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Novel queueing frameworks for performance analysis of urban traffic systems
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Novel queueing frameworks for performance analysis of urban traffic systems
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Novel Queueing Frameworks for Performance Analysis of Urban Traffic Systems by Mohammad Motie 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 (Civil Engineering) December 2018 Dissertation Committee Prof. Ketan Savla (Chair) Prof. Maged Dessouky Prof. Roger Ghanem Prof. Erik A. Johnson Prof. James E. Moore Prof. Amy R. Ward Abstract Performance evaluation of transportation systems and adopting new technolo- gies to improve the existing level of service have significant effects on our society and economy. In this dissertation, we propose novel queueing frameworks to char- acterize traffic performance metrics such as throughput and travel time. Queueing models have been a compelling framework for analyzing traffic systems due to their ability to model congestion and service processes. However, most existing traffic queues rely on simplifying assumptions and study the average evolution of traffic flow. In particular, they often neglect the dependency of the performance of these queues on the state of traffic. Incorporating the state-dependent behavior in queueing models makes their analysis significantly complex. In this disserta- tion, we develop novel state-dependent queueing models for application to traffic systems. In particular, by generalizingprocessor sharing queues, we proposea novel Hor- izontal Traffic Queue (HTQ) in which vehicles on the road can be interpreted as jobs, and the road as a server. In this case, all jobs are processed simultaneously, and the service rate of a given job is time-varying and is equal to its instantaneous speed. The time-varying and state-dependent nature of the service rate also put the proposed HTQ within the class of state-dependent queueing systems. However, I the complex dependence of service rate on the state (i.e., vehicle locations) pre- cludes the use of existing tools from processor sharing queues and state-dependent queues for rigorous analysis. By using this queueing theoretic framework we esti- mate macroscopic performance measures such as throughput and travel time under different microscopic models. The finer properties of the departure process of the proposed queue depend on the dynamics of the minimum inter-vehicular distance. Motivated by the need to study such finer dynamics, we study the evolution of the Kullback-Leibler (K-L) divergence of the inter-vehicular spacings in between and during arrival and departure events. We consider both first and second order car- following models and, by extending busy period calculations for M/G/1 queue to our setting, we provide lower bounds for the throughput of the HTQ for different parameters of the car-following model. We also derive lower and upper bounds on the mean travel time of vehicles for a class of safe car-following behavior. We compare our simulations with real vehicle trajectory dataset. In the next step, we focus on another class of state-dependent queues to model signalized traffic intersections. In traffic intersections, red periods interrupt the service process of queues. By modeling red periods as server breakdowns or vaca- tions, we propose a novel vacation queueing model that allows derivation of the transient probability distribution of the number of queued vehicles. The duration of red periods are determined by the implemented control policy and can depend on the state of the system. In order to improve the accuracy of the model, we con- siderdeterministicinter-departuretimes,asopposedtoexponentialinter-departure times in often used the literature. Therefore, each leg of intersection is modeled as an M/D/1/N queueing model. We extend analytical transient departure process of M/D/1/N queues to our setup to derive the departure process of these queues II and derive transient probability distribution of queue length. Although deter- ministic inter-departure times increases the complexity of our model by causing a dependency between queue length and the departure process, comparison with microscopic simulation suggests improvement in accuracy. III Acknowledgement First and foremost, I owe the deepest gratitude to my advisor, Professor Ketan Savla. He has been immensely patient with all my research adventures. His unique skill for identifying important and challenging research problems, emphasis on the rigour and preciseness, attention for details, sharp insights and deep knowl- edge have greatly helped my research. With his efforts, I was able to observe my transformation from an immature first-year graduate student to an independent researcher. I would like to thank all of the great professors and instructors at USC that taught me difficult concepts. I would also like to thank my amazing friends. I shared tough and happy moments of this PhD with them. Their support and love made this journey very pleasing and enjoyable for me. I would like to extend my gratitude to my family for the encouragement, love, and continued support I have received over the past years. Last but not least, I would like to thank National Science Foundation (NSF) for supporting my research in part by NSF CMMI Project No. 1636377. IV Contents Abstract I Acknowledgement IV List of Figures VIII 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Background 7 2.1 Traffic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Macroscopic Models . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Traffic Queueing Models . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Vertical Queues . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Horizontal Queues . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Existing Traffic Queues for Signalized Traffic Intersections . . . . . 11 2.4 Relevant State Dependent Queueing Models . . . . . . . . . . . . . 14 2.4.1 Service and Arrival Distribution Dependent on Queue Length 15 2.4.2 Processor Sharing Queues . . . . . . . . . . . . . . . . . . . 15 2.4.3 Vacation Queuing Models . . . . . . . . . . . . . . . . . . . 16 3 Dynamical Analysis of a Horizontal Traffic Queue 18 3.1 The Horizontal Traffic Queue Setup . . . . . . . . . . . . . . . . . . 18 3.1.1 Dynamics of Vehicle Coordinates between Jumps . . . . . . 19 3.1.2 Change in Vehicle Coordinates during Jumps . . . . . . . . 20 3.1.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Dynamics in Between Jumps . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.2 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Dynamics at Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 V 4 Throughput and Service Rate Analysis of a Horizontal Traffic Queue 29 4.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Service Rate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.1 Service Rate Monotonicity between Jumps . . . . . . . . . . 35 4.2.2 Change in Service Rate at Jumps . . . . . . . . . . . . . . . 39 4.3 Busy Period Properties of the Horizontal Traffic Queue . . . . . . . 40 4.3.1 Expected Busy Period Duration . . . . . . . . . . . . . . . . 41 4.4 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4.1 Linear Case: m = 1 . . . . . . . . . . . . . . . . . . . . . . . 43 4.4.2 Monotonicity of Throughput in m and x 0 . . . . . . . . . . . 45 4.4.3 Finite Dimensional Approximation . . . . . . . . . . . . . . 48 4.4.4 ThroughputEstimationusingtheFiniteDimensionalApprox- imation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4.5 Throughput Bounds for the Super-linear Case from Busy Period Calculations . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.6 Throughput Bounds under Batch Release Control Policy . . 61 4.4.7 A second order model . . . . . . . . . . . . . . . . . . . . . 70 4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 Horizontal Traffic Queue over Non-periodic Road Segment: Performance Analysis and Case Study 76 5.1 The Basic HTQ Setup . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Release Policy and Car Following Model . . . . . . . . . . . . . . . 78 5.2.1 Release Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.2 Car Following Model . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Theoretical Bounds on Travel Time . . . . . . . . . . . . . . . . . . 83 5.3.1 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4 Simulation Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.1 Brief Description of the Data Set and Preprocessing . . . . . 90 5.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 91 6 Connection between Traffic Flow Capacity and the Throughput of Horizontal Traffic Queues 95 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 Car-following Models . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2.1 First Order Model . . . . . . . . . . . . . . . . . . . . . . . 97 6.2.2 Second Order Model . . . . . . . . . . . . . . . . . . . . . . 98 6.3 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3.1 First Order Model . . . . . . . . . . . . . . . . . . . . . . . 99 6.3.2 Second Order Models . . . . . . . . . . . . . . . . . . . . . . 107 VI 6.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7 Vacation Queueing Models for Analyzing Traffic Intersections 113 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3 Existing Queue Size Computation Techniques for Fixed-Time Policies119 7.3.1 Uninterrupted Queues . . . . . . . . . . . . . . . . . . . . . 119 7.3.2 Webster’s Model . . . . . . . . . . . . . . . . . . . . . . . . 119 7.3.3 Time-dependent Models . . . . . . . . . . . . . . . . . . . . 120 7.3.4 Queues with on/off Service . . . . . . . . . . . . . . . . . . . 121 7.3.5 Double Queue Model . . . . . . . . . . . . . . . . . . . . . . 122 7.4 Transient Queue Length Distribution Over Red and Green Periods . 127 7.4.1 Finite Capacity Queue . . . . . . . . . . . . . . . . . . . . . 128 7.4.2 Infinite Capacity Queue . . . . . . . . . . . . . . . . . . . . 134 7.5 Link-level Queueing Model with Spillbacks . . . . . . . . . . . . . . 138 7.5.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . 139 7.5.2 Computation of Queue Length Distributions . . . . . . . . . 143 7.6 Asymptotic Analysis under Fixed-time and Feedback Control Policies145 7.6.1 Fixed Time Control Policy . . . . . . . . . . . . . . . . . . . 146 7.6.2 Feedback Control Policy . . . . . . . . . . . . . . . . . . . . 148 7.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.7.1 Vertical Queues . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.7.2 Link-level Model . . . . . . . . . . . . . . . . . . . . . . . . 154 7.7.3 Control Policies . . . . . . . . . . . . . . . . . . . . . . . . . 156 8 Conclusions and Future Work 159 Reference List 162 A Appendix 169 A.1 Additional Technical Results for Chapter 4 . . . . . . . . . . . . . . 169 A.2 Additional Technical Results for Chapter 5 . . . . . . . . . . . . . . 173 A.2.1 Derivation of (5.6) . . . . . . . . . . . . . . . . . . . . . . . 173 A.3 Additional Technical Results for Chapter 6 . . . . . . . . . . . . . . 175 A.4 Additional Technical Results for Chapter 7 . . . . . . . . . . . . . . 176 A.4.1 Formulation for platoon arrivals . . . . . . . . . . . . . . . . 176 VII List of Figures 3.1 Illustration of the proposed HTQ with three vehicles. . . . . . . . . 19 4.1 Throughput for various combinations of m, L, and n 0 . The param- eters used in individual cases are: (a) ϕ = δ 0 , ψ = δ L , and n 0 = 0 (b) ϕ = δ 0 , ψ = δ L , and n 0 = 100 (c) ϕ = U [0,L] , ψ = U [0,L] , and n 0 = 0 (d)ϕ =U [0,L] ,ψ =U [0,L] , andn 0 = 100. In all the cases, the locations of initial n 0 vehicles were chosen at equal spacing in [0,L]. 32 4.2 f(y,m) vs. m for a typical y∈S 10 . . . . . . . . . . . . . . . . . . . 39 4.3 (a) Queue length process and (b) workload process during a busy period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 Decomposition of HTQ into HTQ1 and HTQ2 in series. . . . . . . . 62 4.5 Comparison between theoretical estimates of throughput from The- orem 2, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: L = 1, δ = 0.1, and (a) ϕ =δ 0 , ψ =δ L , (b) ϕ =U [0,L] , ψ =U [0,L] . . . . . . 72 4.6 Comparison between theoretical estimates of throughput from The- orem 2, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: L = 100, δ = 0.1, T = 10, and ϕ =δ 0 , ψ =δ L . . . . . . . . . . . . . . . . . . 73 VIII 4.7 Comparison between theoretical estimates of throughput from The- orem 3, and range of numerical estimates from simulations. The parameters used for this case are: L = 1, δ = 0.1, ϕ = δ 0 , ψ = δ L , w 0 = 1 and n 0 = 4, x 1 (0) = 0.6,x 2 (0) = 0.7,x 3 (0) = 0.8,x 4 (0) = 0.9. 73 4.8 Theoretical estimates of throughput from Theorems 4 for different valuesofη. The parameters used for this case are: L = 1,ϕ =U [0,L] , ψ =U [0,L] , andw 0 = 0 . Note that the vertical axis is in logarithmic scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.9 TheoreticalestimatesofthroughputfromTheorem5, andnumerical estimatesfromsimulationsfordifferentTheparametersusedforthis case are: ϕ =U [0,L] , ψ =U [0,L] , and (a) L = 1, (b) L = 100. . . . . 75 4.10 Comparison between the empirical expectation of the queue length andtheupperboundsuggestedbyRemark3. Weletthesimulations run up to time t = 80, 000. The parameters used for this case are: L = 1, m = 1, ϕ =δ 0 , ψ =δ L . For these values, we have λ max = 1. . 75 5.1 Aschematicofthehorizontaltrafficqueue(HTQ)setupshowingthe spatial distributionsϕ andψ for sideway arrivals and departures for vehiclesarrivingatthestartingpointoftheroadsegment, respectively. 77 5.2 A schematic view of storage with infinite queue capacity at the ori- gin. Vehicles are released from the storage on to the road segment under a release policy. . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 The first order car-following model described in (5.1). . . . . . . . . 80 IX 5.4 (a): The study area from NGSIM dataset corresponding to US-101, used in the case study. (Figure is obtained from [1]). (b): Part of trajectories for the left-most lane. Vehicle trajectories that enter at the beginning of the lane and leave at the end of it are shown in blue. Trajectories for the vehicles that enter the lane in the middle are shown in green, and the trajectories of the vehicles that leave the lane in the middle are shown in red lines. Note the backward shockwave due to slower vehicles. In this particular, we did not observe trajectories that join and leave the lane in the middle of section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5 Mean travel time comparison between the NGSIM dataset, lower bound from Theorem 6 and from microsimulations with Model 2 second car following rule; the latter two with and without cooper- ation. The desired speed distribution are modeled to be uniform distributions. The mean is the same 15 m/s in all scenarios, but standard deviations are different. . . . . . . . . . . . . . . . . . . . 93 5.6 Comparison between the bounds from Theorems 6 and 7 under dif- ferent values of θ = U [v min ,vmax] (uniform distribution) and4. The other parameters, not shown in the legends, are: (a) L = 500, v max = 30, and4 = 4; (b) L = 50, v min = 19, and v max = 23. . . . . 93 5.7 Comparison of theoretical bounds from Theorems 6 and 7 with (a) the first order car following model in (5.1) and (b) the second order car following model in (5.2) under different model parameters. The other parameters common to all the simulation scenarios are L = 250, θ = U [v min ,vmax] (uniform distribution) with v min = 20 and v max = 25,4 = 5, and a max = 4. . . . . . . . . . . . . . . . . . . . 94 X 6.1 A schematic view of the fundamental traffic diagram. k jam and q m denote the jam density and maximum capacity, respectively. . . . . 96 6.2 Average queue length in LTM and HTQ for various values of λ. Parameters of simulations are: L = 100,4 cr = 4,v max = 10. . . . . . 112 7.1 A schematic view of the saturation headway of vehicles,H [14]. Red dots are vehicles waiting in the queue. . . . . . . . . . . . . . . . . 114 7.2 A triangular traffic diagram. . . . . . . . . . . . . . . . . . . . . . 122 7.3 A link with its upstream queue (UQ) and downstream queue (DQ). 124 7.4 Two links in tandem with their UQ and DQ. . . . . . . . . . . . . 125 7.5 (a) A sample green period that covers five interval i.e. I 1 ,··· ,I 5 . The length of the last interval is different from others and is equal to G−bG/HcH. (b) A green period that starts with 2 initial vehicles. Arrows show the moments when the first two vehicles leave the queue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.6 (a) Two tandem queues. The queue in left is the storage with infi- nite buffer and zero service time, and the queue in right is a finite capacity queue with deterministic service times. (b) The two tan- dem queues are equivalent to one infinite buffer queue. Since the service time in the storage is zero, the equivalent queue is an M/D/1 queueing model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.7 A link connected to a traffic signal. Link is modeled with upstream and downstream queues, and a dummy storage queue to keep vehi- cles that are waiting to join the link. . . . . . . . . . . . . . . . . . 140 XI 7.8 Comparison between mean queue length under the proposed model (Section 7.4.2) and existing models in Section 7.3. The parameters for this simulation are: H = 2 (sec), ρ = 0.85, C = 50 (sec), and G =R = 25 (sec). . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.9 Comparison between mean queue length under the proposed model (Section 7.4.2) and Webster’s model. In the first 8 cycles, we have ρ = 0.6 and in the second 8 cycles, it is increased to ρ = 0.9 and in the last 8 cycles it is reduced to the initial value ρ = 0.6. Other parameters for this simulation are: H = 2 (sec), C = 50 (sec), and G = R = 25 (sec). The choice of these parameters ensures a small blocking probability. . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.10 Comparison between mean number of vehicles on link obtained through double queue models and microscopic simulations for ρ = 0.6. The parameters for this simulations are H = 1.5 (sec), C = 70 (sec), G = 45 (sec), N = 30, τ f = 10 (sec), and τ ω = 20 (sec). . . . 155 7.11 Comparison between mean number of vehicles on link obtained through double queue models and microscopic simulations for ρ = 0.95. The parameters for this simulations areH = 1.5 (sec),C = 70 (sec), G = 45 (sec), N = 30, τ f = 10 (sec), and τ ω = 20 (sec). . . . 155 7.12 Cumulative distribution of the number of vehicles on link at the end last red period obtained through double queue models and micro- scopic simulations. The parameters for this simulations the same as Figure 7.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.13 Mean queue length at the end of red periods for one street under fixed time control policy. Parameters are ρ = 0.8, G = 30 (sec), C = 60 (sec), and H = 2 (sec). . . . . . . . . . . . . . . . . . . . . 157 XII 7.14 Mean queue length at the end of red under proportionally fair con- trol policy. The system starts from equal green periods for both streets. Parameters are ρ 1 = 0.4, ρ 2 = 0.8, G 1 (0) = 30 (sec), G 2 (0) = 30 (sec), C = 60 (sec), N = 50, ΔG = 4 (sec), G min = 5 (sec), G max = 55 (sec), and H = 2 (sec). . . . . . . . . . . . . . . . 157 7.15 Mean green period for street 1 under proportionally fair control policy. The system starts from equal green periods for both streets. Parameters are ρ 1 = 0.4, ρ 2 = 0.8, G 1 (0) = 30 (sec), G 2 (0) = 30 (sec), C = 60 (sec), N = 50, ΔG = 4 (sec), G min = 5 (sec), G max = 55 (sec), and H = 2 (sec). . . . . . . . . . . . . . . . . . . 158 A.1 Aschematicviewof(a)f i (z),i ={1, 2, 3, 4}and(b)f(z) = P 4 i=1 f i (z) for a y∈S L 4 (L = 1) with y min = y 2 < y 4 < y 3 < y 1 = y max for a m< 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 XIII Chapter 1 Introduction 1.1 Motivation Urban traffic systems play a fundamental role in our society. Traffic congestion and the safety of these systems cause major societal issues, with negative economic and environmental consequences. Therefore, accurate analysis of traffic systems is required to evaluate their performance. The performance of traffic is usually measured through quantities such as travel time, throughput, and safety. The analysis of these systems reveals bottlenecks, and new technologies combined with new policies should be utilized to ease traffic congestions and increase mobility. In this dissertation, we introduce novel frameworks for analyzing and evaluating traffic systems and examine the effects of new technologies such as Vehicle-to- Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communication on improving existing systems. In recent years, our society has witnessed rapid advancements in technologies that can revolutionize our existing traffic systems. For example, the production of autonomousvehiclesbyleadingautomakersispoisedtobringfundamentalchanges in urban transportation systems. Moreover, it is envisioned that a robust V2V and V2I communication infrastructure will allow (semi-) autonomous vehicles to query the state of the traffic system at multiple spatial scales and execute desired control laws in response. Such architecture has the potential to save lives, prevent injuries, 1 ease traffic congestion, and improve the environment. The National Highway Traf- fic Safety Administration (NHTSA) has recently released an advance notice of rule- making that includes the proposal to create a new Federal Motor Vehicle Safety Standard to require V2V communication capability for light vehicles [56]. While NHTSA’s focus on the application of V2V technologies is for safety purposes, these technologies are capable of offering interesting opportunities to design microscopic (vehicle-level) interaction rules. For example, complex car-following models under which vehicles can communicate with one or more vehicles in front and behind can be designed to improve mobility while ensuring safety. From the perspective of analysis and control synthesis, these new possibilities raise several scientific questions. For example, for a given microscopic car-following model, it is necessary to rigorously characterize macroscopic traffic properties such as throughput and travel time. Without this characterization, designers cannot evaluate the performance of novel car-following models. Existing methodologies for studying the impact of autonomous and connected vehicles on traffic flow are mostly simulation-based and hence not suitable to develop fundamental insights and tradeoffs between various aspects of the proposed designed regulations. The available analytical techniques are either applicable only to closed systems i.e., consistingofafixednumberofvehicleswhosemotionsarepossiblycoordinatedbya leader, or they make quasi-static assumptions on the flow dynamics. Motivated by the posed scientific challenges, our objective is to develop a novel framework using the construct of Horizontal Traffic Queues (HTQ) to overcome these shortcomings. Analysis of existing traffic system is of great importance for detecting bottle- necks and evaluating policies. The variability and uncertainty in traffic streams increase the complexity of traffic systems. In particular, signalized traffic intersec- tions are often the bottlenecks of arterial networks, and, due to their complexity, 2 they have been an active field of research for decades. The objective of analyz- ing traffic intersections is to characterize the delay that vehicles incur at traffic intersections and the number of queued vehicles in each leg of the intersection. Each phase of traffic is constantly interrupted due to the existence of red periods. These interruptions, combined with variable and uncertain traffic demand and complex control policies make the analysis of signalized traffic intersections chal- lenging. Most of the existing models study the average evolution of these queues under constant conditions, fixed time policies, and simplifying assumptions. In particular, interruptions caused by red periods are often neglected and an approx- imate uninterrupted model is considered. Furthermore, most models are unable to capture link-level characteristics such as forward and backward travel time and spillbacks and their effects on traffic queues at intersections. In this dissertation, we develop a novel Vacation Queueing Model to explicitly model red periods as server vacations. We also integrate our proposed model into existing link level models to capture traffic behavior over the length of each link. In the next section, we briefly discuss key results presented in the dissertation. 1.2 Research Overview Queueing models have been a compelling framework for analyzing traffic sys- tems due to their ability to model congestion and service processes. The majority of traffic queues consider simplifying assumptions and neglect the dependency of the performance on the state of traffic. For example, the service rate of a traffic queue should depend on local congestion among vehicles. However, incorporating the state-dependent behavior in queueing models makes their analysis significantly 3 complex. In this dissertation, we develop novel state-dependent queueing models for the application to traffic systems. For the first-come first-served policy, existing state-dependent models usually consider service and arrival rates that depend on the queue length. Another, class of state-dependent queues is the processor sharing queueing model in which the service rate of each job is a function of the number of jobs in the system. We relate traffic queues on highways to a generalized processor sharing queueing model. In the proposed queueing model, vehicles on the road can be interpreted as jobs and the road as a server. In this case, all jobs are processed simultaneously, and the service rate of a given job is time-varying and is equal to its instantaneous speed. The time-varying and state-dependent nature of the service rate also put the pro- posed HTQ within the class of state-dependent queueing systems. However, the complex dependence of service rate on the state (i.e., vehicle locations) precludes the use of existing tools from processor sharing queues and state-dependent queues for rigorous analysis. By using this queueing theoretic framework one can estimate macroscopic performance measures such as throughput and travel time under dif- ferent microscopic models. Furthermore, we consider another state-dependent queueing model for the application of analyzing signalized traffic intersections. In particular, we model traffic intersections by using Vacation Queues in which vacations and the depar- tureprocessisstate-dependent. Servervacationsareusedtomodelredperiodsthat depend on the implemented control policy. The server is vacationing modification of a queue with Markovian arrivals and deterministic service times. The deter- ministic service times makes the departure process non-Markovian and increases the complexity of departure processes by introducing the dependency on the queue length. However, despite the increased complexity, this model allows us to capture 4 realistic scenarios and improve the accuracy of existing models. In the rest of this section, we summarize results of main chapters of this dissertation. We consider HTQ as a single traffic lane, where vehicles arrive according to a spatio-temporal Poisson process and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the speed of a vehicle is determined by a car-following model. We consider both first- and second-order car-following models. In Chapter 3, the dynamical analysis of HTQ in between arrivals and departures is provided. The service rate of HTQ is equal to the sum of the speeds of the vehicles and has a complex dependency on the state (vehicle locations) of the system. In Chapter 4, for the first-order model, we show the monotonicity of the service rate in between arrivals and departures. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. InChapter4, analyticalestimatesforthroughputunderfirstandsecond- order car-following models are provided. In the first-order model, the speed of a vehicle is proportional to a powerm> 0 of the distance to the vehicle in front. We provide numerical evidence that there exists a sharp phase transition atm = 1. In particular, throughput is unbounded for m∈ (0, 1) and bounded for m > 1. We develop analytical results that match the throughput profile as closely as possible. Moreover, a release control policy that controls the release of vehicles to the road is designed. The considered release control policy guarantees safety and facilitates our analysis. In Chapter 5, under a class of car-following models, lower and upper bounds on the mean travel time of vehicles are characterized. We also investigate the practical significance of our bounds and simulations using real traffic data sets. In Chapter 6, we relate different first- and second-order car-following models to the macroscopic capacity of the system. 5 In Chapter 7, we develop a novel vacation queueing model for deriving the tran- sient queue length distribution at signalized traffic intersections. In this model, each incoming street to the intersection is modeled to experience non-homogenous Poisson arrivals, and to have finite queue capacity. If a vehicle arrives and finds the queue at capacity, it will wait in an infinite capacity storage and will join the queue as soon as there is space available according to a first-come first-serve pol- icy. Each finite capacity queue is modeled as an M/D/1/N. We extend analytical transient departure process of M/D/1/N queues to our setup to derive the depar- ture process of these queues and derive transient probability distribution of queue length. Simulations suggest consistency between the queue sizes computed by the proposed model and other well-known models such as Webster and time-dependent traffic queue models. In the next step, to capture queue spillbacks, forward and backward shockwaves, and other link-level characteristics of traffic, we integrate our proposed queueing system with the double-queue model that is a link-level traffic model. In the original double-queue model, service times are assumed to be exponential. We formally argue that this assumption leads to over-estimation of queue length. We show, through comparison with microscopic simulations, that integrating M/D/1/N queues with double-queue model significantly improves the accuracy. 6 Chapter 2 Background In this chapter, we briefly review the literature that is related to our work. In particular, we discuss the existing literature on different traffic models and provide some examples of application of queueing theory in traffic engineering. We also relate the proposed HTQ to Processor Sharing (PS) queue and briefly discuss the literature on general processor sharing queues. 2.1 Traffic Models Various traffic models have been proposed in order to evaluate and estimate traffic variables. In this section, we explain two major traffic models: microscopic and macroscopic. 2.1.1 Microscopic Models Microscopic models provide a detailed description of traffic flow at the level of an individual vehicle. This involves specification of car following, lane changing and merging behavior, along with attributes such as type (light vs heavy weight, or transit vs. personal). Car-following models determine dynamics of individual vehicles by means of first[44]orsecond[20]orderordinarydifferentialequations. Thesemodelssimulate human drivers behavior when following a vehicle. However, different models may have different objectives. For example, these models can be designed to keep a 7 constant desired spacing between vehicles [68], a constant headway time between vehicles [30], or a minimum space and headway between vehicles [55]. In different traffic regimes, vehicles may choose different behaviors. Therefore, psycho-physical modelsalsoexist[65]thatincorporatethestateoftrafficinthecar-followingmodel. Inadditiontocar-followingmodels, thereexistvariousmodelsforlanechanging and merging behaviors. For example, the model by Gipps [21] considers different state of traffic and other factors such as the presence of a heavy vehicle or obstruc- tions. In another model [5], every lane changing behavior is divided to mandatory and discretionary lane change. Under a mandatory lane change, the driver must change the lane because of, for example, obstructions. In the discretionary lane change, however, the driver will change the lane if he perceives a better driving condition in the target lane. Microscopic models are at the heart of microscopic traffic simulators, e.g., [2]. Because of their complexity, these models are suitable for detailed evaluation of traffic facilities such as intersections and their applicability is usually restricted to small areas. 2.1.2 Macroscopic Models Macroscopic models capture traffic properties in lesser detail by considering aggregate quantities such as traffic density and flow. As such, macroscopic models are suitable for modeling large scale transportation systems. In the following, we briefly review two well-known macroscopic traffic models. Hydrodynamical Models A well-known class of models for highway traffic are hydrodynamic models, e.g., the LWR model [43] and the corresponding discretized versions, e.g., see [16]. 