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Fundamental limits of caching networks: turning memory into bandwidth
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Fundamental limits of caching networks: turning memory into bandwidth
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Fundamental Limits of Caching Networks: Turning Memory into Bandwidth Copyright 2015 by Mingyue Ji A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2015 Mingyue Ji Dedication This dissertation is dedicated to my beloved Grandmother, Yurong Liu, my parents, Xi uling Zhang and Guangzhen Ji, my parents in law, Fajiang Lu and Lanye Shi, my wife, Ping Lu and my lovely son, Maxwell Yuan-Rui Ji. 11 Acknowledgements I would like to express my sincerest gratitude to my advisor Professor Giuseppe Caire for the insightful advices, guidance and enormous encouragement. I am extremely fortunate and honored to be his student. He opened my mind and gave me great guidance about research, learning, teaching, career planning, and so much else. He has always been very kind, patient, and generous. I enjoyed every meeting with him and enjoyed the moments of deep happiness from conquering difficult problems. I would also like to thank my most important collaborator at USC during my entire PhD study, Professor Andreas F. Molisch, for the valuable discussions. I am truly im pressed by his wisdom. Without him, this work would not have been possible. Although Professor Molisch is not my official advisor, he generously gave me great guidance and spent significant amount of time meeting with me. I enjoyed every insightful discussion with him, which will always be precious to me. I would like to express my special thank my dissertation committee for their guidance, support and reviewing this dissertation, including Professor Giuseppe Caire, Professor Andreas F. Molisch and Professor Larry Goldstein. Thanks to Bell Labs, Alcatel-Lucent for the support and hospitality during my in ternship. I must express my special gratitude to Dr. Antonia M. Tulino, Dr. Jaime Llorca and Dr. Thierry Klein for the help during my stay in Bell Labs and also for the collaboration after I left Bell Labs. I enjoyed every long phone call with them and the moments when we solved open problems. Also thanks to Professor Michelle Effros, Pro fessor Michael Langberg and Ming Fai Wong for the insightful discussions when I visited California Institution of Technology. lll Thanks to all the current and previous faculties at Communication Science Institute (CS!) of Ming Hsieh Department of Electrical Engineering at Viterbi School of Engineer ing of USC, including Professor Alexandros G. Dimakis, Professor Salman Avestimehr, Professor Todd Brun, Professor Keith M. Chugg, Professor Solomon W. Golomb, Pro fessor Urbashi Mitra, Professor Michael J. Neely, Professor Robert A. Scholtz, Professor Alan E. Willner and Professor Zhen Zhang. Thanks to my collaborators and colleagues, including Ozgun Y. Bursalioglu, Hoon Huh, Ansuman Adhikary, Song Nam Hong, Sang-Woon Jeon, Bill Ntranos, Karthikeyan Shanmugam, Ryan Rogalin, Dilip Bethanabhotla, Harpreet S. Dhillon, Arash Saber, Yi Gai, Mi Zhang, Negin Golrezaei, Hassan Ghozlan, Joonheon Kim, Seun Sangodoyin, Hao Feng, Hao Yu, Daoud Burghal, Sunav Choudhary, Daphney Zois .... This dissertation benefited enormously from all of them. Also special thanks to the wonderful staffs at CS!, including, Gerrielyn Ramos, Corine Wong and Susan Wiedem, for the great help of everything, and to Diane Demetras, you made my life much easier. Words cannot express my thanks to my beloved Grandmother, Yurong Liu, my par ents, Xiuling Zhang, Guangzhen Ji and parents in law, Fajiang Lu, Lanye Shi, for their love and support. And last, but definitely no least, special thanks to my wife Ping Lu and my son Maxwell Yuan-Rui Ji for love, encouragement and patience of helping me to conquer all the difficulties I met. IV Table of Contents Dedication ii Acknowledgements iii List of Tables x List of Figures xi Abstract xv Chapter 1 Introduction 1 1.1 Device-to-Device (D2D) Caching Networks . . . . . . . . . . . . . . . . . 3 1.1.1 Fundamental Limits of Caching in Wireless D2D Networks . . . . 4 1.1.2 Throughput-Outage Trade-off in Wireless One-Hop Caching Networks 5 1.1.3 Wireless Device-to-Device Caching Networks: Basic Principles and System Performance . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Caching in Wireless Multihop Device-to-Device Networks . . . . . 7 1.2 Caching in Shared Link Caching Networks . . . . . . . . . . . . . . . . . . 9 1.2.1 Order-Optimal Rate of Caching and Coded Multicasting with Ran- dom Demands g 1.2.2 Caching and Coded Multicasting: Multiple Groupcast Index Coding 11 1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Part I: Device-to-Device (D2D) Caching Networks Chapter 2 Fundamental Limits of Caching in Wireless D2D Networks 2.1 Network Model and Problem Definition ......... . 2.2 Deterministic Caching, Achievability and Converse Bound 2.2.1 Transmission range r 2 V2 2.2.2 Transmission range r < V2 2.2.3 An Example ..... . 2.2.4 Discussions ........ . 2 .3 Decentralized Random Caching . . 2.3.1 Transmission range r 2 V2 2.3.2 Transmission range r < V2 2.3.3 Discussions 2.4 Summary ............. . 14 15 17 22 22 25 27 30 32 32 40 40 42 v Chapter 3 The Throughput-Outage Tradeoff of Wireless One-Hop Caching Networks 45 3.1 Network Model and Problem Formulation 48 3.2 Outer Bounds . . . . . . . . . . . . . . . . 55 3.3 Achievable Throughput-Outage Tradeoff . 59 3.4 Summary . . . . . . . . . . . . . . . . . . 62 Chapter 4 Wireless Device-to-Device Caching Networks: Basic Principles and Sys tem Performance 64 4.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Conventional Scaling Laws Results of Ad Hoc Networks and D2D communications with Caching . . . . . . . . 4.1.2 Key results for D2D networks with caching 4.1.3 Coded Multicasting From the Base Station 4.1.4 Harmonic and Conventional Broadcasting . 67 70 72 73 4.1.5 Summary: Comparison between Different Schemes 75 4.2 System Design . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.1 Holistic Multi-Frequency D2D System Design . . . 77 4.2.2 Conventional Unicasting, Coded Multicasting and Harmonic Broad- casting Approaches . . . . . 79 4.3 Simulations and Discussions . . . . 82 4.3.1 Deployment Environments . 4.3.2 Channel Models . . . . . . 4.3.2.1 LOS probability . 4.3.2.2 Device-to-device channels at 38GHz 4.3.2.3 Device-to-device channels at 2.4GHz . 4.3.2.4 Channel between the Base Station and Devices . 4.3.2.5 Link Capacity Computation 4.3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . 4.3.3.1 Throughput-Outage Tradeoff . . . . . . . . . . . 82 83 83 85 86 87 88 89 89 4.3.3.2 Holistic Multi-Frequency D2D System Performance 91 4.3.3.3 Effects of the Density of Nodes . . . . . . . . . . . . 92 4.3.3.4 Effects of the Storage Capacity and the Library Size . 94 4.3.3.5 4.4 Summary .. 2.1 GHz In Band Communications Chapter 5 Caching in Wireless Multihop Device-to-Device Networks 5.1 Problem Formulation ..................... . 5.1.1 Caching in Wireless D2D Networks ......... . 5.1.2 Achievable Throughput and System Scaling Regime 5 .2 Main Results . . . . . 5 .3 Achievable Schemes 5.3.1 °'1 - /31 E (0, 1) 5.3.1.1 Distributed file placement . 5.3.1.2 Local multihop protocol . 5.3.2 °'1 - /31 = 1 ............. . 95 96 99 99 99 100 101 102 102 102 102 105 VI 5.3.2.1 Centralized file placement . 5.3.2.2 Globally multihop protocol 5 .4 Achievable Throughput 5.4.1 a1 - /31 E (0, 1) 5.4.2 °'1 - /31 = 1 5 .5 Converse . . . . . . . . 5.5.1 °'1 - /31 > 1 .. 5.5.2 a1 - /31 E (0, 1] 5.6 Summary ...... . Part II: Shared Link Caching Networks 106 106 106 106 110 111 111 112 114 116 Chapter 6 Order-Optimal Rate of Caching and Coded Multicasting with Random Demands 117 6.1 Network Model and Problem Formulation . . . . . . . . . . . . 120 6 .2 Random Fractional Caching and Linear Index Coding Delivery 6.2.1 Random Fractional Caching Placement 6.2.2 Linear Index Coding Delivery . 6.2.3 Achievable Rate ....... . 6.2.4 Random Caching Optimization 6.3 Rate Lower Bound .. 123 124 125 129 132 134 6 .4 Order-optimality . . . 135 6.4.1 Case 0 S I< 1 136 6.4.2 Case I> 1 . . 139 6.4.2.1 Regime of "very large" number of users n = w (m'I) 139 6.4.2.2 Regime of "large" number of users n = 8 (m'I) . . . 140 6.4.2.3 Regime of "small-to-moderate" number of users n = o (m'I) 141 6.4.3 Remark . . . . . . . . . . . 144 6 .5 Discussions and Simulation Results 6.6 Summary ............. . 145 151 Chapter 7 Caching and Coded Multicasting: Multiple Groupcast Index Coding 153 7.1 Network Model and Problem Formulation 154 7 .2 Achievable Scheme . . . . 7 .3 Performance Analysis 7 .4 Converse and Optimality 7 .5 Discussions . . . . . . . . Chapter 8 Conclusions and Open Problems Appendix A 155 159 161 162 164 Proofs of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.1 Deterministic Caching and Delivery Schemes: General Case, r 2' V2 169 A.2 Proof of Theorem 2.1 and Corollary 2.1 174 A.3 Proof of Theorem 2.2 . . . . . . . . . . . 17 4 A.4 Proof of Theorem 2.3 and Corollary 2.3 177 Vll A.4.1 Casen=w(m) ........ . A.4.1.1 When~ SM= o(m) A.4.1.2 When M = 8(m) A.4.1.3 When M < ~ .... . A.4.2 Casen=O(m) ........ . A.4.2.1 When~ SM= o(m) A.4.2.2 When M = 8(m) A.4.2.3 When M < ~ . A.5 Proof of Theorem 2.4 . A.6 Proof of Theorem 2.5 . A. 7 Proof of Theorem 2.6 . A.8 Proof of Theorem 2. 7 . A.9 Proof of Theorem 2.8 and of Lemma 2.1 A.10 Proof of Theorem 2.9 . A.10.1 Case t = w(l) . A.10.2 Case t = 8(1) . Appendix B Proofs of Chapter 3 . . . . . . . .. B.1 Proof of Theorems 3.1 and 3.2 B.2 Proof of Theorem 3.3 . B.3 Proof of Theorem 3.4 ..... . B.4 Proof of Theorem 3.5 ..... . B.4.1 Achievable T(p) when p 2 p~ B.4.2 Achievable T(p) for p < p~ B.4.2.1 Case 9c(m) = w (,;;") and 9c(m) S 1m/M B.4.2.2 Case 9c(m) = p1m/M, where Pl 2 / . B.5 Proof of Lemma B.1 B.6 Proof of Lemma B.2 B.7 Proof of Lemma B.3 B.8 Proof of Lemma B.5 B.9 Proof of Lemma B.6 B.10 Continuity and perturbations Appendix C Proofs of Chapter 6 C .1 Proof of Theorem 6 .1 . C.2 Proof of Lemma 6.1 . C.3 Proof of Theorem 6.2 . C.4 Proof of Theorem 6.3 . C.4.1 Region of n = w(m) C.4.1.1 When 2 c>(l~o(l)) SM= o(m) C.4.1.2 When M = 8(m) .. C.4.1.3 When M < 2 c>(l~o(l)) C.4.2 Region of n = O(m) ..... . 178 178 178 179 179 179 180 180 181 182 184 184 189 193 193 194 196 196 205 206 210 211 220 221 223 226 227 229 233 237 238 243 243 250 251 257 261 262 262 263 263 Vlll C.4.2.1 When :;;;s(l - 1) ;:- 1 263 C.4.2.2 When :;;;s(l - 1) < 1 264 C.5 Proof of Table 6.2 in Theorem 6.6 266 C.5.1 Region of 0 SM< 1 . . . . . . 267 C.5.2 Region of 1 SM< ";.," . . . . . 269 C.5.2.1 When 1 SM= o ( n?~') 269 C.5.2.2 When w(n "~') = M < ";., 7 (see Fig. C.3 for a reminder) 277 C.5.3 Region of M 2 ";.," . . . . . . . . . . . . . . . 283 C.5.3.1 When m7_ 1 = w(l) . . . . . . . . . 283 C.5.3.2 When m7_ 1 = 8(1) (see Fig. C.3) . 286 References 291 IX List of Tables 4.1 The LOS probability models. . . . . . . . . . . . . . . . . . 84 4.2 The channel parameters for 2.4 GHz D2D communications . 86 4.3 The parameters for the three types of transmissions . . . . 89 6.1 Order-optimal choice of iii and the corresponding achievable rate upper bound Ru 1 (n,m,M,q,iii) for RLFU-GCC with 1> 1andn=8(m'1) ... 140 6 .2 Order-optimal choice of iii and the corresponding achievable rate upper bound Ru 1 (n,m,M,q,iii) for RLFU-GCC with I> 1 and n = o(m'1). Here, 0 < "1 < 1 indicates a fixed positive constant. . . . . . . . . . . . . 142 x List of Figures 2.1 a) Grid network with n = 49 nodes (black circles) with minimum sepa rations= Jn· b) An example of single-cell layout and the interference avoidance spatial reuse scheme. In this figure, each square represents a cluster. The gray squares represent the concurrent transmitting clusters. The red area is the disk where the protocol model allows no other con current transmission. r is the worst case transmission range and 6 is the interference parameter. We assume a common r for all the transmitter receiver pairs. In this particular example, the reuse factor is JC = 9. . . . . 17 2.2 Qualitative representation of our system assumptions: each user caches an entire file, formed by a very large number of packets L 1 • Then, users place random requests of segments of L 11 packets from files of the library, starting at random initial points. In the figure, we have L 11 = 4. . . . . . . . . . 18 2.3 Illustration of the example of three users with M = 2, achieving rate R(2) = 1/2 28 2.4 The augmented network when m = 3, n = 3. The three requested vectors are: (A,B,C), (B,C,A) and (C,A,B) ..................... 31 2.5 Illustration of the example of three users with M = 2, m = 3, achieving rate R(2) = 0.77, where p = 0.95 (p* = 0.9510, E: is chosen as 0.001.). a) The first iteration with IUI = b = 3. b) The second iteration with IUl=b=2 .................................... 37 2.6 Rate R(M) as a function of the cache size M for deterministic and random D2D caching. In (a) we let m = 50 and n = 100. In (b) we let m = 500 and n = 50. The rate of deterministic caching is given by (2.6). The rate of Random Caching (Exact) is given by (2.24). The curve "Random Caching (Approximate)" is plotted by using (2.25). The converse of the rate is given by (2.8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Qualitative behavior of the tradeoff between throughput and outage prob ability, by ways of the tradeoff parameter g, which represents the size of the cluster of nodes over which any node can look for its desired requested file and download it by D2D on-hop communication. . . . . . . . . . . . . 55 XI 3.2 Comparison between the normalized theoretical result (solid lines) and normalized simulated result (dashed lines) in terms of the minimum through put per user vs. outage probability, where m = 1000, n = 10000, M = 1 and spatial reuse factor K = 4. The Zipf parameter I varies from 0.1 to 0.6 (from the left (blue) to the right (cyan)). . . . . . . . . . . . . . . 62 4.1 A video file encoded at rate R is split into blocks Sij : j = 1, ... , i, for i = 1, ... , 4, such that the size of Sij is T /i chunks. Each i-th set of blocks is periodically transmitted in a downlink parallel channel of rate R/i. Any user tuning into the multicast transmission can start its playback after at most T chunks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 4.2 The flow chart of the delivery algorithm for the combination of D2D com munications and multicast by the base station. . . . . . . . . . . . . . . . 78 4.3 Simulation results for the throughput-outage tradeoff for conventional uni casting, coded multicasting, harmonic broadcasting and the 2.45 GHz D2D communication scheme under indoor office channel models. For harmonic broadcasting with only the m 1 most popular files, solid line: m 1 = 300; dash-dot line: m 1 = 280; dash line: m 1 = 250. We have n = 10000, m = 300, M = 20 and I= 0.4. . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 (a). Simulation results for the throughput-outage tradeoff by holistic sys- tem design. Black Solid lines: indoor office; blue dashed lines: indoor hotspot. (b). The CDF of the throughput for different outage probabili- ties (cluster size of 600 2 /Q 2 ) under indoor office model. (c). The CDF of the throughput for different outage probabilities (cluster size of 600 2 /Q 2 ) under indoor hotspot model. ( d). The CDF of the throughput for the clus- ter size of lOOm x lOOm under indoor office and indoor hotspot channel model. Solid lines: indoor office; dashed lines: indoor hotspot. . . . . . . . 93 4.5 Solid lines: indoor office; dashed lines: indoor hotspot. (a). The throughput outage tradeoff for different user densities. (b). The throughput-outage tradeoff for different user storage capacity. ( c). The throughput-outage tradeoff for different library size of files. (d). The throughput v.s. band width division between 2.1 GHz communication and the base station under different cluster size, where Bd 2 d is the bandwidth by 2.1 GHz communica tions and Bgs is the bandwidth by the cellular base station. Bd 2 d + Bgs = B = 20MHz. (e). The outage v.s. bandwidth division between 2.1 GHz communication and the base station for the cluster with size 600 2 /19 2 . . 98 5.1 The proposed multihop routing protocol after the source node selection. 102 5.2 TDMA cell size from the protocol model. 5.3 A lower bound on the exclusive area occupied by the multihop transmission 1-(0:1-/31)+c of a SD pair with distance n 2 103 113 Xll 6.1 An illustration of the conflict graph, where n = 3, U = {1, 2, 3}, m = 3, F = {A, B, C} and M = 1. Each file is partitioned into 3 packets. The caching realization C and the packet-level demand vectors are given in Example 1. The color for each vertex in this graph represents the vertex coloring scheme obtained by Algorithm 6.1. In this case, this vertex col- oring is the minimum vertex coloring, and therefore it achieves the graph chromatic number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 .2 Rate versus cache size for UP-GCC and RLFU-GCC with optimized iii (see (6.25)), for n = 50, m = 50000, and Zipf parameter I= 0.9. . . . . . 139 6 .3 The optimal caching distribution p* for a network, where m = 3, M = 1 and n = 3, 5, 10, 15 and the demand distribution is q = [0.7, 0.21, 0.09]. . . 146 6 .4 RGCC( n, m, M, q, p) for different caching distributions p and for a network with m = 3, M = 1 and n = 3, 5, 10, 15 and demand distribution q = [0.7, 0.21, 0.09]. Note that p = [1/3, 1/3, 1/3], p = [0.5, 0.5, O], and p = [1, 0, OJ correspond to RLFU with iii= 3, iii= 2, and iii= 1, respectively. 147 6.5 Simulationresultsfor1=0.6. a) m = 5000,n = 50. b) m = 5000,n = 500. c) m = 5000, n = 5000. d) m = 500, n = 5000. RLFU in this figure corresponds to the RLFU with optimized iii given by (6.26). . . . . . . . 148 6.6 Simulationresultsfor1=1.6. a) m = 5000,n = 50. b) m = 5000,n = 500. c) m = 5000, n = 5000. d) m = 500, n = 5000. RLFU in this figure corresponds to the RLFU with optimized iii given by (6.26). 149 6.7 Two examples of the simulated expected rate by using RAP-HgC (B = 500). The comparison includes UP-GCC (B-+ oo), RLFU-GCC (B-+ oo and iii given by (6.26)), LFU-NM (B-+ oo). In these simulations, I= 0.6. a) m = n = 5. b) m = n = 8 .......................... 150 7.1 (a). A qualitative illustration of the different delivery schemes (Mis as sumed to be not very small). L·RMN (blue curve) represents rate by using the scheme in [MAN14b] L times. Rrond (black curve) is the rate by ran dom linear coding. R~{; (grey curve) is an upper bound of the achievable rate of the proposed scheme. R 1 c (red curve) represents the rate by the proposed scheme based on directed local chromatic number. (b). An ex am pie of the rate by the proposed scheme. In this exam pie, n = m = 3 and M = 1. In this figure, all the symbols have the same meanings as in Fig. 7.l(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 B.1 Illustration of the fact that the number of small disks intersecting the union of the big disks centered at the active receivers is necessarily an upper bound to the number of active receivers. . . . . . . . . . . . . . . . 199 Xlll B.2 In this figure, u 1 is a transmitter and u is a receiver. v is another receiver corresponding to another transmitter. The diagonal of each squarelet is ~r. The maximum area that are consumed by receiver u is the disk cen tered at u, with radius (1 + 6)r. The blue squarelets are the maximum activated squarelets that are caused by the active receiver u. A indicates the square let containing receiver v. . . . . . . . . . . . . . . . . . . . . . . 228 B.3 In this figure, u and v are receivers. The radius of each grey sector is ~r. Each grey sector is ~ of each disk with radius ~r. The maximum area that are consumed by receiver u is the disk centered at u, with radius (1 + 6)r. The blue sectors are the maximum activated sectors that are caused by the active receiver u. A indicates the sector containing receiver v. . 230 C.1 The sub-cases of the regimes of M when n = w(m). 260 C.2 The sub-cases of the regimes of M when n = O(m). 261 C.3 The sub-cases of the regimes of M when 1 S M < ~ 7 , where r;, r; 1 are some constants which will be given later. . . . . . . . . . . . . . . . . . . . 267 C.4 The sub-cases of the regimes of M when M 2 ~ 7 , where r;1, O", 6 are some constants which will be given later. . . . . . . . . . . . . . . . . . . . . . . 268 XIV Abstract In classical communications/information theoretic problems, the messages (contents) are generated at the sources, and requested at the destinations. Location of the messages and demands are known beforehand. One key feature that makes the study of caching networks different from the classical communication problems is that in this scenario, only the set of possible contents is given and user demands are not known a priori. Some nodes have (constrained) storage capacity, and some nodes make requests. We can build the cached contents a priori (part of the code setup) together with the designed (coded) delivery such that user demands can be satisfied. One important application of caching networks is video streaming in the wireless networks. In this thesis, we present a novel transmission paradigm based on the following two key properties: (i) video shows a high degree of asynchronous content reuse, and (ii) storage is the fastest-increasing quantity in modern hardware. Based on these properties, we suggest caching at nodes in networks, especially, at wireless edge, namely, caching in helper stations (femto-caching) and/or directly into the user devices. We study two fundamentally different network structures: device-to-device (D2D) caching networks and shared link caching networks. D2D caching networks consists of n destination nodes (users), each with a cache of size M files. There is no central controller (base station) in the network and nodes can directly communicate with each other. One the other hand, the shared link network is formed by a source node (base station), hosting a library of m information messages (files), connected via a noiseless common link ton destination nodes with a cache of size M files. The users are not allowed to communicate directly. xv We design content placement and delivery schemes, based on both coded multicasting (in shared link caching networks and D2D caching networks) and D2D transmission (in D2D caching networks), that show a "Moore's law" for throughput: in a certain regime of sufficiently high content reuse and/or sufficiently high aggregate cache capacity, the per-user throughput can increase linearly or even super-linearly with the local cache size, and it is independent of the number of users. This means that under realistic parameter regimes, we can turn memory into bandwidth. In this thesis, we prove that the proposed schemes are information-theoretic order-optimal and identify the regimes in which they exhibit order gains compared to state-of-the-art approaches under different channel models. XVI Chapter 1 Introduction Content distribution services such as video on demand (VoD), catch-up TV, and inter- net video streaming are premier drivers of the exponential traffic growth experienced in today's wireless networks [Cis13]. It is noteworthy that current methods for on demand video streaming treat video like individual data sources with (possibly) adaptive rate. Namely, each video streaming session is handled as a unicast transmission, where users successively download video "chunks" 1 as if they were web-pages, using HTTP, with possible adaptation the video quality according to the conditions of the underlying TCP /IP connection (e.g., Microsoft Smooth Streaming and Apple HTTP Live Stream ing [SdlFSH+11b,SdlFSH+11a,BAB1l]) This approach does not exploit one of the most important properties of video, namely, a constrained request pattern. In other words, the same video is requested by different users, though the requests usually occur at different times. For example, video services such as Amazon or Netflix provide a finite (albeit large) library of video files to the users and, in some cases, may shape the request pattern by making some videos available free of charge. It should also be noted that naive mul- ticasting by overhearing, i.e., by exploiting the broadcast nature of the wireless medium, as implemented in Media Broadcasting Single Frequency Networks (MBSFN) [Luo09], is basically useless for wireless video on-demand. In fact, while the users' requests exhibit 1 Typically a video chunk corresponds to 0.5s to ls of encoded video, i.e., to a group of pictures (GOP) between 15 and 30 frames for a typical video playback rate of 30 frames per second. 1 a very significant content reuse (i.e., the same popular files are requested over and over), the asynchronism between such requests is so large that the probability that two users are streaming the same file at the "same time" (i.e., within a relative delay of a few seconds) is basically zero. We refer to this very peculiar feature of video on-demand as the asynchronous content reuse. Due to the increasing cost and scarcity of wireless bandwidth, an emerging and prom is- ing approach for improving over both naive multicasting and conventional unicasting con- sists of using storage resources to cache popular content directly at the wireless edge, e.g., at small-cell base stations or even end user devices. 2 Caching has been widely studied in several wireline contexts, primarily for web proxy caching systems and content distribution networks (CDNs) [BROl,BRSOS,BGWlO,KRSOO, KD02, Cl97, Wan99, BCF+99]. In these works, a range of interrelated problems, such as accurate prediction of demand, intelligent content placement, and efficient online replace- ment, is considered. The data placement problem was introduced in [BROl], where the objective is to find the placement of data objects in an arbitrary network with capacity constrained caches, such that the total access cost is minimized. It was shown that this problem is a generalization of the metric uncapacitated facility location problem and hence is NP-Hard [BRSOS]. Tractable approaches in terms of LP relaxation [BRS08,BGW10] or greedy algorithms [KRSOO,KD02] have been proposed, by exploiting special assumptions such as network symmetry and hierarchical structures. On the other hand, an exten- sive line of work has addressed the content replacement problem, where the objective is to adaptively refresh the cache(s) content while a certain user data request process evolves in time [Wan99, Cl97, BCF+99]. The most common cache replacement/eviction algorithms are least frequently used (LFU) and least recently used (LRU), by which the least frequently /recently used content object is evicted upon arrival of a new object to a network cache. A combination of placement and replacement algorithms is also possible 2 Note that the storage capacity has become exceedingly cheap: for example, a 2 TByte hard disk, enough to store 1000 movies, costs less than $ 100. 2 and in fact used in today's CDNs, which operate by optimizing the placement of content objects over long time periods for which content popularity can be estimated, and using local replacement algorithms to handle short time-scale demand variations. In a more recent set of works (a non-exhaustive list of which includes [GSD+11, GSD+12,SGD+13,LTGK13,LT13,MAN14b,MAN14a,NMA13,JCM13b,JCM13d,JCM13c, Mco+14, GPT13b]), an information theoretic view of caching has provided insights into the fundamental limiting performance of caching networks of practical relevance. In this framework, the underlying assumption is that there exists a fixed library of m possible information messages (files), each of which consists of F bits and a given network topol ogy that includes n nodes that host a subset of messages (sources), request a subset of messages (users), and/or have constrained cache capacity (helpers/caches) of MF bits. The caching phase is part of the code set-up, and consists of filling up the caches with (coding) functions of the messages whose entropy is constrained to be not larger than the corresponding cache capacity. After this set-up phase, the network is "used" for an arbitrary long time, referred to as the delivery phase. At each request round, a subset of the nodes (users) request subsets of the files in the library and the network must coor dinate transmissions such that these requests are satisfied, i.e., at the end of each round all destinations must decode the requested set of files. In this thesis, we consider two types of caching network structures, namely, distributed Device-to-Device (D2D) caching networks and shared link caching networks. We will briefly discuss both network structures and our contributions in the following. 1.1 Device-to-Device (D2D) Caching Networks We consider the distributed Device-to-Device (D2D) caching network, where there is no base station and users are allowed to communicate with each other. These series of works consists of four parts. In the first part ( [JCM13d] [JCM14] [GJM+12]), by assuming a protocol channel model [GKOO], we study the fundamental limits of the D2D caching 3 networks in terms of throughput in the information theoretic sense. In the second part ( [JCM13a] [JCM13b]), we introduce the concept of outage, formulate and study the fun- damental throughput-outage trade-off problem. The first two parts focus on the protocol channel model [GKOO]. In the third part ( [JCM13c]), we first give a tutorial overview of state-of-the-art schemes for caching networks. Then, we compare our proposed scheme with these schemes in a realistic channel model and present a holistic system design by combing D2D communications and other state-of-the-art transmission technologies. In the first three parts, we constrain the D2D communications to be one-hop transmission. In the fourth part [JHJC15], we extend our result to a multihop D2D caching networks and show that an order gain can be achieved by using the multihop D2D scheme. In the following, we will briefly describe each part in this series of works. 