.34) is continuous inp i for alli2N . Ifp i = 0 for alli2N , the right- hand side of (C.34) is equal to 0. Therefore, as the prices decrease the left-hand side of (C.34) does not decrease and the right-hand side of (C.34) converges to 0, respectively. Without loss of generality, let us assume thatp 1 p 2 ::: p m > 0 andp m+1 = ::: = p N = 0 for somem2N . Then, there exists p2 (0;p m ) such that e 1 V P m i=1 P j2N ij G ij (p)p P m i=1 P j2N ij G ij (p) + P N i=m+1 P j2N ij ! e 1 V P m i=1 P j2N ij G ij (p)p P i;j2N ij G ij (p i ) ! < X i;j2N ij G ij (p i ) 222 m X i=1 X j2N ij G ij (p) + N X i=m+1 X j2N ij : (C.35) Then, we propose the following algorithm: Step 0: i m. Step 1: Decreasep i top continuously. If equality is obtained in (C.34), then stop. Otherwise,i i 1 and repeat Step 1. By (C.35), the algorithm will stop by obtaining equality in (C.34). Let the price vector at the end of the algorithm be denoted by ~ p and ~ x; ~ p; ~ ; ~ e ; ~ y denote the equilibriumE( ~ p). Since X i;j2N ij G ij (~ p i ) = e 1 V P i;j2N ij G ij (~ p i )~ p i P i;j2N ij G ij (~ p i ) ! by construction and ~ is the unique solution of ~ = e 1 V R( ~ ) ~ ! by (C.29), we have ~ = P i;j2N ij G ij (~ p i ) and z(E( ~ p)) = P i;j2N ij G ij (~ p i ). Since ~ p i p i for all i2N ,z(E( ~ p))z(E(p)). Case 2 Suppose that < P i;j2N ij G ij (p i ). Then,z(E(p)) = and there existn2N and 2 [0; 1] such that (C.27) and (C.29) hold. We will increase the prices appropriately such that less customers and more drivers will arrive in the system and we will match the additional drivers with the excess customers. We will stop increasing the prices at the point in which all of the customers are matched and drivers never idle. We consider two different cases. Case 2.1 Suppose that n = N. Then, < 1 because < P i;j2N ij G ij (p i ). Since G ij is contin- uous and nonincreasing for all i;j 2 N , there exists a ~ p N p N such that P j2N Nj G Nj (p N ) = P j2N Nj G Nj (~ p N ). Let ~ be the unique solution of ~ = e 1 V 0 @ ~ 0 @ N1 X i=1 X j2N ij G ij (p i )p i + X j2N Nj G Nj (~ p N )~ p N 1 A 1 A : 223 By (C.29) and the fact that P j2N Nj G Nj (~ p N )~ p N P j2N Nj G Nj (p N )p N , we have ~ . Let ~ p i =p i for alli2f1; 2;:::;N 1g. Then, ~ z(E( ~ p)) = =z(E(p)). If ~ =z(E( ~ p)), then the proof is done. Otherwise, if ~ >z(E( ~ p)), then the desired results follows by Case 1. Case 2.2 Suppose that n < N. Then type ij customers where i2fn + 1;:::;Ng and j 2N are not matched with drivers under the equilibriumE(p). There exists p p 1 such that P i;j2N ij G ij ( p) < e V ( p)=(1). Let us increasep i to p for alli2fn + 1;:::;Ng and order areas from highest price to lowest price such thatp (1) 1 p (1) 2 ::: p (1) N . Since we increase the prices only in the areas where no customer is matched,z(E(p (1) ))z(E(p)). If (1) = P i;j2N ij G ij (p (1) i ), then the proof is done. If (1) > P i;j2N ij G ij (p (1) i ), then the desired results follows by Case 1. Suppose that (1) < P i;j2N ij G ij (p (1) i ). Thenz(E(p (1) )) = (1) and there existn (1) 2N and (1) 2 [0; 1] such that (C.27) and (C.29) hold. Ifn (1) =N, the proof follows by Case 2.1. Otherwise, apply the steps in Case 2.2 and continue with the same way. At some point, we will obtain the desired result because when all the prices are increased to p, we have P i;j2N ij G ij ( p)< e V ( p)=(1) by construction. This implies that there exists a > P i;j2N ij G ij ( p) such that = e 1 V P i;j2N ij G ij ( p) p ! and so >z(E( p)) = P i;j2N ij G ij ( p) where p i := p for alli2N . Therefore, all of the customers will be matched under the constant price p and the desired result will follow by Case 1. Lemma C.5.2 states that among theE(p) equilibriums, the ones in which all of the customers are matched and drivers never idle have the highest total matching rate. Under anE(p) equilibrium, if there are customers who are not matched with drivers, then increasing the prices by sufficiently small amount