8 LWR models use kinetic wave theory combined with fundamental diagram in order to characterize traffic state throughout the time and space domain. It is assumed that stationary fundamental diagrams apply even when traffic is not stationary. In order to estimate traffic density and flow by means of LWR models, computation requires discretization over both space and time. Link Models Another macroscopic model that is widely used is Link Transmission Models (LTM) [80] (see [32] for a continuous time formulation). In LTM model, kinetic wave theory is used to directly evaluate cumulative flow past by any point at any given time. LTM model is based on Newell’s simplified theory of kinetic waves [54]. In particular, by assuming a triangular fundamental diagram, this model enables the computation of cumulative number of vehicles passed by point directly from boundary or initial conditions without computing this quantity at intermediate times and positions. 2.2 Traffic Queueing Models Queueing systems are a compelling modeling traffic framework due to their capabilitytocapturecongestion,toallowentryandexitofvehicles,andtofacilitate study of relevant metrics for system performance such as throughput and system time. 2.2.1 Vertical Queues Vertical queueing models (or point queues) are characterized by the fact that vehiclestravelatmaximumspeeduntiltheyhitacongestionspot(e.g., asignalized 9 traffic intersection) where all vehicles queue on top of each other. Therefore, the queuedoesnotconsumeanyphysicallength, thusthename point queue. Moreover, point queues usually represent the dynamics of queue length in terms of queues as flow of arrivals and departures and exit capacity. In other words, individual vehicles are abstracted by flow of traffic. This abstraction can lead to abnormal behaviros e.g. negative queue length. In queueing theory, such kind of queues is called fluid queue that provides average behavior of a queuing system. Vickery formulated a simple point queue for traffic queues [73]. However, it has been shown that this model can lead to negative queue length. Moreover, queue length is not constrained by the maximum storage capacity of road. In order to address this issues, double-queue model [60] has been proposed. In this model, each link consists of two point queues, one at the end and one beginning of the link. These two point queues are assumed to be finite capacity in order to model blockingandspillbacks. Thesemodelsarewidelyusedtoestimatetheperformance of traffic networks [58, 71, 4] and for traffic signal analyzing and planning [59]. In addition to fluid queues, stochastic queues are also used in modeling vertical queues. In stochastic queues,the complexity of model increases, and a probabilistic characterization of arrival and departure processes combined with queueing policy (e.g. FirstComeFirstServed)isrequired. Researchershavemodeledtrafficqueues in this way; for example, the queue length and waiting time of a minor traffic stream at an unsignalized intersection where major traffic stream has high priority is studied in [70] and [26]. In [27], a vertical single server queue is utilized to model the queue length distribution at signalized intersections. In traffic queues, the service rate highly depends on the state of the system. In [31, 24], a state- dependent queuing system is used to model vehicular traffic flow where the service ratedependsonthenumberofvehiclesoneachroadlink. Althoughtheservicerate 10 in traffic queues depend on the congestion, this dependency is through microscopic congestion and not macroscopic congestion such as queue length. 2.2.2 Horizontal Queues As we discussed, vertical queues are often fluid queues that neglect link-level characteristics. Therefore, their capability to accurately model microscopic vehicle interactions, and hence spillbacks or wave propagation is severely limited. On the other hand, horizontal traffic queues are more accurate at modeling spatial interac- tions between the vehicles, and hence are able to incorporate blocking, congestion and spill-backs [36, 81]. However, the literature on horizontal traffic queues is very limited, and the horizontal traffic queue terminology has been primarily used to study macroscopic traffic flow, e.g., see [28]. While such models capture the macroscopic relationship between traffic flow and density, a rigorous description and analysis of an underlying queue model is lacking. Indeed, to the best of our knowledge, there is no prior work on the analysis of a traffic queue model that explicitly incorporates car-following behavior. 2.3 Existing Traffic Queues for Signalized Traffic Intersections In Sections 2.2.1 and 2.2.2, we reviewed existing traffic queues for modeling urban networks that incorporate highways and traffic junctions. In this section, we narrow down our focus on queueing models specifically proposed for analyzing signalized traffic intersections. Analysisofqueuelengthatsignalizedintersections, underagivencontrolpolicy has been the topic of research for decades. Delay and the number of queued 11 vehicles in the intersections are some of the quantities of interest in analyzing the performance of intersections. Uncertain and time-varying traffic conditions, interruptions in service due to red periods, and queue spillbacks due to finite capacity are some of the factors that make analysis of these systems difficult. In many of existing results in the literature, the mean queue size (or queue length) or delay is analyzed under steady state conditions, and stationary assump- tions on arrival rate. In a seminal work by Webster [76], in order to capture randomness in vehicle arrivals, an uninterrupted queueing model is considered. Webster’s formula characterizes the steady state mean delay under fixed time con- trol policy and stationary traffic conditions. There has been many other studies, e.g. [8, 50, 27], concerned with steady state performance measures of traffic signals under fixed time control policy. In traffic signals, however, it may take a very long time before queues reach the steady state, therefore, relaxation times are large [14] and may not be comparable to signal cycles. Furthermore, red periods introduce service interruption which naturally leads to intra-cycle fluctuation even under sta- tionary arrival processes [52]. However, most of existing time-dependent models in literature that capture transient behavior of traffic condition usually characterize queue length at specific points in a cycle. For example, in Akcelik’s model [6], the time-dependent mean value of over-flow queue (i.e. the number of vehicles at the end of green periods.). However this model is limited to stationary arrival processes. A full characterization of fluctuations of queue length over the entire cycle has significant value for evaluating implemented control policies [74]. Moreover, uncertainty of traffic flows and the interruptions caused by traffic signalscreateshighrandomnessandvariabilityintheperformanceofthesesystems [18]. Therefore, mean values of performance measures of traffic signals are not 12 enough for a proper evaluation and design of control policies. In [27], the steady- state distribution of queue length at fixed-time traffic signals is derived, but the temporal distribution is not studied. [9] uses Markov chain theory to derive the queue length distribution at a specific point of a cycle under fixed-time policies. [75] also modeled queue length process in fixed time signals with a Markov chain, and obtained temporal distribution of queue length. However, in this work [75], the departure process is not fully characterized, and it is assumed that during each cycle a pre-determined and constant number of departure occurs. In a recent work [45], server vacation (or red periods) is modeled using a continuos-time Markov chain that alternates between green and red periods. However, as it is described in Section 7.3.4 and 4.5, this model leads to random green and red periods with high variancethatcausessignificantover-estimationofaveragequeuelength. Inanother line of work [62], vacation queues with pre-determined and deterministic red and green periods are considered to derive the long run distribution of queue length. However, in this model, it is assumed that service start and completion epochs can only occur at the start or end of contiguous and coarse time intervals. This implies that if a vehicle arrives in the middle of one of these intervals, even if the intersectionisidle, ithastowaituntiltheendofthatintervalforitsservicetostart. Moreover, inthiswork, finitecapacityqueuesandqueuespillbacksarenotmodeled and analysis is limited to fixed time control policy. In most of the aforementioned models, a fixed time control policy is considered and very little consideration has been given to the computation of queue length distribution under adaptive control polices. Moreover, spillbacks that significantly affect the performance of traffic signals are not fully captured in these models. Interruptions of service at traffic signals caused by red periods are either pre- determined, under fixed time policy, or have complex dependency to the state of 13 traffic intersection, under adaptive control policies. This adds another layer of complexity to the dynamics of traffic queues at intersections. In fixed time policy, the simplest analytical approach to incorporate this aspect is to model each street to have a continuous stream of departure at a reduced rate which is equal to the product of the saturation headway and the ratio of the allocated green time to the total cycle time. While this approach is widely used, e.g., in Webster’s formula or in store-and-forward model [3], it fails to provide insight into the intra-cycle variations in queue sizes. Therefore, such approximations typically underestimate maximum queue sizes during a cycle, and hence underestimate spillbacks [52]. In order to model finite capacity of queues and spillbacks, double-queue model [60] has been proposed. Double-queue is a link model that is capable of capturing free-flow travel time delay when the link is in free-flow state, and backward shock- wave time delay when the queue is in congested state. Therefore, it is capable of modeling spillbacks. In this link model, there are two queues at the upstream and downstream of link. In Section 7.3 we further discuss details of models used in analyzing traffic intersections and are closely related to our work. 2.4 Relevant State Dependent Queueing Models In section, we briefly review the existing state dependent traffic queues that are relevant to this research. In particular, we first discuss processor sharing queues and its connection to horizontal traffic queues; then, we review vacation queueing models that are relevant to our work on analyzing traffic intersections. 14 2.4.1 Service and Arrival Distribution Dependent on Queue Length Simplest state-dependent queues have exponential service system, but arrival andserviceratesdependonthequeuelength. In[47],M n /M n /1queuesarestudied and the fluid limit of these queues are derived. A similar queueing model is used in [24] to model traffic flows. However, the dependency of state in real traffic queues is through the local congestion and position of vehicles; therefore, aggregate quantities such queue length cannot capture this dependency. 2.4.2 Processor Sharing Queues The considered HTQ in this research has an interesting connection with pro- cessor sharing (PS) queues. A characteristic feature of PS queues is that all the outstanding jobs receive service simultaneously, while keeping the total service rate of the server constant. The simplest model is where the service rate for an indi- vidual job is equal to 1/N, where N is the number of outstanding jobs. In our proposed system, one can interpret the road segment as a server simultaneously providing service to all the vehicles, with the service rate of an individual vehicle equal to its speed. This natural analogy between HTQ and PS queues, to the best of our knowledge, was reported for the first time in [51]. The 1/N rule applied to our setting implies that all the vehicles travel with the same speed. Clearly, such a rule, or even the general discriminatory PS disciplines, e.g., see [39], are not applicable to the car following models considered in this chapter. Indeed, the proposed HTQ is best described as a state-dependent PS queue. In the PS queue literature, the focus has been on the sojourn time and queue length distribution. For example, see [61] and [79] for M/G/1-PS queue and [22] 15 for G/G/1-PS queue. Fluid limit analysis for PS queue is provided in [11] and [23]. However, relativelylessattentionhasbeenpaidtothethroughputanalysisofstate- dependent PS queues. In [53, 35, 12], throughput analysis for state-dependent PS queues is provided, where throughput is defined as the quantity of work achieved by the server per unit of time. 2.4.3 Vacation Queuing Models Queueing models have been an appealing framework for analyzing signalized traffic intersections. Traffic queues, due to red periods, experience service inter- ruptions. Most of existing models, however, consider uninterrupted models as an approximation for traffic queues. In order to explicitly model service interruptions due to red periods, we note that each queue in a signalized traffic intersection is reminiscent of vacation queues [17]. In vacation queueing models, server takes vacationforaspecificperiodoftime. Thesemodelsariseinavarietyofapplications such as communication, manufacturing and other stochastic systems. The perfor- mance of vacation queues highly depends on the implemented policy. Vacation policies can be categorized into two main classes of exhaustive and non-exhaustive policies [72]. In exhaustive policies, the server is allowed to take a vacation only when the system becomes empty. On the other hand, when the server follows a non-exhaustive policy, it may take a vacation when some customers are still in the system. The work by [15] characterizes a queueing model under exhaustive pol- icy; however, queues with exhaustive policies cannot model regular traffic signal policies such as fixed-time control policy. There exist a variety of non-exhaustive policies. For example, in the gated policy [10], the server only serves those cus- tomers present in the system at the beginning of the service period and then takes vacation after they are served. However, this policy is also not applicable to traffic 16 queues at the intersections. Non-gated Time-limited (NT) policy is the closest policy to fixed-time control policy for traffic signals. Under NT policy, the server takes a vacation when either 1) its service time reaches a given maximum value or 2) the system becomes empty [72]. Although traffic signals may not switch to red when there is no vehicle, NT policy may provide reasonable evaluation for heavy regimes. [41] analyzed vacation queues under NT policy and derived the Laplace transform of the amount of work in the queue. However, the their solution cannot be evaluated analytically and requires careful approximations for computation purposes. Moreover, the transient behavior of the system and the queue length distribution is not studied. Although this model can be applicable to traffic queues under heavy regimes, it is limited to fixed time control policies. Recently [52, 62], vacation queues are used to analyze fixed-time signalized traffic intersections. In these models, an M/D/1 queue with pre-determined period of vacations is considered to obtain the distribution of queue length. 17 Chapter 3 Dynamical Analysis of a Horizontal Traffic Queue We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the motion of vehicles is governed by a car following model which is described shortly. The finer properties of the departure process of the proposed queue depend on the dynamics of the minimum inter-vehicular distance. Motivated by the need to study such finer dynamics, we study the evolution of the Kullback-Leibler (K-L) divergence of the inter-vehicular spacings in between and during arrival and departure events. Our analysis here is facilitated by the fact that the dynamics in between jumps is reminiscent of consensus dynamics on directed graphs. While non-quadratic Lyapunov functions for such dynamics have been studied before, we derive explicit rate of convergence with respect to K-L divergence. 3.1 The Horizontal Traffic Queue Setup Consider a periodic road segment of length L; without loss of generality, we assume it be a circle. Starting from an arbitrary point on the circle, we assign coordinates in [0,L] to the circle in the clock-wise direction (See Figure 3.1). 18 Vehicles arrive on the circle according to a spatio-temporal process: the arrival process{A(t),t≥ 0}, is assumed to be a Poisson process with rate λ > 0, and the arrival locations are sampled independently and identically from a spatial dis- tribution ϕ and mean value ¯ ϕ. Without loss of generality, let the support of ϕ be supp(ϕ) = [0,`] for some `∈ [0,L]. Upon arriving, vehicle i travels distance d i in a counter-clockwise direction, after which it departs the system. The travel distances{d i } ∞ i=1 are sampled independently and identically from a spatial distri- bution ψ with support [0,R] and mean value ¯ ψ. Let the set of ϕ and ψ satisfying the above conditions be denoted by Φ and Ψ respectively. The stochastic pro- cesses for arrival times, arrival locations, and travel distances are all assumed to be independent of each other. HTQ1 Release Control Policy HTQ2 HTQ1 HTQ2 (a) (b) 1 x 2 x 3 x 1 y 2 y 3 y 0 L HTQ1 HTQ2 HTQ1 Release Control Policy HTQ2 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L Figure 3.1: Illustration of the proposed HTQ with three vehicles. 3.1.1 Dynamics of Vehicle Coordinates between Jumps Let the time epochs corresponding to arrival and departure of vehicles be denoted as{τ 1 ,τ 2 ,...}. We shall refer to these events succinctly as jumps. We now formally state the dynamics under this car-following model. We describe the dynamics over an arbitrary time interval of the kind [τ j ,τ j+1 ). Let N ∈ N be 19 the fixed number of vehicles in the system during this time interval. Define the inter-vehicle distances associated with vehicle coordinates x∈ [0,L] N as follows: y i (x) = mod (x i+1 −x i ,L), i∈{1,...,N} (3.1) where we implicitly let x N+1 ≡ x 1 (See Figure 3.1 for an illustration). Note that the normalized inter-vehicle distances y/L are probability vectors. When inside the queue, the speed of every vehicle is proportional to a power m > 0 of the distance to the vehicle directly in front of it. We assume that this power m > 0 is the same for every vehicle at all times. Then, starting with x(τ j )∈ [0,L] N , the vehicle coordinates over [τ j ,τ j+1 ) are given by: x i (t) = mod x i (τ j ) + Z t τ j y m i (x(z))dz,L ! , ∀i∈{1,...,N}, ∀t∈ [τ j ,τ j+1 ), (3.2) Remark 1. It is easy to see that the clock-wise ordering of the vehicles is invariant under (3.1)-(3.2). The dynamics in inter-vehicle distances is given by: ˙ y i =y m i+1 −y m i , i∈{1,...,N} (3.3) where we implicitly let y N+1 ≡y 1 . 3.1.2 Change in Vehicle Coordinates during Jumps Let x(τ − j ) = x 1 (τ − j ),...,x N (τ − j ) ∈ [0,L] N be the vehicle coordinates just before the jump at τ j . If the jump corresponds to the departure of vehicle k∈ 20 {1,...,N}, then the coordinates of the vehicles x(τ j ) = (x 1 (τ j ),...,x N−1 (τ j ))∈ [0,L] N−1 after re-ordering due to the jump, for i∈{1,...,N− 1}, are given by: x i (τ j ) = x i (τ − j ) i∈{1,...,k− 1} x i+1 (τ − j ) i∈{k + 1,...,N− 1}. Analogously, if the jump corresponds to arrival of a vehicle at locationz∈ [0,`] in between the locations of the k-th and k + 1-th vehicles at time τ − j , then the coordinates of the vehicles x(τ j ) = (x 1 (τ j ),...,x N+1 (τ j )) ∈ [0,L] N+1 after re- ordering due to the jump, for i∈{1,...,N + 1}, are given by: x k+1 (τ j ) =z x i (τ j ) = x i (τ − j ) i∈{1,...,k} x i−1 (τ − j ) i∈{k + 2,...,N + 1}. 3.1.3 Notations LetR,R + , andR ++ denote the set of real, non-negative real, and positive real numbers, respectively. LetN be the set of natural numbers. Ifx 1 andx 2 are of the same size, then x 1 ≥x 2 implies element-wise inequality between x 1 and x 2 . If x 1 andx 2 are of different sizes, thenx 1 ≥x 2 implies inequality only between elements which are common tox 1 andx 2 – such a common set of elements will be specified explicitly. For a setJ, let int(J ) and|J| denote the interior and cardinality of J, respectively. Given a∈R, and b> 0, we let mod (a,b) :=a−b a b cb. LetS L N be the N− 1-simplex over L, i.e.,S L N = n x∈R N + | P N i=1 x i =L o . When L = 1, we shall use the shorthand notationS N . When referring to the set{1,...,N}, for brevity, we let the indices i =−1 and i = N + 1 correspond to i = N and i = 1 21 respectively. Also, for p,q∈S N , we letH(p) andD(p||q) denote the entropy of p, and K-L divergence of q from p, respectively, i.e., H(p) :=− N X i=1 p i log(p i ), D(p||q) := N X i=1 p i log p i q i ! . (3.4) Also, we define two permutation matrices, P + ∈{0, 1} N×N , P − ∈{0, 1} N×N , as follows: P + = 0 N−1 I N−1 1 0 T N−1 , P − = 0 T N−1 1 I N−1 0 N−1 , (3.5) where 0 N and 1 N stand for vectors of size N, all of whose entries are zero and one respectively. We shall drop N from 0 N and 1 N whenever it is clear from the context. Dynamical Analysis In the rest of this chapter, we study the dynamics in inter-vehicular spacing in between and during jumps, i.e., events corresponding to arrival and departure of vehicles. In particular, we characterize the dynamics in K-L divergence of inter- vehicular spacings in between and during jumps under dynamics in (3.3) when m = 1. For the simplicity of our analysis, without loss of generality, we assume that L = 1. 22 3.2 Dynamics in Between Jumps In between the jumps, the system is close i.e. N(t) =n is the constant number of vehicles in the system. In this case, the dynamics in (3.2), with m = 1, can be succinctly rewritten as: ˙ x =Ax +B, (3.6) where A∈{−1, 0, 1} n×n is given by A = −1 1 0 ··· 0 0 −1 1 ··· 0 . . . . . . . . . . . . . . . 1 0 0 ··· −1 , B = 0 0 . . . 1 . By observation, one can see thatA =P + −I, whereP + is defined in (3.5). In this case, since there is no arrival or departure, we can simplify the indices of x and y in (3.2) such that for i∈{1,··· ,n− 1} and x i (0)≤ x i+1 (0). Therefore, when m = 1, the speed of i-th vehicle is given as ˙ x i = y i . The inter-vehicle spacings {y i } n i=1 constitute a probability vector ( P n i=1 y i = 1, when L6= 1 we shall consider the normalized vector). The dynamics in y i is given by ˙ y =Ay, (3.7) whose solution is given by y =e tA y(0). The dynamics in (3.7) is reminiscent of linear consensus dynamics on directed graphs. It is easy to see that the only equilibrium for (3.7) is y i = 1/n for all i∈{1,...,n}. The stability analysis of this equilibrium has attracted consider- able attention from the research community, primarily using quadratic Lyapunov 23 functions. For example, [57] shows that||y(t)− 1 n || 2 2 ≤e λ 2 t ||y(0)− 1 n || 2 2 whereλ 2 is thesecondlargesteigenvalueofA+A T . Inourcase,λ 2 =−4 sin 2 (2π/n). However, this result readily does not provide a useful lower bound on the minimum inter- vehicular distance. Motivated by the recent interest in non-quadratic Lyapunov functions for linear consensus dynamics, e.g., see [48], we study the dynamics in the K-L divergence associated with the inter-vehicular distances, i.e., D(y|| 1 n ). 3.2.1 Stability Analysis The following result establishes asymptotic stability of y ∗ =1/n. Proposition 1. Under the dynamics in (3.7), we have lim t→+∞ y(t) = y ∗ for all y(0)∈S n . Proof. Consider the following candidate Lyapunov function V (y) :=D (y||y ∗ ) = n X i=1 y i log (ny i ) = n X i=1 y i log(y i ) + logn. (3.8) The time derivative of V (y) is given by: ˙ V (y) = n X i=1 (1 + log(y i )) ˙ y i = n X i=1 ˙ y i + n X i=1 log(y i ) (y i+1 −y i ) = n X i=1 y i log y i−1 y i ! =−D y||P − y , (3.9) where, P − is a permutation matrix defined in (3.5). Now, D (y||P − y)≥ 0 for all y∈S n , where the equality holds true if and only if y is such that y = P − y, i.e., when y =y ∗ . This establishes the proposition. 24 3.2.2 Rate of Convergence While Proposition 1 establishes thatV (y(t)) goes to zero asymptotically under (3.7), we now derive rate of convergence. For this purpose, we need the following two lemmas. Lemma 1. For all y∈S n we have, D(y||y ∗ )≤D(y||P − y) + logn (3.10) Proof. Let g(y) = D(y||y ∗ )−D(y||P − y)− logn. It suffices to show that g(y) is non-positive. If we expand g(y), we get g(y) =D(y||y ∗ )−D(y||P − y)− logn = n X i=1 y i log (ny i )− n X i=1 y i log (y i /y i−1 )− logn = n X i=1 y i log (y i−1 )≤ 0 (3.11) where the third equality follows by noting that log (ny i ) = log (n) + log (y i ) and the last inequality follows since y i ∈ [0, 1],i∈{1,··· ,n}. This establishes this lemma. The following proposition characterizes the convergence rate of V (y(t)). Proposition 2. Under the dynamics in (3.7), we have V (y(t))≤e −t (V (y(0))− logn) + logn ∀t≥ 0 ∀y(0)∈S n Proof. By Lemma 1, (3.8) and (3.9), ˙ V (y(t)) =−D(y(t)||P − y(t))≤−D(y(t)||y ∗ ) + logn =−V (y(t)) + logn (3.12) 25 Integrating the above inequality gives this proposition. We now derive a relationship between the minimum inter-vehicle spacing and V (y(t)), and then utilize Proposition 2 to get an estimate of the dynamics in y min . Let the minimum and maximum inter-vehicle distances be denoted by y min (t) and y max (t) respectively, i.e., y min (t) := min i∈{1,...,n} y i (t) and y max (t) := max i∈{1,...,n} y i (t). Further, let4 y (t) :=y max (t)−y min (t)≤ 1. Note that4 y (t) = 0 corresponds to equilibrium. The following proposition gives an upper bound for 4 y (t). Proposition 3. Under the dynamics in (3.7), we have 4 y (t)≤ min 1, q 2 (e −t (V (0)− logn) + logn) ∀t≥ 0. (3.13) Proof. We recall Pinsker’s inequality, (e.g., see [13]), adapted to our setting as, ky−y ∗ k 2 1 2 ≤V (y). (3.14) Also, note that 4 y (t)≤|y max − 1 n | +|y min − 1 n |≤ky−y ∗ k 1 (3.15) This combined with (3.14) implies that4 y (t)≤ q 2V (y(t)) and this combined with Propostion 2 gives the proposition. 3.3 Dynamics at Jumps So far, our analysis has focused on the case when n is fixed. In other words, this analysis is aimed at the evolution of the system in between jumps, i.e., arrival 26 and departure epochs. Now, we give bounds on the change in the value of V (y) due to arrival and departure. The next lemma gives such an upper bound for an arrival epoch. Lemma 2. Consider an arrival epoch, when the number of vehicles in the system increases from n to n + 1. Let the inter-vehicle distances just before and upon arrival be y∈S n and y + ∈S n+1 respectively. Then, V (y + )−V (y)≤ log 1 + 1 n Proof. Without loss of generality, assume that, at the moment of arrival, y 1 gets split into y + 1,1 and y + 1,2 , such that y + 1,1 +y + 1,2 = y 1 , and all other y i , i∈{2,...,n} remain unchanged. Therefore, V (y + )−V (y) = y + 1,1 log y + 1,1 y 1 ! +y + 1,2 log y + 1,2 y 1 ! + log n + 1 n = y 1 y + 1,1 y 1 log y + 1,1 y 1 ! + y + 1,2 y 1 log y + 1,2 y 1 !! + log 1 + 1 n =−y 1 H y + 1,1 y 1 , y + 1,2 y 1 ! + log 1 + 1 n ≤ log 1 + 1 n . Similarly, the following lemma provides a bound when the jump corresponds to departure of a vehicle. 27 Lemma 3. Consider a departure epoch when the number of vehicles in the system decreases from n to n− 1. Let the inter-vehicle distances just before and after departure be y and y − respectively. Then, V (y − )−V (y)≤ 2y max + log 1− 1 n Proof. Without loss of generality, assume that, at the moment of departure, y 1 and y 2 merge to give y − 1 . Other y i , i∈{3,...,n} remain unchanged. Therefore, V (y − )−V (y) =−y − 1 y 1 y − 1 log y 1 y − 1 ! + y 2 y − 1 log y 2 z 1 ! + log 1− 1 n = y − 1 H y 1 y − 1 , y 2 y − 1 ! + log 1− 1 n ≤ (y 1 +y 2 ) + log 1− 1 n ≤ 2y max + log 1− 1 n It should be noted that Lemmas 2 and 3 hold true for arbitrary arrival and departure locations on the circle. 28 Chapter 4 Throughput and Service Rate Analysis of a Horizontal Traffic Queue In this chapter we study the throughput of HTQ. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. We consider the similar HTQ as described in Section 3.1. When inside the queue, the motion of vehicles is determined by a car-following model. We consider both first and second order models. In the first order model (3.2), the speed of a vehicle is proportional to a power m > 0 of the distance to the vehicle in front. We characterize throughput of HTQ under different values of m. In the second order model, the acceleration is a function of the deviate of the inter-vehicle distances from desired distance. In the linear case (m = 1), i.e., when the speed of every vehicle is proportional to the distance to the vehicle directly in front, the periodicity of the road segment implies that the sum of the speeds of the vehicles is proportional to the total length of the road segment, i.e., it is constant. This feature allows us to exploit the equivalence between workload and queue length to show that, independent of the initial condition and almost surely, the throughput is the inverse of the time required by a solitary vehicle to travel average distance. 29 We prove the remaining bounds on the throughput for the nonlinear case as follows. The standard calculations for joint distributions of duration and num- ber of arrivals during a busy period for M/G/1 queue are extended to the HTQ setting, including for non-empty initial conditions. These joint distributions are used to derive probabilistic upper bounds on queue length over finite time hori- zons for HTQ for them> 1 case. Such bounds are optimized to get lower bounds on throughput defined over finite time horizons. Simulation results show good comparison between such lower bounds and numerical estimates. We also analyze throughput in the sub-linear (m< 1) and super-linear (m> 1) and the second order model under perturbation to the arrival process, which is attributed to the additional expected waiting time induced by a release control policy that adds appropriate delay to the arrival times to ensure a desired min- imum inter-vehicle distance4 > 0 at the time of a vehicle joining the HTQ. Since the minimum inter-vehicle distance is non-decreasing in between arrivals and jumps, this implies an upper bound on the queue length which is inversely proportional to4. We derive a lower bound on throughput for a given combina- tion of maximum allowable perturbation. In particular, in the first order model, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded, asm→ 0 + . Also, for the second order car-following model, we design a release control policy for which we quantify a lower bound on the throughput of the system. The designed release control policy ensures a minimum inter-vehicle distance among vehicle inside the queue. Therestofthechapterisorganizedasfollows. Theformaldefinitionofthrough- put is provided in Section 4.1. Key busy period properties for the M/G/1 queue are extended to the HTQ case in Section 4.3. Throughput analysis is reported in 30 Section 4.4. Simulations are presented in Section 4.5. In this chapter, we use the same set of notations introduce in Section 3.1.3. 4.1 Problem statement Let x 0 ∈ [0,L] n 0 be the initial coordinates of n 0 vehicles present at t = 0. An HTQ is described by the tuple (L,m,λ,ϕ,ψ,x 0 ). Let N(t;L,m,λ,ϕ,ψ,x 0 ) be the corresponding queue length, i.e., the number of vehicles at time t for an HTQ (L,m,λ,ϕ,ψ,x 0 ). For brevity in notation, at times, we shall not show the dependence of N on parameters which are clear from the context. In this chapter, our objective is to provide rigorous characterizations of the dynamics of the proposed HTQ. A key quantity that we study is throughput, defined below. Definition 1 (Throughput of HTQ). Given L > 0, m > 0, ϕ ∈ Φ,ψ ∈ Ψ, x 0 ∈ [0,L] n 0 , n 0 ∈N and δ∈ [0, 1), the throughput of HTQ is defined as: λ max (L,m,ϕ,ψ,x 0 ,δ) := sup{λ≥ 0 : Pr (N(t;L,m,λ,ϕ,ψ,x 0 )< +∞, as t→∞)≥ 1−δ}. (4.1) Figure 4.1 shows the complex dependency of throughput on key queue param- eters such as m and L. In particular, it shows that for every L, ϕ, ψ, x 0 and ϕ, the throughput exhibits a phase transition from being unbounded form∈ (0, 1) to being bounded for m> 1. Moreover, Figure 4.1 also suggests that, for sufficiently smallL, throughput is monotonically non-increasing inm, and that it is monoton- ically non-decreasing in m > 1, for sufficiently large L. Also, it can be observed that initial condition can also affect the throughput. We now develop analytical 31 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 m 0 1 2 3 4 5 6 7 +∞ λ max L=0.5 L=1 L=10 L=100 (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m 0 0.5 1 1.5 2 2.5 3 +∞ λ max L=0.5 L=1 L=10 L=100 L=300 (b) m 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 6 m a x 0 2 4 6 8 10 12 14 + 1 L = 0 : 5 L = 1 L = 1 0 L = 1 0 0 (c) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 m 0 2 4 6 8 10 12 +∞ λ max L=0.5 L=1 L=10 L=100 L=300 (d) Figure 4.1: Throughput for various combinations ofm,L, andn 0 . The parameters used in individual cases are: (a) ϕ = δ 0 , ψ = δ L , and n 0 = 0 (b) ϕ = δ 0 , ψ = δ L , and n 0 = 100 (c) ϕ =U [0,L] , ψ =U [0,L] , and n 0 = 0 (d) ϕ =U [0,L] , ψ =U [0,L] , and n 0 = 100. In all the cases, the locations of initial n 0 vehicles were chosen at equal spacing in [0,L]. results that match the throughput profile in Figure 4.1 as closely as possible. To that purpose, we will make extensive use of novel properties of busy period of the proposed HTQ, which could be of independent interest. 