1.1.1 Fundamental Limits of Caching in Wireless D2D Networks In this work, we consider the D2D wireless networks [JCM13a, JCM13b, JCM14], with a caching and delivery scheme inspired by [MAN14b], based on subpacketization in the caching phase and (inter-session) coding in the delivery phase. Inspired by the current standardization of a D2D mode for LTE (the 4-th generation of cellular sys tems) [WTS+lQ], we restrict to one-hop communication. Our main contributions are as follows: 1) if each node in the network can reach in a single hop all other nodes in the network, the proposed scheme achieves almost the same throughput of [MAN14b], without the need of a central base station for large D2D (or "ad-hoc") wireless networks 2) if the transmission range of each node is limited, such that concurrent short range transmissions can co-exist in a spatial reuse scheme, then the throughput has the same scaling law ( 8 (min { 1:,, ~}), with possibly different leading term constants) of the reuse only case [JCM13a,JCM13b] or the coded-only case [MAN14b]. 3 This result holds even 3 Scaling law order notation: given two functions f and g, we say that: 1) f ( n) = 0 (g( n)) if there exists a constant c and integer N such that f(n) S cg(n) for n > N. 2) f(n) ~ o(g(n)) if limn~= :i~i ~ 0. 3) f(n) ~ 0 (g(n)) if g(n) ~ 0 (f(n)). 4) f(n) ~ w (g(n)) if g(n) ~ o (f(n)). 5) f(n) ~ 8 (g(n)) if f(n) ~ 0 (g(n)) and g(n) ~ 0 (f(n)). 4 if one optimizes the transmission range and therefore the spatial reuse of the system. Counterintuitively, this means that it is not possible to cumulate the spatial reuse gain and the coded multicasting gain, and that these two albeit different type of gains are equivalent as far as the throughput scaling law is concerned. Beyond scaling laws, in order to establish the best combination of reuse and coded multi casting gains, trading off the rate achieved on each local link (decreasing function of distance) with the number of users that can be reached by a coded multicast message (increasing function of distance), must be sought in terms of the actual throughput in bit/s/Hz (i.e., in the coefficients of the dominant terms of the throughput scaling for large n, m and finite M, and not just in the scaling law itself). We consider both deterministic caching and (decentralized) random caching (as done m [MAN14a] for the single bottleneck link case). In both cases, we show that for most regimes of the system parameters (apart from the regime of very small caches, which is not really relevant for applications), the throughput achieved with both the proposed deterministic and random caching schemes is optimal within a constant factor. 1.1.2 Throughput-Outage Trade-off in Wireless One-Hop Caching Networks In this work, instead of creating multicasting opportunities by coding in D2D caching networks, we exploit the spatial reuse provided by concurrent multiple short-range D2D transmissions inspired by the fact that spatial reuse-only scheme can achieve order optimal throughput, which is discussed in last section. However, instead of deterministic caching placement, we focus on an alternative approach that involves random independent caching whole files at the user nodes. Under the restriction of one-hop communication, requiring that all users must be served for any request configuration is too constraining. Therefore, we introduce the possibility of outages, i.e., that some requests are not served, because of some network admission control policy (to be discussed in details later on). For the system described in Chapter 3.1, we define the throughput-outage region and obtain 5 achievability and converses that are sufficiently tight to characterize the throughput outage scaling laws within a small gap of the constants of the leading term. Furthermore, our analysis shows very good agreement with with finite-dimensional simulation results. In the relevant regime of small outage probability, the throughput of the D2D one-hop caching network behaves in the same near-optimal way as the throughput of coded multi casting [MAN14b, MAN14a], while the system architecture is significantly more straight forward for a practical implementation. In particular, for fixed cache size M, as the number of users n and the number of files m become large with nM » m, the through put of the D2D one-hop caching network grows linearly with M, and it is inversely proportional to m, but it is independent of n. Hence, D2D one-hop caching networks are very attractive to handle situations where a relatively small library of popular files (e.g., the 500 most popular movies and TV shows of the week) is requested by a large number of users (e.g., 10,000 users per km 2 in a typical urban environment). In this regime, the proposed system is able to efficiently turn memory into bandwidth, in the sense that the per-user throughput increases proportionally to the cache capacity of the user devices. We believe that this conclusion is important for the design of future wireless systems, since bandwidth is a much more scarce and expensive resource than storage capacity. 1.1.3 Wireless Device-to-Device Caching Networks: Basic Principles and System Performance Given the first two parts of the work in the domain of D2D caching networks, the pur pose of this work is two-fold. On one hand, we provide a tutorial overview of the schemes and recent results on wireless on-demand video streaming, in terms of their throughput vs. outage probability tradeoff, in the regime where both the number of users in the system and the size of the library of video files grow large. While the results presented in Chapter 4.1 are not new, they have been established mostly in individual papers with different assumptions and notations; the tutorial summary presented in Chapter 4.1 is intended to allow a fast and fair comparison under idealized settings. On the other hand, 6 looking at throughput-outage tradeoff scaling laws for idealized network models does not tell the whole story about the relative ranking of the various schemes. Hence, in this work we present a detailed and realistic model of a single cell system with n users, each of which has a cache memory of M files, and place independent streaming requests to a library of m files. Requests can be served by the cellular base station, and/or by D2D links. We make realistic assumptions on the channel models for the cellular links and the D2D links, assuming that the former uses a 4th generation cellular standard [STB09] and the latter use either microwave or mm-wave communications depending on availabil ity [Aww+13, DMRHlO]. By means of extensive simulations, this work relaxes some restrictive assumptions of the theoretical scaling laws analysis based on the "protocol model" of [GKOO], and provides more in-depth practical results with the goal of assess ing the true potential of the various methods in a realistic propagation environment, where the actual transmission rate of each link depends on physical quantities such as pathloss, shadowing, transmit power and interference. Furthermore, we study how the use of short-range mm-wave links can influence the overall capacity. Such links can pro vide very high rates but suffer from high outage probability in some environments such as office environment (see Chapter 4.3). We investigate a composite scheme that com bines robust microwave D2D links with high-capacity mm-wave links in order to achieve, opportunistically, excellent system performance. We also show that the type of environ ment in which we operate, while irrelevant for the asymptotic scaling laws analysis, plays a major role for the actual system throughput and outage probability. Eventually, we shall show that, in such realistic conditions, the D2D caching scheme largely outperforms all other competing schemes both in terms of per-user throughput and in terms of outage probability. 1.1.4 Caching in Wireless Multihop Device-to-Device Networks We study a natural extension of the one-hop D2D network discussed in Section 1.1.1-1.1.3. In this work, we consider multihop transmission [JHJC15]. A related work is presented in 7 [GPT13a], where a multihop transmission scheme for wireless caching networks has been studied under the protocol model. The key difference between this work and [GPT13a] is as follows. First, the objective of [GPT13a] is to minimize the average number of flows passing through each node, which is the reciprocal of average per-user throughput under only certain network model; one the other hand, we study and provide the optimal scaling law of the average per-user throughput directly. Second, [GPT13a] proposed a centralized and deterministic caching placement according to the demand distribution; in contrast, we proposed a completely decentralized, independently at random caching placement according to a uniform distribution on the whole file library, which is "universal" since it is independent of the specific demand distribution and is robust when a small portion of the users move in and out of the network. Third, in [GPT13a], the distance (or number of hops) between the source-destination pairs can traverse the whole network; while in this work, we propose a more practical scheme called local multihop protocol, where the number of hops between any source-destination pairs are independent with the number of users and decreases when the storage capacity per-node increases. The proposed caching placement and delivery scheme yield per-node throughput scaling 8( ~). This result shows that multihop yields a much better throughput scaling than single-hop networks. In fact, 8(~)/8(M/m) = 8(VmfM), with m/M » 1 in any reasonably practical situation. Furthermore, when the demands follows a Zipf distribution with Zipf exponent I E (0, 1), we show that the proposed policy is order-optimal (in terms of scaling law) if m < Mn. 4 Hence, in this case, tight centralized coordination of the user caches and an accurate knowledge of the popularity distribution are not required in order to achieve order-optimal throughput. 4 Throughout this work, an 'order-optimal' scheme means that it achieves the optimal throughput scaling law within the multiplicative gap of nc for any c > 0. 8 1.2 Caching in Shared Link Caching Networks In these series of works, we consider a different caching network structure, named shared link caching networks, which is formed by a single source node (a server or base station) with all m files, connected via a shared noiseless link to n user nodes, each with cache of size M files. For the shared link structure, the performance metric is the number of time slots necessary to satisfy all the demands, which is the inverse of the throughput of the network. In the case of symmetric links, the number of time slots can be normalized by the number of times lots necessary to send a single file across a point to point link. Therefore, the performance metric is rate defined as in the index coding setting [BK98, BYBJK11,ERSG10,LS09,BKL10,CASL11,Jaf13,HL12,ABK+13,uw13], i.e., the number of equivalent file transmissions. First ( [JTLC14b] [JTLC15]), we consider the case when each user makes one requests and study the fundamental limits of this network, i.e., the optimal average rate as a function of all the system parameters (storage capacity, number of users, number of files, demand distribution). Second [JTLC14a], we let each user make L 2 1 requests and study the fundamental limits of the network in terms of the worst-case rate, which is defined as the minimum of the maximum number of equivalent files transmission over all the demand patterns. 1.2.1 Order-Optimal Rate of Caching and Coded Multicasting with Random Demands First, let's discuss the history of the shared link caching networks. Focusing on the subset of current works directly relevant to this work, in [GSD+ll] (see also the successively published papers [GSD+12, SGD+13]) a bipartite network formed by helper nodes (with caches), user nodes (without caches), and capacitated noiseless links, was studied in the case of random i.i.d. requests according to some known demand distribution. This is a special case of the data placement problem with trivial routing [BRSOS]. The problem in g [GSD+11, GSD+12, SGD+13] consists of minimizing the average rate necessary to satisfy all users, where averaging is over the random requests. In [LTGK13,LT13], the data placement problem is generalized to the (coded) content distribution problem (CDP), where information can, not only be stored and routed, but also coded, over the network. The authors showed an equivalence between the CDP and the network coding problem over a so-called caching-demand augmented graph, which proved the polynomial solvability of the CDP under uniform demands (each user re quests the same subset of files), and the hardness of the CDP under arbitrary demands. The authors further showed via simulations on simple networks, the potential of network coding to enable cache cooperation gains between caches sharing a multicast link from a content source. While this work suggested the benefit of cooperative caching via both the direct exchange of information between neighbor nodes as well as via coded multi cast transmissions from a common source, the analytical characterization of the optimal caching performance in arbitrary networks remains a hard and open problem. To this end, significant progress has been made by considering specific network models that capture scenarios of practical relevance, especially in wireless networks. In [MAN14b, MAN14a], the authors considered a network topology called shared link network. In [MAN14b, MAN14a], the authors addressed the min-max rate problem, i.e., minimizing (over the coding scheme) the worst-case rate (over the user demands). Both deterministic and random caching schemes, with corresponding coded multicast delivery schemes, were shown to provide approximately optimal min-max rate, i.e., within a mul tiplicative constant, independent of n, m, M, from an information theoretic lower bound. Interestingly, when translating the results of [MAN14b, MAN14a] in terms of per-user throughput, for the case nM 2 m this scales as 8 ( 1;;'.). While several variants and extensions of these basic setups have been recently consid ered [PMAN13,NMA13,STC13,JTLC14a,KNMAD14,HKD14b,HKD14a,NMA14,BBD14b, BBD14a, Mco+14, APVG14], in this work we focus on the combination of the random requests aspect (as in [GSD+11, JCM13b, LTGK13, LT13]) and the single source shared 10 link network (as in [MAN14b, MAN14a]). This problem has been treated in [NMA13], which considered a strategy based on partitioning the file library into subsets of ap proximately uniform request probability, and applying to each subset the strategy for the min-max approach of [MAN14a]. This is motivated by observing that the average rate with random uniform demands is related, within a constant factor, to the min-max rate under arbitrary demands. Then, by partitioning the set of files and allocating the cache memory across such subsets, the problem is decomposed into subproblems, each of which can be separately addressed by reusing the arbitrary demand strategy. Due to the difficulty of finding the optimal file partitioning and corresponding cache memory allocation, [NMA13] restricts its analysis to a scheme in which for any two files in the same partition, the file popularities differ by at most a factor of two. While in [NMA13] this approach is studied for a general demand distribution, our scaling order-optimality results apply to the specific case of a Zipf demand distribution. This is a very relevant case in practice since the popularity of Internet content has been shown, experimentally, to follow a Zipf power law [BCF+99, CKR+07] (or its variations [HSOS]). In this context, our objective is to characterize the scaling laws of the optimal average rate and provide simple and explicit order-optimal schemes. 1.2.2 Caching and Coded Multicasting: Multiple Groupcast Index Coding In this work, we consider the same setting of [MAN14b] (one server, n users, one-hop multicast transmission from the server to the users) where users make multiple requests instead of a single request. This scenario may be motivated by a Femto Caching network [GMDC13] formed by n small-cell base stations receiving data from a controlling "macro" base station via the cellular downlink. Each small-cell base station has a local cache of M file units and serves L users through its own local high-rate downlink. Hence, each small-cell base station (which is identified as a "user" in our network model) makes L requests to the macro base station at once. A related work is presented in [MAN14a], under the constraint that each user's storage capacity scales linearly with the number 11 of the requests per user. This is referred to as Shared Caches. In addition, [MAN14a] only studied the case that m > nL or every user requests distinct files for the worst-case demands. This can be easily violated if the library is considered as the most popular files due to asynchronous content reuse. Another related relevant work is presented in [UW13], where an exact solution of the 3-user index coding problem for assigned side information and multiple requests is given. We study the fundamental limits of this type of network for the general case of n users, m possible messages, storage capacity M and L requests per user without any constraint, where in contrast to the general index coding problem, the side information (i.e., the cache content) at each user is designed as part of the "code" rather than given a priori. Our contribution is two-fold. First, by using the same combinatorial caching phase of [MAN14b], we generalize the coded delivery phase by using the directed (fractional) local chromatic number proposed in [SDL13] to the case of multiple groupcasting, where multiple means that each user makes L 2 1 requests and groupcasting means that one file or packet (see later) may be requested by several users. We show that order gains can be obtained by using the proposed scheme compared to the naive approach of using L times the delivery phase of [MAN14b]. Second, we present an information theoretical lower bound of the rate, and show that the proposed scheme meets this lower bound within a constant factor of at most 18. 1.3 Dissertation Outline The rest of this dissertation is organized as follows. The first part, which studies the D2D caching networks, is presented in Chapter 2, 3, 4 and 5. In Chapter 2, we present the fundamental limits of D2D caching networks. In Chapter 3, we study the problem of the throughput-outage trade-off problem in D2D caching networks. Then, in Chapter 4, we provide a tutorial overview of the schemes and recent results on wireless on-demand video streaming, com pare the performance of our 12 proposed D2D caching scheme with all the state-of-the-art schemes in caching networks under realistic channel condition, and propose a holistic system design of cellular caching networks based on the D2D caching networks by combining various types of communi cation technologies. Finally, we extend our result of one-hop D2D caching networks to mutihop caching networks in Chapter 5. The second part, which studies the shared link networks, is presented in Chapter 6 and 7. In Chapter 6, we studied the fundamental limits of shared link network with a demand distribution, and with one request per user. Then, in Chapter 7, we extend the result to the case when each user request multiple files. 13 Part I: Device-to-Device (D2D) Caching Networks 14 Chapter 2 Fundamental Limits of Caching in Wireless D2D Networks In this chapter, we study the fundamental limits of D2D caching networks as described in Chapter 1.1.1. Before proceeding in the presentation of our results, we would like to make a few remarks to clarify obvious questions and anticipate possible concerns that this line of work (see for example [MAN14b,MAN14a,PMAN13,NMA13,GSD+12,GMDCl3, GPT12,LTG+13,LTGK13,LT13,JCM13a,JCM13b,JCM13c,JCM13d,JTLC15,JTLC14a, STC13, KND14, HKD14c]) may raise. First, we would like to point out that in this work we refer to "coding" in the sense of "inter-session network coding", i.e., when the codeword is a function of symbols (or, "subpackets") from different source messages (files in the library). Often, coding at the application layer [Sho06,LWG+06,AK11] or right on top of the transport layer [SSM+og] is used in an intra-session only mode, in order to send linear combinations of subpackets from the same source message and cope with packet losses in the network, but without mixing subpackets of different messages. We point out that this intra-session "packet erasure coding" has conceptually little to do with "network coding", although it has been sometimes referred to as "random linear network coding" when the linear combi nations are generated with randomly drawn coefficients over some finite field. In line with [MAN14b] and with the protocol model of our previous work [JCM13a, JCM13b], 15 also in this work any transmission within the appropriate range is assumed to be "noise less" (i.e., perfectly decoded) and therefore we will not consider packet erasure coding against channel impairments. Then, it is important to notice that this work, as well as [JCM13a,JCM13b,MAN14b], is based on an underlying time-scale decomposition for which the caching phase (i.e., placing information in the caches) is done "a priori", at some time scale much slower than the delivery phase (i.e., satisfying the users demands). For example, we may imagine that the caches content is updated every day, through a conventional cellular network used during off-peak time, such that the content library is refreshed by inserting new titles and deleting old ones. This scenario differs significantly with respect to the conventional and widely studied "on-line" caching policies, where the cache content is updated along with the delivery process [MMPOl, DF82, DSS96, KPR99, BRSOS, BGWlO, AA96]. Finally, we would like to mention here that the considerations made in [JCM13a, MAN14b, MAN14a] about handling asynchronous demands holds verbatim in this work, and shall not be repeated for the sake of brevity. It should be clear that although we consider (for simplicity of exposition) files of the same length, the schemes described in this work generalize immediately (with the same fundamental performance) to the case of unequal length files and asynchronous demands. The chapter is organized as follows. Section 2.1 presents the network model and the formal problem definition. We illustrate all the main results based on the deterministic caching scheme and its implications in Section 2.2. In Section 2.3, we discuss the de centralized caching scheme and the corresponding coded delivery approach. Section 4.4 contains our concluding remarks and the main proofs are given in Appendices in order to keep the flow of exposition. 16 (a) (b) Figure 2.1: a) Grid network with n = 49 nodes (black circles) with minimum separation s = Jn· b) An example of single-cell layout and the interference avoidance spatial reuse scheme. In this figure, each square represents a cluster. The gray squares represent the concurrent transmitting clusters. The red area is the disk where the protocol model allows no other concurrent transmission. r is the worst case transmission range and ~ is the interference parameter. We assume a common r for all the transmitter-receiver pairs. In this particular example, the reuse factor is JC = 9. 2.1 Network Model and Problem Definition We consider a grid network formed by n nodes U = {1, ... , n} placed on a regular grid on the unit square, with minimum distance 1/-jn. (see Fig. 2.l(a)). Users u EU make arbitrary requests fu E F = {1, ... , m}, from a fixed file library of size m. The vector of requests is denoted by f = (Ji, ... , fn)· Communication between user nodes obeys the following protocol model: if a node i transmits a packet to node j, then the transmission is successful if and only if: a) The distance between i and j is less than r; b) Any other node k transmitting simultaneously, is at distance d(k,j)?: (1 + ~)r from the receiver j, where r, ~ > 0 are protocol parameters. In practice, nodes send data at some constant rate Cr bit/s/Hz, where Cr is a non-increasing function of the transmission range r. Unlike live streaming, in video on-demand, the probability that two users wish to stream simultaneously a file at the same time is essentially zero, although there is a large redundancy in the demands when n » m. We refer to this feature of video on- demand streaming as the asynchronous content reuse. In order to model the asynchronous 17 I I I I I I I 1---=:I I I I .I •••••• . . I •• • • • . . • • • • . . I I • • . • • . • • . • • • • • •• • • • • ~1.mok· ,,.---.. !:~·: ...... L:.) ,.. ,...,., 1 .,, • w• . ·1.1ff : """ n< 1 ,.,.. 1 ••• • • • • • • l .-, ~: ..... .:J ~·=~: I lW\i. 11 M'1• I llC.\11 I 11~1'1 I • 1ill - ! ....,....., • • • Figure 2.2: QuiliWive represen~on of our system ~umptions: ~ user c.\Ches an entire file, Wrmed by * wry lMge nwnber of packet.s L'. Then> USEfi pl~ r~ requests of segments of L" packets from files of the libr.)l'y> st.&rting *t r.mdom irutial pcGntG. Inthefigure,v1eM>teL" =4. cont.mt reuse built into the problem> M\d forbid any rorm of ~iMve multicast.ins~'> i.e.> achieving ~odtd m~arting g®i by OV6"hmring ~~r fret' tr.ms:missions ~ted to other users> we .assume the following st.remUng model: 1) Ea.ch file in the libruy is formed by If --; 1 2) E.ch -= downlo>.ds >.n .rbitr..-ily selo>toi segmont, of leJ\gth L" -- of the Nqueste:I file (Sae Fig. 2.2 k>r ~quiliWive illustr.tion); g;i We sh>ll consider the system p;a-form..mce in t.he case of luge Lf M\d .n'bitr~wy> but finite> L". In addition> vie cons:ide: the worst.-cne systeln throughput over the US>?l'S~ delm.nds. ~ for sufficiettly la.rge L' ..md finite L 11 and mit is alw~s possible to ha.ve non-overhppin.g segments, even though \lSE:I'S l'n'W n?q1.1e:st the s~ file index /. As *consequence of the above model> a. de:mMid v~tor f is a.ssoci.ua:i v1ith *list of pcGnV-J'S s with elemmts s...: E { t, ... ,L 1 - LN + 1} such tha.t, for ea.ch ~ the dern...md to file 1~ implies that the~ s...:.s...: + 1, ... ,s\: +L" -1 of file/\: a.re SEqUE:ntiilly ruq~&i by user u Fbr simplicity, the e:plicit depe:ndency on s is omitted whenever 1 'Thk k ~t with C\IJ'Jcnt video rlr~ pro~~~ ~:::DASH pMI::C131 wlwe tl\~ video l\kU::i:plit into ?~t::: y;}.~ <MC ?.>q~~do~ byfu51:f~i'Mg'L?.:~. there is no ambiguity of notation. We let Wj denote packet j of file f E F. Without loss of generality, we assume that each packet contains F information bits, such that {Wj} are i.i.d. random variables uniformly distributed over {1, 2, 3, · · · , 2F}. As said before, we are generally interested in the case of large L 1 and finite L 11 • We have: Definition 2.1 (Caching Phase) The caching phase is a map of the file library F onto the cache of the users in U. Each cache has size M files. For each u E U, the function <Pu : IF2FL'-+ !F!J'1FL' generates the cache content Zu ~ ¢u(Wj,f = 1, ·· · ,m,j = 1, ·· · ,L 1 ). 0 Definition 2.2 (Coded Delivery Phase) The delivery phase is defined by two sets of functions: the node encoding functions, denoted by {1/!u : u EU}, and the node decoding functions, denoted by {.\u: u EU}. Let RJ denote the number of coded bits transmitted by node u to satisfy the request vector f. The rate of node u is defined by Ru = j;'f;,. The MFL 1 FL"R c, ( f) function 1/!u : IF 2 X Fn -+ IF 2 u generates the transmitted message Xu,! = 1/!u Zu, of node u as a function of its cache content Zu and of the demand vector f. Let Vu denote the set of users whose transmit messages are received by user u (accord- . t t . . l' . D fi 't. 2 3) Th f t. \ !FF £U Lu ED Ru ing o some ransmission po icy in e ni ion . . e unc ion /iu : 2 u x !F!J'1 FL' x Fn -+ IF~ L" decodes the request of user u from the received messages and its own cache, i.e., we have (2.1) 0 The worst-case error probability is defined as Letting R = LuEU Ru, the cache-rate pair (M, R) is achievable if V E: > 0 there exist a sequence indexed by the packet size F-+ co of cache encoding functions {¢u}, delivery 19 functions {1/Ju} and decoding functions {.