will decrease the customer demand and increase the driver supply in the system which will increase the total matching rate. Similarly, if some drivers are idle, then decreasing the prices by sufficiently small amount will increase the customer demand and decrease the driver supply which will increase the total matching rate. Consequently, Lemma C.5.2 follows. Next, by Lemmas C.5.1 and C.5.2, we will introduce an optimization problem which determines the optimal total matching rate when the customers are patient. C.5.2 An Optimization Problem Characterizing Optimal Equilibriums Lemmas C.5.1 and C.5.2 state that when =e and the origin and destination based pricing is excluded, the optimal objective function value of the optimization problem (3.11) is equal to the highest total matching rate 224 among theE(p) equilibriums in which all of the customers are matched and drivers never idle. Consequently, the optimization problem (3.11) simplifies into the following pricing optimization problem: max p X i;j2N ij G ij (p i ) (C.36a) such that X i;j2N ij G ij (p i ) = e 1 V P i;j2N ij G ij (p i )p i P i;j2N ij G ij (p i ) ! ; (C.36b) p i 0; 8i2N: (C.36c) The optimization problem (C.36) is the same of the optimization problem (3.14) except that it has destination independent prices. Letfx;p;; e ;yg be anE(p) equilibrium such that = P i;j2N ij G ij (p i ). Then, the constraint (C.36b) is the same of the equality (C.29) and the objective function (C.36a) is equal to z(E(p)). Therefore, the optimization problem (C.36) maximizes the total matching rate among theE(p) equilibriums in which all of the customers are matched and drivers never idle. The optimization problem (C.36) has the following properties. The feasible region of (C.36) is nonempty. For any given price vectorp =fp i ;2Ng, there exists a price vector ~ p such that the constraint (C.36b) is satisfied by Lemma C.5.2 and the construction of the equilibriumE( ~ p). Moreover, ifp is a feasible point of (C.36) with the objective function value z, then z(E(p)) = z. Since there are multiplef k ;k2Ng satisfying (C.30), there are multiple equilibriums associated withE(p) and they all have the total matching ratez(E(p)). Therefore, for each feasible point of (C.36), there exist multiple equilibriums with the total matching rate equal to the objective function value of that feasible point. Furthermore, under each of those equilibriums, all of the customers arriving in the system are served, drivers never idle, and each individual driver is assigned to a customer equally likely. When =e and the origin and destination based pricing is excluded, the optimal objective function value of (C.36) is equal to the one of the optimization problem (3.11) by Lemmas C.5.1 and C.5.2. Furthermore, the optimal objective function value of (C.36) is an upper bound on the total matching rate under any equilibrium with destination independent prices under any parameter by Lemma 3.5.1. Ifp i =p for alli2N wherep is the constant price defined in Proposition 3.4.1, thenp is a feasible point of (C.36) by the fact thatp solves (3.12). The associated objective function value isz CP , which is defined in (3.13). 225 C.5.3 Extension of the Results with Origin and Destination Based Pricing In this section, we will extend our results by considering origin and destination based pricing and prove Proposition 3.5.1. For any given ridesharing network, let us first consider the associated hypothetical net- work (see Section C.4.1 for construction). Since at most a single customer type arrives at each area in the hypothetical network, origin and destination based pricing is irrelevant. Therefore, when the customers are patient, in order to determine the optimal equilibrium in the hypothetical network, we can solve the optimization problem (C.36): max ~ p X i2 ~ N ~ i ~ G i (~ p i ) (C.37a) such that X i2 ~ N ~ i ~ G i (~ p i ) = e 1 V P i2 ~ N ~ i ~ G i (~ p i )~ p i P i2 ~ N ~ i ~ G i (~ p i ) ! ; (C.37b) ~ p i 0; 8i2 ~ N: (C.37c) Notice that the optimization problem (C.37) satisfies the properties in Proposition 3.5.1 for the hypothetical network as proven in Section C.5.2. Since P i2 ~ N ~ i ~ G i (~ p i ) = P i;j2N ij G ij (p ij ) and P i2 ~ N ~ i ~ G i (~ p i )~ p i = P i;j2N ij G ij (p ij )p ij , the opti- mization problem (C.37) is equivalent to the optimization problem (3.14). Therefore, since the optimization problem (C.37) has a nonempty feasible region, so does (3.14). Furthermore, the optimization problem (3.14) satisfies Proposition 3.5.1 Part 2 by Theorem C.4.1 and Lemma 3.5.1. Lastly, letp be a feasible point of (3.14) and ~ p be the associated feasible point of (C.37). Then, there exist multiple equilibriums in the hypothetical network that satisfy the properties in Proposition 3.5.1 Part 1. By construction, the associated equilibriums in the original network also satisfy the properties in Proposition 3.5.1 Part 1. C.6 Proofs of Theorems 3.4.1 and 3.5.1 The proofs of Theorems 3.4.1 and 3.5.1 are presented in Sections C.6.1 and C.6.2, respectively. 226 C.6.1 Proof of Theorem 3.4.1 Let G(p) := 1G(p) for all p2 R + . Suppose thatp is an arbitrary feasible point of (3.14) with the objective function valuez. Then, it must be the case that P i;j2N ij G(p ij ) > 0, otherwise the constraint (3.14b) will be violated. By Lemma C.3.1, we have P i;j2N ij G(p ij )p ij P i;j2N ij G(p ij ) P i;j2N ij p ij P i;j2N ij : (C.38) Let p := P i;j2N ij p ij = P i;j2N ij and consider the constant pricing and no cross matching equi- librium associated with the constant price p defined in Proposition C.2.1. By (C.12), the total matching rate under that equilibrium is z CP ( p) = 0 @ X i;j2N ij G( p) 1 A ^ e 1 V ( p) : (C.39) Sincep satisfies (3.14b), we have z = X i;j2N ij G(p ij ) = e 1 V P i;j2N ij G(p ij )p ij P i;j2N ij G(p ij ) ! e 1 V ( p); (C.40) where the inequality is by (C.38). Second, since G is concave, z = X i;j2N ij G(p ij ) X i;j2N ij G( p): (C.41) Hence, z CP ( p) z by (C.39), (C.40), and (C.41). Therefore, for any feasible point of (3.14), the total matching rate under the constant pricing and no cross matching equilibrium defined in Proposition C.2.1 is greater than or equal to the objective function value of that feasible point. This implies that the total matching rate under the CP equilibriums specified in Proposition 3.4.1 is greater than or equal to the objective function value of any feasible point of the optimization problem (3.14). Consequently, the CP equilibriums specified in Proposition 3.4.1 are optimal. 227 C.6.2 Proof of Theorem 3.5.1 By Theorems 3.3.1 and 3.3.3,z NCM = z CP under the conditions stated in Theorem 3.5.1. By definition, z CM = maxfz OP ;z ODP +CM g. Hence, in order to prove Theorem 3.3.3, we will construct an OP equilib- rium such that the difference between the total matching rate under that OP equilibrium and the one under the best CP equilibrium is bounded below by the right-hand side in (3.15). We will make the proof by replacing condition C3 with weaker conditions C4 and C5 below: C4 p <A, which implies that the total matching rate is not maximized trivially under some CP equi- libriums. C5 Consider the equality (1) n X i=1 i = e (1)A P n i=1 i a i P n i=1 i ^A ; (C.42) where 2 [0; 1]. Let the unique solution of (C.42) be denoted by . We have P n i=1 i a i = ( P n i=1 i ) < A, which implies that the total matching rate is not maximized triv- ially under some OP equilibriums. By conditions C1 and C3, we have A a 1 a n > p . Hence, conditions C1 and C3 imply condition C4. Moreover, since 2 [0; 1], we have A a 1 P n i=1 i a i = ( P n i=1 i ). Hence, conditions C1 and C3 also imply condition C5. Therefore, given condition C1, conditions C4 and C5 are weaker than condition C3. Since the left and right-hand sides of (3.12) are both continuous in price, and nonincreasing and nonde- creasing in price, respectively, condition C1, or equivalently (3.16), implies thata n >p a n+1 for some n2f2; 3;:::;Ng and vice versa. Thus there are at least two areas in which customers arrive under the best constant price. Ifp A, then by (3.13), z CP = X i;j2N ij G ij (p ) = e 1 V (p ) = e 1 : By (3.2), (3.6), and (3.9), total matching rate under any equilibrium is bounded above by e =(1). Therefore, the total matching rate is maximized trivially under the best CP equilibrium. 