32 4.2 Service Rate Analysis In this section, we prove interesting properties of the service rate of HTQ. Service rate of HTQ is defined as the sum speed of vehicles on the road. This quantity has a complex dependency on the location of vehicles and the parameter m in the car-following model (3.2). We show that, when m = 1, the service rate is constant and equals to length of the road, L. In the non-linear case (m6= 1), the cumulative service rate of HTQ queue is constant if and only if all the inter-vehicle distances are equal. For all other inter-vehicle configurations, we show that the service rate is strictly decreasing (resp., strictly increasing) in the super-linear, i.e., m > 1 (resp., sub- linear, i.e.,m< 1) case. The service rate exhibits an another contrasting behavior in the sub- and super-linear regimes. In the super-linear case, the service rate is maximum (resp., minimum) when all the vehicles are co-located (resp., when the inter-vehicle distances are equal), and vice-versa for the sub-linear case. Using a combination of these properties, we prove that, when the length of the road segment is at most one, the throughput in the super-linear (resp., sub-linear) case is upper (resp., lower) bounded by the throughput for the linear case. For every y ∈ S L N , N ∈ N, L > 0, we let y min := min i∈{1,...,N} y i , and y max := max i∈{1,...,N} y i denote the minimum and maximum inter-vehicle distances respectively. It is easy to establish the following monotonicity properties of y min and y max . Lemma 4 (Inter-vehicleDistanceMonotonicityBetweenJumps). For anyy∈S L N , N∈N, L> 0, under the dynamics in (3.3), for all m> 0 d dt y min ≥ 0 & d dt y max ≤ 0. 33 Proof. Let y min (t) = y j (t), i.e., the j-th vehicle has the minimum inter-vehicle distance at time t≥ 0. Therefore, (3.3) implies that ˙ y min (t) = ˙ y j (t) = y m j+1 (t)− y m j (t)≥ 0. One can similarly show that y max is non-increasing. Duetothecomplexstate-dependenceofthedepartureprocess, thequeuelength process is difficult to analyze. We propose to study a related scalar quantity, called workload formally defined as follows, where we recall the notations introduced in Section 3.1.3. Definition 2 (Workload). The workload associated with the HTQ at any instant is the sum of the distances remaining to be travelled by all the vehicles present at that instant. That is, if the current coordinates and departure coordinates of all vehicles are x∈ [0,L] N and q∈R N + respectively, with q≥x, then the workload is given by: w(x,q) := N X i=1 (q i −x i ). Since the maximum distance to be travelled by any vehicle from the time of arrival to the time of departure is upper bounded by R, we have the following simple relationship between workload and queue length at any time instant: w(t)≤N(t)R, ∀t≥ 0. (4.2) An implication of (4.2) is that unbounded workload implies unbounded queue length in our setting. We shall use this relationship to establish an upper bound on the throughput. However, a finite workload does not necessarily imply finite queue length. In order to see this, consider the state of the queue withN vehicles, all of whom have distance 1/N remaining to be travelled. Therefore, the workload at this instant is 1/N×N = 1, which is independent of N. 34 When the workload is positive, its rate of decrease is equal to service rate in between jumps, defined next. Definition 3 (Service Rate). When the HTQ is not idle, its instantaneous service rate is equal to the sum of the speeds of the vehicles present in the system at that time instant, i.e., s(x) = P N i=1 y m i (x). Since the service rate depends only on the inter-vehicle distances, we shall alternately denote it as s(y). For m = 1, s(y) = P N i=1 y i ≡ L, i.e., the service rate is independent of the state of the system, and is constant in between and during jumps. This property does not hold true in the nonlinear (m6= 1) case. Nevertheless, one can prove interesting properties for the service rate dynamics. We start by deriving bounds on service rate in between jumps. 4.2.1 Service Rate Monotonicity between Jumps The following result derives bounds on the service rate in between jumps. Lemma 5 (Bounds on Service Rates). For any y∈S L N , N∈N, L> 0, under the dynamics in (3.3), 1. L m N 1−m ≤s(y)≤L m if m> 1; 2. L m ≤s(y)≤L m N 1−m if m∈ (0, 1). Proof. Normalizing the inter-vehicular distances by L, the service rate can be rewritten as s(y) =L m N X i=1 y i L m . (4.3) Therefore, for m > 1, s(y)≤ L m P N i=1 y i L = L m . One can similarly show that, for m∈ (0, 1), s(y)≥ L m . In order to prove the remaining bounds, we note that 35 P N i=1 z m i is strictly convex in z = [z 1 ,...,z N ] for m > 1, and that the minimum of P N i=1 z m i over z∈S N occurs at z = 1/N, and is equal to N 1−m . Similarly, for m∈ (0, 1), P N i=1 z m i is strictly concave in z, and its maximum over z∈S N occurs at z =1/N, and is equal to N 1−m . Combining these facts with (4.3), and noting that y/L∈S N , gives the lemma. Lemma 6 (Service Rate Monotonicity Between Jumps). For anyy∈S L N , N∈N, L> 0, under the dynamics in (3.3), d dt s(y)≤ 0 if m> 1 & d dt s(y)≥ 0 if m∈ (0, 1), where the equality holds true if and only if y = L N 1. Proof. The time derivative of service rate is given by: d dt s(y) = d dt N X i=1 y m i =m N X i=1 y m−1 i ˙ y i =m N X i=1 y m−1 i y m i+1 −y m i (4.4) where the second equality follows by (3.3). The result then follows by application of Lemma 20, and by noting thatg(z) =z m is a strictly increasing function for all m> 0, andh(z) =z m−1 is strictly decreasing if m∈ (0, 1), and strictly increasing if m> 1. The following lemma will facilitate generalization of Lemma 6. In preparation for the lemma, let f(y,m) := m P N i=1 y m−1 i y m i+1 −y m i be the time derivative of service rate, as given in (4.4). 36 Lemma 7. For all y∈ int(S L N ), N∈N\{1}, L> 0: ∂ ∂m f(y,m)| m=1 =−LD y L ||P − y L ≤ 0 (4.5) Additionally, if L<e −2 , then ∂ 2 ∂m 2 f(y,m)| m=1 ≥ 0 (4.6) Moreover, equality holds true in (4.5) and (4.6) if and only if y = L N 1. Proof. Taking the partial derivative of f(y,m) with respect to m, we get that ∂ ∂m f(y,m) = f(y,m) m +m N X i=1 y m−1 i y m i+1 (logy i + logy i+1 )− 2y 2m−1 i logy i In particular, for m = 1: ∂ ∂m f(y,m)| m=1 =f(y, 1) + N X i=1 (y i+1 (logy i + logy i+1 )− 2y i logy i ) =L N X i=1 y i L log y i−1 /L y i /L ! =−LD y L ||P − y L where, for the second equality, we used the trivial fact that f(y, 1) = 0. Taking second partial derivative of f(y,m) w.r.t. m gives: ∂ 2 ∂m 2 f(y,m) = N X i=1 y m−1 i logy i y m i+1 −y m i + N X i=1 y m−1 i y m i+1 logy i+1 −y m i logy i + N X i=1 y m−1 i y m i+1 (logy i + logy i+1 )− 2y 2m−1 i logy i +m N X i=1 y m−1 i y m i+1 (logy i + logy i+1 ) 2 − 4y 2m−1 i log 2 y i 37 In particular, for m = 1: ∂ 2 ∂m 2 f(y,m)| m=1 = N X i=1 (y i+1 −y i ) logy i + N X i=1 (y i+1 logy i+1 −y i logy i ) + N X i=1 (y i+1 (logy i + logy i+1 )− 2y i logy i ) + N X i=1 y i+1 (logy i + logy i+1 ) 2 − 4y i log 2 y i = N X i=1 log 2 y i (y i+1 −y i ) + 2 N X i=1 logy i (y i+1 logy i+1 +y i+1 −y i logy i −y i )≥ 0 (4.7) It is easy to check that, logz, log 2 z and z +z logz are strictly increasing, strictly decreasing and strictly decreasing functions, respectively, for z∈ (0,e −2 ). There- fore, Lemma 20 implies that each of the two terms in (4.7) is non-negative, and hence the lemma. Lemma 7 implies that, for sufficiently small L, f(y,m) is locally convex in m. One can use this property along with an exact expression for ∂ ∂m f(y,m) in Lemma 7 at m = 1, and the fact that f(y, 1) = 0 for all y, to develop a linear approximation in m of f(y,m) around m = 1. The following lemma derives this approximation, as also suggested by Figure 4.2. Lemma 8. For a giveny∈ int(S L N ),n∈N,L∈ (0,e −2 ), there exists m(y)∈ [0, 1) such that d dt s(y)≥ 2 (1−m) L (y max −y min ) 2 , ∀m∈ [m(y), 1] Proof. For a given y∈ int(S L N ), the local convexity of f(y,m) := d dt s(y) in m, and the expression of ∂ ∂m f(y,m) at m = 1 in Lemma 7 implies that d dt s(y)≥ (1−m)LD y L ||P −y L for sufficiently small m < 1. Pinsker’s inequality implies 38 m 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 2 2.5 f ( y ; m ) ( 1 ! m ) L D ( y j j P ! y ) 2 ( 1 ! m ) L ( y m a x ! y m i n ) 2 Figure 4.2: f(y,m) vs. m for a typical y∈S 10 . D y L ||P −y L ≥ ky−P − yk 2 1 2L 2 . This, combined with the fact thatky−P − yk 1 ≥ 2(y max − y min ) for all y∈ int(S L N ), gives the lemma. 4.2.2 Change in Service Rate at Jumps The following lemma quantifies the change in service rate due to departure of a vehicle. Lemma 9 (Change in Service Rate at Departures). Consider the departure of a vehicle that changes inter-vehicle distances from y∈S L N to y − ∈S L N−1 , for some N∈N\{1}, L> 0. If y 1 ≥ 0 andy 2 ≥ 0 denote the inter-vehicle distances behind and in front of the departing vehicle respectively, at the moment of departure, then the change in service rate due to the departure satisfies the following bounds: 1. if m> 1, then 0≤s(y − )−s(y)≤ (y 1 +y 2 ) m (1− 2 1−m ); 2. if m∈ (0, 1), then 0≤s(y)−s(y − )≤ min{y m 1 ,y m 2 }. Proof. Ifm> 1, then y 1 y 1 +y 2 m + y 2 y 1 +y 2 m ≤ y 1 y 1 +y 2 + y 2 y 1 +y 2 = 1, i.e.,s(y − )−s(y) = (y 1 +y 2 ) m −y m 1 −y m 2 ≥ 0. Onecansimilarlyshowthats(y)−s(y − )≥ 0ifm∈ (0, 1). 39 In order to show the upper bound on s(y − )−s(y) form> 1, we note that the minimum value ofz m + (1−z) m overz∈ [0, 1] form> 1 is 2 1−m , and it occurs at z = 1/2. Therefore, s(y − )−s(y) = (y 1 +y 2 ) m −y m 1 −y m 2 = (y 1 +y 2 ) m 1− y 1 y 1 +y 2 ! m − y 2 y 1 +y 2 ! m ! ≤ (y 1 +y 2 ) m 1− 2 1−m The upper bound on s(y)−s(y − ) for m∈ (0, 1) can be proven as follows. Since y m 1 ≤ (y 1 +y 2 ) m , s(y)−s(y − ) = y m 1 +y m 2 − (y 1 +y 2 ) m ≤ y m 2 . Similarly, s(y)−s(y − )≤ y m 1 . Combining, we get s(y)−s(y − )≤ min{y m 1 ,y m 2 }. Note that, in proving this, we nowhere used the fact that m∈ (0, 1). However, this bound is useful only for m∈ (0, 1). Remark 2 (Change in Service Rate at Arrivals). The bounds derived in Lemma 9 can be trivially used to prove the following bounds for change in service rate at arrivals: 1. if m> 1, then 0≤s(y)−s(y + )≤ (y 1 +y 2 ) m (1− 2 1−m ); 2. if m∈ (0, 1), then 0≤s(y + )−s(y)≤ min{y m 1 ,y m 2 }, where y 1 and y 2 are the inter-vehicle distances behind and in front of the arriving vehicle respectively, at the moment of arrival. 4.3 Busy Period Properties of the Horizontal Traffic Queue The system is called busy when there is at least one vehicle on the road, or equivalently, the workload is positive. Once the system gets empty, it becomes 40 idle up to the time of next arrival. Thus, the system alternates between busy and idle periods. Accordingly, while the first busy period might start from a non-zero initial condition, if the first busy period terminates, then the subsequent busy periods will start from the zero initial condition. In this chapter, unless otherwise stated explicitly, we shall implicitly assume a zero initial condition when referring to a busy period. 4.3.1 Expected Busy Period Duration The next lemma provides an expression for the expectation of the busy period duration in the linear case. Lemma 10. For any λ<L/ ¯ ψ, L> 0, m = 1, ϕ∈ Φ, ψ∈ Ψ, the mean value of the busy period duration is equal to ¯ ψ/(L−λ ¯ ψ). Proof. A busy period, say of duration B, is initiated by the arrival of a vehicle, say j, when the system is idle. Let the number of vehicles that arrive during the busy period be N bn . Note that N bn does not include the vehicle initiating the busy period. Therefore, the workload brought into the system during the busy period is equal to w B = P j+N bn i=j d i . The expected value of N bn can be obtained by conditioning on the duration of the busy period: E[N bn ] =E [E[N bn |B]] =E[λB] =λE[B] (4.8) where the second equality follows from the fact that the arrival process is a Poisson process. Since the event{N bn + 1 =n} is independent of{d j+i ,i>n}, N bn + 1 is a stopping time for the sequence{d j+i ,i≥ 1}. Therefore, using Wald’s equation, 41 e.g., see [63, Theorem 3.3.2], and (4.8), the expected value of the workload w B added to the system during the busy period B is given by: E[w B ] = (E[N bn ] + 1) ¯ ψ = (λE[B] + 1) ¯ ψ. (4.9) 1 d 2 d 3 d 4 d 5 d L d d d d d / ) ( 5 4 3 2 1 w Figure 4.3: (a) Queue length process and (b) workload process during a busy period. Since the workload decreases at a constant rate L during a busy period, we have B =w B /L (see Figure 4.3 for an illustration). Therefore, E[B] =E[w B ]/L, which when combined with (4.9), establishes the lemma. Remark 3. Since the mean busy period duration is an upper bound on the mean waiting time, Lemma 10 also gives an upper bound on the mean waiting time. One can then use Little’s law [38] 1 to show that the mean queue length is upper bounded by λ ¯ ψ/(L−λ ¯ ψ). LetI(t) := R t 0 δ {w(s)=0} ds be the cumulative idle time up to time t. The fol- lowing result characterizes the long run proportion of the idle time in the linear case. 1 Little’s law has previously been used in the context of processor sharing queues, e.g., in [7]. 42 Proposition 4. For any λ < L/ ¯ ψ, m = 1, L > 0, ϕ∈ Φ,ψ∈ Ψ, the long-run proportion of time in which HTQ is idle is given by the following: lim t→∞ I(t) t = 1− λ ¯ ψ L > 0 a.s. Proof. HTQ alternates between busy and idle periods. Let Z = I +B be the duration of a cycle that contains an idle period of length I followed by a busy period of lengthB. Idle period,I, has the same distribution as inter-arrival times i.e. an exponential random variable with mean 1/λ, and the mean value of B is given in Lemma 10. Note that duration of cycles, Z, are i.i.d. random variables. Thus, the busy-idle profile of the system is an alternating renewal process where renewals correspond to the moments at which the system gets idle. Suppose the system earns reward at a rate of one per unit of time when it is idle (and thus the reward for a cycle equals the idle time of that cycle i.e. I). Then, the total reward earned up to time t is equal to the total idle time in [0,t] (orI(t)), and by the result for renewal reward process (see [63], Theorem 3.6.1), with probability one, lim t→∞ I(t)/t =E[I]/(E[B] +E[I]). 4.4 Throughput Analysis 4.4.1 Linear Case: m = 1 Inthissection, weprovideanexactcharacterizationofthroughputforthelinear case, i.e., whenm = 1. Recall that, form = 1, the service rates(y) = P N i=1 y i ≡L is constant. 43 Proposition 5. For any L> 0, ϕ∈ Φ, ψ∈ Ψ, x 0 ∈ [0,L] n 0 , n 0 ∈N and : λ max (L,m = 1,ϕ,ψ,x 0 ,δ = 0)≤L/ ¯ ψ. Proof. By contradiction, assumeλ max >L/ ¯ ψ. Letr(t) := P A(t) i=1 d i be the workload added to the system by the A(t) vehicles that arrive over [0,t]. Therefore, w(t) =w 0 +r(t)−L(t−I(t)) (4.10) where w 0 is the initial workload. The process{r(t), t≥ 0} is a renewal reward process, where the renewals correspond to arrivals of vehicles and the rewards cor- respond to the distances{d i } ∞ i=1 that vehicles wish to travel in the system upon arrival before their departures. Inter-arrival times are exponential random vari- ableswithmean 1/λ, andtherewardassociatedwitheachrenewalisindependently and identically sampled from ψ, whose mean is ¯ ψ. Therefore, e.g., [63, Theorem 3.6.1] implies that, with probability one, lim t→∞ r(t) t =λ ¯ ψ (4.11) Thus, for allε∈ 0,λ ¯ ψ−L , there exists at 0 ≥ 0 such that, with probability one, r(t) t ≥λ ¯ ψ−ε/2>L +ε/2 ∀t≥t 0 . (4.12) Sincew 0 andI(t) are both non-negative, (4.10) implies thatw(t)≥r(t)−Lt for all t≥ 0. This combined with (4.12) implies that, with probability one, w(t)≥εt/2 for all t≥ t 0 , and hence lim t→∞ w(t) = +∞. This combined with (4.2) implies that, with probability one, lim t→∞ N(t) = +∞. 44 Theorem 1. For any L> 0, ϕ∈ Φ, ψ∈ Ψ, x 0 ∈ [0,L] n 0 , n 0 ∈N: λ max (L,m = 1,ϕ,ψ,x 0 ,δ = 1) =L/ ¯ ψ. Proof. Assume that for some λ<L/ ¯ ψ, there exists some initial condition (x 0 ,n 0 ) such that the queue length grows unbounded with some positive probability. Since the workload brought by every vehicle is i.i.d., and the inter-arrival times are exponential, without loss of generality, we can assume that the queue length never becomes zero. That is, the idle time satisfiesI(t)≡ 0. Moreover, (4.11) implies that, for everyε∈ 0,L−λ ¯ ψ , there existst 0 ≥ 0 such that, with probability one, r(t) t ≤λ ¯ ψ +ε/2<L−ε/2 ∀t≥t 0 (4.13) Combining (4.10) with (4.13), and substitutingI(t)≡ 0, we get w(t) < w 0 − εt/2, which implies that workload, and hence queue length, goes to zero in finite time after t 0 , leading to a contradiction. Combining this with the upper bound proven in Proposition 5 gives the result. Remark 4. Theorem 1 implies that the throughput in the linear case is equal to the inverse of the time required to travel average total distance by a solitary vehicle in the system. In the linear case, the throughput can be characterized with probability one, independent of the initial condition of the queue. 4.4.2 Monotonicity of Throughput in m and x 0 In this section, we show the following monotonicity property of λ max with respect to m for small values of L: for given x 0 ∈ [0,L] n 0 , n 0 ∈ N, L∈ (0, 1), ϕ∈ Φ, and ψ∈ Ψ, throughput is a monotonically decreasing function of m. For 45 this section, we rewrite (3.2) inR N + , i.e., without projecting onto [0,L] N . Specifi- cally, let the vehicle coordinates be given by the solution of ˙ x i =y m i , x i (0) =x 0,i , i∈{1,...,N} (4.14) Let X(t;x 0 ,m) denote the solution to (4.14) at t starting from x 0 at t = 0. We will compareX(t;x 0 ,m) under different values ofm and initial conditionsx 0 , over an interval of the kind [0,τ), in between arrivals and departures. We recall the notation that, if x 1 0 and x 2 0 are vectors of different sizes, then x 1 0 ≤ x 2 0 implies element-wise inequality only for components which are common to x 1 0 and x 2 0 . In Lemma 11 and Proposition 6, this common set of components corresponds to the set of vehicles common between x 1 0 and x 2 0 . Lemma 11. For any L∈ (0, 1], x 1 0 ∈R n 1 + , x 2 0 ∈R n 2 + , n 1 ,n 2 ∈N, x 1 0 ≤x 2 0 , n 2 ≤n 1 , 0<m 2 ≤m 1 =⇒ X(t;x 1 0 ,m 1 )≤X(t;x 2 0 ,m 2 ) ∀t∈ [0,τ) Proof. The proof is straightforward when n 1 = n 2 . This is because, in this case, since y i ≤ L≤ 1, m 2 ≤ m 1 implies y m 2 i ≥ y m 1 i for all i∈{1,...,n 1 }. Using this with Lemmas 18 and 19 gives the result. In order to prove the result for n 2 < n 1 , we show that X(t;x 1 0 ,m 1 ) ≤ X(t;x 2 0 ,m 1 ) ≤ X(t;x 2 0 ,m 2 ). Note that the second inequality follows from the previous case. Therefore, it remains to prove the first inequality. Let (i 1 ,...,i n 2 ) be the set of indices of n 2 vehicles such that 0≤ x 2 0,i 1 ≤ ...≤ x 2 0,in 2 ≤ L. Sim- ilarly, let (i 1 ,i 1 + 1,...,i 2 ,i 2 + 1,...) be the indices of n 1 vehicles in the order of increasing coordinates in x 1 0 . Our assumption on the initial condition implies that x 1 0,i k ≤ x 2 0,i k for all k∈{1,...,n 2 }. For brevity, let x 1 (t)≡ X(t;x 1 0 ,m 1 ), 46 and x 2 (t) ≡ X(t;x 2 0 ,m 1 ). It is easy to check that, for all t ∈ [0,τ), and all k∈{1,...,n 2 }, ˙ x 1 i k = x 1 i k +1 −x 1 i k m 1 ≤ x 1 i k+1 −x 1 i k m 1 (4.15) Lett∈ [0,τ)bethefirsttimeinstantwhenx 1 i k (t) =x 2 i k (t)forsomek∈{1,...,n 2 }. Then, recalling x 1 i k+1 (t)≤ x 2 i k+1 (t), (4.15) implies that ˙ x 1 i k (t)≤ x 2 i k+1 −x 2 i k m 1 = ˙ x 2 i k (t). The result then follows from Lemma 18. Lemma 11 is used to establish monotonicity of throughput as follows. Proposition 6. For any L∈ (0, 1], ϕ∈ Φ, ψ ∈ Ψ, δ ∈ (0, 1), x 1 0 ∈ [0,L] n 1 , x 2 0 ∈ [0,L] n 2 , n 1 ,n 2 ∈N: x 1 0 ≤x 2 0 , n 2 ≤n 1 , 0<m 2 ≤m 1 =⇒ λ max (L,m 1 ,ϕ,ψ,x 1 0 ,δ)≤λ max (L,m 2 ,ϕ,ψ,x 2 0 ,δ) Proof. For brevity in notation, we refer to the queue corresponding to m 1 , and initial condition x 1 0 as HTQ-S. We refer to the other queue as HTQ-F. Let λ, ϕ and ψ common to HTQ-S and HTQ-F be given. Let x 1 (t)≡ X(t;x 1 0 ,m 1 ) and x 2 (t)≡ X(t;x 2 0 ,m 2 ), and let N s (t) and N f (t) be the queue lengths in the two queues at time t. It suffices to show that N s (t)≥ N f (t) for a given realization of arrival times, arrival locations, and travel distances. In particular, this also implies that the departure locations are also the same for every vehicle, including the vehicles present at t = 0, in both the queues. Indeed, it is sufficient to show thatx 1 (τ)≤x 2 (τ) andN s (τ)≥N f (τ) whereτ is the time of first arrival or departure from either HTQ-S or HTQ-F. Accordingly, we consider two cases, corresponding to whetherτ corresponds to arrival or departure. Sincex 1 (t)≤x 2 (t)forallt∈ [0,τ)fromLemma11, andthedeparturelocations of all the vehicles in HTQ-S and HTQ-F are identical, the first departure from HTQ-S can not happen before the first departure in HTQ-F. Therefore, N s (τ)≥ 47 N f (τ). Since x 1 (τ − ) ≤ x 2 (τ − ), and x 2 (τ) is a subset of x 2 (τ − ), we also have x 1 (τ)≤x 2 (τ). When τ corresponds to the time of the first arrival, since the arrivals happen at the same location in HTQ-S and HTQ-F, and since x 1 (τ − )≤ x 2 (τ − ), rear- rangement of the indices of the vehicles to include the new arrival att =τ implies that x 1 (τ)≤ x 2 (τ). Moreover, since N s (τ − )≥ N f (τ − ), and the arrivals happen simultaneously in both HTQ-S and HTQ-F, we have N s (τ)≤N f (τ). Remark 5. Proposition 6 establishes monotonicity of throughput only for L∈ (0, 1]. This is consistent with our simulation studies, e.g., as reported in Figure 4.1, according to which, the throughput is non-monotonic for large L. For the analysis of the linear car following model, we exploited the fact that the total service rate of the system is constant. However, for the nonlinear model, i.e., m6= 1, the total service rate depends on the number and relative locations of vehicles. The state dependent service rate of nonlinear models makes the through- put analysis much more complex. In the next section, we find probabilistic bound on the throughput in the super-linear case. 4.4.3 Finite Dimensional Approximation In this section, we construct two queuing systems, whose analyses will give tight approximations to the throughput of HTQ, for m > 1. We provide these descriptions for the special case when ϕ = δ 0 and ψ = δ L , i.e., when the arrival and departure locations of all the vehicles are the same, and every vehicle traverses the entire circle before departing. The key idea is to project the vehicle locations to the space of workloads. Consider a vector of vehicle locations x ∈ [0,L] n , n ∈ N. The corresponding 48 projection onto the space of N-dimensional workload, for N ∈ N, denoted as v N (x) = (v 1 (x),...,v N (x)), is given by: v 1 (x) =nL− n X i=1 x i v k (x) = L−x k−1 2≤k≤ min{n + 1,N} 0 n + 2≤k≤N (4.16) The first entry ofv N is the total workload associated with all the vehicles. The rest N− 1 entries are the workloads associated with the last N− 1 vehicles to arrive (and hence the lastN−1 vehicles to depart). The states of the two queuing systems, which we henceforth refer to as slow and fast queuing systems, will be denoted as (n(t),v N ) and (n(t),v N ). Let x 0 ∈ [0,L] n 0 be the initial condition for the actual queuing system. Then the initial condition for both the slow and fast systems is (n 0 ,v N (x 0 )) (cf. (4.16)). Furthermore, the two systems have the same jump dynamics, but differ in the dynamics between jumps. If an arrival happens at time t, then the corresponding dynamics is given by n(t + ),v N (t + ) = (n(t) + 1, (v 1 (t) +L,L,v 2 (t),...,v N−1 (t))) (4.17) Unlikearrivals, thedeparturesarestate-dependent. Inparticular, adeparturehap- pens whenever any of the components of v N become zero. Let t be a time instant such that v k (t − ) > 0 and v k (t) = 0, for some k∈{1,...,N}. The corresponding jump dynamics is n(t),v N (t) = k− 1, (v 1 (t − ),...,v k−1 (t − ), 0,..., 0) (4.18) 49 In reading this dynamics, we recall that v k (t) = 0 implies v j (t) = 0 for all j∈ {k,k + 1,...,N}. We now specify the dynamics between jumps for the two systems. Consider a time interval between jumps when the queue length is n. Let these two dynamics be denoted as ˙ v N = F (n,v N ) and ˙ v N = ¯ F (n,v N ). In order to specify F and F succinctly, let us define a couple of polytopes: Z(n,v N ) := n z∈ [0,L] n+1 |A(n)z≥b(v N ); [0 1 ... 1] T z =L o Z(n,v N ) := n z∈ [0,L] n+1 |A(n)z≤b(v N ); [0 1 ... 1] T z =L o (4.19) where A(n)∈R N×(n+1) and b(v N )∈R N are given by: A(n) = n n− 1 n− 2 n− 3 ... 1 0 1 0 0 0 ... 0 0 1 1 0 0 ... 0 0 . . . . . . . . . . . . ... . . . . . . 1 1 1 1 ... 0 0 , b(v N ) = nL−v 1 L−v 2 L−v 3 . . . L−v N (4.20) The variable z in (4.19) is to be interpreted as (x 1 ,y 1 ,...,y n ). F and F are defined in terms of appropriate optimization problems over the polytopes in (4.19) as follows. F 1 (n,v N ) :=− min z∈Z(n,v N ) n+1 X i=2 z m i , F k (n,v N ) :=− min z∈Z(n,v N ) z m k , k = 2,...,N F 1 (n,v N ) :=− max z∈Z(n,v N ) n+1 X i=2 z m i , F k (n,v N ) :=− max z∈Z(n,v N ) z m k , k = 2,...,N (4.21) 50 Remark 6. P n+1 i=2 z m i is convex in z for m > 1. Therefore, with the exception of F 1 , all the optimizations appearing in (4.21) are convex. Proposition 7. For a givenN∈N, let (n(t),v N (t)) be the evolution of HTQ, and let (n(t),v N (t)) and (n(t),v N (t)) respectively be the evolution of dynamical system specified by (4.19)-(4.21). If n(0),v N (0) ≥ n(0),v N (0) ≥ n(0),v N (0) , then n(t),v N (t) ≥ n(t),v N (t) ≥ n(t),v N (t) for all t≥ 0. Proof. For brevity, we do not show dependence of various quantities on 0. Without loss of generality assume that n > N, and that the time of first arrival, say τ, is strictly positive. We provide the proof over [0,τ]. Recursive application of the argument with initial condition at τ then completes the proof. Note that F =−[ P n i=1 y m i y m 1 ... y m N−1 ] T . Moreover, for notational conve- nience, let z := [x 1 y 1 ... y n ] T . It is sufficient to show that F ≤ F ≤ F, for which, it is sufficient to show that z∈Z(n,v N ), z∈Z(n,v N ) (4.22) If n =n =n, then it is easy to see that: (i) [0 1 1 ... 1] T z = P n i=1 y i =L, and (ii)b(v N )≤b(v N ) =A(n)z =b(v N )≤b(v N ), where the first and last inequalities follow from the definition of b in (4.20). These facts collectively imply (4.22). If n ≥ n ≥ n, then consider ˜ z := h x 1 y 1 ... y n y n+1 ... y n i T = [x 1 y 1 ... y n 0 ... 0] T , with corresponding ˜ F := − h P n i=1 y m i y m 1 ... y m N−1 i T = − h P n i=1 y m i y m 1 ... y m N−1 i T = F. Simple substitution verifies that ˜ z ∈ Z(n,v N ). In order to prove the second identity in (4.22), consider ˆ z := [x 1 y 1 ... y N−1 0 ... 0 P n i=N y i ] T , with corresponding ˆ F :=− h P N−1 i=1 y m i + ( P n i=N y i ) m y m 1 ... y m N−1 i T ≤− h P n i=1 y m i y m 1 ... y m N−1 i T = F. Therefore, it suffices to show that ˆ F ≥ F, for which it suffices to show that 51 ˆ z∈Z(n,v N ). Trivially, [0 1 1 ... 1] T ˆ z = L. Therefore, it remains to show that A(n)ˆ z≤b(v N ), i.e., nx 1 + (n− 1)y 1 +... + (n−N + 2)y N−2 + (n−N + 1)y N−1 ≤nL−v 1 x 1 ≤L−v 2 x 1 +y 1 ≤L−v 3 . . . x 1 +y 1 +... +y N−2 ≤L−v N The lastN− 1 inequalities are true becausev N ≤v N . The first inequality follows from the fact that v 1 ≤ N X i=2 v i ≤ (N− 1)L− ((N− 1)x 1 + (N− 2)y 1 +... +y N−2 ) ≤ (N− 1)L− ((N− 1)x 1 + (N− 2)y 1 +... +y N−2 ) + (n−N + 1) (L−x N ) =nL− (nx 1 + (n− 1)y 1 +... + (n−N + 1)y N−1 ) The next result establishes monotonicity with respect to N. Proposition 8. Consider N 1 ,N 2 ∈N such that N 1 ≤N 2 . If n N 1 (0),v N 1 (0) ≥ n N 2 (0),v N 2 (0) ≥ (n(0),v(0))≥ n N 2 (0),v N 2 (0) ≥ n N 1 (0),v N 1 (0) then v N 1 (t)≥v N 2 (t)≥v(t)≥v N 2 (t)≥v N 1 (t), ∀t≥ 0 Proof. We provide proof for v N 1 (t)≥ v N 2 (t); the proof for v N 2 (t)≥ v N 1 (t) is symmetrical. Similar to proof of Proposition 7, it is sufficient to prove that 52 F N 1 (0)≥F N 2 (0). For the rest of the proof, we drop the argument 0 for brevity in notation. If n N 1 =n N 2 , then it suffices to show thatZ n N 2 ,v N 2 ⊆Z n N 1 ,v N 1 . This is true because b v N 2 ≥b v N 1 . For the case when n N 1 > n N 2 , for brevity in notation, we sim- plify the notation as n N 1 as n 1 , and so on. Let ˜ Z n 2 ,v N 2 := n z∈ [0,L] n 1 | ˜ A(n 2 )z≥b(v N 2 ); [0 1 ... 1] T z =L; z i = 0, i =n 2 + 1,...,n 1 o , where ˜ A(n 2 )∈R N 2 ×(n 1 +1) is given by: ˜ A(n 2 ) = n 2 n 2 − 1 n 2 − 2 n 2 − 3 ... 1 0 ... 0 1 0 0 0 ... 0 0 ... 0 1 1 0 0 ... 0 0 ... 0 . . . . . . . . . . . . ... . . . . . . ... . . . 1 1 1 1 ... 0 0 ... 0 It is easy to see that F N 2 can be equivalently written as: F N 2 1 :=− min z∈ ˜ Z(n 2 ,v N 2 ) n 1 +1 X i=2 z m i , F N 2 k :=− min z∈ ˜ Z(n 2 ,v N 2 ) z m k , k = 2,...,N 2 Moreover, ˜ Z(n 2 ,v N 2 )⊆Z(n 1 ,v N 1 ). This is because A(n 1 )≥ ˜ A(n 2 ), which com- bined with z≥ 0, implies that A(n 1 )z≥ ˜ A(n 2 )z≥b(v N 2 ), where the last inequal- ity follows from the definition of ˜ Z n 2 ,v N 2 . Combining these facts, we get that F N 1 ≥F N 2 . 53 4.4.4 ThroughputEstimationusingtheFiniteDimensional Approximation In this section, we use the finite dimensional approximation proposed in Sec- tion 4.4.3 to get estimates on throughput. We recall the of throughput as being the supremum among all arrival rates under which the expected hitting time to the state (0, 0) starting from the same state is finite. For brevity in notation, let λ max be the throughput for the actual HTQ, and let λ max (N) and λ max (N) be the throughput estimates for the slower and the faster systems, respectively, for a given approximation order N. We now describe a procedure to compute λ max (N) and λ max (N). We focus on λ max (N); the procedure for computing λ max (N) is symmetrical. Let T be the random variable denoting the hitting time to (0, 0). We provide a procedure to compute E(T ). Since E(T ) = R ∞ 0 Pr(T ≥ s) ds, we now focus on computing Pr(T ≥ s). This quantity is the same as Pr(v 1 (s) > 0). Let n arr (s) denote the random variable representing the number of arrivals during the interval [0,s]. Therefore, Pr(v 1 (s)> 0) = ∞ X k=1 Pr (v 1 (s)> 0|n arr (s) =k) Pr(n arr (s) =k) (4.23) The second term in the summand in (4.23) is given by Poisson distribution. In order to compute the first term, we recall the fact that, conditional on the number of arrivals over [0,s], the arrival times, say a 1 ,a 2 ,...,a k , have the same distribution as the order statistics corresponding to n arr (s) independent random 54 variables uniformly distributed on the interval (0,s), e.g. see [63, Theorem 2.3.1]. Therefore, the first term in (4.23) can be computed as: Pr(v 1 (s)> 0|n arr (s) =k) = Z s a k−1 ··· Z s a 2 Z s 0 Pr(v 1 (s)> 0|n arr (s) =k,a 1 ,··· ,a k ) k! s k da 1 ... da k The next result, which provides an important monotonicity property with respect to N, follows directly from Proposition 8. Proposition 9. For any N 1 ,N 2 ∈N such that N 1 ≤N 2 , we have λ max (N 1 )≤λ max (N 2 )≤λ max ≤λ max (N 2 )≤λ max (N 1 ) Proof. Following Propositions 8, conditional on n arr (s) =k,a 1 ,...,a k , we have Pr(v N 1 1 (s)> 0)≥ Pr(v N 2 1 (s)> 0)≥ Pr(v 1 (s)> 0)≥ Pr(v N 2 1 (s)> 0) ≥ Pr(v N 1 1 (s)> 0) Therefore, using (4.23), for a given λ, E(T ) also obeys the same monotonic rela- tionship. This implies the proposition. 4.4.5 Throughput Bounds for the Super-linear Case from Busy Period Calculations In this section, we derive lower bound on the throughput for the super-linear case by using computing busy periods. If the service rate were constant for HTQ, the busy periods computations would be similar to an M/G/1 queueing model. However, since HTQ may start from non-empty initial condition w 0 > 0, its first 55 busyperiodcanhaveadifferentdistribution. Therefore, onecanuseresultsrelated to M/G/1 queues with exceptional first service to compute the distribution of busy periods. IfweletH p (t,n,θ) := Pr{B≤t,N bn =n−1}whentheservicerateequals p, the work of the first vehicle that starts a busy period has distribution θ, and N bn vehicles arrive during the busy period; then, by using results in e.g. [63, 77], it can be shown that G p (t,n,θ) := d dt H p (t,n,θ) = e −λt (λt) n−1 w 0 t(n−1)!p ˜ ψ n−1 (t−w 0 /p) θ =δ w 0 e −λt (λt) n−1 n! ˜ ψ n (t) θ =ψ (4.24) For r∈ N, let G r,p (t,n,θ) be the r-fold convolution of G p (t,n,θ), defined in (4.24), with respect to t. In words, G r,p (t,n,θ) is the probability that the number of new arrivals in each of (any)r busy periods is equal ton− 1, and that the sum of durations of all the busy periods is equal to t. Similarly, for non-zero initial condition, letG p 1 (θ 1 )∗G r−1,p 2 (θ 2 )(t,n) be the probability that the number of new arrivals in each of (any) r busy periods is equal to n− 1, and that the sum of durations of all the busy periods is equal to t, when the constant service rate for the first busy period is p 1 and is p 2 for the rest of the r− 1 busy periods. The next result computes a bound on the probability that the queue length of the HTQ satisfies a given upper bound over a given time interval. In Proposi- tions 10 and 11, for the sake of clarity, we add explicit dependence on λ to this probability distribution function. 