\u}, with rate R(F) and probability of error PY) such that Jim supF-+oo R(F) SR and Jim supF-+oo PY) SE:. The optimal achievable rate 2 is given by R*(M) ~ inf{R: (M,R) is achievable}. (2.3) In order to relate this definition of rate to the throughput of the network, defined later, we borrow from [JCM13a, JCM13b] the definition of transmission policy: Definition 2.3 (Transmission policy) The transmission policy IT, is a rule to activate the D2D links in the network. Let£ denote the set of all directed links. Let A c;; 2L the set of all possible feasible subsets of links {this is a subset of the power set of£, formed by all sets of links forming independent sets in the network interference graph induced by the protocol model). Let AC A denote a feasible set of simultaneously active links. Then, II, is a conditional probability mass function over A given f (requests) and the caching functions, assigning probability II,(A) to A EA. 0 All the achievability results of this work are obtained using deterministic transmission policies, which are obviously a special case of Definition 2.3. Suppose that (M, R) is achievable with a particular caching and delivering scheme. Suppose also that for a given transmission policy II,, the RFL 11 coded bits to satisfy the worst-case demand vector can be delivered in ts channel uses (i.e., it takes collectively ts channel uses in order to deliver the required F L 11 Ru coded bits to each user u E U, where each channel use carries Cr bits). Then, the throughput per user, measured in useful information bits per channel use, is given by FL 11 T~- ts (2.4) 2 As a matter of fact, this is the min-max number of packet transmissions where min is over the caching/ delivery scheme and max is over the demand vectors, and thus intuitively is the inverse of the "rate" commonly used in communications theory. We use the term "rate" in order to stay compliant with the terminology introduced in [MAN14b]. 20 The pair (M, T) is achievable if (M, R) is achievable and if there exists a transmission policy IT, such that the RF L 11 encoded bits can be delivered to their destinations m ts S (F L 11 ) /T channel uses. Then, the optimal achievable throughput is defined as T*(M) ~ sup{T: (M, T) is achievable} (2.5) In the following we assume that t "' ~n ;:> 1. Notice that this is a necessary condition in order to satisfy any arbitrary demand vector. In fact, if t < 1, then the aggregate cache in the entire network cannot cache the file library, such that some files or part of files are missing and cannot be delivered. This requirement is not needed when there is an omniscient node that can supply the missing bits (as in [MAN14b]) or in the case of random demands, as in [JCM13a, JCM13b], by defining a throughput versus outage probability tradeoff, where the outage probability is defined as the probability that a user demand cannot be satisfied. However, this work focuses on deterministic (worst case) demands and has no omniscient node, such that t 2 1 is necessary. We observe that our problem includes two parts: 1) the design of the caching, delivery and decoding functions; 2) scheduling concurrent transmissions in the D2D network. For simplicity, we start by focusing on the case where only a single link can be simultaneously active in the whole network and let the transmission range r such that any node can be heard by all other nodes (i.e., we let r 2 V2). In this case, scheduling of concurrent transmissions in the D2D network is irrelevant and we shall focus only on the caching and delivery schemes. Then, we will relax the constraint on the transmission range r and consider spatial reuse and D2D link scheduling. 21 2.2 Deterministic Caching, Achievability and Converse Bound 2.2.1 Transmission ranger 2 v'2 The following theorem yields the achievable rate of the proposed caching and coded multicasting delivery scheme. Theorem 2.1 For r 2 y'2 and t = ~n E z+, the following rate is achievable: R(M) = : ( 1 - ~) . (2.6) Moreover, when t is not an integer, the convex lower envelope of R(M), seen as a function of M E [O : m]. is achievable. D The caching and delivery scheme achieving (2.6) is given in Appendix A.1 and an illustrative example is given in Section 2.2.3. The proof of Theorem 2.1 is given in Ap pendix A.2. The corresponding achievable throughput is given by the following immediate corollary: Corollary 2.1 For r 2 V2, the throughput T(M) =Rf~)' (2.7) where R(M) is given by {2.6) is achievable. D Proof In order to deliver F L 11 R(M) coded bits without reuse (at most one active link transmitting at any time) we need ts = FL 11 R(M)/Cr channel uses. Therefore, (2.7) follows from the definition (3.1). A lower bound (converse result) for the achievable rate in this case is given by the following theorem: 22 Theorem 2.2 For r 2 V2, the optimal rate is lower bounded by R*(M);:>max{ max (1- l!JM),-n-(1-M) xl{n>l,m>l}}, lE{l,2, ,mm{ m,n)) T n - 1 m (2.8) where 1 {-} denotes an indicator function. D The proof of Theorem 2.2 is given in Appendix A.3. Given the fact that activating a single link per channel use is the best possible feasible transmission policy, we obtain trivially that using the lower bound (2.8) in lieu of R(M) in (2.7) we obtain an upper bound to any achievable throughput. The order optimality of our achievable rate is shown by: Theorem 2.3 As n, m -+ co, for t ~n > 1, the ratio of the achievable over the optimal rate is upper bounded by 4, t = w(l), ~SM= o(m) 4t n = O(m), t = 8(1), ~SM= o(m) ru· R(M) 6, M = 8(m) < R*(M) 2 n=w(m),M<~ M• (2.9) t 2 n= O(m),n> m,M < ~ 1iJ M• 2, n= O(m),nS m,M < ~ where, for all t 2 1, we have l;J S 2. D The proof of Theorem 2.3 is given in Appendix A.4. Obviously, the same quantity upper bounds the optimal/achievable throughput ratio 1;/f::f/. From (2.9), we can see that except the case when n > m and the storage capacity is very small (M < ~. less than a half of a file), our achievable result can achieve the lower bound within a constant factor. The reason why in the regime of redundant requests (n > m) and small caches (M <~),the (multiplicative) gap is not bounded by a constant is because in our problem definition we force asynchronous requests (i.e., we let L 1 -+ co 23 with finite L 11 ). This prevents the possibility of "naive multicasting", i.e., sending then files directly, such that each file transmission is useful to multiple users that requested that particular file. This fact is evidenced by considering the special case of L 11 = L 1 • In this case, naive multicasting becomes a valid scheme and we have: 3 Corollary 2.2 For r 2 y'2 and t = 11nn E z+, the following rate is achievable: R(M) =min {: ( 1 - ~) , m} . (2.10) Moreover, when t is not an integer, the convex lower envelope of R(M), seen as a function of M E [O : m], is achievable. D The first term in the minimum in (2.10) follows from Theorem 2.1, while the second term is the rate obtained by using naive multicasting, where all the bits of all the files in the library are multicasted to all nodes, thus automatically satisfying any arbitrary request. This requires a total length of mL 1 F, i.e., a rate equal to m. Putting together Corollary 2.2 and Theorem 2.2 we have: Corollary 2.3 For r 2 V2 and t = 11nn 2 1, as n, m -+ co, the ratio of the achievable over the optimal rate is upper bounded by where l;J S 2. _R~(M~) < R*(M) 4, 4t 1iJ' 6, 2, t = w(l), ~SM= o(m) n = O(m), t = 8(1), ~SM= o(m) M = 8(m) M<~ (2.11) D Corollary 2.3 is also proved in Appendix A.4. Corollary 2.3 implies that, when all the users request a whole file (L 1 = L 11 ), our achievable rate achieves the lower bound with a constant multiplicative factor in all the regimes of the system parameters. 3 All the definitions in Section 2.1 will be changed accordingly to the case when L 11 = L'. 24 Beyond the theoretical interest of characterizing the system throughput in all regimes, we would like to remark here that, in practice, caching is effective in the regime of large asynchronous content reuse (i.e., n > m) and moderate to large cache capacity (i.e., 1 « M < m). In this relevant regime, we can focus on the asynchronous content delivery (no naive multicasting) letting L 1 -+ co and fixed L 11 , and still obtain a constant multiplicative gap from optimal. 2.2.2 Transmission ranger< J2 In this case, the transmission range can be chosen in order to have localized D2D commu- nication and therefore allow for some spatial reuse. In this case, we need to design also a transmission policy to schedule concurrent active links. The proposed policy is based on clustering: the network is divided into clusters of equal size gc, independently of the users' demands. Users can receive messages only from nodes in the same cluster. Therefore, each cluster is treated as a small network. Assuming that gcM ;:> m, 4 the total cache capacity of each cluster is sufficient to store the whole file library. Under this assumption, the same caching and delivery scheme used to prove Theorem 2.1 can be used here. A sim pie achievable transmission policy consists of partitioning the set of clusters into JC reuse sets, such that the clusters of the same reuse set do not interfere and can be active simultaneously. In each active cluster, a single transmitter is active per time-slot and it is received by all the nodes in the cluster, as in classical time-frequency reuse schemes with reuse factor JC currently used in cellular networks [Molll, Ch. 17]. An example of a reuse set is shown in Fig. 2.l(b). In particular, we can pick JC= ([ V2(1+6)1+1) 2 . This scheme achieves the following throughput: 4 If the condition gcM 2: m is not satisfied, we can choose a larger transmission range such that this condition is feasible. 25 Theorem 2.4 Let r such that any two nodes in a "squarelet" cluster of size Ye can communicate, and let t = g~ E z+. Then, the throughput Cr 1 T(M) = JC R(M)' (2.12) is achievable, where R(M) is given by Theorem 2.1, r is the transmission range and JC is the reuse factor. Moreover, when t ¢ z+, T(M) is given by the expression {2.12} with R(M) replaced by its lower convex envelope over ME [O: m]. D The proof of Theorem 2.4 is given in Appendix A.5. Notice that whether reuse is con venient or not in this context depends on whether C V2 (the link spectral efficiency for communicating across the network) is larger or smaller than Cr/IC, for some smaller r which determines the cluster size. In turns, this depends on the how the link spectral effi- ciency varies as a function of the communication range. This aspect is not captured by the protocol model, and the answer may depend on the operating frequency and appropriate channel model of the underlying wireless network physical layer [JCM13c]. An upper bound on the throughput with reuse is given by: Theorem 2.5 When r < V2 and the whole library is cached within radius r of any node, the optimal throughput is upper bounded by Cr r 4(21f )21 T*(M) S -----~--~---- ,min{m,rKr2nlJJ (1- l~JM)' (2.13) maxlE{l,2, where r is the transmission range and~ is the interference parameter. D The proof of Theorem 2.5 is given in Appendix A.6. Furthermore, we have: 26 Theorem 2.6 When r < V2, fort= M';;; 2 n 2 1, as n, m-+ co, the ratio of the optimal throughput over the achievable throughput is upper bounded by T* (M) I 4(2 + L'.)21 T(M) s JC I f'.2 x where, for all t 2 1, l;J S 2. 4, 4t ru· 6, _;z_ M• t 2 1iJ M• 2, t = w(l), ~SM= o(m) 7rr 2 n = O(m), t = 8(1), ~SM= o(m) M = 8(m) 7rr 2 n =w(m),M < ~ n = O(m),7rr 2 n > m,M < ~ n = O(m),7rr 2 n S m,M < ~ ' (2.14) D The proof of Theorem 2.6 is given in Appendix A. 7. Similar to the case of r 2 V2, when L 1 = L 11 (i.e., when naive multicasting is possible), we can show that 1;/f::f/ is upper bounded by the constant factor, independent of m, n and M. 2.2.3 An Example The proposed caching placement and delivery scheme and the techniques of the proof for the converse are illustrated through a simple example. Consider a network with three users (n = 3). Each user can store M = 2 files, and the library has size m = 3 files, which are denoted by A, B, C. Let r 2 V2. Without loss of generality, we assume that each node requests one packet of a file (L 11 = 1). We divide each packet of each file into 6 subpackets, and denote the subpackets of the j-th packet as { Aj,c : £ = 1, ... , 6}, {Bj,c: £ = 1, ... , 6}, and {Cj,c: £ = 1, ... , 6}. The size of each subpacket is F/6. We let user u stores Zu, u = 1, 2, 3, given as follows: Z1 =(Aj,1, Aj,2, Aj,3, Aj,4, Bj,1, Bj,2, Bj,3, Bj,4, (2.15) 27 Cj,1, Cj,2, Cj,5, Cj,6), j = 1, · · · , L 1 • (2.16) (2.17) In this example, we consider the demand f = (A, B, C). Since the request vector contains distinct files, specifying which segment of each file is requested (i.e., the vector s) is irrel evant and shall be omitted. In the coded delivery phase (see Fig. 2.3), user 1 multicasts B3 + C1 (useful to both user 2 and 3), user 2 multicasts As+ C2 (useful to both users 1 and 3) and user 3 multicasts A 6 + B 4 (useful to both users 1 and 2). It follows that R(2) = Ri + R2 + R3 = i · 3 = ~ is achievable. wants A [ Uem 1] .t:;:::::::;===:::s-i 11---===---~ Ai ,A2,A3,A4, B1,B2,B3, B4, C1, C2, C3, C4, A3, A4, A5, A6, B3, B4, B5, B6, C3, C4, C5, C6, Ai , A2 , A5,A6, B1,B2,B5 ,B6, C1, C2, C5, C6, Figure 2.3: Illustration of the example of three users with M = 2, achieving rate R(2) = 1/ 2. Next, we illustrate the idea of the general rate lower bound of Theorem 2.2. Without loss of generality, we assume that L 1 / L 11 is an integer and let s denote the segment index. For any scheme that satisfies arbitrary demands f, with arbitrary segments s, we denote 28 by RZ:,s,I the number of coded bits transmitted by user u, relative to segment s and request vector f. Since the requests are arbitrary, we can consider a compound extension for all possible request vectors. For example, we let the first request be f = (A, B, C), the second request be f = (B, C, A) and the third request be f = (C, A, B). Then, the augmented compound-extended graph is shown in Fig. 2.4 where, consistently with our general notation defined in Section 2.1, Zu denotes the cached symbols at user u = 1, 2, 3, Xu,I denotes the transmitted message from user u in correspondence of demand f, and Wu,t is the decoded message at user u relative to file f. Considering user 3, from the cut that separates (X1,(A,B,C), X2,(A,B,c), X1,(B,C,A), X2,(B,C,A), Xl,(c,A,B), X2,(C,A,B), Z3) and (W3,C, W3,A, W3,B ), and by using the fact that the sum of the entropies of the received messages and the entropy of the side information (cache symbols) cannot be smaller than the number of requested information bits, we obtain that L' y;rr ~(~ +~ +~ +~ L,, 1,s,(A,B,C) 2,s,(A,B,C) 1,s,(B,C,A) 2,s,(B,C,A) s=l + RL,( C,A,B) + RL,( C,A,B)) + MF L' 2 3F L" . L' IL"- (2.18) Similarly, from the cut that separates (X1,(A,B,c), x3,(A,B,c), X1,(B,C,A), x3,(B,C,A), Xl,(c,A,B), X 3 ,(c,A,B), Z2) and (W2,B, W2,c, W2,A), and from the cut that separates (X 2 ,(A,B,C), x3,(A,B,c), X2,(B,C,A), x3,(B,C,A), X2,(C,A,B), x3,(C,A,B), Z1) and (W1,A, W1,B, W1,c ), we obtain analogous inequalities up to index permutations. By summing (2.18) and the other two corresponding inequalities and dividing all terms by 2, we obtain L' L" ~(RT +RT +RT L,, 1,s,(A,B,C) 2,s,(A,B,C) 3,s,(A,B,C) s=l +RT +RT +RT 1,s,(B,C,A) 2,s,(B,C,A) 3,s,(B,C,A) T T T )3 ,9, +R1,s,(C,A,B) + R2,s,(C,A,B) + R3,s,(C,A,B) + 'jMFL 2 'jFL · (2.19) 29 Since we are interested in minimizing the worst-case rate, the sum R[s, 1 + RL, 1 + RI,s, 1 must yields the same min-max value RT for any s and f. This yields the bound 3L 1 T 9 1 3 1 -R >-FL - -MFL. L 11 -2 2 (2.20) Finally, by definition of rate R(M), we have that R(M) =RT /(F L 11 ). Therefore, dividing both sides of (2.19) by 3FL 1 , we obtain that the best possible achievable rate must satisfy *( ) 3 1 R M >---M. - 2 2 (2.21) In the example of this section, for M = 2 we obtain R*(2) 2 ~· Therefore, in this case the achievability scheme given before is information theoretically optimal. In the same case of n = 3 users, per-node storage capacity M = 2 and library size m = 3, the coded multicasting scheme of [MAN14b] where a single codeword is sent to all users through a common bottleneck link achieves R(2) = ~· Then, in this case, the relative loss incurred by not having a base station with access to all files is 3/2. 2.2.4 Discussions The achievable rate of Theorem 2.1 can be written as the product of three terms, R(M) = n (1- ;';;)Mn with the following interpretation: n is the number of transmis- sions by using a conventional scheme that serves individual demands without exploiting the asynchronous content reuse; ( 1 - ;';;) can be viewed as the local caching gain, since any user can cache a fraction M /m of any file, therefore it needs to receive only the re- maining part; Mn is the global caching gain, due to the ability of the scheme to turn the individual demands into a coded multicast message, such that transmissions are useful to many users despite the streaming sessions are strongly asynchronous. An analogous interpretation can be given for the terms appearing in the rate expression achievable with the scheme presented in [MAN14b] (see Chapter 1), where the base station has access to all the files. Comparing this rate with our Theorem 2.1, we notice that they differ only in 30 ·····•W3 ,B x3 (C,A,B) Figure 2.4: The augmented network when m = 3, n = 3. The three requested vectors are: (A, B, C), (B, C, A) and (C, A, B). the last term (global caching gain), which in the base station case is given by (1 + n;;!)- 1 . For nM » m, we notice that these factors are essentially identical. As already noticed, Theorem 2.4 shows that there is no fundamental cumulative gain by using both spatial reuse and coded multicasting. Under our assumptions, spatial reuse may or may not be convenient, depending whether ~ is larger or smaller than C,/2. A closer look reveals a more subtle tradeoff. Without any spatial reuse, let t = ":n,n E z+, the length of the codewords in coded subpackets for each user, related to the size of the subpacketization, is t (~) . This may be very large when n and M are large. At the other extreme, we have the case where the cluster size is the minimum able to cache the whole library in each cluster. In this case, we can just store M different whole files into each node, such that all m files are present in each cluster, and for the delivery phase we just serve whole files without any coding as in [J CM13a]. In this case, the achieved throughput is CJc ~ bits/s/Hz, which is almost as good as the coded scheme, which achieves JC(~-l). This simple scheme is a special case of the general setting treated in this work, where 31 spatial reuse is maximized and codewords have length 1. If we wish to use the achievable scheme of this work, the codewords length is M:;,, ( J;,). Hence, spatial reuse yields a m reduction in the codeword length of the corresponding coded multi casting scheme. Further, we also notice that even though our scheme is design for a D2D system, it can also be used in a peer-to-peer (P2P) wired network, where each peer is allowed to cache information with limited storage capacity. By using our approach in the case of r 2 V2, peers can exchange multicast messages which are useful for a large number of other peers, provided that the network supports multicasting. 2.3 Decentralized Random Caching The main drawback of the deterministic caching placement in the achievability strategy of Theorem 2.1 is that, in practice, a tight control on the users caches must be implemented in order to make sure that, at any point in time, the files subpackets are stored into the caches in the required way. While this is conceptually possible under our time-scale decomposition assumption (see comment in the beginning of this chapter), such approach is not robust to events such as user mobility and nodes turning on and off, as it may happen in a D2D wireless network with caching in the user devices. In this section, we present a decentralized random caching and coded delivery scheme that allows for more system robustness. 2.3.1 Transmission ranger 2 v'2 Decentralized random caching with coded multicast delivery has been considered in [MAN14a]. However, there is an important difference between our network model and that of [MAN14a], where, thanks to the central omniscient server (base station), possibly missing packets due to the decentralized random caching can always be supplied by the server by uni casting (or naive multicasting). In our system this is not possible, since no node has generally access to the whole file library. Hence, in order to ensure a vanishing 32 probability of error, we shall use an additional layer of Maximum Distance Separable (MDS) coding and consider the limit of large packet size F. We distinguish between two regimes: t "' 11nn > 1 and t = 1. As already noticed before, t < 1 is not valid since in this case the file library cannot be cached into the network and therefore the worst case user demands cannot be satisfied. For t > 1, we consider the following scheme: each file segment of F bits is divided into K blocks of F / K bits each, hereafter referred to as "subpackets". These subpackets are interpreted as the elements of the binary extension field IF 2 F/K, and are encoded using a (K,K/p)-MDS code, for some p < 1 the choice of which will be discussed later. Notice that this expands the size of each packet from F to F / p. The resulting K / p encoded blocks of F / K bits each will be referred to as "MDS-coded symbols". The definition of worst-case error probability and achievable rates given in Section 2.1 hold verbatim in this case. In particular, since the definition of achievable rate considers a sequence of coding schemes for F -+ co, we can choose K as a function of F such that both K and F/K grow unbounded as F increases. This ensures that, for any fixed p, (K,K/p)-MDS codes exist [LC04, YeuOS]. The MDS-coded symbols are cached at random by the user nodes according to Algorithm 2 .1. Algorithm 2.1 Decentralized random caching placement scheme 1: Encode the K subpackets of each packet of each file by using a (K,K/p)-MDS code over IF 2 F/K, for some MDS coding rate p < 1. 2: The MDS-coded symbols for each packet of each file are indexed by {1, 2, · · · ,K/p}. 3: for all u = 1, · · · , n do 4: User u, independently of the other users, chooses with uniform probability an index set Su over all possible sets obtained by sampling without replacement ~K elements from {1,2, ,K/p}. 5: User u caches the MDS-coded symbols indexed by Su for each packet of each file. 6: end for Next, we describe a delivery scheme that provides to each requesting user enough MDS-coded symbols such that it can recover the desired file segments. For k = 1, · · · , L 11 , let Su + k - 1 denote the index of the k-th requested packet of file fu by user u. Let { zy : j = 1, ... 'KI p} denote the block of MDS-coded symbols from packet i of file f. 33 For b = n, n - 1, ... , 2, and each user subset U c;; U of size IUI = b, we define z;:,t\{~J as the symbol sequence obtained by concatenating (in any pre-determined order) the symbols needed by u E U, present in all the caches of users v E U\ { u }, and not present in the cache of any other user v 1 ~ U\ { u }. We refer to z;:,t\{~J as the symbols relative ot u E U and exclusively cached in all nodes v E U\ { u }. Formally, the index set of the symbols forming z;:,t\{~J is given by Jfu,U\{u)= n {jE{l,,K/p}:z;:+k-l,jEZv\{ u Zu•}}' vEU\{u) u'¢U\{u) where the index set does not depend on the packet index Su + k - 1 since the same caching rule is used for all packets (see Algorithm 2.1). Then, z;:t\{~J is the sequence of MDS-coded symbols formed by concatenating the symbols {zj:+k-l,j: j E Jfu,U\{uj}· By construction, each user v E Uhas one local replica of z;:t\{~J (common symbols) for each u E U\{v}. We also let JLJ' = maXuEU IJtu,U\{u)I, such that all sequences z;:,t\{~J can be zero-padded 5 to the common maximum length JU=. In order to deliver the MDS coded symbols, each user v E U\ { u} sends distinct (i.e., non-overlapping) segments of length b~l . JLJ'. ~ of the sequence of XO Red MDS-coded symbols EBuEU,ufv z;:t\{~J. The delivery phase is summarized in Algorithm 2.2. Algorithm 2.2 Decentralized random caching delivery scheme 1: for all k = 1, · · · , L 1 do 2: for all b = n, n - 1, · · · , 2 do 3: for all U c U with I U I = b do 4: Ju== maXuEU IJtu,U\{u) I 5: for all v E U do 6: User v transmits a non-overlapping segment of length b~l · JLJ' · ~ of the zero-padded and XORed MDS-coded symbol sequence EfluEU,ufv z;:t\{~J. 7: end for 8: end for 9: end for 10: end for 5 With a slight abuse of notation we indicate by z;:t~{~} also the zero-padded version. 34 Since the scheme is admittedly complicated and its general description relies on a heavy notation, we provide here an illustrative example (see Fig. 2.5). As in the case of deterministic caching, consider the case of 3 users denoted by 1,2,3. Neglecting the packet superscript (irrelevant in this example), for the first round of the scheme, with b = 3 (see Fig. 2.5(a)), let zfi,{ 2 , 3 ) be the sequence of MDS-coded symbols useful to user 1 (requesting file Ji) and present in the caches of users 2 and 3. Also, let Zh{l, 3 } and Zfs.{l, 2 } have similar and corresponding meaning, after permuting the indices. Then, user 1 forms the XORed sequence Zh{l, 3 } Ell Zfs.{l, 2 }, user 2 forms the XORed sequence Zfi,{ 2 , 3 ) Ell Zfs.{l, 2 }, and user 3 forms the XORed sequence Zfi,{ 2 , 3 1 Ell Zh{l, 3 }. Finally, each user transmits to the other two users 1/2 of its own XORed sequence. For the second round of the scheme, with b = 2 (see Fig. 2.5(b)), let zfi,{ 2 } and z 1 ,,{ 3 ) denote the sequence of MDS-coded symbols useful to user 1 and cached exclusively by user 2 and user 3, respectively. Similarly, let Zh{l}' Zh{ 3 ), Zh{l) and Zfs.{ 2 } have corresponding meaning. Since b = 2, there is no multicasting opportunity. User 1 will just transmit sequence Zh{l} to user 2 and Zfs.{l} to user 3. Users 2 and 3 perform similar operations. Focusing on decoding at user 1, after the first round, half of zfi,{ 2 , 3 ) is recovered from user 2 transmission and the other half from user 3 transmission, by using the side information of its own cache. After the second round, z 1 ,,{ 2 ) is directly received from user 2, and zf,,{ 3 ) from user 3. Finally, if the MDS coding rate p is chosen appropriately, user 1 is able to recover the desired file Ji with high probability from the MDS-coded symbols zf,,{2,3), zf,,{2) and zf,,{3) and the symbols relative to file Ji already present in its cache. The following result yields a sufficient condition for the MDS coding rate p such that, in the general case, all files can be decoded with high probability from the MDS coded symbols cached in the network: Theorem 2. 7 Let p = (1 - E: )p*, where E: > 0 is an arbitrarily small constant and p* is the non-zero solution of the fixed point equation: x = 1- exp(-tx). (2.22) 35 Then, the random caching scheme of Algorithm 2.1 with MDS coding rate p yields, for allf=l, ... ,m, '!!(File f can be decoded from the cached MDS-coded symbols in the network) (2.23) where 61(0) is a term independent of K, such that 61(0) > 0 for all o > 0. D Theorem 2.7 is proved in Appendix A.8. From Theorem 2.7 and the union bound, we have immediately that all the files can be successfully decoded from the cached MDS-coded symbols in the network with arbitrarily high probability for sufficiently large K. In our example, i.e., for n = 3, m = 3 and M = 2, by choosing o in Theorem 2. 7 as 0.001, we obtain p = 0.95, yielding an achievable rate R(2) = 0.77 (see (2.24) in Theorem 2.8). When t = 1, the scheme based on MDS codes given by Algorithms 2.1 and 2.2 cannot be applied since (2.22) has no finite positive solution. In this case, we propose a different caching and delivery scheme given as follows. Each packet of each file is divided into K subpackets of size F / K bits, and each block of K subpackets, interpreted as symbols over lF 2 F; K, is separately and independently encoded by each user u by using a random linear hashing function that compresses the K symbols into MK/m symbols as follows: each user u generates independently a matrix Gu of dimension K x MK /m over lF 2 F/K with i.i.d. components. Representing Wj (the i-th packet of file f) as a 1 x K vector wj over lF 2 F/K, the hashing transformation at user u is given by cf,u = wjGu. Then, each user u caches c 1 i for all f = 1, ... , m and i = 1, ... , L 1 • Notice that, in this way, the sum of the ,u lengths of these codewords is L 1 m x MK/ m x F / L 1 = L 1 MF bits, such that the cache size constraint is satisfied with equality. For the delivery phase, each user unicasts n~l (1-1:,) K coded symbols for each requested packet of all other users. Hence, at the end of the delivery phase, each requesting user collects (1-1:,) K coded symbols from the other n - 1 users and MK/m symbols 36 Z1t,{ 1,2}, Z 11.(1,3}, Z1,, (1} . z h.( 1,2) , zh. (1,3}, z h.{1} Z1,,(1,2}, Z13 (1,3}, Zh.{1} wants ft wants h ' z •z [ u,m 2 ] 2 t..{2.3}© /,,{1.2} z" ,{1,3), z 1 d2,3}, z /1,{3}, z f2.{1,3 1 , z 1 2,{2,3},z 12 • 1 3 1 Z1a,{1,3}, zh.{2,3}, z 13 ,{3}, (a) 11 ,{ 1,2}, z 1 ,,(2,3}, z ,,,{2}, Zh. (1,2) , Z 1,,(2,3}, Z 12 ,(2} zh.(1,2}, z" ,{2,3}, Z1s,{2} ~ -~ ...... Zt_ .. _ 11_ }~~~~~~~ ~~- L.:..:::...:J ~~~-z ~ 1~ "~ 11~ }ti, ~r--~~~~~~--, Z1t,{1,2}, Z11.(1,3} ,Ztdi) Z 12.{1,2}, Z12.(1,3}• Z12,{1} Zh.{1,2),Zh.(13)•Zt. {1}> z,, ,{1 ,3}> z 1 1.{2,3}, z ,d3l, Z12,(1,3),Zh.(2,s},Z12.{s}, Zt..(1,3), Z1.,{2,3}, Zt,,{3), (b) /t ,{1,2}, Z1d2,3}, Z1..{2) Z12.{1,2}. Z 12 .{2,3} , Z12,{2) Zh.{1,2).Zt. {2,3J•Z/3,(2} Figure 2.5: Illustration of the example of three users with M = 2, m = 3, achieving rate R(2) = 0.77, where p = 0.95 (p* = 0.9510, E is chosen as 0.001.). a) The first iteration with I U I = b = 3. b) T he second iteration wit h I U I = b = 2. 