228 Suppose thatp <A (see condition C4). Then, by Proposition 3.4.1, X i;j2N ij G ij (p ) = n X i=1 i 1 p a i = e 1 V (p ) = e 1 p A ; which gives us p = P n i=1 i C 1 + P n i=1 i =a i ; z CP = C 1 P n i=1 i C 1 + P n i=1 i =a i ; (C.43) whereC 1 := e =(A(1)). Next we will construct a destination independent price vectorp =fp i ;2Ng which is a feasible point of (3.14) and then consider the equilibriumE(p) defined in Section C.5.1. Letp i := a i for alli2fn + 1;:::;Ng and (p 1 ;p 2 ;:::;p n ) be such that p 1 a 1 = p 2 a 2 =::: = p n a n =; for some2 [0; 1]: (C.44) Then, underp, the equality (3.14b) becomes equivalent to the equality (C.42). Notice that the left and right-hand sides of the equality (C.42) are strictly decreasing and nondecreasing in on the interval [0; 1], respectively. Hence, the equality (C.42) has a unique solution on the interval [0; 1] which is denoted by . In order for the price vectorp to be a feasible point of (3.14), in (C.44) must be equal to . Notice that the equilibriumE(p) has origin based prices by condition C2 and (C.44). If P n i=1 i a i = ( P n i=1 i ) A, then = 1 e 1 n X i=1 i ! 1 : If e =(1) > P n i=1 i , then < 0 so this case is not possible. If e =(1) P n i=1 i , then the objective function value of the optimization problem (3.14) associated with the price vectorp is e =(1). Hence,z(E(p)) = e =(1) by construction and the total matching rate is maximized trivially under the equilibriumE(p). Suppose that P n i=1 i a i = ( P n i=1 i )<A (see condition C5). By (C.42) and (3.14a), = ( P n i=1 i ) 2 ( P n i=1 i ) 2 +C 1 P n i=1 i a i ; z(E(p)) = C 1 ( P n i=1 i ) ( P n i=1 i a i ) ( P n i=1 i ) 2 +C 1 P n i=1 i a i : (C.45) Then, by (C.43), (C.45), and some algebra, z CM (e)z NCM z OP (e)z CP z(E(p))z CP =C n X i=j+1 n1 X j=1 i j (a i a j ) 2 a i a j > 0; 229 where C :=C 1 n X i=1 i ! 2 4 0 @ n X i=1 i ! 2 +C 1 n X i=1 i a i 1 A C 1 + n X i=1 i a i ! 3 5 1 ; and the strict inequality is by condition C2. C.7 Proof of Theorem 3.5.2 Let > 0 be an arbitrary constant andfx;p;; e ;yg be an arbitrary equilibrium with the total matching ratez under a given. Ifz = 0, then the desired result holds trivially so we assume thatz> 0. We consider two cases. Case 1: ~ ik ik for alli;k2N . If all non-diagonal components of are 0, then ~ = and so the desired result holds trivially. Suppose that some of the non-diagonal components of are strictly positive. Let := minf ik : ik > 0;i;k 2Ng; i.e., is the smallest strictly positive component of. Let (;;z) be an arbitrary number on the interval (0;(1^=z)). Thus, ~ <(;;z) implies that if ik > 0, then ~ ik > 0. Let := min i;k2N : ik >0 ~ ik ik > 0; ~ x ijk := 8 > > > > > < > > > > > : x ijk ; ifi =k, x ijk ik = ~ ik ; ifi6=k and ik > 0, 0; otherwise; 8i;j;k2N: Then, ~ ik ~ x ijk = ik x ijk for alli;j;k2N . Notice that if = 1, then ~ = and so the desired result holds trivially. Suppose that< 1. By (3.6), (3.8), and (3.9), we have = e 1 V 0 @ X i;j;k2N ij G ij (p ij ) ik x ijk p ij 1 A ; and notice that is the unique solution of the above equality whenfx;p;g are fixed. Consider the equality ~ = e 1 V 0 @ ~ X i;j;k2N ij G ij (p ij ) ~ ik ~ x ijk p ij 1 A = e 1 V 0 @ ~ X i;j;k2N ij G ij (p ij ) ik x ijk p ij 1 A ; (C.46) which has a unique solution denoted by ~ such that ~ . Let ~ k := k ~ = for allk2N . Next, we will construct an equilibrium by invoking Lemma 3.2.2 onf~ x;p; ~ g. 230 First, notice thatf~ x;p; ~ g is nonnegative by construction. Let the revenue rate in area k under ~ be denoted by ~ R k for allk2N . For allk2N , if ~ k > 0, then ~ R k = ~ k X i;j2N ij G ij (p ij ) ~ ik ~ x ijk p ij = k ~ X i;j2N ij G ij (p ij ) ik x ijk p ij = ~ R k = ~ R; (C.47) where the first and last equalities in (C.47) are by (3.7) and (3.8), respectively. Therefore,f~ x;p; ~ g satisfies (3.8) by (C.47); i.e., the revenue rates in the areas in which idle drivers arrive are the same. Second, for allk2N , X i;j2N ij G ij (p ij ) ~ ik ~ x ijk = X i;j2N ij G ij (p ij ) ik x ijk k = ~ k ~ ~ k ; where the first inequality is by (3.2). Then,f~ x;p; ~ g satisfies (3.2). Lastly, ~ x ijk x ijk for alli;j;k2N by construction, which implies thatf~ x;p; ~ g satisfies (3.3). Lastly, by (3.2), (C.46), and (C.47), ~ = e 1 V 0 @ ~ X k2N : ~ k >0 X i;j2N ij G ij (p ij ) ~ ik ~ x ijk p ij 1 A = e 1 V 0 @ 1 ~ X k2N : ~ k >0 ~ R k ~ k 1 A = e 1 V 0 @ 1 ~ ~ R X k2N : ~ k >0 ~ k 1 A = e 1 V 1 ~ ~ R ~ = e 1 V ~ R = e 1 V max k2N ~ R k ; which implies that f~ x;p; ~ g satisfies (3.10). Therefore, there exists a vector n ~ e ; ~ y o such that n ~ x;p; ~ ; ~ e ; ~ y o is an equilibrium with total matching ratez by Lemma 3.2.2. Moreover, (1) 0 @ X i;k2N : ik >0 1 ~ ik ik 2 2 ik 1 A 0:5 ~ <(;;z)<(1^=z): (C.48) We have> 1=z by (C.48), which implies thatzz<, which gives the desired result. Case 2: ~ ik > ik for some i;k2N . Let ^ be such that ^ ik := ik ^ ~ ik for all i;k2N . From Case 1, we know that if ^ < 0:5(1^=z), then there exists an equilibrium under ^ , denoted by n ^ x;p; ^ ; ^ e ; ^ y o , with total matching rate ^ z such thatj^ zzj<. By Lemma 3.5.1, there exists a matching policy ~ x such that n ~ x;p; ^ ; ^ e ; ^ y o is an equilibrium with the total matching rate ^ z under ~ . Therefore, by 231 Lemma 3.5.1 and the fact that ^ ~ , if ~ < 0:5(1^=z), then there exists an equilibrium under ~ with total matching rate ^ z such thatj^ zzj<. C.8 Solution of Example 3.4.1 and Proof of Proposition 3.4.2 The solution of Example 3.4.1 and the proof of Proposition 3.4.2 are presented in Sections C.8.1 and C.8.2, respectively. C.8.1 Solution of Example 3.4.1 We have G 8 > < > : U[0; 1]; with prob. 1; U[2; 3]; with prob.; implies G(p) = 8 > > > > > < > > > > > : (1)p; if 0p< 1; 1; if 1p 2; 1(3p); if 2<p 3; 8p2 [0; 3]: We will construct OP equilibriums with total matching rate greater thanz CP by the optimization problem (3.14). Let us fixp 2 = 2 and assume that 0p 1 1. Then, the constraint (3.14b) becomes ((1p 1 )(1) + 2) 2 = 2 ((1p 1 )(1)p 1 +p 1 + 2): (C.49) The uniquep 1 2 [0; 1] which solves the equality (C.49) is p 1 = 1 + 2 p 3 2 2 1 2 ; (C.50) and the associated objective function value of the optimization problem (3.14) is 2 +(1) p 3 2 2 1 1 2 > 2 =z CP : (C.51) Therefore, the OP equilibrium specified in Proposition 3.5.1 Part 1 has a strictly greater total matching rate than z CP . If = 0:25, then p 2 = 2 and p 1 defined in (C.50) are the optimal prices by numerical enumeration, thusz OP (e)=z CP =z CM (e)=z NCM = 128% by (C.51). 232 C.8.2 Proof of Proposition 3.4.2 We will construct an example in which the ratioz CM (e)=z NCM is unbounded. Let us consider Example 3.4.1 but with the following modified parameters: G 8 > < > : U[0; 1]; with probability 1; U[2; 3=]; with probability; ; V U[0; 1=]; e 1 = 2: The expected customer valuation for a ride is bounded above by 2.25 for all 2 (0; 0:5), whereas the expected driver valuation for the maximum revenue rate increases to infinity as# 0. Therefore, if is very small, the firm should charge high prices in some areas in order to attract drivers to the system. However, if the price in an area is greater than or equal to 1, at most fraction of the customers will arrive at that area. Consequently,p = 1 andz CP () = 2 by Propostion 3.4.1. Notice that G(p) = 8 > > > > > < > > > > > : (1)p; if 0p< 1; 1; if 1p 2; 35 32 + 2 32 p; if 2<p 3=; 8p2 [0; 3]: We will construct an OP equilibrium withO( p ) total matching rate by the optimization problem (3.14). Let us fixp 2 = 3=(2) and assume that 0p 1 