56 Proposition 10. For any m> 1, M∈N, L> 0, λ> 0, ϕ∈ Φ, ψ∈ Ψ, and zero initial condition x 0 = 0, the probability that the queue length is upper bounded by M over a given time interval [0,T ] satisfies the following bound: Pr N(t)≤M ∀t∈ [0,T ] ≥ sup r∈N M X n=1 Z ∞ T G r,L m M 1−m(t,n,ψ,λ) dt (4.25) Proof. Let us denote the current queueing system as HTQ-f. We shall compare queue lengths between HTQ-f and a slower queueing system HTQ-s, which starts from the same (zero) initial condition, and experiences the same realizations of arrival times, locations and travel distances. Let every incoming vehicle into HTQ- s and HTQ-f be tagged with a unique identifier. At time t, letJ (t) be the set of identifiers of vehicles present both in HTQ-s and HTQ-f,J s/f (t) be the set of identifiers of vehicles present only in HTQ-s, andJ f/s (t) be the set of identifiers of vehicles present only in HTQ-f. Letv f i denote the speed of the vehicle in HTQ- f with identifier i∈J (t)∪J f/s (t), as determined by the car-following behavior underlying(3.2). ThevehiclespeedsinHTQ-sarenotgovernedbythecarfollowing behavior, but are rather related to the speeds of vehicles in HTQ-f as: v s i (t) = v f i (t) p v f (t) |J (t)| |J (t)| +|J s/f (t)| i∈J (t) p |J (t)| +|J s/f (t)| i∈J s/f (t) (4.26) wherev f (t) := P i∈J (t) v f i (t) is the sum of speeds of vehicles in HTQ-f that are also present in HTQ-s at timet, andp is a parameter to be specified. Indeed, note that P i∈J (t)∪J s/f (t) v s i (t)≡p, i.e., p is the (constant) service rate of HTQ-s. Consider a realization where the number of arrivals into HTQ-s with p = L m M 1−m during any busy period overlapping with [0,T ] does not exceed M. We 57 refer to such a realization as event in the rest of the proof. Since the maximum queue length during a busy period is trivially upper bounded by the number of arrivals during that busy period, conditioned on the event, we have N s (t)≤M, t∈ [0,T ] (4.27) Consider the union of departure epochs from HTQ-s and HTQ-f in [0,T ]: 0 = τ 0 ≤ τ 1 ≤ .... IfJ f/s (τ k ) =∅ for some k ≥ 0, thenJ f/s (t) =∅ for all t∈ (τ k ,τ k+1 ). Hence, the service rate for HTQ-f over the interval (τ k ,τ k+1 ) is v f (t), which, conditioned on the event, is lower bounded by L m M 1−m = p by Lemma 5. Therefore, p/v f (t)≤ 1 over (τ k ,τ k+1 ), and hence (4.26) implies that all the vehicles with identifiers inJ f will travel slower in HTQ-s in comparison to HTQ- f. In particular, this implies thatJ f/s (τ k+1 ) =∅. This, combined with the fact thatJ f/s (τ 0 ) =∅ (both the queues start from the same initial condition), we get that, conditioned on the event,J s/f (t)≡∅, and hence N(t)≤ N s (t) over [0,T ]. Combining this with (4.27) gives that, conditioned on the event, N(t)≤ M over [0,T ]. We now compute the probability of the occurrence of the event using busy period calculations from Section 4.3. The event can be categorized by the max- imum number of busy periods, say r∈ N, that overlap with [0,T ], i.e., the r-th busy period ends after time T (and each of these busy periods has at most M arrivals). Since these busy periods are interlaced with idle periods, the probability of the r-th busy period ending after time T is lower bounded by the probability that the sum of the durations of r busy periods is at least T. (4.24) implies that the latter quantity is equal to P M n=1 R ∞ T G r,L m M 1−m(t,n,ψ,λ) dt. The proposition then follows by noting that this is true for any r∈N. 58 Remark 7. In the proof of Proposition 10, when deriving probabilistic upper bound on the queue length over a given time horizon [0,T ], we neglected the idle periods in [0,T ]. This introduces conservatism in the bound on the right hand side of (4.25). Since the idle period durations are distributed independently and iden- tically according to an exponential random variable (since the arrival process is Poisson), one could incorporate them into (4.25) by taking convolution of G with idle period distributions. Our choice for not doing so here is to ensure concise- ness in the presentation of bounds in (4.25). The resulting conservatism is also present in Proposition 11, and carries over to Theorems 2 and 3, as well as to the corresponding simulations reported in Figures 4.5, 4.6 and 4.7. The next result generalizes Proposition 10 for non-zero initial condition. Note that the non-zero initial condition only affects the first busy period; all subsequent busy periods will necessarily start from with zero initial condition. Proposition 11. For any m > 1, M∈N, L > 0, λ > 0, ϕ∈ Φ, ψ∈ Ψ, initial condition x 0 ∈ [0,L] n 0 , n 0 ∈ N, with associated workload w 0 > 0, the probability that the queue length is upper bounded by M +n 0 over a given time interval [0,T ] satisfies the following: Pr N(t)≤M +n 0 ∀t∈ [0,T ] ≥ sup r∈N M X n=1 Z ∞ T G L m (M+n 0 ) 1−m(δ w 0 )∗G r−1,L m M 1−m(ψ)(t,n,λ) dt Proof. The proof is similar to the proof of Proposition 10; however, since we con- sider M number of new arrivals in each of the busy periods, the event of interest is when the queue length in HTQ-s does not exceedM +n 0 andM in the first and subsequent busy periods, respectively, while operating with constant service rates L m (M +n 0 ) 1−m and L m M 1−m , respectively. 59 We shall use Propositions 10 and 11 to establish probabilistic lower bound for a finite time horizon version of the throughput defined in Definition 1: for T > 0, let λ max (L,m,ϕ,ψ,x 0 ,δ,T ) := sup{λ≥ 0 : Pr (N(t;L,m,λ,ϕ,ψ,x 0 )< +∞, ∀t∈ [0,T ])≥ 1−δ}. Theorem 2. For L > 0, m > 1, ϕ∈ Φ, ψ∈ Ψ, δ∈ (0, 1), T > 0, zero initial condition x 0 = 0, λ max (L,m,ϕ,ψ,x 0 ,δ,T )≥ sup M∈N sup λ≥ 0 sup r∈N M X n=1 Z ∞ T G r,L m M 1−m(t,n,ψ,λ) dt≥ 1−δ (4.28) Proof. Follows from Proposition 10. Theorem 3. ForL> 0,m> 1,ϕ∈ Φ,ψ∈ Ψ,δ∈ (0, 1),T > 0, initial condition x 0 ∈ [0,L] n 0 , n 0 ∈N, with associated workload w 0 > 0, λ max (L,m,ϕ,ψ,x 0 ,δ,T ) ≥ sup M∈N sup λ> 0 sup r∈N M X n=1 Z ∞ T G L m (M+n 0 ) 1−m(δ w 0 )∗G r−1,L m M 1−m(ψ)(t,n,λ)≥ 1−δ Proof. Follows from Proposition 11. Remark 8. In Theorems 2 and 3, we implicitly assume the rather standard con- vention that supremum over an empty set is zero. 60 4.4.6 Throughput Bounds under Batch Release Control Policy In this section, we consider a time-perturbed version of the arrival process. For a given realization of arrival times,{t 1 ,t 2 ,···}, consider a perturbation map t 0 i ≡t 0 i (t 1 ,...,t i ) satisfying t 0 i ≥t i for all i, which prescribes the perturbed arrival times. The magnitude of perturbation is defined as η := E (t 0 i −t i ), where the expectation is with respect to the Poisson process with rate λ that generates the arrival times. We prove boundedness of the queue length under a specific perturbation map. This perturbation map is best understood in terms of a control policy that governs the release of arrived vehicles into HTQ. In order to clarify the implementation of the control policy, we decompose the proposed HTQ into two queues in series: denoted as HTQ1 and HTQ2, both of which have the same geometric charac- teristics as HTQ, i.e., a circular road segment of length L (see Figure 4.4 for illustrations). The original arrival process for HTQ, i.e. spatio-temporal Poisson process with rateλ and spatial distributionϕ is now the arrival process for HTQ1. Vehicles remain stationary at their arrival locations in HTQ1, until released by the control policy into HTQ2. Upon released into HTQ2, vehicles travel according to (3.2) until they depart after traveling a distance that is sampled from ψ, as in the case of HTQ. The time of release of the vehicles into HTQ2 correspond to their perturbed arrival times t 0 1 ,t 0 2 ,.... The average waiting time in HTQ1 under the given release control policy is then the magnitude of perturbation in the arrival times. We consider the following class of release control policy, for which we recall from the problem setup that supp(ϕ) = [0,`] for some `∈ [0,L]. 61 HTQ1 Release Control Policy HTQ2 HTQ1 HTQ2 (a) (b) 1 x 2 x 3 x 1 y 2 y 3 y 0 L HTQ1 HTQ2 HTQ1 Release Control Policy HTQ2 Figure 4.4: Decomposition of HTQ into HTQ1 and HTQ2 in series. Definition 4 (Batch Release Control Policy π b 4 ). Divide [0,`] into sub-intervals, each of length4, enumerated as 1, 2,...,d ` 4 e. Let T 1 be the first time instant when HTQ2 is empty. At time T 1 , release one vehicle each, if present, from all odd-numbered sub-intervals in{1, 2,...,d ` 4 e} simultaneously into HTQ2. Let T 2 be the next time instant when HTQ2 is empty. At timeT 2 , release one vehicle each, if present, from all even-numbered sub-intervals in{1, 2,...,d ` 4 e} simultaneously into HTQ2. Repeat this process of alternating between releasing one vehicle each from odd and even-numbered sub-intervals every time that HTQ2 is empty. Remark 9. 1. Under π b 4 , when vehicles are released into HTQ2, the inter- vehicle distances in the front and rear of each vehicle being released is at least equal to4. 2. The order in which vehicles are released into HTQ2 from HTQ1 under π b 4 may not be the same as the order of arrivals into HTQ1. In the next two sub-sections, we analyze the performance of the batch release control policy for sub-linear and super-linear cases. The Sub-linear Case In this section, we derive a lower bound on throughput when m∈ (0, 1). We firstderiveatriviallowerboundinProposition12impliedbyLemma9andRemark 62 2. Next, we improve this lower bound in Theorem 4 under a under a batch release control policy, π b 4 . Proposition 12. For anyL> 0, m∈ (0, 1), ϕ∈ Φ, ψ∈ Ψ, x 0 ∈ [0,L] n 0 , n 0 ∈N: λ max (L,m,ϕ,ψ,x 0 ,δ = 0)≥L m / ¯ ψ Proof. Remark 2 implies that, for m∈ (0, 1), the service rate does not decrease due to arrivals. Therefore, a simple lower bound on the service rate for any state is the service rate when there is only one vehicle in the system, i.e., L m . Therefore, the workload process is upper bounded as w(t) = w 0 +r(t)− R t 0 s(z)dz≤ w 0 + r(t)−L m (t−I(t)), ∀t≥ 0, where r(t) andI(t) denote the renewal reward and the idle time processes, respectively, as introduced in the proof of Proposition 5. Similar to the proof of Proposition 5, it can be shown that, if λ<L m / ¯ ψ, then the workload, and hence the queue length, goes to zero in finite time with probability one. Next, we establish better throughput guarantees than Proposition 12, under a batch release control policy, π b 4 . The next result characterizes the time interval between release of successive batches into HTQ2 under π b 4 . Lemma 12. For given λ > 0,4 > 0, ϕ∈ Φ, ψ∈ Ψ with supp(ψ) = [0,R], R> 0, m∈ (0, 1), x 0 ∈ [0,L] n 0 , L> 0, n 0 ∈N, let T 1 , T 2 , ... denote the random variables corresponding to time of successive batch releases into HTQ2 under π b 4 . Then, T 1 ≤ n 0 R L m , T i+1 −T i ≤R/4 m for all i≥ 1, and y min (t)≥4 for all t≥T 1 . Proof. Since the maximum distance to be traveled by every vehicle is upper bounded byR, the initial workload satisfiesw 0 ≤n 0 R. Since the minimum service rate for m∈ (0, 1) is L m (see proof of Proposition 12), with no new arrivals, it 63 takes at mostw 0 /L m =n 0 R/L m amount of time for the system to become empty. This establishes the bound on T 1 . Lemma 4 implies that, underπ b 4 , the minimum inter-vehicle distance in HTQ2 is at least4 after T 1 . This implies that y min (t)≥4 for all t≥ T 1 , and hence the minimum speed of every vehicle in HTQ2 is at least4 m after T 1 . Since the maximum distance to be traveled by every vehicle is R, this implies that the time between release of a vehicle into HTQ2 and its departure is upper bounded by R/4 m , which in turn is also an upper bound on the time required by all the vehicles released in one batch to depart from the system. LetN 1 (t)andN 2 (t)denotethequeuelengthsinHTQ1andHTQ2, respectively, at timet. Lemma 12 implies that, for every4> 0,N 2 (t) is upper bounded for all t≥T 1 . The next result identifies conditions under which N 1 (t) is upper bounded. For F > 0, let Φ F := n ϕ∈ Φ| sup x∈[0,`] ϕ(x)≤F o . For subsequent anal- ysis, we now derive an upper bound on the load factor, i.e., the ratio of the arrival and departure rates, associated with a typical sub-queue of HTQ1 among {1, 2,...,d ` 4 e}. It is easy to see that, for every ϕ∈ Φ F , F > 0, the arrival pro- cess into every sub-queue is Poisson with arrival rate upper bounded by λF4. Lemma 12 implies that the departure rate is at least4 m /2R. Therefore, the load factor for every sub-queue is upper bounded as ρ≤ 2RλF4 4 m = 2RλF4 1−m (4.29) In particular, if 4<4 ∗ (λ) := (2RλF ) − 1 1−m , (4.30) 64 then ρ< 1. It should be noted that for n 0 < +∞, by Lemma 12, T 1 < +∞. The service rate is zero during [0,T 1 ]; however, since T 1 is finite, this does not affect the computation of load factor. Proposition 13. For any λ > 0, ϕ∈ Φ F , F > 0, ψ∈ Ψ with supp(ψ) = [0,R], R> 0, m∈ (0, 1), x 0 ∈ [0,L] n 0 , L> 0, n 0 ∈N, for sufficiently small4, N 1 (t) is bounded for all t≥ 0 under π b 4 , almost surely. Proof. By contradiction, assume that N 1 (t) grows unbounded. This implies that there exists at least one sub-queue, say j∈{1, 2,...,d ` 4 e}, such that its queue length, say N 1,j (t), grows unbounded. In particular, this implies that there exists t 0 ≥ T 1 such that N 1,j (t)≥ 2 for all t≥ t 0 . Therefore, for all t≥ t 0 , the ratio of arrival rate to departure rate for the j-th sub-queue is given by (4.29), which is a decreasing function of4, and hence becomes strictly less than one for sufficiently small4. A simple application of the law of large numbers then implies that, almost surely, N 1,j (t) = 0 for some finite time, leading to a contradiction. The following result gives an estimate of the mean waiting time in a typical sub-queue in HTQ1 under the π b 4 policy. Proposition 14. Forϕ∈ Φ F ,F > 0,ψ∈ Ψ,m∈ (0, 1), there exists a sufficiently small4 such that the average waiting time in HTQ1 under π b 4 is upper bounded as: W≤R(2RλF ) m 1−m 2 m m 1−m + m m m 1−m −m 1 1−m ! . (4.31) Proof. It is easy to see that the desired waiting time corresponds to the system time of an M/D/1 queue with load factor given by (4.29) along with the arrival and departure rates leading to (4.29). Note that, by Lemma 12, for finite n 0 , the value ofT 1 is finite and does not affect the average waiting time. Therefore, using 65 standard expressions for M/D/1 queue [38], we get that the waiting time in HTQ1 is upper bounded as follows for ρ< 1: W≤ 2R 4 m + R 4 m ρ 1−ρ ≤ 2R 4 m + R 4 m 1 1−ρ ≤ 2R 4 m + R 4 m − 2RλF4 (4.32) It is easy to check that the minimum of the second term in (4.32) over 0,4 ∗ (λ) occursat4 = m 2RλF 1 1−m . Substitutionintherighthandsideofthefirstinequality in (4.32) gives the result. Remark 10. (4.31) implies that, for everyR> 0,F > 0,λ> 0, we haveW→ 2R as m→ 0 + . We extend the notation introduced in (4.1) to λ max (L,m,ϕ,ψ,x 0 ,δ,η) to also show the dependence on maximum allowable perturbation η. This is not to be confused with the notation for λ max used in Theorems 2 and 3, where we used the notion of throughput over finite time horizons. We choose to use the same notations to maintain brevity. In order to state the next result, for givenR> 0,F > 0,m∈ (0, 1) andη≥ 0, let ˜ W (m,F,R,η) be the value ofλ for which the right hand side of (4.31) is equal toη, if such aλ exists and is at leastL m / ¯ ψ, and let it be equal toL m / ¯ ψ, otherwise. The lower bound ofL m / ¯ ψ in the definition of ˜ W is inspired by Proposition 12. The next result formally states ˜ W as a lower bound on λ max . Theorem 4. For any ϕ∈ Φ F , F > 0, ψ∈ Ψ with supp(ψ) = [0,R], R > 0, m∈ (0, 1), x 0 ∈ [0,L] n 0 , n 0 ∈N, L > 0, and maximum permissible perturbation η≥ 0, λ max (L,m,ϕ,ψ,x 0 ,δ = 0,η)≥ ˜ W (m,F,R,η) 66 In particular, if η> 2R, then λ max (L,m,ϕ,ψ,x 0 ,δ = 0,η)→ +∞ as m→ 0 + . Proof. Consider any λ≤ ˜ W (m,F,R,η), and4≤ m 2RλF 1 1−m . Under policy π b 4 , Lemma 12 and Proposition 13 imply that, for finite n 0 , N 2 (t) and N 1 (t) remain bounded for all times, with probability one. Also, for λ = ˜ W (m,F,R,η), by Proposition 14 and the definition of ˜ W (m,F,R,η), the introduced perturbation remains upper bounded by η. Since the right hand side of (4.31) is monotonically increasing in λ, perturbations remain bounded by η for all λ≤ ˜ W (m,F,R,η). In particular, by Remark 10, we have W → 2R as m→ 0 + . In other words, as m→ 0 + , the magnitude of the introduced perturbation becomes independent of λ. Therefore, when η > 2R, and m→ 0 + throughput can grow unbounded while perturbation and queue length remains bounded. Remark 11. We emphasize that the only feature required in a batch release con- trol policy is that, at the moment of release, the front and rear distances for the vehicles being released should be greater than4. The requirement of the policy in Definition 4 for the road to be empty at the moment of release makes the control policy conservative, and hence affects the maximum permissible perturbation. In fact, for special spatial distributions, e.g., whenϕ is a Dirac delta function and the support ofψ is [0,L−4]), one can relax the conservatism to guarantee unbounded throughput for arbitrarily small permissible perturbation. The Super-linear Case In this section, we study the throughput for the super-linear case under per- turbed arrival process with a maximum permissible perturbation of η. For this purpose, we consider the batch release control policy π b 4 , defined in Definition 4, for our analysis. Time intervals between release of successive batches, under π b 4 , are characterized the same as Lemma 12. However, in the super linear case, by 67 Lemma 5, the initial minimum service rate isL m n 1−m 0 . Therefore, the time of first release is bounded asT 1 <n m 0 R/L m . Moreover, similar to the proof of Lemma 12, it can be shown that y min (t)≥4 for all t≥T 1 . During [0,T 1 ], the service rate of all sub-queues remain zero; however, when n 0 < +∞,T 1 is finite and for the computation of load factor this time interval can be neglected. Therefore, the load factor for each sub-queue will be the same as the sub-linear case (4.29). In this case, however, in order to have ρ < 1, we get the counterpart of (4.30) as: 4>4 ∗ (λ). (4.33) It should be noted that since the batch release control policy iteratively releases from odd and even sub-queues, we need at least two sub-queues to be able to implement this policy. As a result,4 cannot be arbitrary large and4 < `/2. This constraint gives the following bound on the admissible throughput under this policy λ<λ ∗ := (`/2) m−1 /2RF (4.34) Thefollowingresultshowsthatfortheaboverangeofthroughput, thequeuelength in HTQ1, N 1 (t), remains bounded at all times. Proposition 15. For any λ<λ ∗ ,4∈ 4 ∗ (λ),`/2 i , ϕ∈ Φ F , F > 0, ψ∈ Ψ with supp(ψ) = [0,R], R > 0, m > 1, x 0 ∈ [0,L] n 0 , L > 0, n 0 ∈N, N 1 (t) is bounded for all t≥ 0 under π b 4 , almost surely. Proof. The proof is similar to proof of Proposition 13. In particular, by (4.33) and (4.34), one can show that load factor (4.29) remains strictly smaller than one. This implies that no sub-queue in HTQ1 can grow unbounded, and N 1 (t) remains bounded for all times, with probability one. 68 Proposition 16. For any λ < λ ∗ , ϕ∈ Φ F , F > 0, ψ∈ Ψ, m > 1, the average waiting time in HTQ1 under π b 4 for4 =`/2 is upper bounded as: W≤ 2R (`/2) m + R (`/2) m 2RλF (`/2) 1−m 1− 2RλF (`/2) 1−m (4.35) Proof. The proof is very similar to the proof of Proposition 14. Thus, we get the following bounds: W≤ 2R 4 m + R 4 m ρ 1−ρ ≤ 2R 4 m + R 4 m 2RλF4 1−m 1− 2RλF4 1−m The right hand side of the above inequality is a decreasing function of4; therefore, 4 =`/2 minimizes it, and gives (4.35). Let ˆ W (m,F,R,η) be the value of λ for which the right hand side of (4.35) is equal to η, if such a λ≤ λ ∗ exists, and let it be equal to λ ∗ otherwise. Note that since the right hand side of (4.35) is monotonically increasing in λ, for all λ≤ ˆ W (m,F,R,η) the introduced perturbation remains upper bounded by η. Theorem5. For anyϕ∈ Φ F ,F > 0,ψ∈ Ψ with supp(ψ) = [0,R],R> 0,m> 1, x 0 ∈ [0,L] n 0 , n 0 ∈N, L> 0, and maximum permissible perturbation η≥ 0, λ max (L,m,ϕ,ψ,x 0 ,δ = 0,η)≥ ˆ W (m,F,R,η). Proof. For anyλ< ˆ W (m,F,R,η), underπ b 4 , Lemma 12 and Proposition 15 imply that, for finite n 0 , N 2 (t) and N 1 (t) remain bounded for all times, with probability one. Also, by Proposition 16 and the definition of ˆ W (m,F,R,η), the introduced perturbation remains upper bounded by η. 69 4.4.7 A second order model In this section, we investigate throughput under a second order car following model in which vehicles try to keep a desired inter-vehicle distance4 des . Let e i :=4 des −y i represent the deviation from the desired distance for the i-th vehi- cle. We adopt the following model from [49]: ¨ x i =−c|e i | ˙ e i ∀i (4.36) where c > 0 is a design variable. Let d c <4 des be the minimum allowed inter- vehicle distance. 4 des is a design variable and a function of speed limit, v max , maximum deceleration capability, a max , and d c . It is shown in [49] that by the following choice of c and4 des c = 27a 2 max 8v 3 max , 4 des = s 16 27 v 2 max a max +d c (4.37) and starting from a specific set of initial conditions, we will have: 1) y min >d c , 2) ˙ x i ≤ v max , and 3) ¨ x i ≥−a max for all times. In particular, for the choice of c in (4.37), if the initial condition satisfies ˙ x i (0) + (c/2)e 2 i (0)≤v max if e i (0)≥ 0 ˙ x i (0)≤v max if e i (0)< 0 , ∀i (4.38) then, the aforementioned three constraints will be satisfied for all times. Remark 12. Under (4.36), acceleration has the opposite sign of ˙ e i , and ˙ e i = −( ˙ x i+1 − ˙ x i ). Using these two facts, one can show that, under (4.36), the min- imum (resp. maximum) speed among all vehicles is non-decreasing (resp. non- increasing). 70 We consider a similar release control policy as π b 4 defined in Definition 4, and denote it as ˜ π b 4 (v 4 , ¯ v 4 ) where in this case4 > d c and it cannot be arbitrary small. In essence,4, in this case, incorporates the vehicle length. The difference between π b 4 and ˜ π b 4 (v 4 , ¯ v 4 ) is that vehicles that are about to enter on the road sample their initial speed, independent from each other, from a distribution with support [v 4 , ¯ v 4 ] where v 4 and ¯ v 4 (0<v 4 ≤ ¯ v 4 <v max ). Note that, by Remark 12, the speed of every vehicle remains lower bounded by v 4 . Therefore, the time difference between successive instants of vehicle release from each sub-queues, is at most 2R/v 4 . Similar to (4.29), one can show that the load factor for each sub-queue is upper bounded as ρ≤ (2RλF4)/v 4 when traveling on the road. Therefore, if we setλ<λ ∗ (4) :=v 4 /(2RF4), thenρ< 1 for all sub-queues, and N 1 (t) will remain bounded, almost surely. Proposition 17. For any F > 0,4>d c , ψ∈ Ψ and ϕ∈ Φ F , under ˜ π b 4 (v 4 , ¯ v 4 ), λ max ≥λ ∗ (4), almost surely. Proof. Under ˜ π b 4 (v 4 , ¯ v 4 ), N 2 (t) is always bounded. Also, as we discussed, for λ<λ ∗ (4), N 1 (t) is also bounded for all times, almost surely. The following result shows that under ˜ π b 4 (v 4 , ¯ v 4 ), initial conditions in (4.38) can be satisfied and safety is guaranteed. Proposition 18. For any F > 0,4 > d c , λ > 0, ψ∈ Ψ and ϕ∈ Φ F , there exists a ¯ v 4 > 0 under which y min (t)>d c , ˙ x i (t)≤v max , and ¨ x i (t)≥−a max for all t≥ 0 and all i∈{1,··· ,N(t)}. In particular, if4 >4 des , then ¯ v 4 ∈ (0,v max ], otherwise ¯ v 4 ∈ (0,v max −c/2(4 des −4) 2 ]. Proof. If4 >4 des , then e i (0) < 0 for all i, and any ¯ v 4 ∈ (0,v max ] ensures that all vehicles satisfy (4.38). On the other hand, if4 <4 des , then e i (0)≥ 0 for some i, but since4 > d c , by (4.37), e i (0) < q 16 27 v 2 max amax for all i. Therefore, there 71 m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 λ max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Estimate from Theorem 2, T =500 Estimate from Theorem 2, T =20 Lower Range of Numerical Estimate Upper Range of Numerical Estimate (a) m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 λ max 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Estimate from Theorem 2, T =500 Estimate from Theorem 2, T = 20 Lower Range Numerical Estimate Upper Range Numerical Estimate (b) Figure 4.5: Comparison between theoretical estimates of throughput from Theo- rem2, andrangeofnumericalestimatesfromsimulations, forzeroinitialcondition. The parameters used for this case are: L = 1,δ = 0.1, and (a)ϕ =δ 0 ,ψ =δ L , (b) ϕ =U [0,L] , ψ =U [0,L] . exists a ¯ v 4 ∈ (0,v max −c/2(4 des −4) 2 ] such that (4.38) is satisfied for all i. This establishes this proposition. Remark 13. Proposition 17 characterizes a lower bound on throughput that depends on4. By considering the two cases of4 >4 des and4∈ (d c ,4 des ] mentioned in Proposition 18 and assuming v 4 = ¯ v 4 , one can show that, at 4 = q 4 2 des − 2v max /c ∈ (d c ,4 des ), the lower bound on throughput is maximized and safety is guaranteed. 4.5 Simulations In this section, we present simulation results on throughput analysis, and com- pare with our theoretical results from previous sections. Figures 4.5, 4.6 and 4.7 show comparison between the lower bound on through- put over finite time horizons, as given by Theorems 2 and 3, and the corresponding 72 m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 λ max 0 1 2 3 4 5 6 7 Estimate from Theorem 2 Upper Range of Numerical Estimate Figure4.6: ComparisonbetweentheoreticalestimatesofthroughputfromTheorem 2,andrangeofnumericalestimatesfromsimulations,forzeroinitialcondition. The parameters used for this case are: L = 100, δ = 0.1, T = 10, and ϕ =δ 0 , ψ =δ L . m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 λ max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Estimate from Theorem 3 Lower Range of Numerical Estimate Upper Range of Numerical Estimate Figure 4.7: Comparison between theoretical estimates of throughput from The- orem 3, and range of numerical estimates from simulations. The parameters used for this case are: L = 1, δ = 0.1, ϕ = δ 0 , ψ = δ L , w 0 = 1 and n 0 = 4, x 1 (0) = 0.6,x 2 (0) = 0.7,x 3 (0) = 0.8,x 4 (0) = 0.9. numerical estimates from simulations. Figures 4.5 and 4.6 are for zero initial con- dition, and Figure 4.7 is for non-zero initial condition. 73 Figures 4.8 and 4.9 show comparison between the lower bound on throughput as given by the bacth release control policy, as per Theorems 4 and 5, respectively, under a couple of representative values of maximum permissible perturbationη. In particular, Figure 4.8 demonstrates that the lower bound achieved from Theorem 4 increases drastically asm→ 0 + . Both the figures also confirm that the throughput indeed increases with increasing maximum permissible perturbation η. It is instructive to compare Figures 4.5(b) and 4.9(a), both of which depict throughputestimatesforthesub-linearcase,howeverobtainedfromdifferentmeth- ods, namely busy period distribution and batch release control policy. Accordingly, one should bear in mind that the two bounds have different qualifiers attached to them: the bound in Figure 4.5(b) is valid probabilistically only over a finite time horizon, whereas the bound in Figure 4.9(a) is valid with probability one, although under a perturbation to the arrival process. m 0 0.2 0.4 0.6 0.8 l og ( 6 m a x ) 0 5 10 15 20 25 30 35 40 2 = 15 2 = 5 Figure 4.8: Theoretical estimates of throughput from Theorems 4 for different values of η. The parameters used for this case are: L = 1, ϕ = U [0,L] , ψ = U [0,L] , and w 0 = 0 . Note that the vertical axis is in logarithmic scale. Finally, Figure 4.10 shows a good agreement between queue length bound sug- gested by Remark 3, and the corresponding numerical estimates in the linear case. 74 m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 λ max 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Estimate from Theorem 5, η=8 Estimate from Theorem 5, η=20 Upper Range of Numerical Estimate (a) m 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 λ max 0 5 10 15 Estimate from Theorem 5, η=3 Estimate from Theorem 5, η=10 Upper Range of Numerical Estimate (b) Figure 4.9: Theoretical estimates of throughput from Theorem 5, and numerical estimates from simulations for different The parameters used for this case are: ϕ =U [0,L] , ψ =U [0,L] , and (a) L = 1, (b) L = 100. λ 0.9 0.92 0.94 0.96 0.98 Queue Length 5 10 15 20 25 30 35 40 45 50 Upper Bound by Remark 3 Empirical Expectation Figure 4.10: Comparison between the empirical expectation of the queue length and the upper bound suggested by Remark 3. We let the simulations run up to time t = 80, 000. The parameters used for this case are: L = 1, m = 1, ϕ = δ 0 , ψ =δ L . For these values, we have λ max = 1. 75 Chapter 5 Horizontal Traffic Queue over Non-periodic Road Segment: Performance Analysis and Case Study 5.1 The Basic HTQ Setup In this chapter, we investigate the relationship between car following model and travel time using a queueing theoretic framework. We consider a horizontal traffic queue (HTQ) consisting of a single lane road segment of length L. The geometry of the considered HTQ is slightly different from the ones considered in Chapters 3 and 4. Arrivals to the queue are in two stages. In the first stage, vehicles arrive, according to a spatio-temporal stochastic process, to a system of storage associated with the road segment. In particular, every point on the road segment has an infinite capacity storage associated to it. In the second stage, vehicles move from the storage to the same location on the road segment, under a release policy. Such a release policy embodies safety rules that vehicles typically follow when merging into a traffic stream, and will be made explicit in Section 5.2. We distinguish between two types of arrivals to the storage system, either arriving at the starting point of the road segment, or arriving along the length of 76 the road segment. The first one is due to vehicles arriving from upstream. Let it be modeled as a Poisson process with rate λ o > 0. The second one is due to arrivals from on-ramps or due to change of lanes by vehicles from adjacent lanes. Let it be modeled as a spatio-temporal Poisson process with rate λ s > 0, and spatial distribution ϕ with support on [0,L]. Upon entering the traffic stream, a vehicle moves towards its destination on the road segment, under a car following model. These car following models, which are described in detail in Section 5.2 give the instantaneous speed and acceleration of a vehicle in terms of the relative distance and speed with respect to the vehicle in front. Each vehicle wishes to travel with a desired speed v i , which is sampled independently and identically from a distributionθ with support [v min ,v max ], where v max is the maximum speed limit specified by traffic regulations. The departure locations are determined as follows. Since our focus is on the regime when L is relatively small, the departure location for all the vehicles arriving sideways is at the end of the road. On the other hand, the departure point of a vehicle entering from the starting point of the road segment is sampled independently and identically from a distribution ψ with support on [0,L]. These departure points are intended to model off-ramps or movement to an adjacent lane. A schematic of the HTQ setup is provided in Figure 5.1. HTQ1 Release Control Policy HTQ2 HTQ1 HTQ2 (a) (b) 1 x 2 x 3 x 1 y 2 y 3 y 0 L HTQ1 HTQ2 HTQ1 Release Control Policy HTQ2 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 0 1 i x i x 1 i x L 0 1 i x i x 1 i x L HTQ2 HTQ1 0 1 i x i x 1 i x L 0 1 i x i x 1 i x L 1 x 2 x 3 x 0 L 0 L 1 i x i x 1 i x 0 1 i x i x 1 i x L 0 L 1 i x i x 1 i x Figure 5.1: A schematic of the horizontal traffic queue (HTQ) setup showing the spatial distributionsϕ andψ for sideway arrivals and departures for vehicles arriv- ing at the starting point of the road segment, respectively. 77 Our objective is to provide a tight quantification of the travel time for given λ 0 , λ s , ϕ, ψ, release policy, and car following model. We also intend to compare the bounds obtained from our analysis with simulation and case studies. For the latter, we consider vehicle trajectories from the NGSIM dataset. 5.2 Release Policy and Car Following Model In this section, we describe frameworks for release policies and car following models. 