37 from its own hashed codeword, such that it has a total of K coded symbols. If the K x K system of linear equations corresponding to these symbols has rank K, then the packet can be retrieved. This condition is verified with arbitrarily large probability for sufficiently large F / K [HMK+06, Yeu08]. Furthermore, we observe that the caching and delivery scheme for t = 1 can be applied, trivially, also for t > 1. Eventually, combining the two caching and delivering schemes, we can prove the following general results: Theorem 2.8 For r 2 v2 and t = ~n > 1, as F, K -+ co with sufficiently large ratio F / K, the following rate is achievable by decentralized caching: { 1 n () (M )s-1( M )n-s+l } R(M) =min p; : 8 ~ 1 :: 1 - :: , n - t . (2.24) Consequently, the throughput T(M) = ~'{]) is also achievable. D In order to evaluate the achievable rate of Theorem 2.8, the following lemma is useful: Lemma 2.1 The achievable rate R(M) of Theorem 2.8 is upper bounded by R(M) Smin -- 1- - 1 + 1- 1- - { m ( Mp) ( 3 ( ( Mp)n+l) Mp 2 m (n + 1)%1' m ( M )n 5M ( M )n-1) } -4 1 - :: - 2 ~n 1 - :: , n - t . (2.25) D The proof of Theorem 2.8 and of Lemma 2.1 are given in Appendix A.9. For t = 1, we have: Corollary 2.4 For r 2 v2 and t = ~n = 1, as F, K-+ co with sufficiently large ratio F / K, the following rate is achievable: R(M) = : ( 1 - ~) . (2.26) D 38 Corollary 2.4 is immediately obtained by using the second term in the "min" of (2.24) and letting t = 1. The gap between the achievable rate and the lower bound of Theorem 2.2, which applies to any scheme, also centralized, is given by: Theorem 2. 9 For r 2 V2 and t = ~n 2 1, as F, K -+ co with sufficiently large ratio F/K, then let m, n-+ co, for any MS 1 ~ 6 m, for some arbitrarily small E: > 0, the ratio of the achievable rate with decentralized caching over the optimal rate with unrestricted caching is upper bounded by where _R~(M~) < R*(M) - 1 fg(t) = 2 (1 + fp(t)) p 8 (1-s) 2 ' 6, 4 M(l-s) 2 ' min{ 4t, fg(t) }, t=w(l),~SM=o(m) M = 8(m) n=w(m),M<~ Otherwise 4t TiJ' t 2 1iJ M' n = O(m), t = 8(1), ~SM= o(m) n= O(m),n> m,M < ~ 2, n=O(m),nSm,M<~ (2.27) (2.28) where l;J S 2 and fp(t) = f,- e-pt (ft+ 4 +~pt). Further, fort fc 1, : 2 (1 + fp(t)) can be upper bounded by a positive constant. D The proof of Theorem 2.9 is given in Appendix A.10. The gap between T(M) and T*(M) follows as a consequence. When naive multicasting is allowed, if t > 1, with high probability (given by (2.23) in Theorem 2.7), there are at least K distinct coded symbols for each packet of each file cached it the network. Therefore, by requiring multicasting of at most K distinct coded symbols for each file, a rate m can be achieved. Similarly, when t = 1, by using the linear random hashing scheme, there are again at least K distinct coded symbols for each packet 39 of each file cached in the network with high probability as the field size (2F/ K) grows large. Then, we can achieve a rate of m by naive multicasting at most K distinct coded symbols of each requested packet such that all the users can decode. Hence, similar to Theorem 2.9, (2.27) becomes a constant when naive multicasting is allowed. 2.3.2 Transmission ranger< J2 Based on the scheme developed for the case r 2 V2, by using the same clustering approach for the deterministic caching case, we immediately have: Corollary 2.5 Let r is such that any two nodes in a "squarelet" cluster of size Ye can communicate, as F, K-+ co with sufficiently large ratio F/K, the throughput Cr 1 T(M) = 7( R(M)' (2.29) is achievable with decentralized caching, JC is the clustering scheme reuse factor, and R(M) is given by Theorem 2.8 fort> 1 and by Theorem 2.4 fort= 1. D Furthermore, similar to Theorem 2.6, the ratio T*(M)/T(M) is upper bounded by the terms in (2.27), multiplied by the geometry factor JC r 4 ( 2 1~) 2 1. 2.3.3 Discussions From the above results, we observe that the proposed decentralized random caching scheme achieves a performance very close to that of the deterministic caching scheme. Specifically, in the case of r 2 V2 and t > 1, from Theorem 2.9, we see that our decen tralized approach achieves order optimality (constant multiplicative gap) in the scaling throughput law for large networks, i.e., in the limit of n-+ co. It is important to notice the order in which we have to take the limits here: for any finite n, m, M, we consider the limit for large file size (in packets) L 1 -+ co, and large bits per packet F -+ co. Then, we look at the rate behavior for possibly large network size n and library size m. Taking limits in this order is meaningful if we consider typical applications of caching for video 40 (a) (b) Figure 2.6: Rate R(M) as a function of the cache size M for deterministic and random D2D caching. In (a) we let m = 50 and n = 100. In (b) we let m = 500 and n = 50. The rate of deterministic caching is given by (2.6). The rate of Random Caching (Exact) is given by (2.24). The curve "Random Caching (Approximate)" is plotted by using (2.25). The converse of the rate is given by (2 .8). on-demand delivery. Consider for example a good-quality movie file encoded at 2 MB/s, of total duration of lh. In current Dynamic Adaptive Streaming over HTTP (DASH) standards [SdlFSH+ lla, KLC+ 12] a video packet has typical duration of ls, correspond ing to 2 Mb. In this case we would have L' = 3600 and F = 2 · 10 6 , which justify our assumptions. For finite n, from Theorem 2.9 and its proof, we can see that the multiplicative gap between the achievable rate of the decentralized random caching approach and that of the centralized deterministic caching approach is a function of the system parameters M,m and n. However, from the simulation results (see Figs. 2.6(a) and 2.6(b)), we observe that this gap vanishes as the memory size M increases. In addition, from Theorem 2.9, as n, m -+ CXJ, this gap becomes a constant. Hence, the decentralized random caching scheme performs approximately as well as the centralized deterministic caching scheme in the most interested regimes of the system parameters. 41 2.4 Summary In this work, we have determined constructive achievability coding strategies and infor mation theoretic bounds for a D2D caching network under the constraint of arbitrary (i.e., worst-case) demands. We have considered two caching and (inter-session network coded) delivery schemes: the first is based on deterministic (centralized) caching, and the second is based on random (decentralized) caching. The decentralized nature of the second scheme lies in the fact that each user independently determines the (coded) symbols to cache, without knowing what the others do. Our work differs from concur rent and previous recent works by the fact that we do not consider a central omniscient server that has access to all files in the library. Hence, the proposed schemes are strictly peer-to-peer and infrastructureless, and therefore they are suited to a wireless D2D net work [WTs+10, CMGSS12, DRw+og, JCM13c]. In the case where all nodes in the network are in the reach of each other (range r 2 V2, under our normalizations), under the ass um pt ion of asynchronous content reuse, i.e., when naive multicasting is forbidden or useless by our model, we showed that the deterministic caching scheme is optimal within a constant multiplicative factor in almost all system regimes, with the exception of the regime of large content reuse (number of users n larger than the number of library files m) and very small cache capacity M < 1/2. This regime is arguably not very interesting for applications, since the goal of caching is precisely to trade cache memory in the user devices (an inexpensive and largely untapped network resource) for bandwidth (a very expensive and scarce commodity). In any case, allowing for naive multicasting fills also this small gap. Interestingly and somehow counterintuitively, we found that when we restrict the transmission range to some r < V2 in order to allow multiple concurrent transmissions in the network under the protocol model (spatial spectrum reuse), the throughput does not improve in terms of the scaling law with respect ton, m and M. Spatial reuse has therefore only a possible gain in terms of actual rates, due to the possible improvement of the actual 42 transmission rates due to the shorter link distance. It follows that, in order to assess whether spatial reuse is beneficial or not, one has to consider an accurate model for the underlying physical layer and propagation channel, and consider actual transmission rates and interference. Evidently, the protocol model considered in this work is too "coarse" to capture these aspects. An example of such analysis is provided in [JCM13c] for a D2D network with realistic propagation channel modeling operating at various frequency bands, from cellular microwave to mm-wave bands, as envisaged in the forthcoming 5G standardization [RGBD+11, Aww+13, DMRHlO, ALs+13, RRE+13, RRE14]. Moreover, for the deterministic caching scheme, the trade-off between coded multi casting and spatial reuse is reflected by the code length, which depends on the communication cluster size. In the proposed decentralized random caching scheme, we have used MDS coding (or random linear intra-session network coding in the form of random linear hashing of the files subpackets) in order to ensure, with high probability, that all files can be recovered by the (coded) symbols cached into the network. This was not necessary in the setting of [MAN14a], since the missing symbols can be always supplied by the omniscient broadcasting node. We showed that also this random caching scheme achieves order optimality when the network size n become large. Overall, the decentralized random caching scheme appears to be more attractive for practical applications since it allows all users to cache at random and independently their assigned fraction of coded symbols of the library files, without knowing a priori which symbols have been already cached by other nodes. As a final remark, we wish to stress the fact that a decentralized D2D caching scheme may effectively provide a very attractive avenue for efficient content distribution over wireless networks, avoiding the cluttering of the cellular infrastructure. For example, consider the results of Figs. 2.6(a) and 2.6(b). In both cases, when M"' m/10 (i.e., 10% of the whole library is cached in each user device), then user requests can be satisfied by sending the coded equivalent of 10 files. This means that if the physical layer link peak 43 rate is (say) a modest 20 Mb/s, each user can stream video at 2 Mb/s, irrespectively of the number of users. 44 Chapter 3 The Throughput-Outage Tradeoff of Wireless One-Hop Caching Networks From Chapter 2, it can be seen that caching entire files at nodes and spatial reuse-only transmission scheme can achieve order-optimal throughput. Based on this fact, instead of deterministic caching placement, we consider an alternative approach that uses random independent caching entire files at the user nodes. As mentioned in Chapter 2, a coded multicasting scheme exploiting caching at the user nodes was proposed in [MAN14b]. In this scheme, the files in the library are divided in blocks (packets) and users cache carefully designed subsets of such packets. Then, for given set of user demands, the base station sends to all users (multicasting) a common codeword formed by a sequence of packets obtained as linear combinations of the original file packets. As noticed in [MAN14b], coded multicasting can handle any form of asynchronism by suitable sub-packetization. Hence, the scheme is able to create multicasting opportunities through coding, exploiting the overlap of demands while eliminating the asynchronism problem. For the case of arbitrary (adversarial) demands, the coded multicasting scheme of [MAN14b] is shown to perform within a small gap, independent of the number of users, of the cache size and of the Ii brary size, from the cut-set bound of the underlying com pound channel. 1 However, the scheme has some significant drawbacks that makes it not easy to be implemented in practice: 1) the construction of the caches is combinatorial and the sub-packetization 1 The compound nature of this model is due to the fact that the scheme handles adversarial demands. 45 explodes exponentially with the library size and number of users; 2) changing even a single file in the library requires a significant reconfiguration of the user caches, making the cache update difficult. In [MAN14a], similar near-optimal performance of coded caching is shown to be achieved also through a random caching scheme, where each user caches a random selection of bits from each file in the library. In this case, though, the combinatorial complexity of the coded caching scheme is transferred from the caching phase to the (coded) delivery phase, where the construction of the multicast codeword requires solving multiple clique cover problems with fixed clique size (known to be NP complete), for which a greedy algorithm is shown to be efficient. The analysis of the capacity scaling laws for large D2D (or "ad-hoc") wireless networks has been the subject of a vast body of literature. Gupta and Kumar [GKOO] proposed a network model where n are randomly placed on some planar region and communicate through multiple hops. They characterized the transport capacity scaling as n -+ co, under the same protocol model considered in our work (see Section 3.1). For random assignment of source-destination pairs, [GKOO] showed that the per-link capacity must vanish as 0( Jn). In addition, a multi-hop relaying scheme was shown to achieve through put scaling O(~). The same results were confirmed, using a somehow simpler and n ogn more general analysis technique, in [KV04]. The multicast capacity of large wireless networks has been investigated in [SLSlO, NGSlO]. Finally, Ftanceschetti, Dousse, Tse and Thiran [FDTT07] closed the .Jlog( n) gap between upper bound and achievability in [GKOO, KV04] by creating an almost deterministically placed grid sub-network with vertical and horizontal "highways" that relay messages with very short hops. The exis- tence of such grid subnetwork is guaranteed with high probability by percolation theory. Given the fact that randomly placed nodes yield the same scaling laws of nodes placed on a deterministic squared grid, in this work we assume a grid network from the start. This allows to focus on the essential aspect of caching at the nodes, and avoid the ana- lytical complication of randomly placed nodes. The same approach is taken in [GPT12], where multi-hop D2D communication is considered under the protocol model for a network 46 of nodes placed deterministically on a squared grid. If the aggregate distributed storage space in the network is larger than the total size of the library, multi-hop guarantees that all user requests can be served by the network. Under the same assumption made here of the user requests distribution, [ G PT12] finds a deterministic replication caching scheme and a multi-hop routing scheme that achieves order-optimal average throughput. Besides the consideration of multihop and single hop, there are several other key differ ences between our work and [GPT12]. First, [GPT12] considers a deterministic caching placement approach, which depends on the files popularity distribution. This approach is not robust when users move between cells. In contrast, mobility is easily handled by our scheme which is based on independent random caching. Next, in [GPT12], the transmission range is fixed, where each node can only reach its four neighbors. Besides the deterministic caching placement, the key aspect of the problem is the design of the routing protocol and analyze the traffic through the bottleneck link of the network. Our work focuses on determining the transmission range within which nodes can access their neighbors caches in one hop. This, in turn, determines the point of the throughput outage tradeoff at which the system operates. Finally, [ G PT12] only gives the order of the throughput as the number of users n goes to infinity, but does not characterizes the multiplicative constant of the throughput leading term in the scaling law. Therefore, it is difficult to understand in which regime of (large but finite n) the scaling laws become relevant. In passing, we notice that this is a common problem in several studies focused on scaling laws of large wireless networks. In our case, we provide outer bounds and inner (achievable) bounds to the throughput-outage tradeoff, which are tight enough to characterize also the constants of the leading terms, within a bounded gap. In particular, the analysis of our achievability scheme matches well with finite-dimensional simulations. Preliminary work of the present work is given m [GDM12], where only the sum throughput was considered irrespectively of user outage probability. The analysis in [GDM12] considers a heuristic random caching policy, while here we find the optimal 47 random caching distribution. More importantly, the total sum throughput is not a suf ficient characterization of the performance of D2D one-hop caching networks: in certain regimes of the number of users and file library size, it can be shown that in order to achieve a large sum throughput only a small portion of the users should be served, while leaving the majority of the users in outage. In contrast, the throughput-outage trade off region considered here is able to capture the notion of fairness, since it focuses on the minimum per-user average throughput and on the fraction of users which are denied service. The chapter is organized as follows. Section 3.1 introduces the network model and the precise problem formulation of the throughput-outage tradeoff in D2D one-hop caching networks. The main results on the outer bound and achievability of the throughput outage tradeoff are presented in Sections 3.2 and 3.3, respectively. In Section 3.4 we presents some concluding remarks. All proofs are relegated in the Appendices, in order to maintain the flow of exposition. 3.1 Network Model and Problem Formulation We consider a network deployed over a unit-area squared region and formed by n nodes U = {1, ... , n} placed on a regular grid with minimum node distance 1/ yn (see Fig. 2.l(a) ). Each user u E U makes a request to a file fu E F = {1, ... , m} in an i.i.d. manner, ac cording to a given request probability mass function { qf : f E F}. In order to model the asynchronous content reuse and forbid any form of "for-free" multicasting by "overhear ing", we consider the following theoretical model. We assume that each file is formed by a sequence of L 1 packets. Each user demand corresponds to a file index f E F and a segment of L 11 < L 1 consecutive packets, starting at some initial index £, uniformly and independently distributed over {1, ... , L 1 -L 11 +1 }. The packets of the requested segment are downloaded sequentially. We measure the cache size in files, and in order to compute the system performance we consider first the limit for large file size (L 1 -+ oo) with L 11 48 finite, and then study the system scaling laws for n, m -+ co. Hence, the probability that users request overlapping segments vanishes for L 1 -+ co for any finite n, m, L 11 , thus preventing the trivial use of naive multi casting (i.e., overhearing common messages). In contrast, the probability that two users request segments of the same file depends on the library size m and on the request distribution q = [q1, · , qm]. We hasten to say that this model is just a way to express in precise mathematical terms the notion of asynchronous content reuse, such that the overlap of the demands and the overlap of concurrent trans missions are decoupled. 2 Fig. 2.2 shows qualitatively our model assumptions. In our system, D2D communication obeys the following protocol model [GKOO]: a node u can receive successfully a packet from node v if d( u, v) S r and if no other node v 1 at distance d(u,v 1 ) < (1+6)r is transmitting. The transmission ranger is a design parameter that can be set as a function of m and n. We consider the following definitions: Definition 3.1 (Network) A network if formed by a set of user nodes U and a set of files F = {1, ... , m }. Nodes in U are placed in the two-dimensional unit-square region, and their transmissions obeys the protocol model. In general, all n( n - 1) directed links between user nodes, subject to the protocol model, define an interference (conflict) graph. Only the links in an independent set in the interference graph can be active simultaneously. 0 Definition 3.2 (Cache placement) The cache placement Ile is a rule to assign files from the library F to the user nodes U with "replacement" (i.e., with possible replication). Let G = {U, F, £} be a bipartite graph with "left" nodes U, "right" nodes F and edges E such that (u, f) EE indicates that file f is assigned to the cache of user u. A bi-partite cache placement graph G is feasible if the degree of each user node is not larger than 2 As a side note, we observe also that the segmentation of large files into smaller packets (or "chunks") to be sequentially downloaded is consistent with current video streaming protocols such as DASH [Sd!FSH+ 11b, Sd!FSH+ 11a, Pan12, Sto11]. 49 the maximum user cache size equal to M files. Let g denote the set of all feasible bi- partite graphs G. Then, Ile is a probability mass function over Y, i.e., a particular cache placement G E g is assigned with probability Ilc(G). 0 Notice that deterministic cache placements are special cases, corresponding to a single probability mass equal to 1 on the desired assignment G. In contrast, we will be interested in "decentralized" random caching placements constructed as follows: each user node u E U selects its cache content in an i.i.d. manner, by independently generating at random M indices fu,1, ... , fu,M according to the same caching probability mass function {Pc(!): f E F}. Definition 3.3 (Random requests) At each request time (integer multiples of L 11 ), each user u EU makes a request to a segment of length L 11 of chunks from file fu E F, se- lected independently with probability q. The vector of current requests f is a random vector taking on values in r, with product joint probability mass function IF(f = (Ji, ... , fn)) = 0 In this work, we assume that q is a Zipf distribution with parameter 0 < I < 1 [BCF+99], i.e., qf = H(~~l~m) for f = 1, ... , m, and H(r, x, Y) "' LT~x A. Definition 3.4 (Transmission policy) The transmission policy IT, is a rule to activate the D2D links in the network. Let£ denote the set of all directed links. Let A c;; 2L denote the set of all feasible subsets of links {this is a subset of the power set of£, formed by all independent sets in the network interference graph). Let A CA denote a feasible set of simultaneously active links according to the protocol model. Then, IT, is a conditional probability mass function over A given f (requests) and G (cache placement), assigning probability IT,(Alf, G) to A EA. 0 We may think of IT, as a way of scheduling simultaneously compatible sets of links (subject to the protocol model). Modeling the scheduling policy in a probabilistic manner allows the analytical convenience of defining the average per-user throughput (see below) 50 as an ensemble average. As a matter of fact, deterministic link activation rules can be included by defining the average throughput as a time-average. For example, a bounded deterministic delay per user can be guaranteed by activating groups of compatible links (forming a maximal independent set in the network interference graph) in a deterministic round-robin sequence, such that each user is served with a deterministic delay. Definition 3.5 (Useful received bits per slot) For given q, Ile and IT,, and user u EU, we define the random variable Tu as the number of useful received information bits per slot unit time by user u at a given scheduling time (irrelevant because of stationarity). This is given by Tu= L Cu,vl{fuEG(v)} (3.1) v (u,v)EA where fu denotes the file requested by user node u, Cu v denotes the rate of the link ( u, v), and G(v) denotes the content of the cache of node v, i.e., the neighborhood of node v in the cache placement graph G. 0 Consistently with the protocol model, Cu,v depends only on the active link ( u, v) E A and not on the whole set of active links A. Furthermore, we shall obtain our results under the simplifying assumption (usually made under the protocol model [GKOO]) that Cu,v = C for all ( u, v) E A. The indicator function 1 {fu E G( v)} expresses the fact that only the bits relative to the file fu requested by user u are "useful" and count towards the throughput. Obviously, scheduling links (u,v) for which fu ¢ G(v) is useless for the sake of the throughput defined above. Hence, we can restrict our transmission policies to those activating only links (u,v) for which fu E G(v). These links are referred to as "potential links", i.e., links potentially carrying useful data. Potential links included in A are "active links", at the given scheduling slot. The average throughput for user u E U is given by Tu = JE[Tu], where expectation is with respect to the random triple (f, G, A) ~ IT?~i q(fi)Ilc(G)IT,(Alf, G). We say that user u is in outage iflE[Tulf, G] = 0. This condition captures the event that no link (u,v) with fu E G ( v) is scheduled with positive probability, for given requests vector f and cache 51 placement G. In other words, a user u for which IE[Tulf, G] = 0 experiences a "long" lack of service (zero rate), as far as the cache placement is G and the request vector is f. Definition 3.6 (Number of nodes in outage) The number of nodes in outage is given by No= L l{JE[Tulf, G] = O}. (3.2) uEU Notice that N 0 is a random variable, function off and G. 0 Definition 3.7 (Average outage probability) The average (across the users) outage probability is given by 1 1 Po= -JE[No] = - L IF (JE[Tulf, G] = 0). n n uEU (3.3) 0 In this work we focus on max-min fairness, i.e., we express the throughput-outage tradeoff in terms of the minimum average user throughput, defined as (3.4) Notice that that the max-min fairness criterion in our setting is essential to make the outage probability p 0 defined in (3.3) meaningful. In fact, for 0 Sp~ < p 0 S 1, consider a system that achieves outage probability p 0 by serving only a fraction 1 - .\ of users with outage probability p~ = P{~:, while leaving the remaining fraction .\ of users permanently idle. In this case, we have T min = 0 since there are .\n > 0 users with identically zero throughput. Hence, a system that permanently excludes some users in favor of others is certainly not optimal in terms of the throughput-outage tradeoff as defined below: Definition 3.8 (Throughput-Outage Tradeoff) For a given network and request probability distribution q, an throughput-outage pair (T, p) is achievable if there exists 52 a cache placement Ile and a transmission policy IT, with outage probability p 0 S p and minimum per-user average throughput T min 2 T. The throughput-outage achievable re- gion T is the closure of all achievable throughput-outage pairs (T, p). In particular, we let T*(p) = sup{T: (T,p) ET}. 0 Notice that T*(p) is the result of the following optimization problem: max1m1ze T min subject to Po s; p, (3.5) where the maximization is with respect to the cache placement and transmission policies Ile, IT,. Hence, it is immediate to see that T* (p) is non-decreasing in p. The range of feasible outage probability, in general, is an interval lPo,min, 1] for some Po,min 2 0. We say that an achievable point (T,p) dominates an achievable point (T 1 ,p 1 ) ifp S p 1 and T ;:> T 1 • The Pareto boundary of T consists of all achievable points that are not dominated by other achievable points, i.e., it is given by {(T*(p),p): p E lPo,min, 1]}. It is also immediate to see that the throughput-outage tradeoff region is convex. In fact, consider two achievable points (T;;]n, p~ 1 )) and (T;,;/n, p~ 2 )) corresponding to caching placements G 1 and G 2 , with probability assignments IT~ 1 ) and IT~ 2 ). For .\ E [O, 1], the caching placement G with mixture probability assignment Ile = .\Il~l) + (1 - .\)IT~ 2 ) achieves p 0 = .\p~l) + (1 - .\)p~ 2 ). For this value of outage probability, the best possible strategy achieves where, by definition, T;;]n = minu T~ 1 ) and T;,;/n = minu T~ 2 ). Hence, the segment joining any two achievable throughput-outage points (T;;]n, p~ 1 )) and (T;,;/n, p~ 2 )) is contained into the achievable throughput-outage region. 53 We conclude this section by providing the intuition behind the tension between out age and throughput, and explaining through an intuitive argument why T* (p) is non decreasing for p E IPo,min, 1]. The key tradeoff quantity here is the cooperation cluster size g, that is, the size of the set of nearest neighbor nodes among which any node u can look for its desired file fu· On one hand, we would like to have g large, in order to take advantage of the content reuse, i.e., the larger g, the larger the probability that any user can find and retrieve its desired file. On the other hand, we would like to have g small, in order to take advantage of the spatial reuse, i.e., the smaller g, the larger the number of simultaneously active links that the network can support. Therefore, g describes the tradeoff between content reuse and spatial reuse. As g increases the probability that user u does not find its desired file decreases. Hence, p 0 is a decreasing function of g (see Fig.3.l(a)). Since nodes can retrieve their desired files within a cluster of size g, then the com munication range of the D2D links must be enough to communicate across such clusters. The average number of active links that can be activated without violating the protocol model is of the order of the number of disjoint clusters in the network, i.e., ~· With a probability 1 - p 0 that any user cannot find its requested file, the average per-user throughput is roughly given by T e<: 1 -;;~ x ~ = l-:~. Since for small g we have p 0 t 1, and for large g we have p 0 + Po,min, where the latter is the probability that a node u does not find its requested file in the whole network, we clearly see that T must be increasing for small g and decreasing for large g (see Fig.3.l(b)). Now, consider the constraint on the outage probability p 0 S p. If p is small, then the constraint must be satisfied with equality, yielding the corresponding value of g (Fig.3.1( a)) which in turns yields a corresponding value of T (Fig.3. l(b)). Asp increases, we obtain the concave increasing part of the throughput vs outage Pareto boundary curve qualitatively depicted in Fig. 3.l(c). However, when p becomes larger than some thresh old, the optimal throughput is obtained by letting g = g*, which is the size that achieves the maximum unconstrained throughput (see Fig. 3.l(b)). This means that for values of 54 outage prob. throughput g (a) (b) throughput n Po,min outoge prob. ( c) Figure 3.1: Qualitative behavior of the tradeoff between throughput and outage prob ability, by ways of the tradeoff parameter g, which represents the size of the cluster of nodes over which any node can look for its desired requested file and download it by D2D on-hop communication. p beyond this threshold value, the throughput curve reaches a bound (horizontal line), equal to the unconstrained ma.