1. Then, the constraint (3.14b) becomes 1 (1)p 1 + 3 6 4 2 = 2 p 1 (1)p 2 1 + 9 12 8 : (C.52) The uniquep 1 2 [0; 1] which solves the equality (C.52) is p 1 = 2 + 6 64 6 2 64 r 2 + 6 64 6 2 64 2 4 (1 2 ) 1 + 6 64 18 128 + 9 2 (64) 2 2 (1 2 ) (C.53) and the associated objective function value of the optimization problem (3.14) is 1 + 3 6 4 (1)p 1 ; (C.54) 233 wherep 1 is defined in (C.53). Since lim #0 1 + 3 64 (1)p 1 p = 1:2247; the objective function value in (C.54) isO( p ). Therefore, the total matching rate under the OP equilibrium specified in Proposition 3.5.1 Part 1 associated with the prices p 2 = 3=(2) and p 1 defined in (C.53) is O( p ). C.9 Information about Simulation Set-up In this section, we present additional information about the simulation set-up. We used Omnet++ v4.6 simulation freeware in our simulation experiments. Initialization At each run associated with each instance, the system starts empty and the warm-up period is 30,000 time units. In the time interval [0; 30000=24], external idle drivers arrive in the system with respect to a homogeneous Poisson process with the raten e =20. After time 30,000/24, external idle drivers arrive in the system as explained below in detail. In the time interval [0; 30000=12], external idle drivers arrive at each area equally likely. After time 30,000/12, external idle drivers choose the areas with the highest average profit as explained below in detail. By initializing the system in this way, we ensure that there will be enough driver mass in the system at the beginning of the warm-up period. After time 30000/12, the system evolves as explained below. Driver decisions If a driver in areak is matched with a typeij customer, then the profit of the driver from that matching is p ij c ki c ij . Let P k (t) denote the cumulative amount of profits earned by the idle drivers in areak2N up to timet2R + . LetA k (t) denote the cumulative number of idle driver arrivals to areak2N up to timet2R + . A k (t) is the sum of the number of external idle driver arrivals to areak up to timet and the number of idle drivers who reposition themselves to areak to look for customers in area k up to timet. We track the ratioP k (t)=A k (t) at all times in all areas. When an idle driver in areak2N makes a repositioning decision at timet2R + , he chooses the area argmax l2N fP l (t)=A l (t)c kl g. When an external idle driver arrives in the system at timet2R + , he chooses the area argmax l2N fP l (t)=A l (t)g. Moreover, the next external idle driver arrival occurs after an an exponentially distributed time with the rate n e V (max l2N P l (t)=A l (t)). When the patience of an idle driver waiting in areak is over before he is matched with a customer, that driver leaves the system with probability 1 and makes a repositioning decision with probability. 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Creator
Özkan, Erhun
(author)
Core Title
Essays on service systems with matching
School
Marshall School of Business
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Doctor of Philosophy
Degree Program
Business Administration
Publication Date
07/09/2018
Defense Date
04/02/2018
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University of Southern California
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asymptotic optimality,dynamic matching,fork-join network,OAI-PMH Harvest,optimization,service system
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Ward, Amy Ruth (
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), Drakopoulos, Kimon (
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), Mikulevicius, Remigijus (
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), Randhawa, Ramandeep (
committee member
)
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eozkan@usc.edu,erhunozkan@gmail.com
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Tags
asymptotic optimality
dynamic matching
fork-join network
optimization
service system