5.2.1 Release Policy We describe release policies for the arrivals at the origin and sideway arrivals separately. Release Policy at the Origin HTQ1 Release Control Policy HTQ2 HTQ1 HTQ2 (a) (b) 1 x 2 x 3 x 1 y 2 y 3 y 0 L HTQ1 HTQ2 HTQ1 Release Control Policy HTQ2 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 0 1 i x i x 1 i x L 0 1 i x i x 1 i x L HTQ2 HTQ1 0 1 i x i x 1 i x L 0 1 i x i x 1 i x L 1 x 2 x 3 x 0 L 0 L 1 i x i x 1 i x 0 1 i x i x 1 i x L 0 L 1 i x i x 1 i x 0 1 i x i x 1 i x L Figure 5.2: A schematic view of storage with infinite queue capacity at the origin. Vehicles are released from the storage on to the road segment under a release policy. The storage at the origin with infinite storage capacity is illustrated in Fig- ure 5.2. Letπ o (4,v) denote the policy that releases a vehicles from the storage on to the road segment if and only if the storage is non-empty, and there is no vehicle in [0,4]; moreover, the release speed is set to be v. While the release speed v is 78 deterministc and determined by the release policy, π o (v, Δ), the desire speed of vehicles is sampled from θ. Sideway Release Policy Let π s (x,θ) denote the policy that releases vehicles from sideway storage at x∈ (0,L) onto the road segment at x if and only if the corresponding storage at x is non-empty, and the relative distance to closest vehicle in front is acceptable, as per the notion of critical gap computed by the lane changing model in [5]; moreover, the release speed is sampled from the distribution θ for desired speeds. Remark 14. The purpose of modeling vehicle arrival as a two stage process is to ensure safety of arriving vehicles when merging into the stream. It is easy to see that a Poisson process can not guarantee this. For example, if the inter- arrival times are exponentially distributed, then, for any4 > 0, the probability of inter-arrival time being smaller than4/v max is non-zero. If4 is the safety distance, then this implies that, under a Poisson arrival process, with non-zero probability, the inter-vehicle distances will not be safe. An alternate approach to considering two-stage arrival is to consider a different arrival process, such as a Matérn process, e.g., see [67], that can be generated by thinning a Poisson process by neglecting the arrivals that their inter-arrival times are smaller than a specific threshold. However, such a modification alters the arrival rate, and does not lend itself to analytical tractability. 5.2.2 Car Following Model Let x i (t)∈ [0,L] be the location of the i-th vehicle at time t. Let y i (t) := x i+1 (t)−x i (t) be the inter-vehicle distance between i-th and (i + 1)-th vehicle; if there is no vehicle in front of the i-th vehicle, then we let y i := +∞. Let the 79 distance between the entry and exit points for vehiclei be denoted byl i . The time needed by the i-th vehicle to traverse distance l i at desired speed v i is T i :=l i /v i . In other words, T i is the desired travel time for the i-th vehicle. Vehicles are not allowed to overtake each other. Therefore, when a vehicle faces a slower vehicle it has to adjust its speed according to a car following model. Let us denote a generic car following model asM(4,θ), where the desired minimal inter-vehicle spacing 4, includes vehicle length as well as desired safe distance between the vehicles. We next provide a few examples ofM(4,θ). First-order Model HTQ1 Release Control Policy HTQ2 HTQ1 HTQ2 (a) (b) 1 x 2 x 3 x 1 y 2 y 3 y 0 L HTQ1 HTQ2 HTQ1 Release Control Policy HTQ2 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 1 x 2 x 3 x 1 y 2 y 3 y 0 L 0 1 i x i x 1 i x L 0 1 i x i x 1 i x L HTQ VTQ i x min v i v i y Figure 5.3: The first order car-following model described in (5.1). Consider the following model, which is also illustrated in Figure 5.3: ˙ x i = v min y i ≤4 min n (v i −v min )y i +4(αv min −v i ) (α−1)4 ,v i o y i >4 (5.1) where α > 1 is a parameter. When implemented along with the release policy π o (v min ,4) (cf. Section 5.2.1) and assuming no sideway arrivals, (5.1) ensures that ˙ x i ∈ [v min ,v i ] and y i ≥4 for all vehicles on the road segment, at all times. 80 Second-order Model We consider two models in this category. • Model 1: Consider the following extension of the model in [49]. The mag- nitude of acceleration and deceleration of vehicles is constrained to be less than a max . Let d s >4 be the nominal safe following distance, which allows enough braking distance to guarantee the minimum relative distance of4 with respect to the front vehicle, under all circumstances. d s depends on4, a max , v min , v max , and the car following model. The driving condition for the i-th vehicle is divided into three distinct zones: i) green zone, corresponding to y i > d s , ii) orange zone, corresponding to y i ∈ [4,d s ], and iii) red zone, corresponding toy i <4. When a vehicle is in green zone, its only constraint is that it cannot travel faster than its desired speed v i . However, when a vehicle is in the orange zone, a specific car following model is required to prevent it from falling into the red zone [49]: ¨ x i = min n 1 { ˙ x i <v i } a max ,−β|e i | ˙ e i o e i < 0 −c|e i | ˙ e i e i ≥ 0 (5.2) wheree i :=d s −y i ,β > 0 is a model parameter, and c andd s are chosen as: c = 27a 2 max 8(v max −v min ) 3 , d s = s 16 27 (v max −v min ) 2 a max +4 (5.3) 81 It can be shown, similar to [49], that if the initial condition of the vehi- cles satisfies the following, then y i ≥4,|¨ x i |≤a max , and ˙ x i ∈ [v min ,v i ] at all times: ˙ x i (0) + (c/2)e 2 i (0)≤v max if e i (0)≥ 0 ˙ x i (0)≤v max if e i (0)< 0 Such an initial condition can be guaranteed by the release policyπ o (v min ,4). • Model 2: Consider the well-studied Intelligent Driver Model (IDM) model [34], which is suitable for modeling connected driving behavior, e.g., see [69]. Under IDM, the acceleration of a vehicle is determined by percep- tive parameters such as desired gap, desired deceleration, relative distance and speed. We consider a slightly modified version of original IDM in [34] by imposing the constraint that vehicles cannot travel faster than their desired speeds. With this modification, the model becomes: ¨ x i =1 { ˙ x i ∈[v min ,v i ]} a max 1− ˙ x i v i δ − y ∗ y ! 2 (5.4) where y ∗ =4 + ˙ x i T + ˙ x i ˙ y i 2 √ ab , and T, a, and b are model parameters. Models used in analysis In Section 5.3, we shall utilize the following abstract car following models for our analysis: • m 0 : Vehicles travel with constant speed, as set by the release policy (i.e., zero acceleration). Overtaking is allowed. • m l : Every vehicles travels at its desired speeds (sampled from θ) until the first time instant that it either reaches its destination or its distance to the 82 vehicle in front becomes zero. In the latter case, the vehicle matches its speed to be equal to the vehicle in front, and hence there is no overtaking. 5.3 Theoretical Bounds on Travel Time In this section, we provide lower and upper bounds on the throughput and average travel time for the HTQ formulated in 5.2. We emphasize that the notion oftraveltimethatweusealsoincludesthewaitingtimeinthestorage. Weleverage the relationship between the proposed HTQ and the standard settings of M/D/1 and M/G/∞ queues to derive our bounds. 5.3.1 Lower Bounds In this section, we derive a couple of lower bounds on the mean travel time and queue length for the proposed HTQ. These bounds are derived by relaxing release policy and car following rules. Lower Bound with Overtaking Our first lower bound is obtained by relaxing the release policies and car fol- lowing rules. Proposition 19. Consider the HTQ setup in Section 5.1. For anyL> 0, λ o > 0, λ s > 0, and distributions θ and ϕ, the mean queue length is lower bounded by (λ s +λ o ) R t 0 ¯ F (s) ds, whereF (s) := Pr(T≤s) is the CDF for free flow travel times at desired speed. Proof. Consider the scenario with the trivial release policy that releases vehicles from the storage to the road segment as soon as they arrive to the storage, and 83 the release speeds are set to be the desired speeds of the vehicles. Moreover, let the vehicles travel to their destination at constant speeds (i.e., no car following). The queue length under such a relaxation is clearly a lower bound to the queue length under the release policy and car following models of Section 5.2. Therefore, we derive mean queue length for this scenario for the lower bound. The relaxation scenario can be modeled as an M/G/∞ queue, i.e., with infinite identical servers. The service time CDF for each server is F. Consider some time t > 0. A vehicle that arrives at time s < t is still on the road segment with probability ¯ F (t−s). Therefore, queue length at any time t > 0 is a Poisson process, with mean (λ s +λ o ) R t 0 ¯ F (t−s) ds. Remark 15. 1. The CDF of F can be computed from the distributions of desired speeds v i and travel distances l i , and noting that T i =l i /v i . 2. Proposition 19 implies that the steady state mean queue length is ¯ N = (λ o + λ s ) ¯ T, where ¯ T is the average free flow travel time at desired speeds 1 . This could also have been obtained from Little’s law. Lower Bounds without Overtaking In this section, we provide a tighter lower bound for the special case when there are no sideway arrivals. This result will be also a lower bound when sideway arrivals exists. Sideway arrivals decrease inter-vehicle distances and consequently speed of vehicles. 1 The expected value T exists because T has a finite support, by assumption. This point is used at several places in this chapter, without mentioning explicitly. 84 Proposition 20. Consider the HTQ setup in Section 5.1 with no vehicles on the road initially. For any L> 0, λ o > 0, λ s = 0, ϕ =δ 0 , ψ =δ L , and distribution θ, the steady state mean queue length is lower bounded by P ∞ k=0 kπ k , where π 0 =e −λoT π k = Z Tmax 0 e −λou (λ o u) k−1 (k− 1)! λ o F (u)e −g(u) du, k = 1, 2,... (5.5) with g(u) :=λ o R ∞ u F (s)ds. Proof. Consider the scenario where vehicles implement the m l (cf. Section 5.2.2) carfollowingmodel. Thesteadystatemeanqueuelengthundersuchacarfollowing rule is clearly a lower bound on the steady state mean queue length under any combination of release policy and car following model from Section 5.2. Therefore, we derive the steady state mean queue length for this scenario to get the lower bound. It should be noted that, since4 = 0, we do not require a release control policy. In the Appendix, we show that the pmf of queue length, under the m l policy, for all t≥ 0, is given by: Pr(N(t) = 0) =e −¯ r(t,t) Pr(N(t) =k) = Z Tmax 0 e −λou (λ o u) k−1 (k− 1)! λ o F (u)e −¯ r(t−u,t) du, k = 1, 2,... (5.6) where T max :=L/v min , ¯ r(z,t) :=λ o R z 0 F (t−s)ds for all z≥ 0 and t≥ 0. We next study the steady state limit of (5.6). lim t→∞ ¯ r(t,t) = lim t→∞ λ o R t 0 F (t−s)ds = lim t→∞ λ o R t 0 F (s)ds = λ o T, where the last equality follows from the definition of 85 expectation. (5.6) then implies that π 0 = lim t→∞ Pr(N(t) = 0) = e −λoT , and, for k = 1, 2,..., π k = lim t→∞ Pr(N(t) =k) = lim t→∞ Z Tmax 0 e −λou (λ o u) k−1 (k− 1)! λ o F (u)e −¯ r(t−u,t) du = Z Tmax 0 e −λou (λ o u) k−1 (k− 1)! λ o F (u)e − limt→∞ ¯ r(t−u,t) du From this, the expression for π k , k = 1, 2,..., in (5.5) follows from the fact that lim t→∞ ¯ r(t−u,t) = lim t→∞ λ o Z t−u 0 F (t−s)ds =λ o Z ∞ u F (s)ds Remark 16. Little’s law implies that the steady state mean travel time is lower bounded by: ¯ W l 1 := 1 λ 0 ∞ X k=0 kπ k (5.7) with{π k : k = 0, 1,...} given by (5.5). Remark 17. In the presence of sideway arrivals, vehicles on the road will face more slow downs because of entrance of sideway arrivals. Therefore, Remark 16 remains a lower bound when sideway arrivals exist. The following example illustrates (5.5) for a special case. Example 1. Consider the setting of no sideway arrivals, as in Proposition 20. Additionally, let the desired speed of all the vehicles be identically equal to v f . Therefore, the desired travel times are also identical, and equal toT i =L/v f =:T m 86 for all i. In this case, g(u) =λ(T m −u) for u∈ [0,T m ], and g(u) = 0 for u>T m . Therefore, (5.5) becomes: π 0 =e −λoTm π k = Z Tm 0 e −λou (λ o u) k−1 (k− 1)! λ o e −λo(Tm−u) du = (λ o T m ) k k! e −λoTm , k = 1, 2,... Indeed, in this case, the pmf corresponds to that of a Poisson random variable with mean λ o T m . The lower bounds developed in Proposition 20 and Remark 16, while valid for arbitrary4≥ 0, are expected to be conservative for large4. Proposition 21. Consider the HTQ setup in Section 5.1. For anyL> 0, λ s = 0, ϕ =δ 0 , ψ =δ L ,4> 0, and distribution θ: (i) ifλ o >v max /4, then the queue length is unbounded with probability one; and (ii) ifλ o <v max /4, then the steady-state mean travel time and steady-state mean queue length, if they exist, are lower bounded by ¯ W l 2 := L +4 v max + λ o 4 2 2(1− 2λ o v max 4) (5.8) and λ o ¯ W l 2 respectively. Proof. Consider the scenario where the vehicles implement the m 0 car following model and the π o (4,v max ) release policy. It is easy to see that, the throughput under such a relaxation is clearly an upper bound on the throughput under any combination of car following and release policy from Section 5.2. Similarly, the steady state mean queue length and travel times, if they exist, are lower bounds to these quantities under any combination of car following and release policy from Section 5.2. 87 Under m 0 and π o (4,v max ), the queue length on the road segment remain bounded, by L/4, at all times, under all λ o ≥ 0; therefore, it is sufficient to focus on the storage at the origin. The storage on the other hand can be modeled as an M/D/1 queue with arrival rate λ o and service times4/v max . Therefore, for λ o <v max /4, almost surely, the queue length in the storage, and hence the total queue length in the HTQ remains bounded at all times, e.g., see [38]. Furthermore, standard treatment of M/D/1 queue, e.g., in [38], implies that, for λ o < v max /4, thesteadystatemeanwaitingtimeforthestorageisequalto 4 vmax + λo 2 4 2 vmax(vmax−λo4) . Combining this with the fact that the travel time on the road segment is at least L/v max gives the lower bound in (5.8) on the total waiting time in HTQ. The lower bound on the queue length is then obtained from Little’s law. For arbitrary4, one can pick the best of the lower bounds in Propositions 20 and 21, as is formally stated next. Theorem 6. Consider the HTQ setup in Section 5.1 with no vehicles on the road initially. For any L > 0, λ s = 0, ϕ = δ 0 , ψ = δ L , and distribution θ, if λ o < v max /4, then the steady state mean travel time and mean queue lengths, if they exist, are lower bounded by max n ¯ W l 1 , ¯ W l 2 o and λ o max n ¯ W l 1 , ¯ W l 2 o respectively, where ¯ W l 1 and ¯ W l 2 are, respectively, given by (5.7) and (5.8). Remark 18. The assumption of empty initial condition in Propositions 20 and 21, and in Theorem 6 is without loss of generality. This is because one can slightly modify the release policies to empty all the initial vehicles first. The lower bounds would still be valid under the additional delay induced by such a modification. 88 5.3.2 Upper Bounds Inthissection, wederiveupperboundstocomplementthelowerboundsderived in Section 5.3.1. Proposition 22. Consider the HTQ setup in Section 5.1. For anyL> 0, λ s = 0, 4> 0, ϕ =δ 0 , ψ =δ L , and distribution θ, if λ o <v min /4, then the queue length is bounded almost surely. Proof. Consider the scenario where the vehicles implement the m 0 car-following model (cf. Section 5.2.2) and theπ(4,v min ) release policy. The throughput under such a scenario is clearly a lower bound on the throughput under any combina- tion of release policy and car following model from Section 5.2. Under m 0 and π(4,v min ), the storage can be interpreted as an M/D/1 queue with arrival rate λ o and service times4/v min , similar to proof of Proposition 21. The boundedness of the queue length for λ o <v min /4 then follows trivially. Theorem 7. Consider the HTQ setup in Section 5.1. For anyλ o > 0,L> 0,λ s = 0,ϕ =δ 0 ,ψ =δ L , and distributionθ, if the steady-state mean travel time and mean queue length exist, then they are upper bounded by ¯ W u := L+4 v min + λo4 2 2(v 2 min −λov min 4) and λ o ¯ W u respectively. Proof. The steady state mean travel time for the storage, if it exists, is lower bounded by 4 v min + λ4 2 2(v 2 min −λv min 4) . Upon release into the road segment, since the minimum speed is v min , the result for steady state mean travel time for the entire HTQ then follows. Mean steady state queue length is then obtained by Little’s law. Remark 19. Theorems 6 and 7, and Example 1 imply that, as 4→ 0 + and (v max −v min )→ 0 + , the lower and upper bounds on the steady state mean travel time coincide. 89 5.4 Simulation Case Study In this section, we numerically compute and compare the upper and lower bounds derived in Section 5.3, and compare these bounds with empirical mean travel times for the sample car following models in Section 5.2.2, as obtained by simulations. Furthermore, we consider real trajectory data in NGSIM dataset to compare the performance of connected vehicles with human driven vehicles. 5.4.1 Brief Description of the Data Set and Preprocessing WestudythefollowingdatasetfromtheNextGenerationSimulation(NGSIM) data base [1]. The study area contains a 640 m segment on the south-bound direction of US-101 (Hollywood Freeway) in the vicinity of Lankershim Avenue in Los Angeles, CA. The segment has five main lanes, and an auxiliary lane is present throughaportionofthecorridorbetweentheon-rampandoff-ramp. SeeFigure5.4 (a) for an illustration. The data was collected on June 15th 2005 between 07:50 am and 08:35 am. The data was processed to obtain trajectories of individual vehicles, see Fig- ure 5.4 (b) for an illustration. From these individual trajectories, we determined when and where vehicles change lane. This information was then used to construct the spatial distributions ϕ and ψ. Next, we compare the travel time analysis of the data set with the bounds from Sections 5.3.1 and 5.3.2. We also evaluate the effect of cooperative driving on travel times. 90 (a) (ft) (b) Figure 5.4: (a): The study area from NGSIM dataset corresponding to US-101, used in the case study. (Figure is obtained from [1]). (b): Part of trajectories for the left-most lane. Vehicle trajectories that enter at the beginning of the lane and leave at the end of it are shown in blue. Trajectories for the vehicles that enter the lane in the middle are shown in green, and the trajectories of the vehicles that leave the lane in the middle are shown in red lines. Note the backward shockwave due to slower vehicles. In this particular, we did not observe trajectories that join and leave the lane in the middle of section. 5.4.2 Simulation Results Figure 5.5 shows comparison of mean travel times as obtained from the dataset, from the lower bound, without cooperation (Theorem 6) 2 , and with cooperation (as defined shortly), and from the microsimulations, implemented using MATLAB, of the second order car following model, Model 2, from Section 5.2.2, without and with cooperation. The microsimulations with cooperation correspond to letting vehicles traveling at their desired speeds until they meet a faster vehicle in the front, at which point, the follower vehicle changes its speed to be the average of its original desired speed and the speed of the leader vehicle. We define this 2 Figure 5.4 (b) suggests that the NGSIM dataset includes sideway arrivals, but, strictly speak- ing, Theorem 6 is valid only with no sideway arrivals. However, we expect it to be true even with sideway arrivals, and therefore, have included in the comparison. 91 behavior as cooperation among vehicles and combine this with car-following model m l in order numerically obtain lower bounds under this cooperative behavior. We compare different scenarios for all the desired speed distributions. The latter is chosen to be a uniform distribution with the same mean, but different standard deviations, across the scenarios. Figure 5.5 suggests that, as variability in desired speed decreases, (e.g., under cooperative driving behavior), the lower bounds from our analysis as well as microscopic simulations decrease. The figure also suggests that the lower bounds are expectedly smaller under cooperation in comparison to the no cooperation scenario. Figure 5.6 compares the bounds in Theorems 6 and 7, under different λ o ,4, and θ. Figure 5.6 (a) plots the bounds vs. λ o for a fixed4 but two different θ. Both of these θ are uniform distributionswith the same v max but different v min . The plots suggest that, the gap between the bounds decrease with decreasing variability in desired speeds. Figure 5.6 (a) plots the bounds vs. λ o for a fixed θ under different4. θ in this case is again chosen to be a uniform distribution. The plots suggest that the gap between the bounds decreases with decreasing4. These trends observed in Figure 5.6 are consistent with Remark 19. Figure 5.7 compares the mean travel times observed in microsimulation of the first and second order car following models with theoretical bounds, under differ- ent values of model parameters. The travel time estimates from simulations are expectedly within the theoretical bounds, and the trends in the plots are consistent with the interpretation of the model parameters. 92 0.5 1 1.5 2 2.5 3 3.5 Standard Deviation of Speed (m/s) 38 40 42 44 46 48 50 52 54 Mean Travel Time (s) NGSIM Dataset Lower Bound without Cooperation Simulations without Cooperation Lower Bound with Cooperation from Simulations Simulations with Cooperation Figure 5.5: Mean travel time comparison between the NGSIM dataset, lower bound from Theorem 6 and from microsimulations with Model 2 second car fol- lowing rule; the latter two with and without cooperation. The desired speed dis- tribution are modeled to be uniform distributions. The mean is the same 15 m/s in all scenarios, but standard deviations are different. 6 0.1 0.7 1.3 1.9 2.3 M e an T r a v e l T i m e 15 25 35 45 55 65 Up p e r B ou n d , v m i n = 1 0 L o w e r B ou n d , v m i n = 1 0 Up p e r B ou n d , v m i n = 2 0 L o w e r B ou n d , v m i n = 2 0 (a) 6 0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 M e a n T r a v e l T i m e 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 U p p e r B o u n d , 4 = 3 L o w e r B o u n d , 4 = 3 U p p e r B o u n d , 4 = 4 L o w e r B o u n d , 4 = 4 (b) Figure 5.6: Comparison between the bounds from Theorems 6 and 7 under differ- ent values of θ =U [v min ,vmax] (uniform distribution) and4. The other parameters, not shown in the legends, are: (a) L = 500, v max = 30, and4 = 4; (b) L = 50, v min = 19, and v max = 23. 93 6 0.1 0.6 1.1 1.6 2.1 2.6 3 M e a n T r a v e l T i m e 10 11 12 13 14 U p p e r B o u n d L o w e r B o u n d F i r s t - o r d e r M o d e l , , = 2 F i r s t - o r d e r M o d e l , , = 3 (a) 6 0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 M e a n T r a v e l T i m e 11 12 13 14 U p p e r B o u n d L o w e r B o u n d S e c o n d - o r d e r M o d e l , - = 3 S e c o n d - o r d e r M o d e l , - = 1 (b) Figure 5.7: Comparison of theoretical bounds from Theorems 6 and 7 with (a) the first order car following model in (5.1) and (b) the second order car following model in (5.2) under different model parameters. The other parameters common to all the simulation scenarios are L = 250, θ =U [v min ,vmax] (uniform distribution) with v min = 20 and v max = 25,4 = 5, and a max = 4. 94 Chapter 6 Connection between Traffic Flow Capacity and the Throughput of Horizontal Traffic Queues In this chapter, we relate the notion of traffic capacity in the traffic engineering literature and throughput of Horizontal Traffic Queues. For this purpose, we use a specific setup of HTQ framework in order to relate throughput of HTQ to traffic capacity. Capacity plays an important role in the Fundamental Traffic Diagram (FDM) as shown in Figure 6.1. FDM is used in the most of popular macroscopic traffic models such as LWR [43] and LTM [80] models. In the existing literature of transportationengineering, parametersofthis diagram (e.g. capacity)is quantified by using experiments. However, this methodology cannot be extended to the next generation of transportation systems where the microscopic interaction rules are part of design variables. HTQ framework allows us to directly bridge between vehicle-scale interaction rules and macroscopic metrics such as throughput. In particular, by leveraging HTQs, we derive estimates of the capacity for a variety of car-following models. In order to examine the performance of our estimates, we use our estimations in macroscopic models and compare the performance of those models with micro- scopic simulations. In particular, we consider Link Transmission Model (LTM) 95 Flow Density jam k m k m q Figure 6.1: A schematic view of the fundamental traffic diagram. k jam and q m denote the jam density and maximum capacity, respectively. that is a macroscopic model and characterizes the flow of vehicles. In LTM, a tri- angular FDM is considered and this, together with the theory developed by Newell [54], gives a macroscopic representation of the traffic state. Since HTQ is a microscopic model, it requires the specifications of car-following behavior. We consider first and second order car following models. The considered car-following models ensures collision avoidance property. 6.1 Problem Formulation We consider the similar HTQ as described in Section 5.1. However, the motion of vehicles is governed by different set of car-following models described in the fol- lowing of this chapter. We also consider the decomposition of HTQ into two serial queues named as HTQ1 and HTQ2. For a detailed description of this decomposi- tion, readers are referred to Section 5.2.1. We consider an incoming flow of vehicles at origin with rateλ. However, for microscopic models, we need exact arrival times of vehicles. In order to model this incoming flow, while recording the exact arrival times of vehicles, we consider that the arrival process of vehicles is governed with a stationary stochastic counting process. In particular, we assume that the tem- poral arrival process of vehicles is governed by a Poisson process with rate λ. In 96 this case, the average inter-arrival time of vehicles is a random variable with an exponential distribution with mean 1/λ. This implies time variant fluctuations in the system which is natural in transportation models. Upon arrival, vehicles start traveling from left to right and leave the system once they reach a common destination point atx =L. We let4 cr denote the average length of vehicles. This implies that the traffic density associated with full congestion is k jam = 1/4 cr . Our primary focus in this chapter is to characterize the maximum possible flow rate (i.e. capacity) for a given car-following model. Since the considered HTQ has no constraint on the right hand side (i.e. the exit point) the throughput of considered HTQ can characterize the capacity. Therefore, using the framework of HTQ, we provide analytical estimates on the capacity of the system for a given car-following model. In the next section, we briefly described the considered first and second order car-following models that are considered in this chapter. 6.2 Car-following Models In this section, we describe car-following models that are used in the analysis of this chapter. We consider both first and second order models that ensure collision avoidance property. 6.2.1 First Order Model Thefollowingcar-followingmodel, denotedbym 1 , describesasafecar-following model. ˙ x i = min (v max ,β(y i −4 cr )) ∀i∈{1,··· ,N(t)} (6.1) where β > 0 is a parameter of this model,4 cr is the length of vehicles. 97 This model ensures that the speed of each vehicle is an increasing function of the inter-vehicular distance and it becomes saturated at the maximum allowed speed. Moreover, when the inter-vehicular distance between two vehicles equals the the length of vehicle, this model makes the follower vehicle stop. 6.2.2 Second Order Model In this chapter, we consider three different second-order car-following models. As we will prove in Section 6.3.2, the implementation of these models, combined with π o (4, 0), ensures collision avoidance property. Model m 2 m 2 is a simple second-order car-following model. Under this model, ¨ x i =1 { ˙ x i <vmax} a max , ∀i∈{1··· ,N(t)} (6.2) where 1 { ˙ x i <vmax} is an indicator function i.e. it returns one if ˙ x i <v max , and zero, otherwise. Under m 2 , vehicles speed up with a constant acceleration of a max until their speed reachesv max . From that point on, they travel with a constant speed of v max . Model m 3 m 3 is another second-order car-following model that incorporates the relative speed of vehicles. In particular, ¨ x i = 1 { ˙ x i <vmax} a max , i = 1 min(a max ,α ˙ y i ), i∈{2,··· ,N(t)} (6.3) 98 where ˙ y i denotesthetimederivativeoftherelativedistancewhichgivestherelative speed. Also, α≥ 0 is a parameter of this model. Note that, under m 3 , the first vehicle implements a different model. 6.3 Throughput Analysis Inthissection, weanalyzethethroughputofHTQ(Definition1)underdifferent car-following models defined in Section 6.2. 6.3.1 First Order Model We consider the first-order car-following model m 1 (6.1), described in Section 6.2.1. In order to prove the main result as stated in Theorem 8, we need several intermediate results. In Lemma 13, by neglecting the maximum speed constraint in m 1 (6.1), we characterize the trajectory of the second vehicle (i.e. the vehicle that follows the leader). Then, in Lemma 14, we show that, in a special case of interest, the maximum speed constraint in m 1 (6.1) never becomes active. Lemma 13. For any v max > 0, L≥4 cr , β > 0, x 1 (0)∈ [4 cr ,L], and4 cr > 0, under m 1 (6.1) and by neglecting the maximum speed constraint, the trajectory of the second vehicle is given as x 2 (t) =v max t + 4 cr + v max β −x 1 (0) ! e −βt − v max β + (x 1 (0)−4 cr ) (6.4) Proof. The first vehicle always travels with the constant speed of v max . Therefore, its trajectory for any t≥ 0 is given as x 1 (t) =v max t +x 1 (0) (6.5) 99 By plugging (6.5) in (6.1) and neglecting the maximum speed constraint, we get ˙ x 2 (t) =β(v max t +x 1 (0)−x 2 (t)−4 cr ) The solution of the above ordinary differential equation leads to the result of this lemma. While the above lemma characterizes the trajectory of second vehicle when the maximumdesiredspeedisneglected, thenextresultshowsthat, whenx 1 (0) =4 cr , the maximum speed constraint in (6.1) never becomes active. It should be noted thatx 1 (0) =4 cr is of special interest for us since if HTQ1 is non-empty, the second vehicle will be released to HTQ2 once the first vehicle reaches x 1 (0) =4 cr . Lemma 14. For any v max > 0, L≥4 cr , β > 0, x 1 (0) =4 cr , and4 cr > 0, under m 1 (6.1), for all t≥ 0 ˙ x 2 (t)<v max Proof. By Lemma 13, the trajectory of second vehicle is given as x 2 (t) =v max t + 1 β (e −βt − 1) ! . By (6.5), the trajectory of the first vehicle is given asx 1 (t) =v max t+4 cr ; therefore, the distance between first and second vehicles is y 2 (t) =4 cr + v max β (1−e −βt ). One can see that, for any β > 0, y 2 (t)<4 cr +v max /β for all times. By plugging this upper bound for y 2 in (6.1) we get the desired result. 100 The above result shows that, in the heavy regime, when HTQ1 is never empty and vehicles are released to HTQ1 every time that a vehicle passes x =4 cr , the maximum speed limit constraint in (6.1) for all non-leader vehicles never is active. Remark 20. When x 1 (0) =4 cr and HTQ1 is non-empty, by using Lemmas 13 and 14, one can determine t ∗ i.e. the time it takes for the second vehicle to reach x 2 (t ∗ ) =4 cr by solving the following equation. v max t ∗ + 1 β (e −βt ∗ − 1) ! =4 cr (6.6) In the next result, we show that, when HTQ2 is initially empty, HTQ2 contains infinite number of vehicles, and the length of the road equals, L ∗ :=v max t ∗ +4 cr , (6.7) the trajectories of all vehicles except for the first vehicle is the same up to a translation in time. Proposition 23. For any v max > 0, β > 0,4 cr > 0, N 1 (0) 1, N 2 (0) = 0, and L =L ∗ , under m 1 (6.1), the trajectory of vehicles is given as x 1 (t) =v max t x i (t) = 0 t≤r i v max (t−r i ) + 1 β (e −β(t−r i ) − 1) r i <t≤r i +t ∗ 4 cr +v max (t−r i −t ∗ ) t>r i +t ∗ , ∀i∈{2, 3,···} where r i denotes the release time of i-th vehicle to HTQ2. 101 Proof. Since N 2 (0) = 0, the first vehicle is released at t =r 1 = 0 and travels with a constant speed of v max ; therefore, its trajectory is given as x 1 (t) = v max t. Once this vehicle reaches x 1 (t = r 2 ) =4 cr , the second vehicle is released to HTQ2. It is easy to see that r 2 =4 cr /v max . By Remark 20, one can see that at time t = r 2 +t ∗ the second vehicle reaches4 cr and third vehicle can be released to HTQ2; therefore r 3 =r 2 +t ∗ . The location of the first vehicle at this moment i.e. t = r 3 is x 1 (t = r 3 ) = v max r 3 = v max (r 2 +t ∗ ) = v max t ∗ +4 cr = L ∗ . This implies that at t =r 3 , the first vehicle reaches the departure point and leaves the system. Now, since no vehicle is in front of second vehicle, it travels with the constant speed ofv max and its trajectory is given asx 2 (t>r 2 +t ∗ ) =4 cr +v max (t−r 2 −t ∗ ). Now, second vehicle will have the same effect on the motion third vehicle as the first vehicle had on the second vehicle once it was released. This shows that third vehicle will have the same trajectory as second vehicle. It can be seen that this sequence continues and trajectory of vehicles released after the first vehicle is the same. In particular, their trajectories is just translated in time by t ∗ . The above result suggests that vehicles depart HTQ1 every t ∗ unit of time. In other words, the maximum departure rate from HTQ1 when HTQ1 is always full is 1/t ∗ . Using this fact, in the following result, we characterize throughput for L =L ∗ . Proposition 24. For anyv max > 0, β > 0,4 cr > 0, andL =L ∗ , underm 1 (6.1), the throughput of the system is given as λ max (L ∗ ) = 1/t ∗ a.s. (6.8) Proof. It suffices to show that, with probability one, : 1) when λ < 1/t ∗ queue length remains bounded, and 2) when λ > 1/t ∗ , queue length grows to infinity. 