-ximum throughput, as shown in Fig. 3.l(c). It follows from the upper bounds developed in Section 3.2 that this intuitive argument, even though it is developed for a cluster-based achievability strategy, holds true also for the upper bounds, despite the fact that the latter do not assume any a priori transmission strategy other than the one-hop constraint and the protocol model. 3.2 Outer Bounds Under the one-hop restriction, network topology and protocol model given in Section 3.1, we can provide an outer bound Tub(p) on the throughput-outage tradeoff, such that the ensemble of points {(Tub(p),p) : p E [O, 1]} dominates the tradeoff region T. In what follows, the quantity (3.6) 55 plays an important role. Notice that 0 S a < 1/2 under the assumption made here that 0 < I< 1. The following results are proved in Appendices B.1 and B.2: Theorem 3.1 When limn-+oo "';," = 0, the throughput-outage region is dominated by the set of points (Tu 1 (p),p) given by: Tub(p) = 16CM 1 + 0 ( 1 1 ) ' p = l - (Mg;,,(m))l-1' £,2m(l-p) T=? m(l-p) T=? min { l 5 CM_ 1 _, fub(p 1 )m-"} + o (m-"), 1- (Mp1) 1 -1'm-a Sp< 1- (Mp*) 1 -1'm-a £,2m(l-p) 1-7 fub(p*)m-a + o (m-"), 1 - (Mp*) 1 -1'm-a Sp S 1, (3.7) where p 1 2 p* and p* is the solution (with respect top) of the equation C,(p) = log(l + (2 -1)((p)) (3.8) with 36 2-7 ( 2 ) 2-'( C,(p)"' ( 1 + 2) p M1-1', (3.9) Yr(m) is any function such that Yr(m) = w (m") and Yr(m) S min{M,n}, and where f ( ) "' 16C (l _ -((p)) ub p 62p e . (3.10) D 56 Theorem 3.2 When there exists a positive constant~ such that~ S limn-+oo "';," S 6 \~,, the throughput-outage region is dominated by the set of points (Tu 1 (p),p) given by: Tub(p) = 1 min { l 5 CM 1 , fu1(p 1 )m-"} + o(m-"), 6 2 m(l-p) T=? fu1(p*)m-a + o(m-"), 1- (Mp1) 1 -'Ym-a Sp< 1- (Mp*) 1 -'Ym-a 1 - (Mp*) 1 -'Ym-a Sp S 1, (3.11) where p* is the solution of {2.6), and p1 E [p*, l~ ,:al· D Theorem 3.3 When limn-+oo "';," > 6 \~, (p* being the solution of {2.6)), the throughput outage region is dominated by the set of points (Tu 1 (p),p) given by (3.12) D Notice that the range of pin Theorems 3.2 and 3.3 is limited to IPo,min, 1] with Po,min = 1- (Mp1) 1 -'Ym-a (for Theorem 3.2) and Po,min = 1- ( 11nn) l-'f (for Theorem 3.3), showing that in these regimes the outage probability cannot be small. As a matter of fact, of all the regimes identified by Theorems 3. 1, 3.2 and 3.3, the only practically interesting one is the first regime of Theorem 3.1. In particular, Theorem 3.2 and 3.3 show that, when limn-+oo "';," is bounded away from zero, any scheme for the one-hop D2D caching network yields outage probability that goes to 1, which is clearly not an acceptable. In contrast, limn-+oo ';; = r; < co, there might exist schemes achieving some fixed target outage probability value p E [O, 1), as n,m-+ co. Intuitively, the function Yr(m) plays the role of the size of the cooperation cluster of neighboring nodes within which each user can find its requested file. For exam pie, choosing Yr ( m) = (3m for some constant f3 S min { k, ~}, both conditions Yr(m) = w(m") and Yr(m) S min {Ji'.}, n} are satisfied, for all sufficiently large n. Notice that for r; SM, the choice f3 = 1/M yields that the outer bound contains 57 points of the type ( O(M /m), 0) (zero outage probability). We shall see in the next section that throughput-outage points with throughput O(M /m) and fixed p bounded away from 1 are achievable. For a conventional unicast system where users are served by a single omniscient node (e.g., a base station) that can store the whole library, the throughput scaling is 0(1/n). 3 Hence, in the case of nM » m, the combination of caching and D2D spatial reuse yields a very large throughput relative gain with respect to a conventional system. It is also interesting to notice that despite the first regime of Theorem 3.1 requires that m grows more slowly than n 1 1", the only practically interesting sub-regime ism= o(n), otherwise conventional unicast from the base station yields better throughput scaling and zero outage probability (all users are served). All other regimes in Theorems 3.1 - 3.3 are included for completeness, in order to prove mathematically a somehow intuitively expected "negative" result: unless the Ii- brary size m is small with respect to the aggregate caching memory nM, caching cannot achieve significant throughput gains with respect to conventional unicast from a single base station. This result is expected since, in this case, the asynchronous content reuse that the D2D caching network tries to exploit is essentially non-existent. It is also important to notice that here we are considering the (realistic) case of a "heavy tail" Zipf request distribution with I E (0, 1). If I > 1, then a finite number of files collects essentially all the request probability mass, and this case is similar to the case of m = 0(1), which is a special case of m = o(n). As a matter of fact, Zipf-distributed requests with IE (0, 1) have been observed experimentally [ZSGK08,ZSGK09,BCF+Qg]. In the next section, we show that the upper bounds obtained here are tight in the seal- ing laws, and that the constants of the leading terms can be determined within constant gaps. This is obtained by exhibiting and analyzing a specific achievability strategy. 3 This is obviously achieved by TDMA, serving users on different time slots in a round-robin fashion. Notice that even if a more refined physical layer including a Gaussian broadcast channel is considered, the throughput scaling remains the same. 58 3.3 Achievable Throughput-Outage Tradeoff Consistently with the outer bounds in Theorems 3.1 - 3.3 and the concluding remarks in Section 3.2, we consider achievability only in the "small library" regime limn-+oo "';," = 0, for which there is hope to achieve some target outage probability strictly less than 1. We obtain an achievable inner bound on the achievable throughput-outage tradeoff region by considering a transmission policy based on clustering and a caching policy based on independent random caching. Clustering: the network is divided into clusters of equal size, denoted by gc(m), and independent of the users' demands and cache placement realizations. A user can only look for the requested file inside its own cluster. For each user whose demand is found inside its cluster, we say that a potential link exists in the cluster. If a cluster contains at least one potential link, we say that this cluster is good. We use an interference avoidance transmission policy rr; for which at most one concurrent transmission is allowed in each cluster, over any time-frequency slot (transmission resource). Furthermore, potential links inside the same cluster are scheduled with equal probability (or, equivalently, in round robin), such that all users have the same throughput Tu = T min· To avoid interference between clusters, we use a conventional TDMA with spatial reuse scheme [Molll, Ch. 17], very similar to the spatial reuse scheme of a cellular network, where each cluster acts as a cell. In short, a "coloring" scheme with K colors is applied to the clusters such that clusters with the same color can be concurrently active on the same time slot, without violating the protocol model. The resulting groups of clusters are assigned to K orthogonal time slots, and are activated in a round-robin fashion. In particular, we use K = ([ V2(1+6)1+1) 2 in order to guarantee that concurrently active clusters do not interference with each other. Fig. 2.l(b) shows an example for K = 9. Random Caching: we consider a caching policy IT~ where each node independently caches M files according to a common probability distribution P;, given by the following result (proved in Appendix B.3): 59 Theorem 3.4 Under the system model assumptions and the clustering scheme described above, the caching distribution P; that maximizes the probability that any user u E U finds its requested file inside its corresponding cluster is given by (3.13) where v = I:m~,-~, Zj = q(j) M(gc(,,;l- 1 1- 1 , and m* = 8 (min{~ Yc(m ), m}). J=i Zj D The following theorem (proved in Appendices B.4) yields an inner bound on the achievable outage-throughput tradeoff region: Theorem 3.5 Assume limn-+oo "';,g = 0. Then, the throughput-outage tradeoff achievable by random caching and clustering behaves as: T(p) = 'k p1;1m + o(l/m), p = (1-1)e'f-p1 ~ M 1 + o ( 1 1 ) , p = 1 - a ( g,~m)) l-'( m(l-p) T=? m(l-p) T=? c;m-"+o(m-"), 0 j!m-"+o(m-"), 1- ap~-'!m-a Sp S 1- ab 1 -'lm-a 1 - ab 1 -'lm-a Sp S 1, (3.14) 1 1 where we define a=/'! M -'( b = __::r A= 1 1 -7 B = 2 D = a and 1 (1- ) 2-1 f}. _i_ f}. ap -i f}. b1-1 ' a ' ' 1 +ap~ I ' 1 +ab2 I where p1 and p2 are positive parameters satisfying p1 2 / and p2 2 b. The cluster size Yc(m) is any function of m satisfying Yc(m) = w (m") and Yc(m) S 1m/M. D In all cases, the achievable throughput scaling law both for p bounded away from 1 and p -+ 1 coincide with the outer bounds of Theorem 3.1. Therefore, these throughput scaling laws are tight up to some gap in the constants of the leading terms. In the rest of this section we compare the achievable throughput scaling law of The- orem 3.5 with the outer bounds of Section 3.2 and with the performance achievable by other schemes. In particular, we focus on the interesting regime of small library (Theo rems 3.1 and 3.5). Since a< 1/2, even in this regime the library size m can grow faster 60 than n 2 . However, we restrict to the practically relevant regime of m = O(n) (linear or sub-linear in n ). Choosing 9c(m) = (3m for some f3 > 0, it is apparent from the first and second line of (3.14) that p strictly bounded away from 1. By fixing a small but positive target outage probability, the per-user average throughput of the D2D one-hop caching network with random (decentralized) caching scales as T*(p) = 8 (max{~,;';;}), where the scaling 8 ( ~) can be trivially achieved by letting the whole network to be a single cluster (e.g., transmission radius r = V2) and serving one demand per unit time. This scaling is equivalent to conventional unicast from a single omniscient node which can be regarded as the state of the art of today's (single cell) systems, with a base station or ac cess point serving individual requests without exploiting the asynchronous content reuse. We notice that, when nM » m, the throughput of the D2D caching network achieves per-user throughput that increases linearly with M. In this regime, caching in the user nodes and exploiting the dense spatial reuse of the D2D network is a very attractive approach, since storage space is much "cheaper" than scarce resources such as bandwidth or dense base station deployment (the reader will forgive this vague statement in this context). It is interesting to notice that our analysis is able to characterize also the constant of the leading term within a bounded gap. This is a fortunate fact that does not happen often for the scaling analysis of wireless network capacity (e.g., see [GKOO,KV04,SLS10,NGS10, FDTT07]). For example, upper and lower bounds (Theorem 3.1 and 3.5, respectively) and finite-dimensional simulation results are compared in Fig. 3.2, which shows both theoretical (solid lines) and simulated (dashed lines) curves of the throughput (y-axis) vs. outage (x-axis) tradeoff for different values of/. In this simulation, the throughput is normalized by C, so that it is independent of the link rate. In particular, the theoretical curves show the dominant term in (3.14) divided by C. In these examples we used m = 1000, n = 10000, M = 1 and the spatial reuse factor K = 4. The Zipf parameter I varies from 0.1 to 0.6, corresponding to the curves from the left (blue) to the right (cyan). 61 x 1 0- 3 2.5 2 8 :0 ~ ~ 1.5 I >-- ~ (ti 1 E ~ z 0.5 0 0.3 0.4 0 .5 0.6 0.7 0.8 0 .9 p Figure 3.2: Comparison between the normalized theoretical result (solid lines) and normalized simulated result (dashed lines) in terms of the minimum throughput per user vs. outage probability, where m = 1000, n = 10000, M = 1 and spatial reuse factor K = 4. The Zipf parameter r varies from 0.1 to 0.6 (from the left (blue) to the right (cyan)). 3.4 Summary In this work we have considered a wireless device-to-device (D2D) network where the nodes have pre-cached information from a fixed library of possible files, users request files at random and, if the requested file is not in the on-board cache, then it is downloaded from some neighboring node via one-hop "local" communication. To model the wireless network, we have followed the simple protocol model, widely considered in the analysis of the transport capacity scaling laws of wireless ad-hoc networks. We have proposed a model that captures mathematically the asynchronous content reuse typical of on-demand video streaming, where the users' requests have strong over- lap and concentrate on a small set of popular movies, but the demands are completely asynchronous, such that "naive multicasting" is not effective. 62 In our model, a user is in outage when its requested file is not found within the allowed transmission range. We have defined the optimal tradeoff between minimum per user average throughput and the average fraction of users in outage, that we refer to as outage probability. Then, we have characterized such optimal tradeoff in terms of tight scaling laws in all the scaling regimes of the system parameters, when both the number of nodes and the number of files in the library grow to infinity. The main result of this work is that, in the relevant regime "small library", i.e., when m = O(n) and the aggregate cache capacity of the network, nM is much larger that the library size m, the throughput of the D2D one-hop caching network is proportional to M /m and independent of n. Hence, D2D one-hop caching networks are very attractive to handle situations where a relatively small library of popular files (e.g., the 500 most popular movies and TV shows of the week) is requested by a large number of users (e.g., 10,000 users per km 2 in a typical urban environment). In this regime, the proposed system is able to efficiently turn memory into bandwidth, in the sense that the per-user throughput increases proportionally to the cache capacity M of the user devices. Since the latter follows the doubling rate of Moore's law, caching in the user devices can achieve orders of magnitude throughput gains without requiring more bandwidth. 63 Chapter 4 Wireless Device-to-Device Caching Networks: Basic Principles and System Performance Over the years, a number of other suggestions have been made to make better use of constrained request patterns: [LSS13,JK10,AK11,BFCP11] considers the case that users want the same video at the same time (e.g., in a live streaming service) but with a different channel quality or requested video quality. In this case, scalable video coding can be coupled with some form of broadcast channel coding [CT06]. Specifically, scalable rateless codes [Sho06,LWG+06, OFTLlO] to support heterogeneous users in a broadcast channel scenario are considered in [LSS13, AK 11, BFCPll]. Another set of recent works considers the case where neighboring wireless users want the same video at the same time, and collaborate in order to improve their aggregate downlink throughput. In particular, [KLC+12] suggests that different users download simultaneously different parts of the same video file from the serving base station and then share them by using device-to device (D2D) communications. The above approaches are suited for synchronous streaming of live events (e.g., live sport events) but yield no gain in the presence of asynchronous content reuse, charac teristic of on-demand video streaming. On the other hand, treating each user request as independent data yields a fundamental bottleneck: in conventional unicasting from a single serving base station, the per-user throughput decreases linearly with the number of users in the system. In [JT97, PCL98a, ES06, PCL98b], a coding scheme referred to 64 as harmonic broadcasting is introduced. This scheme can handle asynchronous users re questing the same video at different times, such that each user can start playback within a small delay from its request time. With harmonic broadcasting, a video encoded at rate R requires a total downlink throughput of R 1og(L 1 /T ), where L 1 is the total length of the video file and T is the maximum playback delay. For T « L 1 (as it is required in on demand video streaming), the bandwidth expansion incurred by harmonic broadcasting can be very significant. Recent work presented in this thesis, as well as by other research groups, has shown that one of the most promising approaches relies on caching, i.e., storing the video files in the users' local caches and/or in dedicated helper nodes distributed in the network coverage area. From the results of [SGD+13,JCM13a, GPT13b, GDM14], we observe that caching can give significant (order) gains in terms of throughput. Intuitively, caching provides a way to exploit the inherent content reuse of on-demand video streaming while coping with asynchronism of the requests. Also, caching is appealing since it leverages the wireless devices storage capacity, which is probably the cheapest and most rapidly growing network resource that, so far, has been left almost untapped. One possible approach consists of "Femtocaching", i.e., of deploying a large number of dedicated "helper nodes", which cache popular video files and serve the users' requests through local short-range links. Essentially, such helper nodes are small base stations that use caching in order to replace the backhaul, and thus obviate the need for the most expensive part of a small cell infrastructure [SGD+13]. Another recently suggested method combines caching of files on the user devices with a common multicast transmis sion of network-coded data [MAN14b]. We refer to this approach as coded multicasting. The third approach, which is at the center of this work, combines caching of files on the user devices with short-range device-to-device (D2D) communications [JCM13a]. In this way, the caches of multiple devices form a common virtual cache that can store a large number of video files, even if the cache on each separate device is not necessarily very large. Both coded multicasting and D2D caching have a common interesting feature: the 65 common virtual cache capacity grows linearly with the number of users in the system. This means that, as the number of users in the network grows, also their aggregate cache capacity grows accordingly. We shall see that, qualitatively, this is the key property that allows for significant gains with respect to the other methods reviewed here, where the content is only stored in the network infrastructure. The purpose of this work is two-fold. On one hand, we provide a tutorial overview of the schemes and recent results on wireless on-demand video streaming summarized above, in terms of their throughput vs. outage probability tradeoff, in the regime where both the number of users in the system and the size of the library of video files grow large. On the other hand, looking at throughput-outage tradeoff scaling laws for idealized network models does not tell the whole story about the relative ranking of the various schemes. Hence, in this work we present a detailed and realistic model of a single cell system with n users, each of which has a cache memory of M files, and place independent streaming requests to a library of m files. Requests can be served by the cellular base station, and/or by D2D links. We make realistic assumptions on the channel models for the cellular links and the D2D links, assuming that the former uses a 4th generation cellular standard [STB09] and the latter use either microwave or mm-wave communications depending on availability [Aww+13,DMRH10]. The chapter is organized as follows. Section 4.1 presents a literature review of the recent results on wireless caching networks, where the system model and the main theoret ical results are summarized. Then the system design approach is presented in Section 4.2 and the simulation results are given in Section 4.3. Conclusions are pointed out in Sec tion 4.4. 4.1 Literature Review In this section we review the most important recent results on the throughput of wireless caching networks. The emphasis lies on results that use caching in combination with D2D 66 communications, though we also review results for caching combined with Base Station (BS)-only transmission, as well as pure D2D communication (without caching). 4.1.1 Conventional Scaling Laws Results of Ad Hoc Networks and D2D communications with Caching The capacity of conventional ad hoc networks, where source-destination pairs are drawn at random with uniform probability over the network nodes, has been studied extensively. Under the protocol model (see Chapter 3.l)and a decode-and-forward (i.e., packet for warding) relaying strategy, the throughput per user of such networks scales as 8( ),,), where n denotes the number of nodes (users) in the network. While the conclusions for realistic physical models including propagation pathloss and interference are more variegate [GKOO, XK06, KV04, FDTT07, OLT07, FMM09], we can conclude that practi cal relaying schemes are limited by the same per-user throughput scaling bottleneck of 8( }nl which holds for the protocol model. Notice that this result assumes that the traffic generated by the network is 8(n), i.e., constant requested throughput per user. This does not take into account the intrinsic content reuse of video on-demand streaming. In other words, when treating each session as independent data, the per-user throughput vanishes as the total demanded throughput increases. Fortunately, the video-aware networks, i.e., networks designed to support video on demand, can behave in a much better way. For this purpose, it is useful to consider another measure of network performance called transport capacity [XK06], which is the sum over each link of the product of the throughput per link times the distance between source and destination. It is known that the transport capacity of ad-hoc dense networks (i.e., networks of fixed area 0(1) with node density that scales as 8(n)), under the protocol model, or under a physical model with decode and forward relaying, scales as 8(yn). For random source-destination pairs, at distance 0(1), the throughput per link scales again as 8( }nl as mentioned before. On the other hand, if we can reduce the distance between the source (requested file) to the destination (requesting user) to the 67 minimum distance between nodes (8( }nl ), which corresponds to one hop, then a constant throughput per user can be achieved. The reason is that many short distance links can co-exist by sharing the same spectrum, which can be used more and more densely as the density of the network grows. In another word, by caching the files into the network such that request can be satisfied by short-range links, the spectrum spatial reuse of the network increases linearly with the number of users. Based on this observation, it is meaningful to consider a system design based on one-hop D2D transmission and caching of the video files into the user devices. In this section, as a reminder, we first recall the formal network model and the detailed problem definition for the uncoded D2D caching networks. We consider a network formed by user nodes U = {1, ... , n} placed on a regular grid on the unit square, with minimum distance 1/y'n (see Fig. 2.l(a)). 1 Each user u EU makes a request for a file f E F = {1, ... , m} in an i.i.d. manner, according to a given request probability mass function q1(f). Communication between user nodes obeys the protocol model [GKOO] : 2 namely, communication between nodes u and v is possible if their distanced( u, v) is not larger than some fixed ranger, and if there is no other active transmitter within distance (1 +6)r from destination v, where 6 > 0 is an interference control parameter. Successful transmissions take place at rate Cr bit/s/Hz, which is a non-increasing function of the transmission ranger [SGD+13]. In this model we do not consider power control (which would allow different transmit powers, and thus transmission ranges). Rather, we treat r as a design parameter that can be set as a function of m and n. 3 All communications are single-hop. We assume that the request probability mass function qf is the same for all users and follows a Zipf distribution with parameter 0 < I< 1 [BCF+99], i.e., qf = I:f," tr. These 1 for some of the later simulations, we will also consider the case that nodes are uniformly and randomly distributed in a square region. 2 In the simulations of Section 4.3, we relax the protocol model constraint and take interference into consideration by treating it like noise. 3 Since the number of possibly requested files m typically varies with the number of users in the system n, and r can vary with n, r can also be a function of m. 68 model assumptions allow for a sharp analytical characterization of the throughput scaling law. We consider a simple "decentralized" random caching strategy, where each user caches M files selected independently from the library F with probability Pc(!). On the practical side, video streaming is obtained by sequentially sending "chunks" of video, each of which corresponds to a fixed duration. The transmission scheduling slot duration, i.e. the duration of the physical layer slots, is generally two to three orders of magnitude shorter than the chunk playback duration (e.g., 2 ms versus 0.5 s [WTS+lO]). Invoking a time scale decomposition, and provided that enough buffering is used at the receiving end, we can always match the average throughput per user (expressed in information bit/s) with the average source coding rate at which the video file can be streamed to a given user. Hence, while the chunk delivery time is fixed, the "quality" at which the video is streamed and reproduced at the user end depends on the user average throughput. Therefore, in this scenario, we are concerned with the ergodic (i.e., long-term average) throughput per user. Referring to Fig. 2.l(b ), the network is divided into clusters of equal size, denoted by gc(m) (number of nodes in each cluster) and independent of the users' requests and cache placement realization. A user can look for the requested file only inside its own cluster. If a user can find the requested file inside the cluster, we say there is one potential link in this cluster. We use an interference avoidance scheme for which at most one transmission is allowed in each cluster on any time-frequency slot (transmission resource). A system admission control scheme decides whether to serve potential links or ignore them. The served potential links in the same cluster are scheduled with equal probability (or, equivalently, in round robin), such that all admitted user requests have the same average throughput IE[Tu] = T min, for all users u, where expectation is with respect to the random user requests, random caching, and the link scheduling policy (which may be randomized or deterministic, as a special case). To avoid interference between clusters, we use a time-frequency reuse scheme [Molll, Ch. 17] with parameter K as 69 shown in Fig. 2.l(b). In particular, we can pick K = ([ V2(1+6)1+1) 2 , where 6 is the interference parameter in the protocol model. Qualitatively (for formal definition see [JCM13a]), we say that a user is in outage if the user cannot be served by the D2D network. This can be caused by: (i) the file requested by the user is not in the user's own cluster, (ii) that the system admission control decides to ignore the request. We define the outage probability p 0 as the average fraction of users in outage. The definition of the throughput-outage trade-off is given by Definition 3.8. Notice that T*(p) is the result of the optimization problem: max1m1ze T min subject to Po s; p, (4.1) where the maximization is with respect to the cache placement and transmission policies. Hence, it is immediate to see that T* (p) is non-decreasing in p, since for given outage probability constraint p 1 , p 2 , the policies satisfying p 2 > p 1 are a superset of the policies satisfying Pl. The range of feasible outage probability, in general, is an interval IPo,min, 1] for some Po,min 2 0. Whether Po,min = 0 or strictly positive depends on the model assumptions. 4.1.2 Key results for D2D networks with caching The following results are proved in [J CM13a] and yield scaling laws of the optimal throughput-outage tradeoff under the clustering transmission scheme defined above. First, we characterize the optimal random caching distribution Pc as shown in Theorem 3.4. From (3.13), we observe a behavior similar to the water-filling algorithm for the power allocation in point-to-point communication [Molll]: if ZJ > v, file f is cached with positive probability (1 - ~ ). Otherwise, file f is not cached. 70 Although the results of [J CM13a] are more general, here we focus on the most relevant regime of the scaling of the file library size with the number of users, referred to as "small library size" in [JCM13a]. Namely, we assume that limn-+oo ~" = 0, where a = ~::} Since I E (0, 1), we have a < 1/2. This means that the library size m can grow even faster than quadratically with the number of users n. In practice, however, the most interesting case is where m is sublinear with respect to n. An example of such sublinear scaling is provided by the following simple model: suppose that user 1 has a set m 0 of highly demanded files, user 2 highly demanded files overlap over mo/2 files with the set of user 1, and consists of mo/2 new files, user 3 requests overlap for 2mo/3 over the union of user 1 and user 2, and contributes with mo/3 new files and so on, such that the union of all highly demanded files of the users is m = mo :>::?~ 1 1/i "' mo log n. Remarkably, any scaling of m versus n slower than n 1 /" is captured by Theorem 3.5. The dominant term in (3.14) can accurately capture the system performance even in the finite-dimensional case, as shown through simulations in [JCM13a]. Notice that the first two regimes of (3.14) are the most relevant ones in practice, providing the throughput for small outage probability. The reason for the different behaviors in these two regimes is that the first regime is achieved by a large cluster size 9c(m), yielding m* = m in the optimal caching distribution. In this case, all files are stored in the common virtual cache with positive probability. In the second regime, m* < m if 9c(m) < 1m/M. The third and fourth regimes in (3.14) correspond to the large outage probability regimes, where the outage probability asymptotically goes to 1 as m -+ co. These regimes are not interesting in practice, and are included here for completeness. In [JCM13a], we show that the throughput-outage scaling laws of Theorem 2 are indeed tight, in the sense that an upper bound on the throughput-outage tradeoff that holds for any one-hop scheme under the protocol model yields the in the same order of the dominant terms with (slightly) different constants. 71 4.1.3 Coded Multicasting From the Base Station In this section, we review the recent work on coded multicast by the base station pro posed in [MAN14b]. This scheme is based on a deterministic cache placement with sub-packetization, where each user cache contains a fraction M /m of packets from each of the files in the library. The scheme is designed to handle arbitrary requests. There fore, its outage probability (under the ideal protocol model where only the base station transmits and all nodes can receive the same rate with zero packet error rate) is zero. Here we start with some simple example. We consider the case of n = 2 users requesting files from a library of m = 3 files denoted by A, B and C. Suppose that the cache size of each user is M = ~ file. Each file is divided into three packets, Ai, A 2 , Bi, B 2 and Ci, C 2 , each of size~ of a file. Each user u caches the packets with index containing u. For example, user 1 caches Ai, Bi, Ci. Suppose that user 1 requests A, user 2 requests B. Then, the base station will send the packets { A2 Ell Bi}, where "Ell" denotes a modulo 2 sum over the binary field), of size ~ files, such that all requests are satisfied. Clearly, the scheme can support (with the same downlink rate) any arbitrary request. For example, suppose that user 1 wants Band user 2 wants C, then the base station will send { B 2 Ell Ci}, which again results in transmitting ~ files. The scheme is referred to as "coded multicasting" since the base station multicasts a common message to all the users, formed by linear combinations of the packets of the requested files. The term "coded" refer to the fact that sending linear combinations of the messages is a instance of linear network coding [LYC03,HMK+06]. By extending this idea to general n, m and M, and letting Nrx(n, m, M) denote the number of equivalent file transmissions from the base station, [MAN14b] proves the following result: 72 Theorem 4.1 For any m, n, M and arbitrary requests, for ~n E z+ and M < m, NTx(n, m, M) = n (1 - Mm) -- 1 ~ 1 +Mn' m (4.2) is achievable. For ~n ¢ z+, the convex lower envelope of the points with coordinates ( n, m, M, NTx ( n, m, M)) for integer ~n is achievable. D Through a compound channel (over the requests) and cut-set argument, [24] proves that the best possible caching and delivery scheme transmitting form the base station requires a number of transmissions not smaller than 1/12 of (4.2). This means that, within a bounded multiplicative gap not larger than 12, the coded multicasting scheme of [24] is information theoretically optimal, under the arbitrary request and zero outage probability constraint. Letting Cro denote the rate at which the base station (BS) can reach any point of the unit-square cell, the corresponding order-optimal per-user throughput achieved by the coded multicast scheme is: * Cr 0 TBS,coded = N ( M) TX n,m, (4.3) Obviously, since the scheme is designed to handle any user request, the outage probability of this scheme is Po = 0. 4.1.4 Harmonic and Conventional Broadcasting In brief, harmonic broadcasting works as follows: fix the maximum waiting delay of T "chunks" (from the time a streaming session is initiated to the time playback is started), and let L 1 denote the total length of the video file, expressed in chunks. In harmonic broadcasting, the video file is split into successive blocks such that for i = 1, '''' r L 1 /Tl. there are i blocks of length T /i (see Fig. 4.1). Then, each i-th set of blocks of length T /i is repeated periodically on a (logical) subchannel of throughput R/i, where R is the 73 transmission rate (in bit/s) of the video playback (see again Fig. 4.1). Users receive these channels in parallel. 81 81 81 81 81 81 81 81 81 Figure 4.1: A video file encoded at rate R is split into blocks Sij : j = 1, ... , i, for i = 1, ... , 4, such that the size of Sij is T /i chunks. Each i-th set of blocks is periodically transmitted in a downlink parallel channel of rate R/i. Any user tuning into the multicast transmission can start its playback after at most T chunks. In this way, each file requires a downlink rate of Rlog(L 1 /T). Hence, the total number of files that can be sent in the common downlink stream is m 1 = min { Rlo~rL' /T) , m}, yielding an average throughput per user of R (1 - Po) with outage probability Po = Lf=m'+l qf, since all requests to files not included in the common downlink stream are necessarily in outage. Finally, the conventional approach of today's technology in cellular and WiFi networks consists of handling on-demand video streaming requests exclusively at the application layer. Then, the underlying wireless network treats these requests as independent individ ual data. In this case, the average throughput per user is e ( n(~-=Po ) (1- Po) ) = e (~) for a system whose admission control serves a fraction l-p 0 of the users, and denies service to a fraction p 0 of users (outage users). 74 4.1.5 Summary: Comparison between Different Schemes In this section, we com pare the schemes reviewed before in terms of theoretical scaling laws. We focus on case where Mn » m, M is a constant and m, n, L 1 -+ co. As indicated in the beginning of this chapter, we consider a single cell of fixed area containing one base station and n user nodes (dense network), and take into account adjacent cell interference into the noise floor level. For the conventional unicasting where no D2D communication is possible and the users don't cache files, the system serves users' requests as if they were independent messages from the BS. Hence, we are in the presence of an information-theoretic broadcast channel with independent messages, whose per-user throughput is known to scale as 8(1/n), i.e., even significantly worse than the ad-hoc networks scaling law. As an intermediate system, we may consider the case of conventional caching (e.g., using prefix caching as advocated in [SRT99]), where users can cache M files, but the system does not handle D2D communication. In the prefix caching, users requesting files with index larger than the cutoff index rh are not served and are in outage. The users that are served, need to download a fraction (1 - AJ) for each file f, with index f < rh, where AJ S 1, II f and Lf~l AJ = M. Thus, the fundamental scaling behavior of this case is again 8(1/n) m small outage regime. In the case of harmonic broadcasting, as mentioned in Section 4.1.4, if we constrain the maximum waiting time to be T chunks, then the throughput per user of harmonic broadcasting scales as 8 ( 1 u (1 - LJ~m' +1 qt)), where m 1 S m. m 1 log T Next, we examine the scaling laws of the throughput for the uncoded D2D scheme for arbitrary small outage probability. By using the first line of (3.14), the average per user throughput scales as 8 ( 1::,), which is very attractive, since the throughput increase linearly with the size of the user cache. Finally, from (4.3) we observe that the throughput of coded multicasting scales also 8 (1::,). This indicates that by one-hop communication (either D2D or multicasting from 75 the base station), the fundamental limit of the throughput in the regime of small outage probability is 8( ;';;). 4 As a conclusion of this section we observe that, in the regime of Mn » m, where the total network storage is larger than the library size, both the uncoded D2D caching scheme and the coded multicasting scheme have an unbounded gain with respect to conventional unicasting as m, n -+ co. Harmonic broadcasting yields also a constant throughput with respect to the number of users n. The gain of the caching schemes over harmonic broadcasting depends critically on the system parameters, L 1 , T and m 1 for har- monic broadcasting, and M, m, for the caching schemes. According to the above model, uncoded D2D caching and coded multicasting are equivalent in terms of throughput seal- ing laws. 5 However, several other factors play a significant role in determining the system throughput and outage in realistic conditions. For example, the availability of D2D links may depend on the specific models for propagation at short range and may significantly differ depending on the frequency band such links operate in. Also, coded multicasting requires to send a common coded message to all the users in the cell. Multicasting at a common rate incurs the worst-case user bottleneck, since in practice users have different path losses and shadowing conditions with respect to the base station. Hence, in order to appreciate the performance of the various schemes reviewed in this work in realistic system conditions, beyond the scaling laws of the protocol model, in the next sections we resort to a holistic system optimization and simulation. 4 Notice that, in practice, also coded multicasting is subject to outages, due to the shadowing of the channel between the base station and the users. Since a common transmission rate has to be guaranteed for all the users, then some users will be in outage if the channel capacity between these users and base station is less than the common transmission rate. 5 Interestingly, in the recent work [JCM14], the authors showed that the gain of spatial reuse from uncoded D2D caching scheme and the gain of the coded multicasting do not accumulate in the order sense. 76 4.2 System Design We assume that devices can operate in multiple frequency bands. For transmission from the BS to mobile stations (MS), we assume operation at 2.1 GHz carrier frequency, corresponding to one of the standard long-term-evolution (LTE) bands. 6 We furthermore assume that D2D communication can occur at 2.45 GHz carrier frequency (specifically in the Industrial, Scientific and Medical (ISM) bands), as well as in the unlicensed mm-wave band at 38 GHz. Note that the 2.45 and 38 GHz bands are not suitable for BS-to-MS communications due to propagation conditions as well as transmit power restrictions imposed by frequency regulators. The 38 GHz band provides the possibility for very high data rates at very short range, due to the large available bandwidth at that frequency and the large pathloss. 4.2.1 Holistic Multi-Frequency D2D System Design In this case, we try to use all the resources in the network. As discussed above, file delivery is most efficiently achieved if it involves a short range communication link, for which the mm-wave frequency band is ideally suited. However, such connections are not robust since the mm-wave can be easily blocked by walls or even human body. Hence, if the mm-wave link is not available, the next best option is then D2D communication in the 2.45 GHz band. Finally, if even this band is not available or if the requested file is not present within the range of the D2D connections, the file may be served from the BS using the cellular downlink, depending on the admission control decisions. For the D2D links, we use clusterization, i.e., D2D communication is possible within a cluster, but not between clusters. For the cache placement, an independent and randomized cache placement of complete files is used as described in Section 4.1.2. The clustering 6 Due to the non-universal availability of sub-lGHz bands for LTE, we do not consider it further in this work. 77 A ll nodes make requret.<J No The base station serves rest of the users by using conventional unicasting approach Yro User u will be served by 38 GHz D2D communication in a ro und robin fashion inside the cluster a nd the cluster will be act ivated by TDMA scheme User u will be served by 2 .45 GHz; DZD conununication in a round ro bin fas hion inside the cluster and the clus ter will be activated by TDMA scherrn Figure 4.2: The flow chart of the delivery algorithm for the combination of D2D commu nications and multicast by the base station. and caching placement algorithm is summarized in Algorithm 4.1, 7 in which we focus on the regime of small outage, where all potential links (requests found in the cluster) are served. The flow chart of the delivery algorithm is shown in Fig. 4.2. Algorithm 4.1 Clustering and uncoded caching placement 1: According to the given outage probability of D2D communications (the probability that any user is not served by the D2D networks), decide the cluster size 9c(m) by using (3.14). 2: for all u EU do 3: Node u randomly caches M files independently according the probability distribu tion given in (3.13). 4: end for 7 In reality, there is a chance that the same file is selected to be cached multiple times in the same cache. Although irrelevant for the sake of the throughput scaling laws, this case should be avoided by practical caching algorithms. We do not further consider this aspect since its impact on the overall system performance is negligible in our simulation setting. 78 4.2.2 Conventional Unicasting, Coded Multicasting and Harmonic Broadcasting Approaches • Conventional Unicasting Approach: For the cache placement, we restrict to the case where we do prefix caching and each node caches a fraction of M /m of each file such that the local caching gain (1- ;';;) can be obtained. As our baseline approach, we consider a fairness constraint subject to all the users having the same outage probability. Consider a link between BS and user u, with log-normal shadowing xif and deterministic distance-dependent pathloss P L(du) not including the shadow ing, 8 where du denotes the distance between the BS and user u, let Cu denote the required individual downlink rate for user u such that for a fixed outage probability p 0 , Vu E {1, · · · ,n}, we have (4.4) where SNR (Signal-to-Noise Ratio) denotes the ratio between the transmit power and the noise power spectral density at the receiver. From ( 4.4), the user u down- link rate Cu can be determined as a function of p 0 and du, given the log-normal distribution of Xif· Now, given Cu, Vu, let Pu denote the fraction of downlink transmission resource dedicated to serve user u. In order to maximize the minimum user rate (our refer ence performance metric, see ( 4.1) ), the downlink transmission resource allocation is the solution of: max1m1ze subject to (4.5) 8 ln Section 4.3.2, PL is defined to include the log-normal shadowing Xa for the ease of presentation. 79 where A is the set of users that are not in outage. It is immediate to obtain the 1 solution of (4.5) as Pu= I: c;; 1 . Thus, let lube the indicator that user u is not u 1 EA Cul in outage, we obtain (4.6) Therefore, for any given p 0 , we obtain the set of points (T min, Po) that yields the throughput-outage tradeoff achievable by conventional unicasting. It is immediate to see that T min= 8(1/n) for any target p 0 in (0, 1). • Coded Multicasting Approach: Since common multicast messages have to be de- coded by all the served users, the downlink rate R is possible if R= Cro NTx(n,m,M) log(l + ~~~d ) ) < Xa u NTx(n, m, M) ' (4.7) where NTx(n,m,M) is given by (4.2). Since for all channel models used in our results, the receiver SNR of users at larger distance from the BS is stochastically dominated by the receiver SNR of users at smaller distance from the BS, the most stringent outage condition is imposed on the worst case users at largest distance r 0 from the BS, namely, (4.8) for du < ro. Hence, we have (4.9) 80 The outage probability is given by 1 ( SNR ) Po=-"'IF x"PL(du)2 c . nL 2ro-1 u (4.10) Note that Cro is the only control parameter, for this scheme. Hence, usmg (4.9) and ( 4.10), the throughput-outage tradeoff of coded multicasting can be obtained by varying the common downlink rate Cr 0 • • Harmonic broadcasting approach: In this case, the common multicasting messages also have to be decoded by all the served users. Let m 1 S m, the downlink rate reliable R is possible if R- Cro - m 1 log L' T log(l + ~~~d ) ) < Xa u m 1 log L' T ( 4.11) Two events can cause outage for harmonic broadcasting: i) the physical channel is not good enough to support the video encoding rate; ii) the requested file is not included in the set of files broadcasted by the BS. Since the two outage events are independent, we have (4.12) The outage probability is given by Po = ~ t (1 - (1 - f qf) u=l f=m 1 +1 (4.13) 81 Also in this case, by varying the common downlink rate Cr°' by (4.12) and (4.13) we can obtain the throughput-outage tradeoff of harmonic broadcasting. 4.3 Simulations and Discussions In this section, we provide the simulation results and some discussions. First, we describe the environments and discuss the channel models for the three types of links mentioned in Section 4.2. We then present our simulation results and discuss implications for de- ployment. 4.3.1 Deployment Environments We perform simulations in two types of environments: (i) office environments and (ii) indoor hotspots. More specifically we assume a cell of dimensions 0.36km 2 (600m x 600m) that contains buildings as well as streets/outdoor environments. We assume n = 10000, i.e., on average, there are 2 ~ 3 nodes, every square 10 x 10 meters for the grid network model. We will also investigate the effect of user density later. For the office environment, we assume that the cell contains a Manhattan grid of square buildings with side length of 50m, separated by streets of width lOm. Each building is made up of offices; of size 6.2m x 6.2m. 9 Corridors are not considered, as they would lead to a further com plication of the channel model. For the "indoor hotspot" model, which describes big factory buildings or airport halls, we also assume that the cell is filled with multiple buildings. The size of these buildings are squares with side length of lOOm and distributed on a grid with street width of 20m. There are no internal partitions (walls) within the building. Within the cell, users (devices) are distributed at random according to a uniform distribution. Due to our geometrical setup, each node is assigned to be outdoors or gThe motivation for this size stems from the line-of-sight (LOS) model (see below); we choose the office size such that it results in a LOS probability of 0.5 if two devices are half the office dimensions apart from each other. 82 indoors, and (in the case of the office scenario) placed in a particular officern This information is exploited to determine which channel model (e.g., indoor-to-indoor or outdoor-to-indoor) is applicable for a particular link. The use of such a virtual geometry is similar in spirit to, e.g., the Virtual Deployment Cell Areas (VCDA) of the COST 259 microcellular model [CorOl]. 4.3.2 Channel Models Corresponding to the three types of transmissions (cellular, microwave D2D, millimeter wave D2D), we use three types of channel models. We only consider pathloss and shad owing, since the effect of small scale fading can be eliminated by frequency /time diversity over the bandwidth and timescales of interest. The channel models are mostly obtained from the Winner II channel models [WIN07]. We note that although these channels are not explicitly defined for device-to-device, the range of parameter validity includes device height of about 1.5 m, which is typical for a user-held device. 4.3.2.1 LOS probability One of the key parameters for any propagation channel is the existence of a line-of sight (LOS) condition: all channel characteristics, including path loss, delay spread, and angular spread, depend on this issue. It is obvious that the existence of a LOS is independent of the carrier frequency; it thus seems straightforward to simply apply the LOS model of Winner at all frequencies. However, as we will see in the following, there are subtleties that depend on the carrier frequency and greatly impact the overall performance. By denoting the distance between each transmitter-receiver pair as d, the LOS probability (Pi'.b 8 ) models by Winner [WIN07] are summarized in Table 4.3.2.1. 10 Note that the division of buildings into offices is only used to determine the "wall penetration loss", while the basic pathloss and LOS probability are determined by the purely distance-dependent model (see below for details). 83 Table 4.1: The LOS probability models. indoor office Plbs = { 1, 1 - 0.9(1 - (1.24 - 0.61 log 10 (d)) 3 )t, d > 2.5 d c; 2.5 indoor hotspots pW _ { 1, d C: 10 LOS - exp ( - d-;-JD l, d > 10 outdoor-to-outdoor P{ 08 = min(18/ d, 1) (1 - exp(-d/36)) + exp(-d/36) outdoor-to-indoor Pws = 0 The LOS probability given in the literature (including the Winner model) usually refers to a LOS connection between users, not necessarily between the antennas on the devices held by the users. In other words, there are situations where a transmit and receive antenna nominally have LOS (according to the model definition), because there are no environmental obstacles between them; however, the bodies of the users and/or the device casings might prevent an actual LOS. We will henceforth refer to this situation as "body-obstructed LOS" (BLOS). It is especially critical at mm-wave frequencies, which to which the human body is essentially impervious. For microwaves, the human body can be taken into account by introducing an additional shadowing term. Let us first consider the case of mm-wave propagation. Considering the way smart- phones are usually held in front of the body, approximately, we assume that each user has a degree of '2 = 250 "free" sector, then half the cases of "nominal" LOS are actually BLOS, while the rest is "true" LOS: 1 lti1os = llos = 2 · Pfcis, (4.14) For the case of BLOS, alternative propagation paths such as reflections by walls, can sustain links, but the resulting path loss and related parameters are different from the "true" LOS; thus separate parameterization has to be used. The case of non-line-of-sight 84 (NLOS) 11 , clearly occurs with probability 1 - P{b 8 . In the case of mm-wave communi- cations, walls constitute an insurmountable obstacle, i.e., penetration of radiation into neighboring rooms, and between inside/outside the building, is negligible. For microwave propagation, the effect of body shadowing is better explained by an additional lognormal fading. In contrast to the "standard" shadowing that describes shadowing by environmental obstacles and that changes as users move laterally, body shadowing variations are created by rotation of the users - resulting in the highest at- tenuation when they are standing back-to-back. In [KJTM08], it is shown that the body shadowing attenuation x"Lb follows log-normal distribution with mean 0 and standard deviation crLb· For D2D communication, we use the hand-to-hand model (HH2HH) as shown in [KJTM08]. 4.3.2.2 Device-to-device channels at 38GHz In this case, the pathloss is given by ( 47rdo) ( d) PL(d) = 20log 10 -.\- + 10alog 10 do + X"' (4.15) where d 0 = 5m is the free-space reference distance, .\ is the wavelength, a is the average pathloss, x" is the shadowing parameter with mean 0 and standard deviation er. We assume that no 38 Hz communication is possible when d > 80 m. From [RGBD+13] [KTMM13], the system parameters are given by: °'LOS= 2.21, °'NLOS = 3.18, er Los= 9.4 and CfNLOS = 11. 11 0ne example of NLOS transmission is that the transmitter and the receiver are in different rooms, with walls between them. 85 4.3.2.3 Device-to-device channels at 2.4GHz For this case we can directly use the Winner II channel model, although we assume that no communication is possible for a distance larger than lOOm. 12 Since 2.4 GHz communication can penetrate walls, we have to account for different scenarios, which are indoor communication (Winner model Al), outdoor-to-indoor communication (B4), indoor-to-outdoor communication (A2), and outdoor communication (Bl). We illustrate the case of the indoor (Al) communication, where the path loss model for both LOS and NLOS is given by [WIN07], (4.16) where fc is the carrier frequency. Ai includes the path loss exponent. A2 is the intercept and A3 describes the path loss frequency dependence. X = 5nw is the (light) wall attenuation parameter, where nw is the number of walls between transmitter and receiver. xif is the shadowing parameter assumed to be a log-normal distribution with mean 0 and standard deviation O", where O'LOS = 3 and O"NLOS = 6. Note that according to our discussion above, we add the body shadowing loss to Eq. ( 4.16), where for LOS, 0'£b = 4.2 and for NLOS, 0'£b = 3.6. All the other parameters for the indoor pathloss channel model in 2.4 GHz are summarized in Table 4.3.2.3. Table 4.2: The channel parameters for 2.4 GHz D2D communications A 1 (LOS) A 2 (LOS) A 3 (LOS) A 1 (NLOS) A 2 (NLOS) A 3 (NLOS) indoor office 18.7 46.8 20 36.8 43.8 20 indoor hotspot 13.9 64.4 20 37.8 36.5 23 For the other three cases, namely outdoor (Bl), indoor-to-outdoor (A2) and outdoor to-indoor (B4), we similarly directly use the respective Winner II channel models with 12 This is a conservative assumption motivated by the fact that at low SNR it is difficult for a D2D link to acquire beacon signals and discover other D2D devices. 86 antenna heights of 1.5m, probabilistic LOS, and with the consideration of body shadow mg. 4.3.2.4 Channel between the Base Station and Devices In this case the Winner II channel model can also be used directly. In particular we use the urban macro-cell (C2) model for outdoor to outdoor communications and the urban macro outdoor to indoor (C4) model for outdoor to indoor communication; the only modification is the addition of the rotational body shadowing x"Lb. As model for the rotational body shadowing, we use the access point to handheld device model (AP2HH [KJTM08]: for the case of LOS, "Lb = 2.3 dB, while for NLOS, it is "Lb = 2.2 dB.) For example, for the urban macro-cell (C2) channel model, the pathloss for LOS of the Winner model (i.e., without body shadowing) is given by P L1os(d) = Ai log 10 (d) + A2 + A3 log 10 (fc[GHz]/5) + x"" lOm < d < dkp 40log 10 (d) + 13.37 - 14log 10 (hk 8 ) (4.17) -14log 10 (h~ 8 ) + 6 log 10 (fc[GHz]/5) + x"" dkp < d < 5000m where dkp = 4hk 8 h~ 8 fc/ c and hks = hgs-1 and h~ 8 = hMs-1. We pick hgs = 25m and h~ 8 = 1.5m. x"' and x" 2 are shadowing attenuations, which are lognormally distributed with mean 0 and standard deviation <Y 1 = 4 and <Y 2 = 6. For NLOS, we have PLN1os(d) = (44.9 - 6.55 log 10 (hgs)) log 10 (d) + 34.46 + 5.83log 10 (hBs) + 23log 10 (fc[GHz]/5) + x"' (4.18) 87 where 50m < d < 5000m. The shadowing xif is zero-mean and has standard deviation er = 8. Similarly, the urban outdoor to indoor (C4) channel model can be found in [WIN07]. Moreover, to simulate the realistic scenario, we also assume a frequency reuse factor Kin this case to avoid the interference between cells [Molll]. 4.3.2.5 Link Capacity Computation Given all the system parameters, the link capacity for a transmitter-receiver pair is given by C = B · log 2 (1 + SINR) (4.19) where SINR = P~ignal/ (Pnoi~e + Flnterference) (Signal to Interference plus Noise Ratio), and B denotes the signal channel bandwidth. Specifically, on a dB scale, P~ignal is given by P~ignal,dB = Prx +Gt+ Gr - P L(d) (4.20) where the Prx is the transmit power, Gt and Gr are the transmit and receive antenna gains. Pinterference is the sum of the all the interference to a receiver . 