102 Whenλ< 1/t ∗ , with contradiction, assume that with a non-zero probability queue length goes to infinity. Since the capacity of HTQ2 is finite, this implies that there exists a t 0 > 0 such that HTQ1 remains non-empty for all t > t 0 . Then, by Proposition 23, the departure rate of the system will be 1/t ∗ ; however, the arrival rate isλ< 1/t ∗ . This implies that after a finite time, the queue length will become zero which leads to a contradiction. Whenλ> 1/t ∗ , by Proposition 23, we see that the arrival rate is greater than the maximum departure rate of HTQ1. By law of large number arguments, this implies that queue length in HTQ1 and consequently in HTQ grows to infinity, with probability one. While the above result characterizes the throughput of HTQ for L = L ∗ , the following result does the same for another special case i.e. L =4 cr . Lemma 15. For any v max > 0, β > 0,4 cr > 0 and L =4 cr ,under(6.1), λ max (L =4 cr ) =v max /4 cr a.s. Proof. WhenL =4 cr , underπ 4cr , there will be always at most one vehicle on the road. Therefore, by (6.1), the travel time of vehicles will be deterministic and equal to4 cr /v max . Therefore, the maximum departure rate from HTQ2 isv max /4 cr . The following result shows the monotonicity of throughput with respect to the length of the road, L. Proposition 25 (Monotonicity of Throughput). For two HTQ with different lengths L s > 0 and L l > 0, if L s <L l , then, with probability one, λ max (L s )>λ max (L l ) 103 Proof. For brevity in notation, we refer to the HTQ corresponding toL s , as HTQ- S. We refer to the other queue as HTQ-L. It suffices to show that queue length in HTQ-S is always smaller than queue length in HTQ-L i.e. N s (t)≤N l (t),∀t≥ 0. Let fix the same realization of arrival process (i.e. arrival times of vehicles) for both systems. Let an event represent either an arrival to HTQ-S and HTQ-L (arrivals to both systems are synchronized) or departure from either HTQ-S or HTQ-L. Let τ i denote the time of i-th event. We claim that immediately after any event until immediately after the next event the queue length in HTQ-S remains smaller or equal to the queue length of HTQ-L, and vehicles in HTQ-S are ahead of corre- sponding vehicles in HTQ-L. In particular, this claim means that N s (t)≤N l (t) andx s j (t)≥x l j (t) for anyj∈I HTQ−S ,t∈ [τ + i ,τ + i+1 ] andi∈{1, 2,···}. Since both systems start from empty initial conditions, the first event corresponds to the first arrival that is synchronized in both systems. Until the time of the second event, there will be only one vehicle in both systems that travels with constant speed of v max . Thus, the claim holds, with equality, for all t∈ [τ + 1 ,τ 2 ). Since HTQ-S has a shorter road, the first departure always happens from HTQ-S. Therefore, the second event could be either another arrival or a departure from HTQ-S. In both cases, it is easy to see that claim holds for all t∈ [τ + 1 ,τ + 2 ]. Assume that, for some i > 2, claim holds for all t∈ [τ + i ,τ + i+1 ]. We show that it also holds true for any t ∈ [τ + i+1 ,τ + i+2 ]. By Lemmas 24, the claim holds for t∈ [τ + i+1 ,τ i+2 ). The (i + 2)-th event can be 1) an arrival, 2) a departure from HTQ-S, or 3) a departure from HTQ-L. If it is either an arrival or departure from HTQ-S, it is easy to see that claim holds for t∈ [τ + i+1 ,τ + i+2 ]. Since, immediately before (i + 2)-th event, all vehicles in HTQ-S are ahead of the corresponding vehicles in HTQ-L, the (i + 2)-th event can be a departure from HTQ-L only if N s (τ − i+2 )<N l (τ − i+2 ). In this case, immediately after a departure from HTQ-L still 104 claim holds. Since the i-th event was chosen arbitrary, this implies that the claim holds for all times which consequently establishes the result of this proposition. Now, we investigate dynamics of inter-vehicular distances in between events (i.e. departure or arrivals of vehicles). Let y T = (y 2 ,y 3 ,··· ,y N(t) ) (6.9) be a vector that keeps track of the inter-vehicular distances of all the vehicles except for the first vehicle,y i =x i−1 −x i ; therefore, ˙ y i = ˙ x i−1 − ˙ x i . Thus, by (6.1) and neglecting the speed limit constraint, when N(t)≥ 2, dynamics of y can be written as, ˙ y =Ay +B (6.10) where A∈R (N(t)−1)×(N(t)−1) , B∈R N(t)−1 and they are expressed as A = −β 0 0 ··· 0 0 β −β 0 ··· 0 0 0 β −β ··· 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ··· β −β , B = v max +β4 cr 0 0 . . . 0 (6.11) The following result establishes asymptotic stability of y ∗ := v max β +4 cr ! 1 N(t)−1 , in between events. 105 Proposition 26. For any v max > 0, β > 0, and4 cr > 0, under the dynamics in (6.10), we have lim t→∞ y =y ∗ . Proof. It is easy to verify that for y ∗ , ˙ y = 0; therefore, y ∗ is an equilibrium point. Also, by the following similarity transformation (See Appendix, Definition 7), matrix A can be transformed to ˜ A. ˜ A :=T −1 AT = −β 1 −β . . . . . . 1 −β , T = β 0 β 1 . . . β N(t)−1 Note that ˜ A is in canonical Jordan form; therefore, all of its eigenvalues are the same and equal to−β. This implies that eigenvalues of A are also the same and equal to−β < 0. This shows that y ∗ is asymptotically stable. Remark 21 (Headways in Steady State). At y =y ∗ , under m 1 (6.1), the speed of all vehicles is v max . Therefore, in the steady state, the headways also converge and the steady state value of headway for all vehicles will be h ∗ := v max +β4 cr βv max . (6.12) Proposition26suggeststhat, startingfromanyinitialcondition, inter-vehicular distances converge to their steady state values exponentially fast. However, it was assumed that system remains closed (i.e. no departure or arrival). However, under m 1 (6.1), an arrival (and consequently a change of size of the system from N(t) to N(t) + 1) does not perturb the convergence of the first N(t) vehicles. Also, if we assume L→∞, there will be no departure and system can be considered closed for the first N(t) vehicles. 106 Theorem 8. For any v max > 0,4 cr , and β > 0, under m 1 (6.1), the throughput of system is monotonically decreasing with respect to L. Also, for L =4 cr , λ max = v max /4 cr , for L =L ∗ , λ max = 1/t ∗ , and as L→∞, λ max → 1/h ∗ , with probability one. Proof. The monotonicity of throughput with respect to L follows by Proposition 25. Also, the result forL =4 cr andL =L ∗ follows by Lemma 15 and Proposition 24, respectively. Also, when L → ∞, by Remark 21 the headway of vehicles approaches h ∗ (6.12) and the rate at which vehicles pass a point is the inverse of this quantity. Remark 22. Theorem 8 implies that, when L ∈ [4 cr ,L ∗ ], we have λ max ∈ [v max /4 cr , 1/t ∗ ], and when L > L ∗ , we have λ max ∈ [1/t ∗ , 1/h ∗ ]. Moreover, since dynamics of y in (6.10) converges exponentially fast, as L becomes larger than 2v max the throughout will be very close to the limit value when L→∞. 6.3.2 Second Order Models In this section, we analyze the throughput of HTQ under second-order car- following models defined in Section 6.2.2. Model m 2 In this section, we provide throughput analysis for the second-order model,m 2 , defined in (6.2). In the following result, we show that this model, combined with π 0 4cr , guarantees safety. Lemma 16. For any a max > 0, v max > 0,4 cr ≥ 0, and L≥4 cr , under m 2 (6.2) and π 0 4cr , y i (t)≥4 cr for all i and all times. 107 Proof. We show that for any two consecutive vehicles, say (i− 1)-th and i-th, y i (t)≥4 cr for all times. Under π 0 4cr , the initial relative distance is greater or equal to4 cr . Also, if bothi-th and (i− 1)-th vehicles travel with the same type of motion (i.e. constant acceleration or constant speed), their relative distance does notdecrease. Whenithappensthatpartoftheirmotionisnotofthesametype, we let designatet = 0 to the time at which (i−1)-th vehicle switches to constant speed motion, buti-th vehicle is still traveling with constant acceleration. Also, without loss of generality, let assumex i (t = 0) = 0. Therefore,x i (t> 0) = 1 2 a max t 2 + ˙ x i (0)t and x i−1 (t> 0) =v max t +y i (0) and y i (t> 0) = (v max − ˙ x i (0))t +y 1 (0)− 1 2 a max t 2 . At t = 0, y i (t) = y i (0) ≥ 4 cr and y i (t) is an increasing function in t for all t∈ [0, (v max − ˙ x i (0))/a max ] which implies that y i (t)≥4 cr for all t∈ [0, (v max − ˙ x i (0))/a max ]. It is easy to see that at t = (v max − ˙ x i (0))/a max , the i-th vehicle will also switch to constant speed motion. Therefore, y i (t) will not decrease after t> (v max − ˙ x i (0))/a max . Itshouldbenotedthat, underm 2 (6.2), themotionofeachvehicleonlydepends on its own speed and does not rely on the position, speed, or acceleration of other vehicles. Therefore, when v max and a max are common for all vehicles, the trajectories of vehicles will look like the same. The following result characterizes the time it takes for a vehicle to travel from origin to x =4 cr . Proposition 27. For any a max > 0, v max > 0, L≥4 cr and4 cr > 0, under m 2 (6.2) and π 0 4cr , the time it takes for a vehicle to travel from origin to x =4 cr is given as t ∗ := q (24 cr )/a max if 4 cr ≤v 2 max /(2a max ) v max /(2a max ) +4 cr /v max otherwise (6.13) 108 Proof. Let, without loss of generality, designate t = 0 to the time at which a vehicle, say i-th, is released to HTQ2 i.e. x i (0) = 0 and ˙ x i (0) = 0. Since vehicles speed up with a constant acceleration, the time it takes fori-th vehicle to speed up to v max is ˜ t :=v max /a max . Also, the position of i-th vehicle at t = ˜ t will bex i ( ˜ t) = a max ˜ t 2 /2 =v 2 max /(2a max ). If x i ( ˜ t)≥4 cr , then i-th vehicle travels interval [0,4 cr ] with a constant acceleration motion; therefore, it takes q (24 cr )/a max amount of time for it to reach x =4 cr . On the other hand, if x i ( ˜ t) <4 cr , then in [0,x i ( ˜ t)] the vehicle travels with constant acceleration of a max , and in (x i ( ˜ t),4 cr ] it travels with constant speed ofv max . Therefore, in this case, the time it takes for it to reach x =4 cr can be easily computed and is equal to v max /(2a max ) +4 cr /v max . Theorem 9. For any a max > 0, v max > 0,4 cr ≥ 0, and L≥4 cr , under m 2 (6.2) and π 0 4cr , λ max = 1/t ∗ , where t ∗ is defined in (6.13). Proof. By Lemma 16, the number of vehicles in HTQ2 always remains upper bounded by L/4 cr . By Proposition 27, under m 2 (6.2) and π 0 4cr , the shortest inter-release time can be t ∗ . Therefore, the maximum departure rate from HTQ1 is 1/t ∗ which implies that for λ > 1/t ∗ , queue length in HTQ1 grows unbounded and for λ< 1/t ∗ , it remains bounded. Model m 3 The following result shows that the implementation ofm 3 , combined withπ 0 4cr , guarantees safe car-following behavior. Lemma 17. For any a max > 0, v max > 0, α > 0,4 cr ≥ 0, and L≥4 cr , under m 3 (6.3) and π 0 4cr , y i (t)≥4 cr for all i and all times. Proof. Under π 0 4cr , we have y i (0)≥4 cr and ˙ x i (0) = 0 for all i. Under m 3 (6.3), i-th vehicle speeds up, but, as its speed gets closer to the speed of front vehicle, it 109 decreasesitsacceleration. Thisimpliesthatitsspeedneverbecomeslargerthanthe speed of (i−1)-th vehicle. This implies that relative speed i.e. ˙ y(t) = ˙ x i−1 − ˙ x i ≥ 0 for all times which establishes this lemma. The following result characterizes the steady state headway of vehicles. Proposition 28. For any a max > 0, v max > 0, α ∈ (0,a max /v max ], 4 cr ≥ 0, and L 4 cr , under m 3 (6.3) and π 0 4cr , the headways of vehicles reaches h ∗ = 1/α +4 cr /v max . Proof. Similar to the proof of Theorem 8, we study dynamics of y (6.9). We have y i =x i−1 −x i ; therefore, ¨ y i = ¨ x i−1 − ¨ x i . Thus, by (6.3), whenN(t)≥ 2, dynamics of y can be written as, ¨ y =A 1 ˙ y (6.14) where A 1 ∈R (N(t)−1)×(N(t)−1) , B 1 ∈R N(t)−1 and they are expressed as A 1 = −α 0 0 ··· 0 0 α −α 0 ··· 0 0 0 α −α ··· 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ··· α −α (6.15) Similar to the proof of Proposition 26, one can show that the eigenvalues of A 1 are the same and equal to−α < 0. Therefore, dynamics in (6.14) converges to a asymptotically stable equilibrium point i.e. ˙ y ∗ = 0. This implies that, in the equilibrium, relative speeds of vehicles is zero; therefore, they all have the same speed of v max . 110 By integrating (6.3), we get ˙ x i (t) =αy i (t) + ( ˙ x i (0)−αy i (0)), ∀i∈{2,··· ,N(t)}. (6.16) Since vehicles are released according toπ 0 4cr , ˙ x i (0) = 0 for alli. Also, since we are interested in the heavy regime performance of the system, with high probability y i (0) =4 cr for all i. Therefore, ˙ y i = α(y i−1 −y i ) for all i∈{3,··· ,N(t)}, and ˙ y 2 =v max −α(y i −4 cr ). Thus, ˙ y =A 2 y +B 2 (6.17) where A 2 ∈R (N(t)−1)×(N(t)−1) , B 2 ∈R N(t)−1 and they are expressed as A 2 = −α 0 0 ··· 0 0 α −α 0 ··· 0 0 0 α −α ··· 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ··· α −α , B 2 = v max +α4 cr 0 0 . . . 0 (6.18) Again, similar to the proof of Proposition 26, one can show that the dynamics converges to y ∗ = (v max /α +4 cr )1 N(t)−1 . This combined with (6.16) gives the result. Theorem 10. For any a max > 0, v max > 0, α∈ (0,a max /v max ],4 cr ≥ 0, and L4 cr , underm 3 (6.3) andπ 0 4cr , the throughput isλ max = 1/h ∗ , with probability one. Proof. The proof is similar to the proof of Theorem 8 and follows by Proposition 28. 111 6.4 Simulations In this section, we report simulation results. In Figure 6.2, one can see that by matching the throughput, obtained from HTQ analysis, to the q m we get good match between LTM and HTQ models. 6 0.1 0.2 0.3 0.4 0.5 0.6 Average Queue Length 0 2 4 6 8 10 12 L = 1 0 0 ; 4 c r = 4 ; v m = 1 0 LTM HTQ Figure 6.2: Average queue length in LTM and HTQ for various values ofλ. Param- eters of simulations are: L = 100,4 cr = 4,v max = 10. 112 Chapter 7 Vacation Queueing Models for Analyzing Traffic Intersections 7.1 Introduction Analysis of signalized intersections, under a given control policy has been the topic of research for decades. Vehicles delay and the number of queued vehicles (referred to as queue length) in intersections are some of the quantities of interest in analyzing the performance of intersections. Uncertain and time-varying traf- fic conditions, interruptions in service due to red periods, and queue spillbacks due to finite capacity are some of the factors that make analysis of these systems challenging. Queueing models have been a compelling framework for analyzing traffic intersections due to their ability to model congestion and service processes (See e.g.[33, 78, 25, 42]). The choice of queuing model can significantly affect the accuracy of predictions. For example, due to the existence of analytical results of M/M/1queues (that arequeueingmodelswith Poisson arrivaland exponential ser- vice times), they are often used for computing transient queue length distribution of traffic queues [60]. The exponential service times in M/M/1 queues introduce a high variance that may be justifiable in highway networks where headways can have a higher variability. However, it is well-known that, at traffic signals, from the onset of green periods, the inter-departure times have a low variance and con- verge to the saturation headway (See Figure 7.1 for an illustration) [14]. The high 113 variance of exponential service times can significantly increase the mean queue length compared to queueing models with smaller variance in service times. To see this, the steady-state mean queue length for M/M/1 and M/D/1 (queues with deterministic service times) queues are given in the following (See e.g. [38]), E[n] =ρ +ρ 2 (2) 2(1−ρ) M/M/1, E[n] =ρ +ρ 2 1 2(1−ρ) , M/D/1 (7.1) where ρ is the degree of saturation of queues. In (7.1), it can be observed that the first terms are identical for both M/D/1 and M/M/1 queues, but the second term in M/M/1 queue is twice the second term in M/D/1 queue. This implies that as the queue approaches the heavy regime (i.e. as ρ→ 1), M/M/1 queues significantly over-estimate the mean queue length. 2 1 x t a b H Figure 7.1: A schematic view of the saturation headway of vehicles, H [14]. Red dots are vehicles waiting in the queue. Although M/D/1 queues are more realistic for traffic queues at intersections, from the perspective of analysis, M/D/1 queues are more complex because, unlike M/M/1 queues, the departure process does not posses Markovian property. In this chapter, we model traffic queues at intersections with finite capacity M/D/1 queues (referred to as M/D/1/N) that can incorporate spillbacks. We charac- terize the transient queue length distribution by deriving differential equations 114 equations. Unlike M/M/1 queues in which departure rates in differential equa- tions are independent of the state of queue, in M/D/1 queues, departure rates are state-dependent. We extend techniques proposed in [19] to fully characterize transient departure process of M/D/1 queues. To the best of our knowledge, this approach has not been used in transportation literature. The characterization of departure process allows us to model red periods as server vacations and compute the intra-cycle fluctuations of queue length and capture a link-level properties by incorporating forward and backward traffic shockwaves. In this chapter, the transient characterization of departure process incorporates service interruptions and leads to a computational method that can determine transient intra-cycle fluctuations of traffic queues. In this chapter, we propose a vacation queuing model for finite capacity M X /D/1 queues to compute the transient queue length distribution at signalized intersections under both fixed time and adaptive control policies. We model each incoming street to experience non-stationary and compound Poisson arrivals. This arrival process allows us to model time-varying traffic conditions and also platoon arrivals of vehicles. When a vehicle arrives and finds the queue at capacity, it will wait in an infinite capacity storage and join the queue as soon as there is space available. Our results extends the work in [19] by introducing server vacations, time-varying and platoon arrivals, and adding an infinite storage to prevent loss of blocked vehicles. Moreover, convergence guarantees to steady-state distribu- tions are provided, including for adaptive control policies. In order to capture link-level characteristics of traffic queues, we integrate our proposed model with double-queue model [60] that is a link-level model for traffic queues. Double-queue model captures the free-flow travel time delay when the link is in free-flow state, and backward shockwave time delay when the queue is in congested state . In [60], 115 downstream queue (DQ) and upstream queue (UQ) are modeled as M/M/1/N queues that are finite capacity queueing systems with exponential inter-arrival and service times. However, as shown in (7.1), in traffic intersections, the assumption of exponential service times leads to over-estimation of queue length. In this chap- ter, we extend double-queue model to traffic intersection setups by considering DQ and UQ as M/D/1/N queues instead of M/M/1/N queues. Microscopic simula- tionssuggestsignificantimprovementintheaccuracyoftheproposeddouble-queue model. Moreover, simulations show consistency between our proposed models and the classical results in the literature such as Webster [76] and Akcelik model [6]. The main contributions of this chapter are as follows. First, we propose a finite capacity M/D/1/N queueing model with storage for signalized traffic intersections in which red periods are explicitly modeled as service vacations. Second, we extend transient analysis of M/D/1/N queues in [19] to characterize the transient depar- ture process of the proposed queueing model. Third, we identify an imbedded Markov Chain for the proposed queueing model and derive differential equations that allow us to compute transient queue length distribution at any point dur- ing the cycle under non-stationary traffic conditions, and also provide sufficient conditions for convergence to steady-state, under fixed-time policy and adaptive control policy. Fourth, in order to capture spillbacks and link-level characteris- tics of queues, we integrate M/D/1/N queueing models into original double-queue model to increase the accuracy in traffic intersection applications. The rest of this chapter is organized as follows. In Section 7.2, we formulate the problem and describe details of the considered model. In Section 7.3, we provide a brief overview of existing models that are relevant to the results of this chapter. We describe the transient queue length distribution of M/D/1/N queues with and without storage over red and green periods in Section 7.4. We apply the proposed 116 queueing models to double-queue model in Section 7.5. In Section 7.6, we provide convergenceguaranteesforqueuelengthdistributionunderfixedtimeandadaptive control policies. We report our simulation results in Section 7.7. 7.2 Problem Formulation We consider single-lane traffic streets 1 meeting at a signalized intersection. A control policy determines the duration of green phases for each traffic street. We consider both fixed time and adaptive control policies. For simplicity, we postpone the discussion of adaptive policies until Section 7.6. Vehicles join queues in each street either individually or in platoons. For each queue, inter-arrival times are determined using a non-homogenous Poisson processes with rate λ(t),t≥ 0. Each arrival brings a platoon of vehicle whose size,X, is a random variable with support {1, 2,···}. Therefore, thearrivalprocessisatime-dependentbatchPoissonprocess that is often used for traffic simulations (see e.g. [40]). Time dependency of arrival rates allow to incorporate temporal traffic dynamics, and random platoon size captures the variability and non-smoothness of arrival process in traffic flows. Let the number of vehicles that are in queue be denoted by{n(t),t≥ 0}. We shall refer to this quantity as the queue length. Let ¯ n(t) denote the mean queue length at time t. Due to physical space constraints, a street has a finite capacity of N vehicles; therefore, finite capacity queueing models are used. If a vehicle arrives when queue is at capacity, it waits in a dummy storage until space becomes available. Let H denote the saturation headway of traffic in one of the street, and μ = 1/H be its saturation flow rate. Different streets may have different saturation headways. At the onset of green period, inter-departure times are not 1 We use street, link and stream interchangeably in the chapter. 117 initiallyH, but they converge toward it with successive vehicles (see Figure 7.1 for an illustration). This process introduces a lost time due to starting [14]. In [76], a (t,x) diagram, similar to Figure 7.1, is used to show that the lost time is the time difference between pointsa andb in Figure 7.1. This means if vehicles accelerated instantaneously and the first one were to start at point b instead of a, then the resulting idealized trajectories would coincide with the original ones. Therefore, in our analysis, whenever we mention a green period duration, we mean the effective green period. We assume that vehicles start departing at the beginning of the effective green period. Although this assumption may perturb the departure time of the first or second vehicle, perturbations diminished with successive vehicles. Thus, we assume that the service time of each vehicle is constant and equals toH. Therefore, we model each traffic queue with and M X /D/1 queueing model in which batch arrivals are Markovian and service times are deterministic and constant. The signal cycle length, denoted as C, and the red and green periods are denoted byR andG, respectively. Under a fixed time control policy and stationary arrival process, the effective saturation flow, μ e , and the degree of saturation, ρ, are defined as (μ e ) =Gμ/C, ρ =λ/μ e (7.2) Let π j (t) := Pr(n(t) =j) be the probability that total number of queued vehicles in one of streets is j ∈ {0, 1, 2,···} at timet ≥ 0. Correspond- ingly, we denote the instantaneous probability distribution of queue length by π(t) = (π 0 (t),π 1 (t),π 2 (t)··· ) T . In this chapter, we develop analytical tools in order to characterize the transient probability distribution of queue length in each street, π(t). The queueing system thus can be interpreted as a vacation queue, 118 however the key difference with respect to time limited policy is that, in the pro- posed system, the vacationing discipline is determined by the control policy (fixed or adaptive). 7.3 Existing Queue Size Computation Tech- niques for Fixed-Time Policies In this section, we provide a brief overview of relevant queue models from the literature along with the associated queue length computation technique. 7.3.1 Uninterrupted Queues The simplest analytical approach to incorporate interruptions due to red peri- ods, under fixed time policy, is to approximate an interrupted departure process by an uninterrupted departure process with an effective service rate [76, 3]. However, this feature does not provide insight into the intra-cycle variations in queue size. We refer to this model as the uninterrupted model and the model that considers interruptions caused by red periods as the on/off model. It can be proven, e.g., see [52], that the the uninterrupted model gives a lower bound on the queue length at the end of red periods for the on/off model. 7.3.2 Webster’s Model Forunder-saturatedqueues, [76]proposedthefollowingapproximateexpression for steady-state average delay of vehicles in queues under fixed-time control policy E[w] = C(1−G/C) 2 2(1−ρG/C) + ρ 2 2λ(1−ρ) − 0.65 C λ 2 1 3 ρ 2+5G/C (7.3) 119 whereρ andμ e are given in (7.2). The first term in (7.3) is derived assuming uni- form arrivals and using a geometric interpretation of cumulative curves of arrivals and departures of vehicles. The second term is the mean waiting time in an unin- terrupted M/D/1 queue. The third term is an empirical correction factor for better fit with simulation results. However, this model does not give higher moment infor- mation about queue length, does not provide insight into intra-cycle queue length variations, and is not applicable to adaptive control policies. 7.3.3 Time-dependent Models Time-dependent models provide non-asymptotic characterization, unlike Web- ster’smodel. Inthesemodelsacoordinatetransformation, proposedby[37], isused to interpolate between steady-state approximation for the under-saturated regime and deterministic over-saturated models; however, there is no rigorous theoretical basis for this method [29]. As an example, [6] used this method to characterize the expected overflow queue lengthn o (queue length immediately after a green phase) over a finite evaluation period T. The corresponding expression for ρ > ρ 0 , with ρ 0 = 0.67 +μG/600, is given as E T [n o ] = CT 4 ρ− 1 + s (ρ− 1) 2 + 12(ρ−ρ 0 ) CT (7.4) Note that this model characterizes queue length only at a specific time in the cycle, does not provide higher moment information about queue length, and does not extend to adaptive policies. Finally, there exists no rigorous theoretical basis for this method, e.g., see [29]. In another time-dependent model [75], by constructing a Markov chain, a prob- abilistic formulation for n o under fixed-time control policy is proposed. In this 120 model, the transition probability of overflow queue from one cycle to another is computed; however, for simplicity, the number of departures in all cycles are assumed to be constant and independent of arrivals. Since queue length cannot be negative, when the number of departures in a cycle is larger than arrivals, the queue is assumed to be zero at the end of cycle. Although this model provides distribution of overflow queue length, it does not specify a procedure to determine the constant number of departure within a cycle. 7.3.4 Queues with on/off Service In a recent work [45], a model is proposed to consider on and off service periods for fixed time traffic signals. In this work, a Markov chain whose state is (n,s) is considered where s∈{on,off} represents the state of server. In this model, a Poisson clock model is used to determine transition rates. For example, when the k-th state is (n k > 0,off), the next state will be (n k + 1,off) w.p. λ/(λ +γ 2 ) (n k ,on) w.p. γ 2 /(λ +γ 2 ) (7.5) where γ 2 (resp. γ 1 ) is the number of switches from off (resp. on) to on (resp. off) service per unit of time . For example, for C = 120 seconds, and G = R = 60 seconds, γ 2 = γ 1 = 1/120 switches per seconds [45]. Under this model, the mean queue length is given by: ¯ n = λγ 2 1 + 2λγ 1 γ 2 +λγ 1 μ +λγ 2 2 (γ 1 +γ 2 )(γ 2 μ−λ(γ 1 +γ 2 ) (7.6) Once the server switches from on to an off state, the number of transition it takes to return to an on state has a geometric distribution with mean γ 2 /(γ 2 +λ). For 121 practical values of C, R, and G this random variable has a very large variance. Therefore, it is reasonable to expect that (7.6) over-estimates queue lengths. Our simulations in Section 7.7 indeed are consistent with this argument. 7.3.5 Double Queue Model Double-queue model proposed in [60] is another class of traffic queue that, unlike aforementioned models, provides a link-level characterization of the queue length. Double-queue model provides a stochastic formulation of link-transmission model (LTM) [80] that is based on Newell’s simplified theory of kinematics waves. In this theory, a triangular fundamental diagram (see Figure 7.2) is considered, and free flow speed is v f , and backward wave speed is w. Maximum flow and jam density are denoted by q max and k jam , respectively. k q max q jam k f v w Figure 7.2: A triangular traffic diagram. LetN(x,t) denote the cumulative number of vehicles that have passed location x by time t. In LTM model, the maximum flow that can leave the downstream (resp. enter into upstream) of link is denoted by S(t) (resp. R(t)) and referred to as sending function (resp. receiving function). In words, S(t) represents the maximum number of vehicles that can leave the downstream of link over interval [t,t + Δt], if the link were connected to a traffic reservoir with an infinite capacity. Similarly, R(t) is the maximum number of vehicles that can join the upstream of link over interval [t,t+Δt] if the upstream was connected to an infinite reservoir of 122 vehicleswaitingtojointhe link. Based onthismodel, thesefunctionsareexpressed as S(t) = min " N 0,t + Δt− L v f ! −N(L,t), q max Δt # (7.7) R(t) = min " N L,t + Δt− L |w| ! +k jam L−N(0,t), q max Δt # (7.8) Now, two abstract queues at the upstream and downstream of link are consid- eredinordertotrackthedifferencesincumulativeflows. Inparticular, downstream queue (DQ), n uq (t) and upstream queue (UQ), n dq (t), are computed as, n dq (t) =N(0,t + Δt−L/v f )−N(L,t) (7.9) n uq (t) =N(0,t)−N(L,t + Δt−L/|w|) (7.10) The first equation, for example, is the difference between cumulative number vehicles that have left link by time t and cumulative number of vehicles that entered into link by time t + Δt−L/vf. This difference shows the number of queued vehicles in the downstream of link over interval [t,t + Δt]. By using queue lengths of UQ and DQ, (7.7) and (7.8) can be re-written as S(t) = min h n dq (t), q max Δt i (7.11) R(t) = min [k jam L−n uq (t), q max Δt] (7.12) It is assumed that both upstream and downstream queues have a finite capacity ofk jam L. The receiving function in (7.12) is limited by available space in upstream 123 queue that is k jam L−n uq (t). An interpretation for upstream queue is that it is embedded within the upstream of link and it behaves as if vehicles trying to enter the link actually tried to enter upstream queue. It should be noted that n uq (t) should not be confused with the total number of vehicles on the link, n(t). The reason is the existence of the time lag due to the backward wave; therefore, a vehicle may leave link and n(t) is decreased by one, but n uq (t) is not yet affected. k q max q jam k f v w UQ DQ ) (t q out ) ( t q out ) ( f in t q ) (t q in i DQ i UQ 1 i UQ 1 i DQ i link 1 link i UQ UQ DQ l d l l d f l l dq ; l uq ; Storage UQ DQ Storage l Figure 7.3: A link with its upstream queue (UQ) and downstream queue (DQ). In [60],n uq (t) andn dq (t) are stochastically modeled. In particular, UQ and DQ are modeled as M/M/1/N queues that are finite-capacity queues. The dynamics of queues are guided by time-dependent arrival and service rates that are set over discrete time periods. For each time step k, incoming and outgoing flows are computed as (see Figure 7.4), q out,k i =μ k i Pr(n dq,k i > 0,n uq,k i+1 <k jam L) (7.13) where μ k i is the capacity of downstream of link i (see Figure 7.4). The proba- bility term in the right hand side of (7.13) is the probability of transmitting one 124 k q max q jam k f v w UQ DQ ) (t q out ) ( w out t q ) ( f in t q ) (t q in i DQ i UQ 1 i UQ 1 i DQ i link 1 link i Figure 7.4: Two links in tandem with their UQ and DQ. vehicle from upstream link into downstream link; therefore, q out,k i in (7.13) is the out flow of link i, and by conservation of mass, we have, q out,k i =q out,k i+1 (7.14) Now, given the inflows and outflows, arrival and service rate of UQ and DQ are set as follows, • Arrival rate of UQ: For k-th time interval, the arrival to UQ in i-th link, λ UQ,k i , is set such that, q in,k i =λ UQ,k i Pr(n uq,k i <k jam L) (7.15) • Arrival rate of DQ: For k-th time interval, the arrival to DQ in i-th link, λ DQ,k i , is set such that, q in,k−k fwd i =λ DQ,k i Pr(n dq,k i <k jam L) (7.16) wherek fwd is the number of time steps required for a forward wave to travel from upstream to downstream of the link. 125 • Service rate of UQ: Fork-th time interval, the service rate of UQ ini-th link, μ UQ,k i , is set such that, q out,k−k bwd i =μ UQ,k i Pr(N uq,k i > 0) (7.17) wherek bwd is the number of time steps required for a backward wave to travel from downstream to upstream of link. • Service rate of DQ: Service rate of DQ is set to the maximum capacity of link at its downstream i.e. μ DQ,k i =μ k i (7.18) In the next step, given arrival and service rates of UQ and DQ, transition matrix rate is used to compute probability distributions of queue lengths over one time step. For the next step, we repeat from the beginning of this procedure by com- puting incoming and outgoing flows for the next time step. Algorithm 1: Computation procedure for queue length distributions in double-queue model [60] input : π u;0 ,π d;0 , for all links i output:π u;l ,π d;l , for all l∈N and all links i 1 Repeat the following for time intervals l = 1, 2,··· 1. Compute node boundaries Pr(n dq,l i > 0,n uq,l i+1 <k jam L) for all nodes by assuming two queues as two M/M/1/N tandem queues. 