13 On a dB scale, the noise power is given by (4.21) where kBTe = -174 dBm/Hz is the noise power spectral density and FN = 6 dB is a typical noise figure of the receiver. We assume this model to hold at all frequencies. 14 The parameters of the three types of transmissions are summarized in Table 4.3.2.5. 13 The model for mm-wave communication is considered to be interference free (Pinterference = 0) since the angle of arrival (AOA) is very narrow (less than 10 degree). 14 While for the same cost, receivers at 2 GHz might provide a better noise figure due to better established fabrication processes, the impact of this effect on the system performance is low, and will be neglected henceforth. 88 Table 4.3: The parameters for the three types of transmissions B f, Prx G, Gr mm-wave D2D transmissions 800 MHz 38 GHz 20 dBm 9 dB 9 dB microwave D2D transmissions 20 MHz 2.45 GHz or 2.1 GHz 20 dBm 12 dB 0 cellular transmissions 20 MHz 2.1 GHz 43 dBm 12 dB 0 4.3.3 Results and Discussions In this section, we will present the simulation results. If not stated otherwise, we will use the following settings: the number of users is n = 10000; the users are uniformly and indepdently distributed in the cell (It can be shown that a negligible difference between regular grid and random distribution by simulation (not shown here); we thus henceforth show only results for the random node distribution). The number of files in the library is m = 300, which is representative of the library size of a video on-demand service. 15 The user cache size is M = 20 files unless specifically mentioned, which even with high definition (HD) quality requires less than the (nowadays) ubiquitous 64 GByte of storage space. We let each user independently make a request by sam piing from a Zipf distribution with I= 0.4; this value is at the lower edge of the range of values that have been measured in practice [BCF+99]; note that the advantages of caching would be more pronounced for larger /. The interference between concurrent D 2D links sharing the same frequency band is treated as noise. For the harmonic broadcasting, we chose a video file size of L 1 = 5400 chunks and T = 10 chunks, then the number of blocks is r 1;" l = 540 [Sd!FSH+ 11a]. 4.3.3.1 Throughput-Outage Tradeoff In Fig. 4.3, we plot the performance of all the discussed schemes separately, where a 2.45 GHz D2D only scheme is implemented. From Fig. 4.3, we can see that the throughput of the D2D scheme is markedly (orders of magnitude at low outages) higher than the conventional unicasting, harmonic broadcasting and even coded multicast scheme. This 15 In practice, the library of titles in such a service would be refreshed every few days. 89 K 4 4 3 shows that in practical situations, the "scaling law" is not the only aspect of importance. Rather, the higher capacity of the short-distance links plays a significant role, and a good throughput-outage tradeoff can be achieved even without the use of a BS connection as "backstop". The main reason lies in the fact that for the coded multicasting or harmonic broadcasting scheme, outage is determined by bad channel conditions, and no diversity is built into the system. For D2D, even though the outage in our scheme is caused by both physical channel and the lack of the requested files in the corresponding cluster, the channel diversity plays a more importance role. Moreover, although not shown in Fig. 4.3 for the ease of presentation, the behaviors of all schemes hold for both the indoor office and the indoor hotspot environment. 16 In Fig. 4.5(a), for the hotspot, we furthermore obtain the interesting result that the throughput-outage tradeoff is non-monotonous if we use the (theoretically derived) cluster size. This behavior is caused by a higher LOS probability when the cluster size becomes small: there is an appreciable probability that the useful signal is NLOS but there exist some LOS interferers. From Fig. 4.5(a), 4.5(b) and 4.5(c), a similar phenomenon can also be observed for the case of the indoor office model but for different parameter settings. Of course this does not mean that the optimum throughput-outage tradeoff in practice is non-monotonous; rather it is a consequence of using a cluster size derived under one specific model in the deployment by using a different model. Besides the performance advantage of the D2D approach (compared to coded multi cast), it also has the advantage of a simpler caching placement and delivery. The coded multicasting approach in [MAN14b] constructs the cache contents and the coded delivery scheme in a combinatorial manner which does not scale well with n. For example, in our 16 In fact, our D2D scheme performs better in the hotspot scenario than in the indoor office case. This is mainly due to the low probability of LOS from interferers and the high probability of LOS for useful signal (note that the LOS probability in the hotspot is unity up to distances of 10 m and decreases exponentially for larger distances). However, for the coded multicast transmission, the performance in indoor hotspot is actually worse than the indoor office model; this is due to the larger size of the buildings so that the pathloss caused by din in the urban macro outdoor-to-indoor model (C4) [WINO?] is very significant, where din is the distance from the wall to the indoor terminal. 90 10" 10- 4 10-<i 10 ....2 10' C:Utaie Probaality Figure 4.3: Simulation results for the throughput-outage tradeoff for conventional unicas ting, coded multicasting, harmonic broadcasting and the 2.45 GHz D2D communication scheme under indoor office channel models. For harmonic broadcasting with only the m 1 most popular files, solid line: m 1 = 300; dash-dot line: m' = 280; dash line: m 1 = 250. We haven= 10000, m = 300, M = 20 and 'Y = 0.4. network configuration, it requires a code length larger than (1~gg 0 ) , which is larger than 10 1s. 4.3.3.2 Holistic Multi-Frequency D2D System Performance In this section, we investigate the performance of the proposed D2D system given by Fig. 4.2 in Section 4.2.1. Fig. 4.4(a) shows that the average throughput per user increases significantly due t o the help of the 38 GHz D2D communications. Consider the CDF (Cumulative distribution function) of the throughput for different outage probabilities shown in Fig. 4.4(b ), in this way, we can see on average, how many users will be served with a throughput that is less than the minimum required rate for video streaming, for example, lOOKbps. Fig. 4.4(b) and 4.4( c) show the throughput as a function of the cluster size. Intuitively, a large cluster size corresponds to a small outage probability, as the probability is high that 91 the desired file is found within a cluster. This is reflected by the throughput CDF: a small cluster size results in a small minimum throughput, but a large maximum throughput (compare, e.g., the red-dotted line in Fig. 4.4(c); this is similar to the effect we have observed in the previous subsection. For example, if we pick a cluster size of lOOm x lOOm, then the number of users whose rate is less than 100 Kbps only around 250. Moreover, about 30% of users are served with a data rate larger than 2 Mbps) due to help by 38 GHz D2D communications. From Fig. 4.4(a), 4.4(b) and 4.4(c), we observe similar behavior under the indoor hotspot model, where interestingly, the performance of our holistic multi-frequency design in terms of average throughput is not very different from that of the indoor office model. The reason is that the variance of the throughput for the indoor model is much larger than that for the indoor hotspot model, which is due to the fact that fewer users can be served by the 38 GHz D2D communications. 17 Both CDFs for the case of lOOm x lOOm cluster size are shown in Fig. 4.4( d), In this case, almost no users have a service rate less than 100 Kbps and about 90% of users can get HD quality services. Moreover, we notice that the role of base station in this scenario is to reduce the outage probability. For example, when cluster size is lOOm x lOOm, the base station can serve 400 ~ 500 users in the indoor office model. 4.3.3.3 Effects of the Density of Nodes From Theorem 2.5, we expect that the throughput-outage tradeoff does not depend on the number of users or user density as long as n and mare large and Mn» m. However, the throughput-outage scaling behavior was obtained under the simplified protocol model, where the relation between the link rate and the link range (source-destination distance) is not specified. In practice, if we want to obtain a high communication rate, the D2D communication distance cannot be very large due to the large pathloss, which reduces the 17 We serve the users in a round robin fashion in one cluster even for 38 GHz communications to avoid interference. 92 0 .1 10-' 10"' 10- 1 10° 10 4 10' 10 8 10 10 OJtage Probabtlity Throughput per user (bps) (a) (b) 01 0 .1 10 4 1if 1d ~o' 10 4 10' ThrooghPJt per user (t;i>s) Throughput per user (bps) (c) (d) Figure 4.4: (a). Simulation results for the throughput-outage tradeoff by holistic system design. Black Solid lines: indoor office; blue dashed lines: indoor hotspot. (b ). The CDF of the throughput for different outage probabilities (cluster size of 600 2 / Q 2 ) under indoor office model. (c). The CDF of the throughput for different outage probabilities (cluster size of 600 2 /Q 2 ) under indoor hotspot model. (d). The CDF of the throughput for the cluster size of lOOm x lOOm under indoor office and indoor hotspot channel model. Solid lines: indoor office; dashed lines: indoor hotspot. 93 per-link capacity. This is especially true for 38 GHz communications under the indoor office environment. Therefore, the user density is also an important parameter for the system performance. In this section, we investigate the system behavior for different user densities by focusing on the case of 2.45 GHz D2D communications only. In Fig. 4.5(a), we observe that there exists a tradeoff between the user density and the throughput, which is because that the impact of the user density on the link rate is twofold: on one hand, a higher user density allows a smaller cluster size, in turn resulting in shorter links and higher SINR. On the other hand, a small cluster size increases the probability for having LOS interference, which can degrade the performance of the system significantly. 4.3.3.4 Effects of the Storage Capacity and the Library Size As already observed in Section 4.1.2, in the regime nM » m the D2D system yields a linear dependence of the throughput on the user storage capacity M. This means that such a system can directly trade cache memory for throughput. Since storage memory is a cheap and rapidly growing network resource, while bandwidth is scarce and very expensive, the attractiveness of the D2D approach is self-evident. The result also holds true in practice, as demonstrated by the simulation results in Fig. 4.5(b). We observe that, when the outage is small, the average throughput per user increases even faster than linearly with M. This is because in practice the link rate Cr is a decreasing function of the link range. Therefore, when M becomes large, we can decrease the D2D cluster size (and therefore the average link range) while maintaining a constant outage probability. Fig. 4.5( c) shows the throughput-outage tradeoff for different library size. As expected from Theorem 2.5, in this case we notice that the throughput decreases roughly inversely proportional to the library size m, for fixed cache capacity M. 94 4.3.3.5 2.1 GHz In Band Communications Sometimes, the D2D communications and the cellular communications by the BS have to share the same spectrum, which raises the question of how to divide the bandwidth for each type of communications. Obviously, this depends on the channel realizations for each type of communications. From our simulations, we obtain that under our channel model (either indoor office or indoor hotspot), the base station cannot support more than about 1000 users in the best case if the minimum video coding rate is 100 Kbps, while, for 2.1 GHz D2D communication, it is very easy to support a much larger number of users at a certain playback rate requirement. Therefore, it is intuitive that there is no tradeoff between the bandwidth division and the average throughput by fixing the cluster size (outage probability in Theorem 2.5). The simulation results are shown in Fig. 4.5(d), which confirm our intuition. On the other hand, if we care more about the outage probability, then there is a clear tradeoff between the outage probability and the division of the bandwidth, especially for the small cluster size. This occurs because the BS is capable of satisfying "costly" links that normally would either increase outage probability or would enforce an increase in cluster size. In Fig. 4.5( e ), under the office channel model, when the area of the cluster is 600 2 /19 2 , the best bandwidth division is when Bd2d/ Bgs = 0.2, which means that we need to only allocate 20% of the bandwidth to the D2D communication to obtain the minimum outage probability. Similar behavior can also be observed for the hotspot model with the difference that now Bd 2 d/ Bgs = 0.1 is the best bandwidth division, which is because that the link rate under the hotspot channel model is better than that for the indoor office model. 95 4.4 Summary In this work we have reviewed in a tutorial fashion some recent results on base station assisted D2D wireless networks with caching for video delivery, recently proposed by the authors [J CM13a], as well as some competing conventional schemes and a recently devel oped scheme based on caching at the user devices but involving only coded multicasting transmission from the base station. We reviewed the throughput-outage scaling laws of such schemes on the basis of a sim pie protocol model which captures the fundamental aspects of interference and spatial spectrum reuse through geometric link conflict con straints. This model allows a sharp characterization of the throughout-outage tradeoff in the asymptotic regime of dense networks. This tradeoff shows the superiority of the D2D caching network approach and of the coded multicasting approach over the conventional schemes, which can be regarded as today current technology. In order to gain a better understanding of the actual performance in realistic envi ronments, we have developed an accurate simulation model and some guidelines for the system design. In particular, we have considered a holistic system design including D2D links at 38 GHz and 2.45 (or 2.1) GHz, and the cellular downlink at 2.1 GHz, represen tative of an LTE network. We compared the schemes treated in the tutorial part of the work on the basis of their throughput-outage tradeoff performance, and we have put in evidence several interesting aspects. In particular, we have shown the superiority of the D2D caching network even in realistic propagation conditions, including all the aspects that typically are expected to limit D2D communications, such as NLOS propagation, limited link range, environment shadowing and human body shadowing. The D2D caching network shows very com peti tive performance with respect to the other schemes. In particular, the proposed system is able to efficiently trade the cache memory in the user devices for the system throughput. Since the former is a rapidly growing, cheap and yet untapped network resource, while the latter is known to be scarce and very expensive, the interest in developing and deploying 96 such D2D caching networks becomes evident. This fact is even more remarkable if we consider the fact that the D2D network requires simple decentralized caching and does not require any sophisticated network coding technique to share the files over the D2D links. 97 fo-• 10-' 10"' 10- 1 10° 10-' 10-' 10- 1 10° OJtage Probabtlity Outage Pr obab lity (a) (b) 14x1 05 12 . cluster,ize.-600 2 /.10~ .. ~ ft 10 :0 ~ e 8- 1 5 :0 8- ~ '5 6 Q. 0) ~ ii 1 {'; 2 4 {'; fo- • 10-3 10"' 10- 1 10° 0 .2 0.4 0.6 0.9 OJtage Probabtlit y 8 d2d 18 ss (c) (d) 0155 (e) Figure 4.5: Solid lines: indoor office; dashed lines: indoor hotspot. (a). The throughput outage tradeoff for different user densities. (b). The throughput-outage tradeoff for different user storage capacity. ( c). The throughput-outage tradeoff for different library size of files. (d). The throughput v.s. bandwidth division between 2.1 GHz commu nication and the base station under different cluster size, where Bd 2 d is the bandwidth by 2.1 GHz communications and BBs is the bandwidth by the cellular base stat ion. Bd2d + BBs = B = 20MHz. (e). The outage v.s. bandwidth division between 2.1 GHz communication and the base station for the cluster with size 600 2 / 19 2 . 98 Chapter 5 Caching in Wireless Multihop Device-to-Device Networks In this chapter, we study an extension of the works discussed in Chapter 2, 3 and 4. Instead of single-hop D2D caching networks, we allow multihop communications. This chapter is organized in the following. Section 5.1 present the network model and problem formulation. The main results of the throughput are presented in Sections 5.2. The achievable scheme and the achievable throughput are presented in Section 5.3 and 5.4, respectively. The converse of the throughput is discussed in Section 5.5. In Section 5.6 we presents some concluding remarks. 5.1 Problem Formulation 5.1.1 Caching in Wireless D2D Networks We consider a wireless D2D network consisting of n nodes distributed uniformly at random over a unit square area [O, 1] 2 . Let d(u,v) denote the distance between nodes u and v. It is assumed that communication between nodes follows the protocol model of [GKOO]: the transmission from node u to node vis successful if and only if: i) d(u, v) Sr, and ii) no other active transmitter must be in a circle of radius (1 + 6)r from the receiver node v. Here, r, 6 > 0 are given protocol parameters. Also, each node sends its packets at some constant rate W bits/sec/Hz, where Wis a non-increasing function of the transmission ranger. 99 As in the current information theoretic literature on caching networks (see Chapter 1), a caching scheme is formed by two phases: caching placement and delivery. The problem consists of placing information in the caches such that the delivery is efficient for any set of user demands (i.e., requested files). Since the demands are not known a priori, 1 the cache placement must made without a priori knowledge of the set of demands although, in general, it can depend on the demands statistics. During the placement phase, each node u stores M files in its local memory Mu from a Ii brary F of size m files, where M S m. During the delivery phase, each node u requests a file fu E F, independently with probability qfu, and the network operates in order to satisfy all the requests. The probability mass function q(-) is referred to in the following as the popularity distribution. 5.1.2 Achievable Throughput and System Scaling Regime We consider m and M expressed as functions of n as m = na 1 and M = nf3 1 ' (5.1) where a1 > 0 and /31 E [O, a1). 2 Let Tn and Sn denote the per-node average throughput (expressed in files per unit time, averaged over the popularity distribution) and the sum average throughput supported by the network during the delivery phase. We have: Definition 5.1 The throughput Tn is said to be achievable if there exist a cache place- ment and a delivery transmission protocol such that all nodes can receive their requested files with average rate at least Tn with high probability { whp ). 3 Accordingly, the achievable sum throughput is at least Sn= nTn. 0 1 They can be either arbitrary (as in [MAN14b,MAN14a]) or random (as in [GSD+13,JCM13a] and in this work. 2 If /3 1 2: a: 1 , then each node is able to store the entire files in F, which make the delivery phase trivial. 3 ln the following, an event An is said to occur "whp" if limn-+= IP'( An) = L 100 5.2 Main Results We have: Theorem 5.1 For the caching wireless D2D network defined before, the optimal through- put satisfies the scaling laws: if °'1 - /31E(0,1], (5.2) where e > 0 is arbitrarily small. • Furthermore, we have: Theorem 5.2 Consider the caching wireless D2D network defined before and assume that demands follow a Zipf popularity distribution 4 with exponent IE (0, 1). Then the throughput of any scheme must satisfy if °'1 - /31 > 1, (5.3) if °'1 - /31 E (0, 1], where e > 0 is arbitrarily small. • The rest of the work is dedicated to proving Theorems 5.1 and 5.2 . For the case of a 1 - (3 1 > 1, we only need to derive an upper bound, which is given in Section 5.5.1. In this case, the total caching memory size in the network is less than the number of files in the library, i.e., Mn < m. Therefore, there may be a node to request a file not stored in the nodes' memories whp, thus resulting in a non-zero outage probability. Since Tn is defined as a rate with no outage, it gives a zero per-node throughput. For a1 - /31 E (0, 1], we prove Theorem 5.1 by presenting and analyzing a specific achievable scheme in Sections 5.3 and 5.4, respectively, and Theorem 5.2 by finding a converse upper bound in Section 5.5.2. 4 The Zipf distribution with exponent I' [BCF+99] is defined by qi(i) = L~=~~ 7 for 1 :Si :Sm. 101 .C"L J>. ~ ' r• - -- - --- !""'- - ~ "'tY Sourc I ~ I I I ~ ; I I I ? I I e I --- - - - ~ .. ; n '!! - Destinatio - c., - ---- .. 1"" - - · - 1 : I I f I I I I I - I I • • - f- ,- -- • • -- - - - - - ~ Figure 5.1: The proposed multihop routing protocol after the source node selection. 5.3 Achievable Schemes In this section, we present a file placement policy and a transmission protocol for a1 - /31 E (0, 1]. 5.3.1 0:1 - /31E(0, 1) In this regime, a distributed file placement and a local multihop protocol are proposed as follows. 5.3.1.1 Distributed file placement Each node u stores M distinct files in its local memory Mu , chosen uniformly at random from F , independently of other nodes. 5.3.1.2 Local multihop protocol We first explain how each node finds its source node having the requested file (source no de selection): 102 TDMA cell size Transmission pair I Vah ~ I I ,,v 1"~ / "" I I '\ I I \ Iv )ah ! ~ ~ \ I J \ A 1+ ~)' j5a f- / ""' '1 / "- ~ v ___./ Figure 5.2: TDMA cell size from the protocol model. • Divide the entire network into square traffic cells of area ac = n-ri for some ry 2 0, where ry will be determined later on. • Each node chooses one of the nodes having the requested file in the same traffic cell as its source node. If there are multiple candidates, choose one of them uniformly at random. From Definition 5.1 and the above source node selection, all nodes should find their source nodes within their own traffic cells whp, in order to achieve a non-zero Tn. Lemma 5.2 below characterizes such a condition of the area of traffic cell ac (i .e., ry) such as ry E [O, 1 - (0:1 - /31)) . For the ease of exposition, we call the pair of a node and its source node source- destination (SD) pair. Notice that in our model, each SD pair is located in the same traffic cell while in the conventional ad hoc network, SD pairs are randomly located over 103 the entire network. Thanks to caching, we can reduce the distance of each SD pair (see Lemma 5.2). Also, differently from the conventional model, each node can be a source node of multiple destinations, which make the throughput analysis more complicated (see Lemma 5.4). Next, we explain the proposed multihop transmission scheme for the file delivery between n SD pairs, see also Fig. 5.1 (multihop transmission): • Divide each traffic cell into square hopping cells of area% = 21 ~gn. • Define the horizontal data path (HDP) and the vertical data path (VDP) of a SD pair as the horizontal line and the vertical line connecting a source node to its destination node, respectively. Each source node transmits the requested file to its destination by first hopping to the adjacent hopping cells on its HD P and then on its VDP. 5 • Time Division Multiple Access (TDMA) scheme is used with reuse factor T for which each hopping cell is activated only once out of T time slots. • A transmitter node in each active hopping cell sends a file (or fragment of a file) to a receiver node in an adjacent hopping cell. Round-robin is used for all transmitter nodes in the same hopping cell. In this scheme, each hopping cell should contain at least one node for relaying as m [GKOO,EMPS06], which is satisfied whp since%= 21 ~gn (see Lemma 5.1 (a)). Lemma 5.1 The following properties hold whp: (a) Partition the network area [O, 11 2 into cells of area 21 ~gn. Then the number of nodes in each cell is between 1 and 4 log n. (b) Partition the network area [O, 11 2 into cells of area n-a, where a E [O, 1). For any 6 > 0, the number of nodes in each cell is between (1 - 6)n 1 -a and (1 + 6)n 1 -a. 5 If a source node and its destination node are in the same hopping cell, then the source node directly transmits to its destination. 104 Proof The proofs of first and second properties are given in [EMPS06, Lemma 1] and [OLT07, Lemma 4.1], respectively. Lemma 5.2 Suppose that a1 -/31E(0,1). !fr] E [O, 1- (a1 -/31)), then all nodes are able to find their source nodes within their traffic cells whp. Proof Let Ai denote the event that node i establishes its source node within its traffic cell, where i E [1 : n]. Then, we have: (5 .4) where the first inequality follows from the union bound and the second inequality is due to the fact that the number of nodes in each traffic cell is lower bounded by (1 - 6)nac whp M (1-8)na whp (see Lemma 5.1 (b)) and hence, Pr(Af) S (m;;, ) '. Therefore, from the fact that na1-/31 Jim (1 - l/n" 1 -li 1 ) = l/e, n~oo (5.5) Pr (niE[ln]Ai) -+ 1 as n-+ co, since r] < 1 - °'1 + /31 is assumed in this lemma. This completes the proof. 5.3.2 a 1 - /31 = 1 In this case, the total number of files that can be stored by n nodes (i.e., the total number of files stored in the network) is equal to the number of files in the library (i.e., nM = m ), which is not the interesting regime in practice. We study this regime for completeness. In this regime, we propose a centralized file placement and a globally multihop protocol schemes. 105 5.3.2.1 Centralized file placement It can be seen that a distributed file placement might result in an outage, as seen from the analysis in Lemma 5.2. Instead, we employ a simple centralized file placement for which all distinct m files (in the library) are randomly stored in the total memories of n nodes. Hence, the network can contain all m files, thus being able to avoid an outage. 5.3.2.2 Globally multihop protocol As explained before, the traffic cell should be equal to the entire network (i.e., rJ = 0 in Section 5.3.1), in order to avoid an outage. Namely, n SD pairs are located over the entire network . Hence, we can expect the same scaling result with the conventional wireless ad hoc network in [GKOO], namely, no caching gain is expected. 5.4 Achievable Throughput We derive an achievable throughput of the proposed schemes in Section 5.3. 5.4.1 ct1 - /31 E (0, 1) In this subsection, we prove that is achievable if °'1 - /31E(0,1). cq-/31 Tn = n __ 2 __ , (5.6) From Lemma 5.2, we assume rJ E [O, 1 - (a1 - /31)) to achieve a non-zero Tn by the proposed scheme in Section 5.3.1. Then, we provide the following useful lemmas for the proof. Lemma 5.3 Suppose that a1 - /31 E (0, 1) and rJ E [O, 1 - (a1 - /31)). Let I?,,., denote the aggregate rate achievable for any hopping cell. If T 2 (2r(1+6)v51+1) 2 , then I?,,., = 1:j'. is achievable. 106 Proof Consider an arbitrary transmission pair consisting of a transmitter node and its re- ceiver node illustrated in Fig. 5.2. Clearly, the hopping distance is upper bounded by y1)% and hence, we choose the transmission range r = y1)% in the protocol model. Thus, the transmission is successful if there is no node simultaneously transmitting within the dis tance of ( 1 + 6) y1)% from the receiver node. This is satisfied if T 2 ( 2 r ( 1 + 6 )Y51 + 1) 2 . That is, the aggregate rate of~ is achievable if T 2 (2r(1+6)Y51+1) 2 . Since this holds for all hopping cells, Rn = ~ is achievable if T 2 (2r(1+6)Y51+1) 2 . This completes the proof. Lemma 5.4 Suppose that cq - /31 E (0, 1) and r] E [O, 1- ( °'1 - /31)). Fore > 0 arbitrarily small, each node can be a source node of at most n 1 -~-(<>i -lii)+c nodes in its traffic cell whp. Proof Let Bi(k) denote the event that node i becomes a source node for less thank nodes. Denote nl = (1 + 6)n 1 -~. Then, we have: Pr (niE[ln]Bi(k)) = 1- Pr (uiE[ln]Bf(k)) whp ni (n 1 ) (M)j ( M)n,-j 2 1-n'\' . - 1-- L__, J m m j~k (5.7) if~ < }:, < 1, where D(allb) =a log(~)+ (1-a) log(i=b) denotes the relative entropy for a, b E (0, 1). Here the first inequality follows from the union bound and holds whp since the number of nodes in each traffic cell is upper bounded by nl whp from Lemma 5.1 (b), and the second inequality is due to the fact that for X ~ B(n,p), Pr(X;:>k)Sexp(-nD(k/nllP)) ifp<k/n<l. (5.8) 107 Suppose that k = nT for T > 0. Then the condition M < -"- < 1 is given by (1 + m nl 6)n 1 -~-(<>1-li1) < nT < (1+6)n 1 -~, which is satisfied as n increases if 1 - rJ - (a1 - Pl) <TS 1 - rJ· Hence, from (5.7), we have: ~A ( ( n" 1 ((1 + 6)nl-~ - nT) ) ) ·exp -( (1 + 6)n 1 -~ - nT) log (1 + 6)nl-~(n"1 - nli1) provided that (5.9) is satisfied. Since rJ > 0, ( nT-1+~+(<>1-1'1) )-ln(2) A=nexp(-nT) l+<l converges to zero as n increases. Furthermore B =exp (-((1+6)n 1 -~ - nT)) . ( n"1((l + 6)nl-~ - nT) )-ln(2) (1 + 6)nl-~(n"1 - nli1) (5.9) (5.10) (5.11) converges to zero as n increases if TS 1 - rJ· In summary, Pr (niE[ln]Bi(n~)) converges to zero as n increases if (5.9) holds. Therefore, Pr (niE[ln]Bi(n 1 -~-(<> 1 -li 1 )+c)) -+ 0 as n -+ co by setting T = 1 - rJ - (a1 - ;31) + f for f > 0 arbitrarily small, implying that each node becomes a source node of at most n 1 -~-(<> 1 -li 1 )+c nodes whp. This completes the proof. 