2. Compute inflowq in,k i and outflowq out,k i for all linksi using (7.13) and (7.14). 3. Compute service and arrival rates for all queues using (7.15), (7.17) and (7.18). 4. Computeπ u;l ,π d;l for all nodesi using Chapman-Kolmogorov equations for M/M/1/N queues. 126 7.4 Transient Queue Length Distribution Over Red and Green Periods In this section, we derive transient probability distribution of queue length over one cycle for given values of G and R. These values can be determined by a fixed policy, and hence do not change from one cycle to another, or they can be determined by an adaptive policy at the beginning of the cycle. We focus only on one of the traffic streams. Also, for the sake of easier presentation of the model, for the formulations in this chapter we assume that vehicles arrive individually and not in platoons. In Appendix A.4.1, we provide the counterpart of our analysis when platoon sizes can assume any random value. Since the service times of vehicles are constant and deterministic, the queue length process,{n(t);t≥ 0}, is not a Markovian process; however, by identifying an embedded Markov chain for queue length at the moment of the departure of vehicles, differential equations [19] can be used to characterizeπ(t). In particular, ˙ π 0 (t) =−λ(t)π 0 (t) +d 0 (t), ˙ π j (t) =−λ(t)π j (t)−d j−1 (t) +λ(t)π j−1 (t) +d j (t), j≥ 1 (7.19) whered j (t) is the departure rate of vehicles that leavej vehicles in the system. Let D j (t,t +h) be the probability that a vehicle departs between timet andt +h, and leavesj vehicles in the queue upon its departure. Given this probability, departure rates are formally defined as, d j (t) = lim h→0 D j (t,t +h) h (7.20) 127 Moreover, letD(t,t+h) be the probability of a departure between timet andt+h. Therefore, the total departure rate is computed as , d(t) = lim h→0 D(t,t +h) h = ∞ X j=0 d j (t) (7.21) Remark 23. Unlike M/M/1 queues in which all of individual departure rates, d j (t), are simply equal to the inverse of expectation of service times, in M/D/1 queues departure rates depend on the state of the system. The reason is the non- Markovian property of departure process in M/D/1 queue. Although a physical interpretation of individual departure rates in M/D/1 queue may be difficult, the summation of individual departure rates gives the total departure rate. Total depar- ture rate is equivalent to the out-flow of the link in traffic engineering. A full characterization of departure rates allows us to use differential equations (7.19) to derive transient queue length distribution In the rest of this section, we divide our analysis into two parts. First, we show results for the case when queue buffer capacity is finite, and a finite number of equations are used to determine π(t). Second, we present our analysis when queue has an infinite buffer and propose a method that requires solving only a finite number of equations in order to compute queue length distribution. 7.4.1 Finite Capacity Queue In this section, we present our results for the case when the buffer capacity is finite. When a vehicle arrives and finds N vehicles in the queue, it will be dropped. Therefore,n(t)≤N for allt≥ 0. This finite capacity queueing model is used as the downstream queue in the double-queue model in Section 7.5 because, in double-queue model, vehicle blocking is handled in the upstream of link. Due 128 to the finite capacity of this model, differential equations for this case is slightly different from (7.19). In particular, in this case, we have ˙ π 0 (t) =−λ(t)π 0 (t) +d 0 (t), ˙ π j (t) =−λ(t)π j (t)−d j−1 (t) +λ(t)π j−1 (t) +d j (t), j∈{1, 2,··· ,N− 1} (7.22) ˙ π N (t) =−d N−1 (t) +λ(t)π N−1 (t) We model this finite capacity queue with a M/D/1/N queue that incurs server vacations during red periods. The constant service time in this model is H that is the saturation headway. During the red phase, queue length can only grow; therefore, queue length process is modeled by a pure birth process. The next two subsections provide derivation of transient queue length distribution, π(t) = (π 0 (t),π 1 (t),··· ,π N (t)) T , over red and green periods. Red Periods Let, without loss of generality, designate t = 0 to the beginning of the red period; therefore this period ends at t = R. During red periods, no vehicle can leave the queue; therefore, the departure rates in (7.22) are all zero. The next result exploits the fact that, during red periods, we have a pure birth process and thereby gives a tractable expression forπ(t). In preparation for this result, let the probability of m arrivals during interval [t 1 ,t 2 ] be denoted as: a m t 1 ,t 2 := R t 2 t 1 λ(s)ds m e − R t 2 t 1 λ(s)ds m! (7.23) 129 which follows since, under a non-homogenous Poisson process, the number of arrivals observed during an interval [t 1 ,t 2 ] is a Poisson random variable with mean R t 2 t 1 λ(s)ds (see e.g. [63]). Theorem 11. Given π(t 0 ), t 0 ∈ [0,R] for all t∈ [t 0 ,R], we have π j (t) = j X i=0 π i (t 0 )a j−i t 0 ,t j∈{0, 1,··· ,N− 1} π N (t) = N X i=0 π i (t 0 ) 1− N−i−1 X m=0 a m t 0 ,t !! (7.24) Proof. For π j (t), j∈{0, 1,··· ,N}, result follows by conditioning on the queue length at the beginning of the red period i.e. t = t 0 , and observing that the probability of havingm arrivals during interval [t 0 ,t] isa m t 0 ,t (See (7.23)). Moreover, for π N (t), conditioned on n(t 0 ) =i, there has to be more than N−i− 1 arrivals during interval [t 0 ,t]. Therefore, the result follows. Remark 24. By rewriting (7.24) in matrix form, we have, π(t) = Φ(t,t 0 )π(t 0 ), Φ(t,t 0 ) = a 0 t 0 ,t 0 ··· 0 0 a 1 t 0 ,t a 0 t 0 ,t ··· 0 0 . . . . . . . . . . . . . . . a N−1 t 0 ,t a N−2 t 0 ,t ··· a 0 t 0 ,t 0 1− P N−1 j=0 a j t 0 ,t 1− P N−2 j=0 a j t 0 ,t ··· 1−a 0 t 0 ,t 1 (7.25) Note that in computing Φ(t,t 0 ), the only integration involved is R t t 0 λ(s)ds which is common for all a m t 0 ,t , m∈{0, 1,··· ,N}. Therefore, this matrix can be efficiently computed using (7.25). 130 Green Periods Let, without loss of generality designate t = 0 to the beginning of the green period. This implies that the green period ends at t = G. Our analysis in this subsection closely follows and extends [19]. During green periods, queued vehicles are allowed to depart the system while new vehicles continue to be added to the queue. Therefore, in (7.22), departure rates are non-zero, andπ(t) is computed via a recursion as follows. As shown in Figure 7.5-a, starting from t = 0, we consider bG/Hc contiguous intervals of length H (where H is the deterministic service time). Additionally, a last interval of length G−bG/HcH is also considered to cover the entire green period. Therefore, them-th interval during the green period is defined as I m := [(m− 1)H,mH) m∈{1,··· ,bG/Hc} [G−bG/HcH,G] m =bG/Hc + 1 Since service times are deterministic and equal to H, one can observe that during I 1 , all departure rates (7.20) are zero because the earliest possible departure epoch can be att =H. The following results allow to computeπ(t) over an intervalI m , 1 G 1 L 1 R new G 1 new L 1 new R 1 2 R 2 G 2 L new R 2 new G 2 new L 2 1 C 2 C 01 q 02 q 11 q 12 q 21 q 22 q 1 G L 1 R 2 R 2 G L C 1 G 1 L 1 R new G 1 new L 1 new R 1 2 R 2 G 2 L new R 2 new G 2 new L 2 C 01 q 02 q 11 q 12 q 21 q 22 q C R G L k r k g 1 k r C k g H H G 4 0 G H H G 4 0 G (a) (b) 0 t 1 t 2 t 2 ) 0 ( n H H Figure 7.5: (a) A sample green period that covers five interval i.e.I 1 ,··· ,I 5 . The length of the last interval is different from others and is equal to G−bG/HcH. (b) A green period that starts with 2 initial vehicles. Arrows show the moments when the first two vehicles leave the queue. 131 given π(t) and departure rates, d j (t), during the pervious intervalI m−1 . The following result characterizes departure rates. Proposition 29. Givenπ 0 (z) andD(z)∀z∈I m−1 and ˜ t :=t−H, departure rates for all t∈I m and m> 1 are given as, d j (t) =λ( ˜ t)a j ˜ t,t π 0 ( ˜ t) + j+1 X i=1 a j+1−i ˜ t,t d i ( ˜ t), j∈{0,··· ,N− 2} (7.26) d N−1 (t) =λ( ˜ t) " 1− N−2 X i=0 a i ˜ t,t # π 0 ( ˜ t) + N−1 X i=1 " 1− N−1−i X k=0 a k ˜ t,t # d i ( ˜ t) (7.27) and d(t) = N−1 X i=0 d i (t) =λ( ˜ t)π 0 ( ˜ t) +d( ˜ t)−d 0 ( ˜ t) (7.28) Proof. A departure will occur during interval (t,t+h) and leavek∈{0,··· ,N−2} vehicles in the system if one of the following two condition holds. 1. The system was empty at time ˜ t =t−H and one vehicle joined the queue in interval (t−H,t−H +h), and during its service, there were k new arrivals. 2. A vehicle departs in (t−H,t−H +h) and leavesi∈{1,··· ,k + 1} vehicles behind in the queue. During the service of the first of these vehicles, there have been k + 1−i new arrivals. The first condition considers the departure of a vehicle that starts a busy period, and the second condition considers the departure of any vehicle that arrives within a busy period. By considering the aforementioned two conditions, one can write an equation for D j (t,t +h) (by recalling that D k (t,t +h) is the probability that a departure happens in (t,t +h) and leaves k vehicles in the queue) in terms of 132 π 0 ( ˜ t), a i ˜ t,t , and D i ( ˜ t, ˜ t +h), j∈{0,··· ,j}. For example, for j∈{0,··· ,N− 1}, this equation is given as D j (t,t +h) =hλ( ˜ t)a j ˜ t,t π 0 ( ˜ t) + j+1 X i=1 a j+1−i ˜ t,t D i ( ˜ t, ˜ t +h) By dividing this equation by h and taking the limit as h→ 0, (7.26) is derived. Similar procedure can be applied in order to compute d N−1 (t) (7.27). Moreover, (7.28) follows by (7.26) and (7.27). WithD(t) := (d 0 (t),··· ,d N−1 (t), 0) T , differential equations in (7.22) can be written in matrix form as: ˙ π(t) =A(t)π(t)−BD(t) (7.29) where A(t)∈ R (N+1)×(N+1) = λ(t)B, with B ∈ R (N+1)×(N+1) being a constant matrix given by B = −1 0 0 ··· 0 1 −1 0 ··· 0 . . . . . . . . . . . . . . . 0 ··· 1 −1 0 0 ··· 0 1 0 (7.30) The next result provides a solution of (7.29) for zero initial condition. Theorem 12. If the queue is empty at t = 0 i.e., π(0) = (1, 0,··· , 0) T , then for any t∈ [0,G], π(t) = Φ(t, 0)π(0)− Z t 0 Φ(t,z)BD(z)dz Proof. The result follows by (7.29). 133 Recall that Φ(t, 0) can be obtained from (7.25). Theorem 12 provides a method to compute π(t) for zero initial condition. We now consider the case when the queueisnon-emptyatthestartofthegreenperiod,i.e.,n(0)> 0. Definet m :=mH for m ={0, 1,··· ,n(0)}, that are also illustrated in Figure 7.5-b. When initial queue is non-empty,t m ,m{1,··· ,n(0)}, are the only departure epochs up to time t n(0) (See Figure 7.5-b for an illustration)-b. In between these points, queue length process is a pure birth process. Therefore, for each intervalI m , queue length distribution is computed similar to Theorem 11 by conditioning on the queue length at the beginning of interval. That is π j (t) = P min(j+m−1,N) i=0 π i (0)a j+m−1−i 0,t , j∈{0, 1,··· ,N− 1} P N i=0 π i (0) 1− P N−2−i+m q=0 a q 0,t j =N ∀t∈I m , m∈{1,··· ,n(0)} (7.31) (7.31) incorporates the fact that by time t∈I m , m≤n(0), there has been m− 1 departures. For the rest of green period, that is for t∈ [t n(0) ,G], Theorem 12 can be used to compute queue length distribution. In particular, π(t) = Φ(t,t n(0) )π(t n(0) )− Z t t n(0) Φ(t,z)BD(z)dz, ∀t∈ [t n(0) ,G] (7.32) 7.4.2 Infinite Capacity Queue In Section 7.4.1, we analyzed a finite capacity queue over red and green periods. In this section, we provide similar results for an infinite buffer queuing model for queue of vehicles at intersections. In this model, if a vehicle finds queue full (i.e. n(t) = N) it will be held in an infinite capacity storage and will join the queue as soon as space becomes available. Therefore, this model represents two queues 134 in tandem (see Figure 7.6-(a) for an illustration). The service time of vehicles in the storage queue is deterministic and equal to zero. Thus, it can be observed that these two queues are equivalent to a single M/D/1 queue, as show in Figure 7.6-(b). Let n eq (t) and n s (t) denote the queue length in the Equivalent Queue (EQ) and storage, respectively. It is clear that when n eq (t)≤ N, it implies that the storage is empty, and when n eq (t) > N, the storage is non-empty. Since the storage has infinite capacity, the transient queue length distribution of EQ, π(t) = (π 0 (t),π 1 (t),··· ) T is an infinite dimensional vector. We are interested in the distribution of number vehicles in the queue in the presence of the storage. The firstN + 1 elements ofπ(t) can characterize the distribution of the number of vehiclesinthequeue. InordertoobtainthefirstN +1elementsofπ(t), differential equations for EQ (7.19) can be solved; however, (7.19) requires integrating infinite number of coupled equations. In the following, we propose an approach to compute the first N + 1 elements ofπ(t) using a finite number of equations. We define, ¯ π j (t) := ∞ X i=j π i (t), and ˆ π(t) := (π 0 ,··· ,π N , ¯ π N+1 ) T (7.33) where ¯ π N+1 (t) represents the probability of a non-empty storage. Moreover, the expected number of vehicles in the storage and equivalent queue can be obtained as ¯ n eq (t) = ∞ X i=0 iπ i (t), ¯ n s (t) = ∞ X i=1 iπ N+i (t) (7.34) It should be noted that ˆ π(t) is a finite dimensional vector that is obtained from π(t) by keeping its first N + 1 elements and aggregating the remaining elements. In this section, we characterize the dynamics of ˆ π(t) without considering infinite number of equations. During red periods we characterize π(t); however, during green periods, we characterize ˆ π(t) and, at the end of green periods, we recover 135 π(t) from ˆ π(t). In the following two subsections, we analyze this system over red and green periods. Infinite buffer Server Storage 0 H (t) Finite buffer UQ N , 0 H Server Infinite buffer Server Equivalent upstream queue 1 / D / M (t) (a) (b) N / 1 / D / M / 1 / D / M Server Storage with infinite buffer 0 H (t) Finite capacity queue N , 0 H Server Infinite buffer Server Equivalent queue 1 / D / M (t) (a) (b) Figure 7.6: (a) Two tandem queues. The queue in left is the storage with infinite buffer and zero service time, and the queue in right is a finite capacity queue with deterministic service times. (b) The two tandem queues are equivalent to one infinite buffer queue. Since the service time in the storage is zero, the equivalent queue is an M/D/1 queueing model. Red Periods Analysisforredperiodsissimilartofinitecapacityqueuesincethequeuelength process during red periods is the same as a regular pure-birth process. Therefore, queue length distribution is obtained as π j (t) = j X i=0 π i (0)a j−i 0,t , j≥ 0, ∀t∈ [0,R] (7.35) Green Periods As mentioned earlier, during green periods, we characterize ˆ π(t) = (π 0 ,··· ,π N , ¯ π N+1 ) T . The following result gives the dynamics of ¯ π N+1 (t) (i.e. the last element of ˆ π(t)) and mean queue length in the storage, ¯ n s (t). 136 Proposition 30. For anyt∈ [0,G], for the infinite buffer capacity queue, dynam- ics of ¯ π N+1 (t), n eq (t), ¯ n s (t), and d(t) are given as ˙ ¯ π N+1 (t) =λ(t)π N (t)−d N (t) (7.36) ˙ ¯ n eq (t) =λ(t)−d(t) (7.37) ˙ ¯ n s (t) = max{ ˙ ¯ n eq (t)−N, 0} (7.38) Proof. We first show (7.36). By (7.19), we have ˙ ¯ π N+1 (t) = m X i=N+1 ˙ π i (t) =λ(t)π N (t)−d N (t)−λπ m (t) +d m (t) (7.39) Moreover, since P ∞ i=0 π i (t) = 1 is a convergent series, for any t∈ [0,G], we have lim m→∞ π m (t) = 0. This also implies that, for any t∈ [0,G], lim m→∞ d m (t) = 0. Therefore, by taking the limit of (7.39) asm→∞, we obtain (7.36). Similarly, by using (7.34), (7.19), and noting that the limit of tail probabilities and departure rates are zero, (7.37) and (7.38) are obtained. Remark 25. In order to compute ˆ π(t), the firstN +1 equations in (7.19) together with (7.36) should be integrated, simultaneously. These equations depend on the firstN departure rates i.e.{d j (t),j = 0,··· ,N} that are computed using (7.26) for each intervalI m , iteratively. This requires computing all departure rates,d j (t),j≥ 0; but since at any time t≥ 0, the total departure rate is computed by (7.28), individual departure rates can be computed up to a finite threshold and the rest can be approximated as zero. When green periods start from empty initial condition, we integrate the N + 1 equations in (7.19), and (7.36), starting from the beginning of the green period. 137 However, for non-empty initial condition, n(0) > 0, similar to the procedure described in Section 7.4.1, we first computeπ(t) until t n(0) as π j (t) = j+m−1 X i=0 π i (0)a j+m−1−i 0,t , j≥ 0, ∀t∈I m , m∈{1,··· ,n(0)} (7.40) Then, we start integrating the firstN +1 equations in (7.19) and (7.36) for the rest of green period. It should be noted that if the initial queue length is larger than maximum possible number of departures during the green period (i.e. bG/Hc), π(t) over the entire green period can be computed using (7.40). Therefore, during green periods, a finite number of equations are solved to compute ˆ π(t). At the end of green periods we recoverπ(t) from ˆ π(t) by computing π j (t) forj >N. For this step we characterize an approximate relationship between total departure rate and individual departure rates. In particular, by assuming independence between a departure and the number of vehicles left in the queue, we get d j (t) =d(t)π j (t) (7.41) Numerical investigations suggests that the above approximation leads to good results. By the end of green periods, the total departure departure rate, d(t), and individual departure rates,d j (t),j≥ 0 are known. Therefore, we obtainπ j (t) from (7.41). 7.5 Link-level Queueing Model with Spillbacks The queuing models developed in Section 7.4, are vertical queues at the down- stream of a link. Therefore, when these queues are full and vehicles are waiting in 138 the storage to join the queue, as soon as one vehicle leaves the queue one vehicle space will become available in the queue and one vehicle from the storage can join the queue. However, in reality, from the onset of green period and after depar- ture of vehicles from downstream of link, there will be a delay until the backward shockwave propagate to the upstream of link. Moreover, in the models developed in Section 7.4, it is assumed that vehicle arrivals happen at the end of queue. How- ever, in reality, vehicles arrive at the upstream of street and it takes some time before they reach the end of queued vehicles in downstream. In this section, we incorporate spillbacks due to congestion and forward and backward traffic waves by integrating the developed models in Section 7.4 with double-queue model [60] described in Section 7.3.5. In the original double-queue model, however, [60], the service times of queues are assumed to be exponential random variables which can have high variances. However, in traffic signals, service times of vehicles have a low variance and a deterministic assumption on service times is closer to reality (See Figure 7.1 for an illustration). As explained in Section 7.1, a high variance in service times can significantly cause over-estimation of queue length, specially in heavy regimes. 7.5.1 Model Description As illustrated in Figure 7.7, the link is connected to an intersection at the downstream node. Similar to [60], UQ and DQ are considered to be finite capacity queues. In this chapter, however, we consider deterministic service times for these queues. Therefore, they are modeled as M/D/1/N queues. Both UQ and DQ have the same capacity ofN that is the maximum number of vehicles that can be fit in the link. For the upstream boundary, similar to the construction proposed in [46], in order to model vehicles that are blocked due to spillbacks, we consider a storage 139 buffer with infinite capacity. The storage and UQ form two queues in tandem that are identical to the model described in Section 7.4.2 . Therefore, they are reduced to an Equivalent Queue (EQ) that is a variant of M/D/1 queueing model (See Figure 7.6 for an illustration). The service times for DQ isH = 1/C out whereC out is the maximum flow that can be sent out from the downstream of the link. k q max q jam k f v w UQ DQ ) (t q out ) ( w out t q ) ( f in t q ) (t q in i DQ i UQ 1 i UQ 1 i DQ i link 1 link i l d l l d f l l dq ; l uq ; UQ DQ Storage l l out q ; l l out q ; f l l in q ; l in q ; UQ DQ Storage l Figure 7.7: A link connected to a traffic signal. Link is modeled with upstream and downstream queues, and a dummy storage queue to keep vehicles that are waiting to join the link. Letn s (t),n uq (t),n eq (t), andn dq (t) denote the number of vehicles in the storage, UQ,EQ,andDQ,respectively. Correspondingly, theirdistributionsaredenotedby π s (t),π uq (t),π eq (t), andπ dq (t). One can obtainπ s (t),π uq (t) fromπ eq (t). In par- ticular, by conditioning on whether or not n eq (t)≤N and using total probability rule, we obtain, π uq i (t) = π eq i (t), i∈{0, 1,··· ,N− 1} 1− P N−1 j=0 π eq j (t), i =N , π s i (t) = P N j=0 π eq j (t), i = 0 π eq N+i (t), i∈{1, 2,···} (7.42) 140 For computing queue length distributions, we discretize time-dependent vari- ables, and use superscriptl to denote the time index. For example,π s;l denotes the queue length distribution of the storage at the beginning of the l-th time interval. Also, letl f ,l ω , andl H denote the number of time intervals needed to coverτ f ,τ ω , and H, respectively. In the following, we define and describe the quantities that are required for computing queue length distributions. Link Demand, λ l We consider λ l as the total demand that wants to enter into the link. The demand process is modeled with a non-homogenous Poisson process whose rate is λ l . Inflow Rate of Link, q in;l The inflow of vehicles into the link is controlled by the state of UQ. Vehicles enter the link either when UQ is not full or a departure happens in UQ and space becomes available for vehicles waiting in storage to join UQ. Therefore, the arrival process into UQ characterizes the inflow of vehicles into the link. This process behaves differently under two different scenarios. • First, when the UQ is not full (i.e. n uq;l <N), the inflow rate is equal to the total link demand i.e. q in;l =λ l . • Second, when UQ is full and storage is non-empty (i.e. n uq;l = N, and n s;l > 0), the inflow rate is equal to the outflow rate of the link (denoted by q out;l ), lagged by the delay due to the travel time of the backward shockwave. By considering the probability of the aforementioned two scenarios, the inflow rate is determined as, 141 q in;l =λ l 1−π uq;l N +q out;l−l ω 1−π s;l 0 (7.43) where 1−π s;l 0 is the probability that storage is not empty, and by (7.42), we have 1−π s;l 0 = ¯ π eq;l N+1 . In Section 7.5.1, we formally characterize departure of link i.e q out;l . Arrival Rate of Downstream Queue, λ dq;l As described in Section 7.3.5, the inflow rate of the downstream queue is that of upstream queue but lagged with the amount of free-flow travel time. Therefore, we set the arrival rate of DQ such that it provides the desired inflow into UQ. In particular, λ dq;l = q in;l−l f 1−π dq;l N (7.44) Outflow Rate of Link, q out;l The outflow process of link is characterized by the departure process of DQ. It should be emphasized that the departure rate for a queuing system is different from its service rate i.e. 1/H = C out . In particularthe departure rate is upper bounded by service rate, 1/H, for all times. Given the arrival rate, λ dq;l , and the service time,H = 1/C out , the departure rates of DQ,d dq;l j ,j∈{0,··· ,N− 1}, can be characterized using Proposition 29. Then, the outflow of link can be computed as q out;l = N−1 X j=0 d dq;l j It is clear that, during red periods, departure rates are zero and consequently, q out;l = 0. 142 Departure Rate of Upstream Queue, q out;l−l ω Based on the double-queue model, the departure rate of UQ is equal to the departure rate rate of DQ, lagged by delay due to the travel time of the backward shockwave. Therefore, the departure rate of UQ is q out;l−l ω . 7.5.2 Computation of Queue Length Distributions In this section, we describe how queue length distributions for UQ and DQ are computed. Transient Queue Length Distribution for DQ DQ is modeled by an M/D/1/N queue; therefore, given the its arrival rate (7.44), and service time H = 1/C out , the results in Section 7.4.1 can be used to compute π dq;l over red and green periods. In particular, Theorem 11 is used to compute π dq;l over red periods, Proposition 29 is used to obtain departure rates over green periods, and Theorem 12, (7.31) and (7.32) are used to compute π dq;l over green periods. Transient Queue Length Distribution for UQ and Storage In this section, we describe how to compute queue length distribution of UQ, π uq;l ,andstorage,π s;l . Asexplainedearlier,UQandthestorageformanequivalent queue whose arrival rate, λ l , is given. Moreover, the total departure rate of EQ is obtainedfromtheoutflowofdownstreamqueuei.e. d eq;l =q out;l−l ω . EQisavariant of an M/D/1 queueing model; therefore, one can use results from Section 7.4.2 in order to obtain π uq;l . However, it should be noted that since the total departure rate is determined from downstream queue, Proposition 29 cannot be used in order 143 to determine individual departure rates. Therefore, to obtain individual departure rates, we use (7.41) with total departure rate equal to q out;l−l ω . In the following, we argue that queue length distribution of the storage can be approximated by a Poisson distribution. Let A l and S l denote the the total number arrivals and departures of the storage up to l-th time interval. Given the initial queue length in the storage at the beginning of a green period,n s;0 , its queue length at any other interval during the green period is given as n s;l =n s;0 +A l −S l (7.45) A vehicle arrives to the storage only if UQ is full; therefore, the arrival process to the storage is a Poisson process with rateλ l π uq;l N . This implies thatA l is a Poisson random variable whose mean is P l k=0 λ k π uq;k N Δt. In light traffic regimes, when the storage is empty most of the times, S l is also very small and close to zero. In heavy traffic regimes; however, the storage is often non-empty and a departure from UQ causes an immediate departure from the storage. Since service times are deterministic, in heavy regimes, S l will have a very low variance. For example, when the storage is non-empty during the entire green period, the total number of departures from storage will be deterministic and equal tobG/Hc. Therefore, we approximate S l by a deterministic process. This assumption, (7.45), and the fact that A l is a Poisson random variable, implies that n s;l is a Poisson random variable whose mean is ¯ n s;l and obtained from (7.38). Therefore, π s;l j = ¯ n s;l e ¯ n s;l /j!, j≥ 0 (7.46) This fully characterizesπ s;l . 144 The steps for computing queue length distributions for storage, UQ, and DQ over a finite time period, T, are summarized in Algorithm 2. Algorithm 2: Computation procedure for obtaining the storage, UQ, and DQ queue length distributions input : λ l ,π eq;0 ,π uq;0 ,π s;0 ,π dq;0 , C out , N, and the schedule of green and red periods output:π uq;l ,π dq;l ,π s;l , q in;l , and q out;l for all l∈{1, 2,··· ,bT/Δtc} 1 for l = 0, 1,··· ,bT/Δtc do 2 compute q in;l from (7.43) 3 compute λ dq;l from (7.44) 4 if green period then compute d dq;l j ,j∈{0,··· ,N− 1} using Proposition 29. 5 else d dq;l j = 0,j∈{0,··· ,N− 1} 6 obtainπ dq;l+1 by integrating (7.22) 7 compute q out,l using q out,l = P N−1 j=0 d dq;l j 8 givenπ s;l andπ uq;l , computeπ eq;l using (7.42) 9 compute d eq;l j ,j≥ 0 using (7.26) i.e. d eq;l j =q out;l−l ω π eq;l j+1 /(1−π eq;l 0 ) 10 obtain first N entries ofπ eq;l+1 by integrating first N equations in (7.19); then, by (7.42) obtainπ uq;l+1 11 obtain ¯ n s;l+1 by integrating (7.38) 12 obtainπ s;l+1 by using (7.46) 13 end 7.6 Asymptotic Analysis under Fixed-time and Feedback Control Policies In this section, we provide asymptotic analysis of the queue length process under stationary arrival, and when each incoming link has a finite queue storage capacity. A vehicle arriving to a full queue is assumed to abandon the queue. This is the same as the setup considered in Section 7.4.1. We provide analysis under fixed-time and a class of feedback control policies. Extension to unbounded queue 145 storage capacity under fixed-time policy is (implicitly) included in [62]; extension under feedback control policies is an ongoing work, and will be reported elsewhere. 7.6.1 Fixed Time Control Policy Consider the embedded Markov chain at the epochs corresponding to beginning of cycles. Denote this Markov chain as{X(k) =n(kC) : k∈{0, 1,···}}. In order to avoid the trivial case, we implicitly assume that G> 0 2 . Proposition 31. Consider the setup from Section 7.4.1. For all λ(t)≡ λ and G> 0, the Markov chain{X(k) =n(kC) : k∈{0, 1,···}} is ergodic. Proof. The state space for the Markov chain is{0, 1,...,N}. Let P denote the state transition matrix of this Markov chain, where P i,j is the probability that the chain transitions from state i∈{0, 1,...,N} to state j∈{0, 1,...,N}. We shall show that the tri-diagonal entries of P are strictly positive. Positivity of the diagonal entries implies aperiodicity of all the states, and positivity of the upper and lower diagonal entries implies irreducibility of the chain. Together, these two properties imply the proposition. For all i∈ [N], we have that if i≥dG/He P i,i−1 ≥e −λG (λG) (dG/He−1) (dG/He− 1)! ·e −λR =e −λC (λG) (dG/He−1) (dG/He− 1)! (7.47) 2 Throughout this section, we allow the maximum number of departures in a time interval to bedG/He, i.e., the last vehicle which sees remaining green time less than H will indeed depart the queue. 146 and if i<dG/He, P i,i−1 ≥e −λiH (λ· min{G,iH}) (dG/He−1) (dG/He− 1)! ·e −λ(G−iH) ·e −λR =e −λC (λ· min{G,iH}) (dG/He−1) (dG/He− 1)! (7.48) The first case in (7.47) corresponds to i being large enough so that the queue willnotemptyduringthegreenperiod, eveniftherearenoarrivals. Theexpression for this case in (7.47) corresponds to the number of arrivals during the green period being equal to one less than the number of departures, and no arrival during the red period. On the other hand, the second case in (7.48) corresponds to queue becoming empty during the green period when there are no arrivals during green. The expression for this case in (7.48) corresponds to the number of arrivals during thedepartureofivehiclestobejustenoughtohavei−1vehiclesattheendofgreen period, and no arrivals during the rest of the green period after the first i vehicles have departed, or during the red period. Similarly, for all i∈{0, 1,...,N− 1}: P i,i+1 ≥ e −λC (λG) (dG/He+1) (dG/He+1)! , i≥dG/He e −λC (λiH) (dG/He+1) (dG/He+1)! , 0<i<dG/He e −λC λ 2 (G−HbG/Hc) 2 2 , i = 0 & G>HbG/Hc e −λCλ 2 H 2 2 , i = 0 & G =HbG/Hc > 0 (7.49) The argument for the first two cases in (7.49) is similar to (7.47). The expression for the third case corresponds to no arrival during the entire cycle, except for the lastG−HbG/Hc time of the green when there are two arrivals. This ensures that 147 by the time the green period ends, queue length will only be one. The expression for the fourth case follows similarly. Along the same lines, for alli∈{0, 1,...,N}: P i,i ≥ e −λC (λG) dG/He dG/He! , i≥dG/He e −λC (λiH) dG/He dG/He! , 0<i<dG/He e −λC λ (G−HbG/Hc), i = 0 & G>HbG/Hc e −λC λH, i = 0 & G =HbG/Hc > 0 7.6.2 Feedback Control Policy Inthissubsection, weconsideracoupleoffeedback(alsoreferredtoasadaptive) control policies that determine green times for a cycle based on the queue lengths of all the incoming links to the intersection, specifically queue lengths at the end of the respective red periods for the links in the previous cycle. Let n r i (k) be the queue length on link i at the end of red period in the k-th cycle. Without loss of generality, we refer to the start of green period on link 1 as the beginning of cycle for the intersection. For simplicity, we consider only two incoming links into the intersection. For each of these links we letG min andG max to be the maximum and minimum green allowable green times. Let G(k) denote the green time for link 1 during the k-th cycle. Naturally, the green time for link 2 during the k-th cycle is thenC−G(k). Consider the Markov chain with stateX(k) = (n r 1 (k),n r 2 (k),G(k)). 148 A Simple Monotone Policy Consider the following policy: for all k = 1, 2,... G(k) = G(k− 1) + Δ G if n r 1 (k− 1)>n r 2 (k− 1) and G(k− 1) + Δ G ≤G max G(k− 1)− Δ G if n r 1 (k− 1)<n r 2 (k− 1) and G(k− 1)− Δ G ≥G min G(k− 1) otherwise (7.50) for some constant Δ G > 0. The policy in (7.50) simply increases the green time by Δ G for the link which has a relatively higher queue length. We implicitly assume that G(0)∈ [G min ,G max ]⊆ [0,C]. LetG :={...,G(0)− Δ G ,G(0),G(0) + Δ G ,...}∩ [G min ,G max ] be the set of admissible values for G(k), k = 0, 1, 2,.... Proposition 32. Consider the setup from Section 7.4.1 for an intersection with two incoming links, and under the feedback control policy in (7.50). Let λ 1 (t)≡ λ 1 > 0 and λ 2 (t) ≡ λ 2 > 0. If G min > 0, then the Markov chain{X(k) = (n r 1 (k),n r 2 (k),G(k)) : k∈{0, 1,···}} is ergodic. Proof. LetX :={0, 1,...,N}×{0, 1,...,N}×G denote the state space for the Markov chain. The proposition follows by showing irreducibility and aperiodicity of the Markov chain. The chain is irreducible if, for every state (i,j,g)∈X, the transitionfrom (i,j,g)to (i−1,j,g), (i+1,j,g), (i,j−1,g), (i,j+1,g), (i,j,g−Δ G ), or (i,j,g+Δ G )(discardingstateswhichdonotbelongtoX)happensinfinitesteps. If i = j, then following (7.50), the green time values do not change for the next cycle. Moreover, positivity of the probability of queue length on each link independently transitioning to +1 or−1 value can be treated along the same lines as in the proof of Proposition 31. 149 In order to treat the i6=j case, without loss of generality, assume that i>j. According to (7.50), if g + Δ G >G max , then the green time for link 1 in the next cycle remains at g, in which case the proof for the positivity of the probabilities of transition to (i− 1,j,g), (i + 1,j,g), (i,j− 1,g), or (i,j + 1,g) in one step is along the lines of the proof of Proposition 31. However, if g + Δ G ≤ G max , then the green time for link 1 in the next cycle is g + Δ G . We establish that the probability of transitioning from (i,j,g) to (i− 1,j,g) in finite steps is positive in this case; the proof for the other cases follows along similar lines. Consider the following finite path from (i,j,g) to (i− 1,j,g): (i,j,g)→ (i− 1,i− 1,g + Δ G )→ (i− 2,i− 2,g + Δ G )→ ...→ (j,j,g + Δ G )→ (j,j + 1,g + Δ G )→ (i− 1,j,g). Following similar arguments as in the proof of Proposition 31, it can be shown that the probability that each transition in this path takes place in one step is strictly positive. The proof for transitioning from (i,j,g) to (i,j,g− Δ G ) follows by a slight modification of the above path towards the end: ...→ (j,j,g + Δ G )→ (j,j + 1,g + Δ G ) → (j,j + 1,g) → (i,j,g− Δ G ). Finally, the probability of transitioning from (i,j,g) to (i,j,g + Δ G ) in one step can be shown to be strictly positive trivially. Since the chain is irreducible, it is sufficient to show aperiodicity of one state. Consider state (i,i,g) for some i∈{0, 1,...}. (7.50) implies that green times do not change for the next cycle. Thereafter, following arguments similar to those in the proof of Proposition 31, the probability that the queue length at the end of red on each link during the next cycle is again i is strictly positive. 150 7.7 Simulations In this section, we report simulation results to compare the proposed mod- els with existing models summarized in Section 7.3, and microscopic simulations using PTV Vissim. We divide this section into three sections. First, we present results related to Section 7.4 that is a model for vertical queues with finite capacity and infinite storage. Second, we present link-level queue length results obtained by integrating M/D/1/N queues with double-queue model (Section 7.5). Third, we focus on a variety of control policies, and, through numerical examples, we illustrated our results in Section 7.6. 7.7.1 Vertical Queues In this subsection, we present numerical results obtained from our results in Section 7.4, and compare it with existing models in Section 7.3. to show the per- formance of the proposed model for computing queue length distribution. Results of this subsection are illustrated in Figures 7.8 and 7.9. We consider one phase of a fixed time intersection. Moreover, for computing queue length distribution, an infinite storage is considered as described in Section 7.4.2. Figure 7.8 depicts the transient mean queue length under a fixed time policy – correspondingredandgreenperiodsareshownonthehorizontalaxis. Asexpected, mean queue length increases over red periods and decreases over green periods, and the proposed model captures these intra-cycle fluctuations. The results suggest that intra-cycle variations achieve a steady-state. The transient behavior of the uninterrupted model is obtained by considering an uninterrupted M/D/1 queue with appropriately scaled saturation headway. The mean queue length at the end of red cycles in the proposed model (which is an on/off model) is lower bounded 151 by the uninterrupted model, as also discussed in Section 7.3.1. We use Little’s law to compute the mean queue length from Webster’s mean delay formula (7.3). Figure 7.8 shows that the mean queue length obtained from Webster’s formula lies almost in the middle of the range of the proposed model. In order to verify this observation, the time-average of mean queue length obtained from the proposed model is computed using, R T 0 ¯ n(s)ds T (7.51) where T is the total time of simulation. It can be observed in Figure 7.8 that the time-average obtained from the proposed model is very close to Webster’s model. Furthermore, the mean overflow queue length (queue length at the end of green periods) obtained from Akcelik model (7.4) is close to the proposed model. The mean queue length using the on/off service model in (7.6), as proposed in [45], was found to be 41.08, which is substantially greater than the output of the other models shown in Figure 7.8. This illustrates our semi-formal argument in Section 7.3.4 for this discrepancy. Figure 7.9 shows the mean queue length under a fixed-time policy but time- varying arrival rate. During the first 8 cycles, the arrival rate is constant; then, it is increased by 50% for the next 8 cycles. For the last 8 cycles, the arrival rate is again reduced to its initial value. Figure 7.9 illustrates that the proposed model is capable of capturing the transient behavior of queue length under dynamic traffic conditions. During each of these time periods, the corresponding mean queue length obtained from Webster’s formula (7.3) is also shown in Figure 7.9. One can observe that stationary models cannot accurately model the transient behavior of traffic signals. 152 0 100 200 300 400 Time 0 2 4 6 8 10 Average Queue Length Proposed Model Webster Model Proposed Model (Time Average) Uninterrupted Model Akcelik Model Figure 7.8: Comparison between mean queue length under the proposed model (Section 7.4.2) and existing models in Section 7.3. The parameters for this simu- lation are: H = 2 (sec), ρ = 0.85, C = 50 (sec), and G =R = 25 (sec). 0 200 400 600 800 1000 1200 Time 0 2 4 6 8 10 Average Queue Length Proposed Model Webster Model Figure 7.9: Comparison between mean queue length under the proposed model (Section 7.4.2) and Webster’s model. In the first 8 cycles, we have ρ = 0.6 and in the second 8 cycles, it is increased to ρ = 0.9 and in the last 8 cycles it is reduced to the initial valueρ = 0.6. Other parameters for this simulation are: H = 2 (sec), C = 50 (sec), and G = R = 25 (sec). The choice of these parameters ensures a small blocking probability. 153 7.7.2 Link-level Model In this subsection, we report the simulation results related to our results in Section 7.5 that is related the double-queue model. We compare the proposed model in Section 7.5 (referred to as DQ-M/D/1 model) with the existing version of double-queue model in the literature [60] (referred to as DQ-M/M/1 model), and microscopic simulations. For microscopic simulations, we use PTV Vissim that is a commercial traffic simulator. We consider one phase of a fixed time intersection. For microscopic simulations, it is assumed that the desired speed of vehicles is uniformly distributed between 52 to 56 km/hr. Corresponding to this distribution of speeds, a saturation headway ofH = 1.5 seconds is considered. The average jam distance between vehicles is 5 (m). Moreover, a link of length 150 (m) is considered; which implies that the link can fit at most 30 vehicles. Thus, the capacity of both UQ and DQ are set to 30. We provide numerical results for both light (ρ = 0.6) and heavy (ρ = 0.95) traffic regimes. Figure 7.10 shows results for the light traffic regime, and Figure 7.11 depicts the same results for the heavy regime. In Figures 7.10 and 7.11, the mean number of vehicles on the link obtained from the DQ-M/M/1 model, DQ-M/D/1 model, and microscopic simulations are compared. For a better com- parison, the time-average of mean queue lengths are also shown in these figures. Time averages are computed using (7.51). In both Figure 7.10 and 7.11, it can be observed that DQ-M/M/1 model over-estimates the queue length. Specifically, this over-estimation is more significant in Figure 7.11 which illustrates the heavy regime (ρ = 0.95). Figure 7.12 shows the comparison between the cumulative distribution function of the downstream queue at the end of last red period obtained from DQ-M/M/1, DQ-M/D/1, and microscopic simulations. Figure 7.12 shows that the distribution 154 0 100 200 300 400 490 Time 0 2 4 6 8 10 Average Queue Length DQ - M/M/1 DQ - M/D/1 Simulation DQ - M/M/1 - Time Average DQ - M/D/1 - Time Average Simulation - Time Average Figure 7.10: Comparison between mean number of vehicles on link obtained through double queue models and microscopic simulations forρ = 0.6. The param- eters for this simulations are H = 1.5 (sec), C = 70 (sec), G = 45 (sec), N = 30, τ f = 10 (sec), and τ ω = 20 (sec). 0 100 200 300 400 490 Time 0 5 10 15 20 25 Average Queue Length DQ - M/M/1 DQ - M/D/1 Simulation DQ - M/M/1 - Time Average DQ - M/D/1 - Time Average Simulation - Time Average Figure 7.11: Comparison between mean number of vehicles on link obtained through double queue models and microscopic simulations for ρ = 0.95. The parameters for this simulations are H = 1.5 (sec), C = 70 (sec), G = 45 (sec), N = 30, τ f = 10 (sec), and τ ω = 20 (sec). in the DQ-M/M/1 model is shifted to the right which implies that queue length 155 in the DQ-M/M/1 model is stochastically larger than the queue length in the DQ-M/D/1 model. 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 DQ - M/D/1 DQ - M/M/1 Simulations Figure 7.12: Cumulative distribution of the number of vehicles on link at the end lastredperiodobtainedthroughdoublequeuemodelsandmicroscopicsimulations. The parameters for this simulations the same as Figure 7.11. 7.7.3 Control Policies In this subsection, we present our simulation results related to different control policies discussed in Section 7.6. Figure 7.13 shows the evolution of mean queue length at the end of cycles. As shown in Proposition 31, It can be observed that after about 10 cycles the steady state is achieved. Similar illustration for proportionally fair control policy is shown in Figure 7.14. Moreover, Figure 7.15, shows that mean green period under this policy converges to the steady-state value. 156 5 10 15 20 25 30 Cycle 6.5 7 7.5 8 8.5 Average Queue Length Figure 7.13: Mean queue length at the end of red periods for one street under fixed time control policy. Parameters areρ = 0.8,G = 30 (sec),C = 60 (sec), andH = 2 (sec). 1 10 20 30 40 50 60 Cycle 3 3.5 4 4.5 5 5.5 6 6.5 7 Average Queue Length Street 1 Street 2 Figure 7.14: Mean queue length at the end of red under proportionally fair control policy. The system starts from equal green periods for both streets. Parameters are ρ 1 = 0.4, ρ 2 = 0.8, G 1 (0) = 30 (sec), G 2 (0) = 30 (sec), C = 60 (sec), N = 50, ΔG = 4 (sec), G min = 5 (sec), G max = 55 (sec), and H = 2 (sec). 157 10 20 30 40 50 60 Cycle 18 20 22 24 26 28 30 Average Green Period Figure7.15: Meangreenperiodforstreet1underproportionallyfaircontrolpolicy. The system starts from equal green periods for both streets. Parameters are ρ 1 = 0.4, ρ 2 = 0.8, G 1 (0) = 30 (sec), G 2 (0) = 30 (sec), C = 60 (sec), N = 50, ΔG = 4 (sec), G min = 5 (sec), G max = 55 (sec), and H = 2 (sec). 158 Chapter 8 Conclusions and Future Work This dissertation aims to develop novel queueing frameworks for analyzing existing traffic systems and provide tools to evaluate the effects of new technologies such as V2V on these systems. The dependency of the performance of transporta- tion systems on different traffic regimes encourages the use of state-dependent queueing models to capture this dependency. Although this feature increases the complexity of our analysis, it improves the accuracy of predictions. Thisdissertationconsidersanovelconnectionbetweenprocessorsharingqueues and traffic queues. In particular, Horizontal Traffic Queues are proposed in which the road is considered as a server that serves the vehicles, simultaneously. The service rate of each vehicle is its instantaneous speed that is determined by a safe car-following model. A periodic road geometry is considered as an abstraction for a large highway system. In this periodic road segment, vehicles can travel their desired distances until they depart from HTQ. This framework allows for rigorous characterization of macroscopic traffic performance metrics such as throughput and travel time for a variety of microscopic interaction rules between vehicles. We provide a lower bound for the throughput of HTQ under first- and second- order car-following models. Our analysis shows a phase transition for throughput of the system under first-order car-following models. We also consider a single lane of traffic with a finite length to compute the travel time and throughput of vehicles under different car-following models. We provide lower and upper bounds for average travel time. We also study real data sets that contain trajectories 159 of vehicles in highways to compare our microscopic simulations and theoretical bounds. This dissertation also tackles the classical problem of analyzing signalized traf- fic intersections. The majority of existing literature focuses on the steady-state performance of these queues and neglect interruptions caused by red periods. We develop a novel vacation queueing model in which state-dependent red periods are explicitly modeled as server vacations. We consider each queue as an M/D/1/N queue, and the vacation period is determined by the implemented control policy. By characterizing the state-dependent departure process of this queueing model, we use differential equations to compute the transient queue length distribution. We integrate the vacation queueing model with link-level traffic models to cap- ture spillbacks, forward and backward traffic waves caused by green and red peri- ods. As microscopic simulations suggest, using deterministic inter-departure times increases the accuracy of model. In the future, we plan to consider different traffic setups of HTQ. For example, in Chapters 3 and 4, we considered HTQ on a periodic road that represents an infinitehighwaywherevehiclestravelontheroadforaslongastheywish. However, in Chapter 5, we consider a finite length traffic length and study its characteristics. We conjecture that an appropriate scaling of a finite length lane can lead to similar performance as the periodic setup. We plan to formally investigate this conjecture in future. Furthermore, in Chapter 7, we mainly focused on the derivation of the transient probability distribution of queue length, which can reveal useful insights that can be utilized for designing efficient control policies. In particular, the model can be used to obtain the optimal green splits for each phase. Our initial investigations have shown that the objective function of this problem is convex near the optimal 160 point. Moving forward, we plan to rigorously use the proposed model in Chapter 7 to optimize traffic signals. 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Theory of Stochastic Processes, 12:162–170, 2006. 168 Appendix A Appendix A.1 Additional Technical Results for Chapter 4 In this section, we gather a few technical results that are used in the main results of Chapter 4. Definition 5 (Type K function). [66] Let g :S7→R n be a function on S⊂R n . g is said to be of type K in S if, for each i∈{1,...,n}, g i (z 1 )≤g i (z 2 ) holds true for any two points z 1 and z 2 in S satisfying z 1 ≤z 2 and z 1,i =z 2,i . Lemma 18. Let g : S→R N and h : S→R N be both of type K over S⊂R N . Let z 1 (t) and z 2 (t) be the solutions to ˙ z =g(z) and ˙ z =h(z), respectively, starting from initial conditions z 1 (0) and z 2 (0) respectively. Let S be positively invariant under ˙ z =g(z) and ˙ z =h(z). If g(z)≤h(z) for all z∈S, and z 1 (0)≤z 2 (0), then z 1 (t)≤z 2 (t) for all t≥ 0. Proof. radiction, let ˜ t≥ 0 be the smallest time at which, there exists, say k∈ {1,...,N}, such that z 1 ( ˜ t)≤z 2 ( ˜ t), z 1,k ( ˜ t) =z 2,k ( ˜ t), and g k (z 1 ( ˜ t))>h k (z 2 ( ˜ t)). (A.1) Since g(z) is of class K, z 1 ( ˜ t)≤ z 2 ( ˜ t) and z 1,k ( ˜ t) = z 2,k ( ˜ t) imply that g(z 1 ( ˜ t))≤ g(z 2 ( ˜ t)). This, combined with the assumption that g(z)≤ h(z) for all z ∈ S implies that g(z 1 ( ˜ t))≤h(z 2 ( ˜ t)), which contradicts (A.1). 169 Lemma 18 is relevant because the basic dynamical system in our case is of type K. Lemma 19. For any L> 0, m> 0, and N∈N, the right hand side of (4.14) is of type K inR N + . Proof. Consider ˜ x, ˆ x∈R N + such that ˜ x≤ ˆ x. If ˜ x i = ˆ x i for some i∈{1,...,N}, then, according to (3.1), y i (˜ x)−y i (ˆ x) = (˜ x i+1 − ˆ x i+1 )− (˜ x i − ˆ x i ) = ˜ x i+1 − ˆ x i+1 if i∈{1,...,N− 1}, and is equal to (˜ x 1 − ˆ x 1 )− (˜ x N − ˆ x N ) = ˜ x 1 − ˆ x 1 if i =N. In either case,y i (˜ x)≤y i (ˆ x), which also impliesy m i (˜ x)≤y m i (ˆ x) for allm> 0. In order to state the next lemma, we need a couple of additional definitions. Definition6 (MonotoneAlignedandMonotoneOppositeFunctions). Two strictly monotone functions h : R→ R and g : R→ R are said to be monotone-aligned if they are both either strictly increasing, or strictly decreasing. Similarly, the two functions are called monotone opposite if one of them is strictly increasing, and the other is strictly decreasing. Lemma 20. Let h : R + → R and g : R + → R be strictly monotone functions. Then, for every y∈S L N , n∈N, L> 0, N X i=1 h(y i ) (g(y i+1 )−g(y i )) (A.2) is non-negative if h and g are monotone-opposite, and is non-positive if h and g are monotone-aligned. Moreover, (A.2) is equal to zero if and only if y = L N 1. Proof. For i ∈ {1,...,N}, let I i be the interval with end points g(y i ) and g(y i+1 ). For i ∈ {1,...,N}, let f i (z) := sgn (g(y i+1 )−g(y i ))h(y i )1 I i (z). Let 170 g min := min i∈{1,...,N} g(y i ), andg max := max i∈{1,...,N} g(y i ). Withf(z) := P N i=1 f i (z), (A.2) can then be written as: N X i=1 h(y i ) (g(y i+1 )−g(y i )) = Z gmax g min f(z)dz. (A.3) We now show that, for every z∈ [g min ,g max ]\{g(y i ) : i∈{1,...,N}}, f(z) is non-negative if h and g are monotone-opposite, and is non-positive if h and g are monotone-aligned. This, together with (A.3), will then prove the lemma. It is easy to see that every z∈ [g min ,g max ]\{g(y i ) : i∈{1,...,N}} belongs to an even number of intervals in{I i : i ∈ {1,...,N}}, say I ` 1 ,I ` 2 ,..., with ` 1 <` 2 <... (see Figure A.1 for an illustration). We now show thatf ` 1 (z)+f ` 2 (z) is non-negative if h and g are monotone-opposite, and is non-positive if h and g are monotone-aligned. The same argument holds true for f ` 3 (z) +f ` 4 (z),.... Assume that g(y ` 1 )≤ g(y ` 2 ); the other case leads to the same conclusion. By definition off i ’s,f ` 1 (z) =h(y ` 1 ) andf ` 2 (z) =−h(y ` 2 ). g(y ` 1 )≤g(y ` 2 ) implies that f ` 1 (z) +f ` 2 (z) =h(y ` 1 )−h(y ` 2 ) is non-negative ifh andg are monotone-opposite, and is non-positive ifh andg are monotone-aligned, with the equality holding true if and only if y ` 1 =y ` 2 . (a) (b) Figure A.1: A schematic view of (a)f i (z),i ={1, 2, 3, 4} and (b)f(z) = P 4 i=1 f i (z) for a y∈S L 4 (L = 1) with y min =y 2 <y 4 <y 3 <y 1 =y max for a m< 1. 171 Lemma 21. For n∈N\{1}, let ψ n be the n-fold convolution of ψ∈ Ψ. Then, Z t 0 zψ(z)ψ n−1 (t−z)dz = t n ψ n (t) ∀t≥ 0 Proof. Let J 1 ,··· ,J n be n random variables, all with distribution ψ. Therefore, the probability distribution function of the random variable V := P n i=1 J i is ψ n . Using linearity of the expectation, we get that t =E " n X i=1 J i |V =t # = n X i=1 E [J i |V =t] =nE [J 1 |V =t] i.e., E [J 1 |V =t] = t n (A.4) Let f J 1 |V (j 1 |t) denote the probability distribution function of J 1 |V. By definition: f J 1 |V (j 1 |t) = f J 1 ,V (j 1 ,t) ψ n (t) = ψ(j 1 )ψ n−1 (t−j 1 ) ψ n (t) (A.5) Therefore, using (A.4) and (A.5), we get that E[J 1 |V =t] = Z t 0 zf J 1 |V (z|t) dz = Z t 0 z ψ(z)ψ n−1 (t−z) ψ n (t) dz = t n Simple rearrangement gives the lemma. The following is an adaptation of [63, Lemma 2.3.4]. Lemma 22. Let a 1 ,··· ,a n−1 denote the ordered values from a set of n− 1 inde- pendent uniform (0,t) random variables. Let ˜ d 0 = z ≥ 0 be a constant and 172 ˜ d 1 , ˜ d 2 ,··· ˜ d n−1 be i.i.d. non-negative random variables that are also independent of{a 1 ,··· ,a n−1 }, then Pr( ˜ d k +··· + ˜ d n−1 ≤a n−k ,k = 1,··· ,n− 1| ˜ d 0 +··· + ˜ d n−1 =t, ˜ d 0 =z) = z/t z <t 0 otherwise A.2 Additional Technical Results for Chapter 5 In this section, we collect couple of technical results required in Chapter 5. Assume that each event of a Poisson process with rate λ can independently be classified as being of type-I or type-II. In particular, if an event occurs at time s, then it is classified as a type-I event with probabilityP (s) and a type-II event with probability 1−P (s). The following result, characterizes the number of type-I and type-II events that have occurred by time t that are denoted byN I (t) andN II (t), respectively. Proposition 33. N I (t) andN II (t) are independent Poisson random variables with respective means p and λt−p, where p =λ Z t 0 P (s)ds Proof. See e.g. Proposition 2.3.2 in [63] A.2.1 Derivation of (5.6) The derivation mimics calculations in the context of M/G/∞ [63, 64]. 173 Fix a time t> 0. Call a vehicle, say i, arriving at time s<t active if it is still on the road segment at t, i.e., if T i > t−s. Therefore, the probability that the vehicle which arrives at s is active is P l (s,t) := Pr(T i >t−s) =F (t−s) Let R(z,t) be the number of active vehicles that have arrived by time z. Propo- sition 33 implies that R(t,t) is a Poisson random variable with mean ¯ r(t,t) = λ R t 0 P l (s,t)ds = λ R t 0 F (t−s)ds. Recall that the initial condition is N(0) = 0; therefore, by definition of R(t,t) we have R(t,t) =N(t), and hence Pr(N(t) = 0) = Pr(R(t,t) = 0) =e −¯ r(t,t) (A.6) Consider now the case when R(t,t) > 0, and let Z < t denote the random variable corresponding to the time at which the first active vehicle arrives. A vehicle that arrives in (Z,t) will be on the road at time t. Thus, conditional on the first active vehicle arriving at Z = z, the number of vehicles on the road at time t is N(t) = 1 +A(z,t) (A.7) where A(z,t) is a Poisson random variable with mean λ(t−z). We compute Pr(N(t) =k),k> 0, by conditioning on Z. For this purpose, we need to compute the distribution of Z. Note the arrival of an active vehicle at Z < t implies that the number of active vehicles is non-zero during [Z,t]. Therefore, H(z) := Pr(Z≤z) = 1− Pr(Z >z) = 1− Pr(R(z,t) = 0) = 1−e −¯ r(z,t) 174 where the last equality follows by (A.6). The pdf corresponding to H is then h(z) := dH(z) dz = λF (t−z)e −¯ r(z,t) z≥t−T max 0 z <t−T max (A.8) By (A.7) and conditioning on Z: for k = 1, 2,..., Pr(N(t) =k) = Z t 0 Pr(N(t) =k|Z =z)h(z)dz = Z t t−Tmax e −λ(t−z) (λ(t−z)) k−1 (k− 1)! λF (t−z)e −¯ r(z,t) dz where the second equality follows from (A.7) and (A.8). A change of variable gives the second expression in (5.6). A.3 Additional Technical Results for Chapter 6 Lemma 23. For any L > 0, β > 04 cr ≥ 0 and N∈N, the right hand side of (6.1) is of type K inR N + . Proof. Consider ˜ x, ˆ x∈R N + such that ˜ x≤ ˆ x and ˜ x i = ˆ x i for some i∈{1,...,N}. If i∈{2,··· ,N} then y i (˜ x)−y i (ˆ x) = (˜ x i+1 − ˆ x i+1 )− (˜ x i − ˆ x i ) = ˜ x i+1 − ˆ x i+1 . Therefore y i (˜ x)≤y i (ˆ x), and, by (6.1), ˙ ˜ x i ≤ ˙ ˆ x i . If i = 1, ˙ ˜ x 1 ≤ ˙ ˆ x 1 =v max . Thus, in either cases, ˙ ˜ x i ≤ ˙ ˆ x i which establishes the result. Remark 26. While Lemma 23 shows that the right hand side of (6.1) which is a type K function inR N + , one can, in a similar way, show that the function associated with the last N s (N s ≤ N) elements of this function is also a type K function in R Ns + . 175 Let X(t;x 0 ) denote the solution to (6.1) at t starting from x 0 at t = 0. We will compareX(t;x 0 ,m) under different values ofm and initial conditionsx 0 , over an interval of the kind [0,τ), in between arrivals and departures. Also, if x 1 0 and x 2 0 are vectors of different sizes (i.e. n 1 0 6= n 2 0 ), then x 1 0 ≤ x 2 0 implies element-wise inequality only for components which are common to x 1 0 and x 2 0 . For example, if n 1 0 ≥n 2 0 , then x 2 0 ≥x 1 0 means that x 2 0,i ≥x 0,i+(n 1 0 −n 2 0 ) . The following result considers two different initial conditions, x 1 0 and x 2 0 , and their evolution when systems are closed (i.e. there is no departure or arrival). Lemma 24. For any L> 0, β > 0,4 cr ≥ 0 and x 1 0 ∈R n 1 + , x 2 0 ∈R n 2 + , n 1 ,n 2 ∈N, x 1 0 ≤x 2 0 , n 2 ≤n 1 =⇒ X(t;x 1 0 ,m 1 )≤X(t;x 2 0 ,m 2 ) ∀t∈ [0,τ) Proof. The result follows by Lemmas 18, 23 and Remark 26. Definition 7 (Similar Matrices). Matrices A∈R n and ˜ A∈R n are called similar if ˜ A =T −1 AT for some invertible T ∈ R n . Also, the transformation A → T −1 AT is called a similarity transformation of matrix A. Two similar matrices have the same characteristic polynomial; therefore, they have the same set of eigenvalues. A.4 Additional Technical Results for Chapter 7 A.4.1 Formulation for platoon arrivals Inthissection, weextendouranalysisinSection7.4tothecaseofbatcharrivals that can incorporate platoon arrivals in traffic flows. As mentioned in Section 176 7.2, we consider a time-dependent batch Poisson process for the arrival process of vehicles. Therefore during green periods queues can be modeled as M X /D/1/N queueing model where X denotes the random variable representing platoon sizes, H is the saturation headway, and N is the finite capacity of queue. When an arrival occurs, it consists of a platoon of vehicles of random size X. Each platoon will have at least one vehicle; therefore, X = 1 +Y where Y is a general discrete random variable. The p.m.f of X is given as p k := Pr(X = k),k∈{1, 2,···}. Given the p.m.f of X, the sum of n independent platoon sizes has a p.m.f that is the n-fold convolution of p.m.f ofX. In this section, without loss of generality, we assume that Y has a Poisson distribution; therefore, p k = Pr(Y =k− 1) = λ k−1 b e −λ b (k− 1)! , k∈{0, 1, 2,···}. (A.9) where 1+λ b is the mean size of platoons. Consequently, sum ofn platoons isn+Z where Z is Poisson variable with mean nλ b . Red Periods The differential equations in (7.22) are modified to incorporate batch arrivals. Therefore, these equations are written as ˙ π 0 (t) =−λ(t)π 0 (t), ˙ π j (t) =λ(t) j−1 X i=0 p j−i π i (t) −λ(t)π j (t), j = 1,··· ,N− 1 ˙ π N (t) =λ(t)π N−1 (t) +λ(t) N X i=2 " π N−i (t) 1− i−1 X k=1 p k !# and in matrix form, ˙ π(t) = ˆ A(t)π(t), (A.10) 177 Solution to (A.10) is given as π(t) = ˆ Φ(t,r k )π(r k ) where ˆ Φ(t,r k ) is the state transition matrix for the dynamics in (A.10). In order to compute the solution to (A.10)inasimilarwayasinTheorem11, weneedtoconsideramodifiedexpression tocomputeprobabilityofarrivalofatotalofmvehiclesin [t 1 ,t 2 ]. First,weconsider arrival epochs that may happen within [t 1 ,t 2 ], and then compute the total number of vehicles arrived in all the platoons. Therefore, (7.23) can be re-written as, a m t 1 ,t 2 = m X i=1 R t 2 t 1 λ(s)ds i e − R t 2 t 1 λ(s)ds i! . (iλ b ) m−i e −iλ b (m−i)! Now, given these modified probabilities, Theorem 11 can be used to compute π(t) over red periods. Green Periods Platoon arrivals change the computation method of departure rates in Proposi- tion 29. In the following, we mention the two scenarios that are used for computing the probability of a departure in (t,t + Δt) that leaves k vehicles in the system. 1. Thesystemwasemptyattimet−D andoneplatoonofsizem∈{1, 2,··· ,k+ 1} joined the queue in interval (t−D,t−D + Δt), and during its service, there were k− (m− 1) new vehicle arrivals. 2. A vehicle departs in (t−D,t−D +Δt) and leavesi∈{1,··· ,k + 1} vehicles behind in the queue. During the service of the first of these vehicles, there have been k + 1−i new arrivals. In the above two cases, in order to compute the probability of, say m, arrivals within a time interval, we first condition on the number of arrival epochs and then compute the distribution of the total size of platoons. Therefore, in a similar way 178 as in Proposition 29 departure rates can be computed, and Theorem 12 can be used to compute queue length distribution over green periods. 179
Abstract (if available)
Abstract
Performance evaluation of transportation systems and adopting new technologies to improve the existing level of service have significant effects on our society and economy. In this dissertation, we propose novel queueing frameworks to characterize traffic performance metrics such as throughput and travel time. Queueing models have been a compelling framework for analyzing traffic systems due to their ability to model congestion and service processes. However, most existing traffic queues rely on simplifying assumptions and study the average evolution of traffic flow. In particular, they often neglect the dependency of the performance of these queues on the state of traffic. Incorporating the state-dependent behavior in queueing models makes their analysis significantly complex. In this dissertation, we develop novel state-dependent queueing models for application to traffic systems. ❧ In particular, by generalizing processor sharing queues, we propose a novel Horizontal Traffic Queue (HTQ) in which vehicles on the road can be interpreted as jobs and the road as a server. In this case, all jobs are processed simultaneously, and the service rate of a given job is time-varying and is equal to its instantaneous speed. The time-varying and state-dependent nature of the service rate also put the proposed HTQ within the class of state-dependent queueing systems. However, the complex dependence of service rate on the state (i.e., vehicle locations) precludes the use of existing tools from processor sharing queues and state-dependent queues for rigorous analysis. By using this queueing theoretic framework we estimate macroscopic performance measures such as throughput and travel time under different microscopic models. The finer properties of the departure process of the proposed queue depend on the dynamics of the minimum inter-vehicular distance. Motivated by the need to study such finer dynamics, we study the evolution of the Kullback-Leibler (K-L) divergence of the inter-vehicular spacings in between and during arrival and departure events. We consider both first and second order car- following models and, by extending busy period calculations for M/G/1 queue to our setting, we provide lower bounds for the throughput of the HTQ for different parameters of the car-following model. We also derive lower and upper bounds on the mean travel time of vehicles for a class of safe car-following behavior. We compare our simulations with real vehicle trajectory dataset. ❧ In the next step, we focus on another class of state-dependent queues to model signalized traffic intersections. In traffic intersections, red periods interrupt the service process of queues. By modeling red periods as server breakdowns or vacations, we propose a novel vacation queueing model that allows derivation of the transient probability distribution of the number of queued vehicles. The duration of red periods are determined by the implemented control policy and can depend on the state of the system. In order to improve the accuracy of the model, we consider deterministic inter-departure times, as opposed to exponential inter-departure times in often used the literature. Therefore, each leg of intersection is modeled as an M/D/1/N queueing model. We extend analytical transient departure process of M/D/1/N queues to our setup to derive the departure process of these queues and derive transient probability distribution of queue length. Although deterministic inter-departure times increases the complexity of our model by causing a dependency between queue length and the departure process, comparison with microscopic simulation suggests improvement in accuracy.
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Motie Share, Mohammad Ali
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Novel queueing frameworks for performance analysis of urban traffic systems
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