108 Based on Lemma 5.4, we will derive an upper bound on the number of data paths that should be carried by each hopping cell in the following lemma. Lemma 5.5 Suppose that cq -/31 E (0, 1) and r] E [O, 1- (a1 - /31)). Fore> 0 arbitrarily 3(1-") ( f3 ) small, each hopping cell is required to carry at most n-2-- " 1 - 1 +c data paths whp. Proof First consider the number of HDPs that must be carried by an arbitrary hopping cell, denoted by Nhdp· By assuming that all HDPs of the nodes in the hopping cells located at the same horizontal line pass through the considered hopping cell, we have an upper bound on Nhdp· Since the total area of these cells is given by ~ -v _ 2logn _ '-=" 1 2logn v ucun - n 17--- - n 2 n y2logn n ' the number of nodes in that area is upper bounded by l=2l_ 1 l=2l_ ~ n 2 ~4 log n = n 2 y 2 log n y 2 logn (5.12) (5.13) whp from Lemma 5.1 (a). Moreover, each of these nodes may become a source node of multiple nodes within the same traffic cell. Therefore, from Lemma 5.4 and (5.13) whp ( ) , '-=" Nhdp S nl-~- " 1 -(3 1 +c n 2 .j2 log n 3(1-") ( f3 ) ' ~ = n-2-- cq- i +c V 2logn (5.14) for e 1 > 0 arbitrarily small. The same analysis holds for VDPs. In conclusion, each hopping cell carries at most n 311 ;-" 1 -(<> 1 -f3 1 )+c data paths whp for e > 0 arbitrarily small, which completes the proof. We are now ready to prove our main result in (5.6). Let e 1 > 0 be an arbitrarily small constant satisfying 1- (a1 - /31) - e 1 > 0, which is valid since a1 - /31 E (0, 1). Set r] = 1 - (a 1 - (3 1 ) - e 1 . From Lemma 5.2, every node can find its source node within its 109 traffic cell whp. From Lemma 5.3, setting T = (2r(1+6)Y51+1) 2 , each hopping cell is able to achieve the aggregate rate of (5.15) Furthermore, from Lemma 5.5, the number of data paths that each hopping cell needs to perform is upper bounded by (5.16) whp, where we used rJ = 1 - (a1 - !"1) - E 1 . Since each hopping cell serves multiple data paths using round-robin fashion, each data path is served with a rate of at least (5.15) divided by (5.16) whp. Therefore, an achievable per-node throughput is given by W cq -/31 5 I cq -/31 Tn = -------~n--2--2t ~ n __ 2 __ , (2r(1 + 6)Y51 + 1 ) 2 (5.17) for f > 0 arbitrarily small. In conclusion, (5.6) holds. 5.4.2 a 1 - /31 = 1 In this regime, the following rate is achievable: Tn = n-~-E (5.18) We briefly explain how to achieve the above rate, since the procedures of proof are almost similar to the case of a 1 - ;3 1 E (0, 1). Similarly to Lemma 5.4, we can show that each node is able to be a source node of at most n' nodes whp for f > 0 arbitrarily small. Then, following the analysis in Section 5.4.1, we can easily prove that (5.18) is achievable. 110 5.5 Converse In this section, we assume qi = i-'Y /CLj~ 1 j-'Y) (Zipf distribution), with IE (0, 1). We first introduce the following two technical lemmas. Lemma 5.6 Let X follow a binomial distribution with parameters n and p, i.e., X ~ B(n,p). Then, fork E [O: np], Pr(XSk)Sexp( l(np-k)2)· 2p n Lemma 5. 7 For anyµ E [O, a1), the following holds: n" Jim '\'qi = 0. n---+ooL i=l (5.19) • Proof Letting f(q) = :>::f~ 1 i-'Y, we have :>::7~ 1 qi= f(nu)/ f(n"' ). Then, we obtain the lower and upper bounds: Using them, we have: where both bounds converge to zero since a1 > u. 5.5.1 a 1 - /31 > 1 Lemma 5.8 Suppose that a1 - /31 > 1. Let N 0 "' 1 denote the number of nodes that they cannot find their requested files in the entire network. Then, for any e > 0, we have whp N 0 "t,1 2 (1 - e)n. 111 Proof The total number of files that are able to be stored by the entire network is given by nM = nl+lii. Hence the probability that each node cannot find its requested file in the entire network is lower bounded by Then, forµ E [O,p 0 ",,1], we have: ni+/31 Pout,1 := 1 - L qi. i=l ( ) (a) ~ (n) i ( )n-i Pr Nout,1 ~ µn ~ _L i Pout,1 1 - Pout,1 i=µn 2 1 - ; ( 7) P~"'·1 (1 - Po"t,1r-i (>b) l ( (Po"t,1 - /.' )2 ) -exp - n - 2Pout,1 ' (5.20) (5.21) where (a) follows from (5.20) and the fact that each node requires a file independent of other nodes and (b) follows from Lemma 5.6. Notice that the condition µ E [O, Po"t,1] is required to apply Lemma 5.6. From Lemma 5.7, Po"t,1 -+ 1 as n-+ co. Hence, setting µ = 1 - f for f > 0 arbitrarily small in (5.21), which satisfies µ E [O,p 0 "t,1] as n -+ co, yields that No"t,1 2 (1 - E)n whp. This completes the proof. As said before, The non-vanishing outage probability implied by Lemma 5.8 yields that Tn = 0 in this regime. 5.5.2 a1 - /31 E (0, l] From Lemma 5.9 below, for f > 0 arbitrarily small, there are at least cin SD pairs whose distances are larger than n 1-(0:1-/31)+c 2 whp, where c1 > 0 is some constant and independent of n. Then, we restrict only on the delivery of the requests of such SD pairs, obtaining clearly an upper bound on the per-node throughput. First, we consider the exclusive area (i.e., the area to prohibit the transmission for other SD pairs) occupied by the multihop transmission of a SD pair with distance n 1-(cq -;3 1 )+c 2 In order to obtain a 112 n Destination Figure 5.3: A lower bound on the exclusive area occupied by the multihop transmission . . . 1-(<> 1-Jl1)+e of a SD pair with distance n- 2 • lower bound of such area, we assume that Ll = 0 and each receiver node is located at the distance of r from its transmitter node along with the SD line (see Fig. 5.3). Then, the exclusive area is lower bounded (i.e., only taking the blue-colored areas in Fig. 5.3) such as 2 1-(<> 1- Jl 1)+e l-(c"! - il 1 )+e 2Irr n- 2 / 2r = Irrn- 2 . (5.22) Hence, the maxim um number of SD pairs guaranteeing a rate of W over the entire network 1 1-( <>1-Jl1)+e W 1-( <>1-Jl1)+e of a unit area is upper bounded by n n 2 whp. As a result, Sn ::; 7 rr n 2 and accordingly, we have: (5.23) Notice that the per-node throughput in (5.23) increases as r decreases. On the other hand, it was shown in [GKOO, Section VJ that the absence of isolated nodes is a necessary condition for a nonzero Tn requiring that (5.24) for some constant c 2 > 0 independent of n. From (5.23) and (5.24), we derive an upper - (<>1 - il1) bound on the per-node throughput as Tn ::; n 2 +E for t: > 0 arbitrarily small, when a1 - /31 E (0, l]. 113 Lemma 5.9 Suppose that a1 - /31 E (0, 1]. Fore> 0 arbitrarily small, let No"t,2 denote the number of nodes that they cannot find their requested files within the distance of 1-(cq -/31 )+c whp n 2 from their positions. Then, we have Nout,2 ~ (1 - c)n. Proof For simplicity, denote ( = l-(<> 1 ~li 1 )+c. Let N1,ie denote the total number of files that are able to be stored by the area ofradius n-c. From Lemma 5.1 (b), the number of nodes in that area is upper bounded by (1 + 6)n 1 - 2 c whp. Hence N 1 , 1 e S (1 + 6)n 1 - 2 c1 = (1 + 6)n 1 - 2 c+li 1 whp. Then the probability that each node cannot find its requested file within the radius of n-( is lower bounded by (1+8)n 1 - 2 <:+/31 (1+8)n°'1-c Po"t,2 :=1 - L Pi = 1 - L Pi (5.25) i=l i=l whp. Then similarly to (5.21), we have: Pr(No"t,2 2 µn) 2 1 - exp (- (Po~'· 2 - µ)2 n) Pout,2 (5.26) for µ E [O,p 0 "t,2]. From Lemma 5.7, Po"t,2 -+ 1 as n -+ co. Hence settingµ= 1 - e in (5.26), which satisfies µ E [O, Po"t,1] as n -+ co, yields that No"t,2 2 (1 - e)n whp. This completes the proof. 5.6 Summary We considered a wireless D2D network in which nodes have cached information from a library of possible files. For such network, we proposed an order-optimal caching policy (i.e., file placement policy) and multihop transmission protocol. Interestingly, we showed that a distributed uniform random caching is order-optimal for the parameter regimes of interest as long as the total number of files in the library is less than the overall caching memory size in the network. i.e., a 1 - (3 1 E (0, 1]. Also, it was shown that a multihop transmission provides a significant throughput gain over one-hop direct transmission as in the conventional wireless ad hoc networks. As a future work, we will completely 114 characterize the throughput scaling of this network for an arbitrary Zipf exponent I (e.g., I 2 1). In this regime, a uniform random caching might not be an order-optimal policy since a subset of popular files are highly requested. 115 Part II: Shared Link Caching Networks 116 Chapter 6 Order-Optimal Rate of Caching and Coded Multicasting with Random Demands In this chapter, we focus on the shared link caching network. It is worthwhile to illustrate the contribution of this work in the following: 1. We recognize that the sub-optimality of the scheme analyzed in [NMA13] is due to the fact that files are partitioned according to their local popularity without considering the effects of the remaining system parameters ( n, m, M) on the "ag gregate user demand distribution". In particular, the probability with which each user requests files can be very different from the probability with which each file is requested by the aggregate users. The other limitation of [NMA13] is that the scheme for coded delivery (see details in Section 6.2) is applied separately for each file group, resulting in missed coding opportunities between different groups. We propose a different way to optimize the random caching placement, according to a caching distribution that depends on all system parameters, and not just the "local" demand distribution q. Also, we consider "chromatic number" index coding deliv ery scheme applied to all requested packets. We refer to this scheme as RAndom Popularity-based (RAP) caching, with Chromatic-number Index Coding (CIC). 2. For the proposed RAP-CIC, we provide a new upper bound on the achievable rate by bounding the average chromatic number of the induced random conflict graph. 117 By numerically optimizing this bound, we demonstrate the efficiency of our method and the gains over the method of [NMA13] by simulation. However, a direct analysis of the proposed scheme appears to be elusive. 3. For the sake of analytical tractability, we further focus on a simpler caching place- ment scheme where the caching distribution is a step function (some files are cached with uniform probability, and others are not cached at all) and a polynomial-time approximation of CIC, referred to as greedy constrained coloring (GCC). We refer to this scheme as Random Least-Frequently-Used (RLFU) caching, given its analogy with the standard LFU caching policy, 1 with GCC delivery, or RLFU-GCC. 4. We provide an information theoretic lower bound on the (average) rate achieved by any caching scheme, and show the order-optimality of the proposed achievabil- ity schemes for the special case of a Zipf demand distribution. To the best of our knowledge, these are the first order-optimal results under this network model for nontrivial popularity distributions. In addition, our technique for proving the con- verse is not restricted to the Zipf distribution, such that it can be used to verify average rate order-optimality in other cases. 5. Our analysis identifies the regions in which conventional schemes (such as LFU with naive multicasting) can still preserve order-optimality, as well as exposes the wide range of opportunities for performance improvements via RAP or RLFU, combined with CIC or GCC. We show that, as in the D2D setting of [JCM13b], when the Zipf parameter is 0 S I < 1, the average rate with random demands and the min-max rate with arbitrary demands are order-equivalent. On the other hand, when I> 1, the average rate can exhibit order gains with the respect to the min-max rate. 1 LFU discards the least frequently requested file upon the arrival of a new file to a full cache of size M files. In the long run, this is equivalent to caching the M most popular files. 118 We remark that while we consider simultaneous requests, the argument made in [MAN14b, MAN14a] to handle streaming sessions formed by multiple successive requests starting at different times holds here as well. Finally, it is interesting to note that, while RLFU-GCC becomes a special case of the general scheme described in [NMA13], the optimization carried out in this work and the corresponding performance analysis are new and non-trivial extensions. As pointed out in [NMA13], one would think that an approach based on uniformly caching only the iii S m most popular files has the disadvantage that " ... the difference in the popularities among the iii cached files is ignored. Since these files can have widely different popularities, it is wasteful to dedicate the same fraction of memory to each one of them. As a result, this approach does not perform well in general." In contrast, we show that for the Zipf case, the approach is order-optimal provided that the cache threshold iii is carefully optimized as a function of the system parameters. Also, in [NMA13] it was also pointed out that "Another option is to dedicate a different amount of memory to each file in the placement phase. For example the amount of allocated memory could be proportional to the popularity of a file. While this option takes the different file popularities into account, it breaks the symmetry of the content placement. As a result, the delivery phase becomes intractable and the rate cannot be quantified ... " In contrast, using the proposed RAP caching optimization and CIC delivery across all requested packets, it is possible to find schemes that significantly outperform previous heuristics and, again for the Zipf case, are provably order-optimal for all regimes of the system parameters n, m, M, /. The chapter is organized as follows. In Section 6.1, we present the network model and the problem formulation. The random caching and coded multicasting scheme is introduced in Section 6.2 and a general converse result for the achievable average rate is given in Section 6.3. In Section 6.4, we prove and discuss the order-optimality of the proposed scheme for the Zipf request distribution. Further results, simulations and conclusive remarks are presented in Section 6.5 and 6.6. 119 6.1 Network Model and Problem Formulation Consider a shared link network [MAN14b,MAN14a,NMA13] with file library F = {1, · · · , m }, where each file (i.e., message) has entropy equal to F bits, and user set U = {1, · · · , n }, where each user has a cache (storage memory) of capacity MF bits. Without loss of generality, the files are represented by binary vectors Wt E lFr The system setup is as follows: 1. At the beginning of time, a realization {Wt : f E F} of the library is revealed to the encoder. 2. The encoder computes the cached content, by using a set of IUI functions {Zu : JF2F-+ JF~F : u EU}, such that Zu( {Wt : f E F}) denotes the codeword stored in the cache of user u. The operation of computing { Zu : u EU} and filling the caches does not cost any rate, i.e., it is done once for all at the network setup, referred to as the caching phase. 3. After the caching phase, the network is repeatedly used. At each use of the network, a realization of the random request vector f = (fi, ... , fn) E Fn is generated. We assume that f has i.i.d. components distributed according to a probability mass function q = (q1, ... , qm), referred to as the demand distribution. This is known a priori and, without loss of generality up to index reodering, has non-increasing components qi 2 · · · 2 qm. 4. We let f = (!1, ... , fn) denote the realization of the random request vector f. This is revealed to the encoder, which computes a multicast codeword as a function of the library files and the request vector In this work we consider a fixed-to-variable almost-lossless framework. Hence, the multicast encoder is defined by a fixed-to variable encoding function X : lF2F x Fn -+ JF~ (where JF~ denotes the set of finite length binary sequences), such that X( {Wt : f E F}, f) is the transmitted 120 codeword. We denote by L( {Wt : f E F}, f) the length function (in binary symbols) associated to the encoding function X. 5. Each user receives X( {Wt : f E F}, f) through the noiseless shared link, and decodes its requested file Wtu as Wtu = .\u(X, Zu, f), where Au : IF~ x IF~F x Fn-+ IF~ denotes the decoding function of user u. 6. The concatenation of 1) demand vector generation, 2) multicast encoding and trans mission over the shared link and 3) decoding, is referred to as the delivery phase. Consistently with the existing information theoretic literature on caching networks (see Chapter 1), we refer to a content distribution scheme, formed by both caching and delivery phases, directly as a caching scheme, and measure the system performance in terms of the rate during the delivery phase only. In particular, we define the rate of the scheme as R(F) = sup {WJtE.F) JE[L( {Wt : f E F}, f)] F where the expectation is with respect to the random request vector. 2 (6.1) This definition of rate has the following operational meaning. Assume that the down load of a single file through the shared link takes one "unit of time". Then, (6.1) denotes the worst-case (over the library) average (over the demands) download time for the whole network, when the users place i.i.d. random requests according to the demand distribution q. The underlying assumption is that the content library (i.e., the realization of the files) changes very slowly in time, such that it is generated or refreshed at a time scale much slower than the time scale at which the users download the files. Hence, it is meaningful to focus only on the rate of the delivery phase, and disregard the cost of filling the caches (i.e., the cost of the caching phase), which is included in the code construction. Users make requests, and the network satisfies them by sending a variable length transmission 2 Throughout this work, we directly use "rate" to refer to the average rate defined by (6.1) and explicitly use "average (expected) rate" if needed for clarity. 121 until every user can successfully decode. After all users have decoded, a new round of requests is made. This forms a renewal process where the recurrent event is the event that all users have decoded their files. In the spirit of fixed-to-variable source coding, R(F) is the coding rate (normalized coding length) expressed in file "units of time". Also, by the renewal theorem, it follows that 1/R(F) yields (up to some fixed proportionality factor) the channel throughput in terms of per-user decoded bits per unit time. Finally, since the content library changes very slowly, averaging also over the realization of the files has little operational meaning. Instead, we take the worst-case over the file library realization. Consider a sequence of caching schemes defined by cache encoding functions {Zu}, multicast coding function X, and decoding functions {Au}, for increasing file size F = 1, 2, 3, . . .. For each F, the worst-case (over the file library) probability of error of the corresponding caching scheme is defined as PYl({Zu},X, {.\u}) = sup {WjfE.F) (6.2) A sequence of caching schemes is called admissible if limF-+oo Pe(F) ( { Zu}, X, { .\u}) = 0. Achievability for our system is defined as follows: Definition 6.1 A rate R(n, m, M, q) is achievable for the shared link network with n users, library size m, cache capacity M, and demand distribution q, if there exists a sequence of admissible caching schemes with rate R(F) such that Jim sup R(F) S R( n, m, M, q). F-+oo 0 We let R* ( n, m, M, q) denote the infimum (over all caching schemes) of the achievable rates. The notion of "order-optimality" for our system is defined as follows: 122 Definition 6.2 Let n, M be functions of m, such that limm-+oo n(m) = co. A sequence of caching schemes for the shared link network with n users, library size m, cache capacity M, and demand distribution q, is order-optimal if its rate R( n, m, M, q) satisfies 1 . R(n,m,M,q) im sup ( ) ~ v, m--+oo R* n, m, M, q (6.3) for some constant 1 S v < co, independent of m, n, M. 0 Notice that in the definition of order-optimality we let first F-+ co (required by the definition of achievable rate) and then we let m -+ co. In this second limit, we let n and M be functions of m, indicating that the notion of "order", throughout this work, is with respect to the library size m. Depending on how n and/or M vary with respect tom, we can identify different system operating regimes. 6.2 Random Fractional Caching and Linear Index Coding Delivery In this section we focus on a particular class of admissible schemes where the caching functions {Zu} are random and independent across the users [JCM13b, MAN14a] and the multicast encoder is based on linear index coding [BK98, BKLlO]. With random coding functions, two flavors of results are possible: 1) by considering the average rate with respect to the random coding ensemble, one can prove the existence of deterministic sequences of caching schemes achieving rate not worse than average; 2) by considering the concentration of the rate conditioned on the random caching functions, we obtain a stronger result: namely, in the limit of large file size F, the (random) rate is smaller than a given threshold with high probability. This implies achievability of such rate threshold by the random scheme itself (not only in terms of a non-constructive existence argument based on random coding). Here, we prove achievability in this second (stronger) sense. 123 6.2.1 Random Fractional Caching Placement The caching placement phase works as follows: 1. For some given integer B, each file Wt is divided into packets of equal size F / B bits, denoted by {WJ,b: b = 1, ... , B}. 3 2. Each user randomly selects and stores in its cache a collection of PJM B distinct packets from each file f E F, where p = (p1, ... ,Pm) is a vector with components 0 S Pf S 1/M, such that Lt~iPJ = 1, referred to as the caching distribution. 4 It follows that, for each user u, (6 .4) where bj,i is the index of the i-th packet of file f cached by user u, and where the tuples of indices (bj,i, ... , bj,p 1 MB) are chosen independently across the users u E U and the files f E F, with uniform probability over all (P/lrnJ distinct subsets of size PJM B of the set of packets of size B. The collection of the cached packet indices (over all users and all files) is a random vector, denoted in the following by C. For later use, a given cache configuration, i.e., a realization of C, will be denoted by C. Also, we shall denote by Cu,t the vector of indices of the packets of file f cached by user u. Finally, for the sake of notation simplicity, we shall not distinguish between "vectors" (ordered lists of elements) and the corresponding "sets" (unordered lists of elements), such that we write b ~ Cu,t (resp., b E Cu,t) to indicate that the b-th packet of file f is not present (resp., present) in the cache of user u. Observe that a random fractional caching scheme is completely characterized by the caching distribution p, where PJM denotes the fraction 3 Since we eventually take the limit for F --+ oo, for simplicity we neglect the fact that B may not divide F. 4 Note that Pi represents the fraction of the memory M allocated to file f. Hence, we let Pi be a function of n, m, M, q, but not a function of B. 124 of file f cached by each user. In Section 6.2.4, we shall describe how to design the caching distribution as a function of the system parameters. 6.2.2 Linear Index Coding Delivery Finding a delivery scheme for the caching problem in the shared link network is equivalent to finding an index code with side information given by the cache configuration C. It is clear that under the caching functions defined before, each user u requesting file fu needs to obtain all the packets Wfu,b with b ¢ Cu.Ju. It follows that a demand vector f, given the cache configuration C, can be translated into a packet-level demand vector Q, containing the packets needed by each user. Symmetrically with the notation introduced for the cache configuration, we denote by Q the corresponding random vector, and by Qu,f the packet-level demand restricted to user u and file f. In particular, if user u requests file fu, then Qu,f is empty for all f fc fu and it contains the complement set of Cu.Ju for f = fu· Following the vast literature on index coding (see for example [Jaf13,BK98,BKL10]), we define the side-information graph Sc,Q corresponding to the index coding problem defined by (C, Q) as follows: • Vertices of Sc,q: For each packet in Q (i.e., requested by some user), form all possi ble distinct labels of the form v = {packet identity, user requesting, users caching}, where "packet identity" is the pair (!, b) of file index and packet index, "user re questing" is the index of some user u such that b E Qu,f, and "users caching" is the set of all users u 1 such that b E Cu• ,f. Then, a vertex in Sc, Q is associated to each of such distinct labels. For simplicity of notation, we do not distinguish between label and vertex, and refer to "label v" or "vertex v" interchangeably, depending on the context. Notice that while "packet identity" and "users caching" are fixed by the packet identity and by the cache realization, the second label component ("user requesting") can take multiple values, since several users may request the same packet. 125 • Edges of Sc,q: For each vertex v, p(v), µ(v) and r)(v) denote the three fields in its label (namely, "packet identity", "user requesting", and "users caching"). Any two vertices v1 and v2 are connected by an edge if at least one of the following two The complement graph Hc,Q of the side information graph Sc,Q is known as the conflict graph. In particular, Hc,Q has the same vertices of Sc,Q and any two vertices v 1 and v 2 in Hc,Q are connected by an edge if both the following conditions are satisfied: 1) Example 1 We consider a network with n = 3 users denoted as U = {1, 2, 3} and m = 3 files denoted as F = {A, B, C}. We assume M = 1 and partition each file into B = 3 packets. For example, A= {A1,A2,A3}. Let p = g, ~,O}, which means that two packets of A, one packet of B and none of C will be stored in each user's cache. We assume a caching realization C is given by: C1,A = {A1, A2}, C1,B = {Bi}, C1,c = 0; Suppose that user 1 request A, user 2 request Band user 3 request C {f = {A,B,C}), such that Q = {A3, B1, B3, C1, C2, C3}. The corresponding conflict graph 1ic,Q is shown in Fig. 6.1. 0 A well-known general index coding scheme consists of coloring the vertices of the con- flict graph Hc,Q and transmitting the concatenation of the packets obtained by EXOR-ing the packets corresponding to vertices with same color. For any vertex coloring of the con- flict graph, vertices with the same color form (by definition) an independent set. Hence, the corresponding packets can be EXOR-ed together and sent over the shared link at the cost of the transmission of a single packet. Letting x(Hc,Q) denote the chromatic number of1ic,Q, the corresponding normalized code length is L( {Wt : f E F}, f) F x(Hc,q) B (6.5) 126 Figure 6.1: An illustration of the conflict graph, where n = 3, U = {1, 2, 3}, m = 3, F = {A, B, C} and M = 1. Each file is partitioned into 3 packets. The caching realization C and the packet-level demand vectors are given in Example 1. The color for each vertex in this graph represents the vertex coloring scheme obtained by Algorithm 6.1. In this case, this vertex coloring is the minimum vertex coloring, and therefore it achieves the graph chromatic number. since each coded packet corresponds to F / B coded binary symbols, and we have a total of x(1-lc,Q) coded packets (i.e., colors). For ease of reference, we denote this coding scheme as Chromatic-number Index Coding (CIC). In passing, we observe that, by design, the CIC scheme allows coding over the full set of requested packets Q, unlike t he scheme proposed in [NMA13], where coding is allowed within packets of specific file groups. Notice also that, by construction, CIC allows all users to decode their requested packets. Therefore, any sequence of CIC schemes yields probability of error identically zero, for all file lengths F, and not just vanishing probability of error in the limit. Hence, while in our problem definition we have considered a fixed-to-variable length "almost-lossless" coding framework, this class of algebraic coding schemes are fixed-to-variable length exactly lossless. 127 The graph coloring problem is NP-complete and hard to approximate in general [Zuc06]. However, it is clear from the above presentation that any coloring scheme (possi bly using a larger number of colors), yields a lossless caching scheme with possibly larger coding length. In particular, exploiting the special structure of the conflict graph orig- inated by the caching problem, we present in the following an algorithm referred to as Greedy Constrained Coloring (GCC), which has polynomial-time complexity of n, Band achieves asymptotically smaller or equal rate with respect to the exponentially complex greedy coloring algorithm proposed in [MAN14a] for the case of arbitrary demands. The proposed GCC is the composition of two sub-schemes, referred to as GCC1 and GCC2, given in Algorithms 6.1 and 6.2, respectively. Eventually, GCC chooses the coloring with the smallest number of colors between the outputs of GCC1 and GCC2 (i.e., the shortest codeword). Algorithm 6.1 GCC1 1: Initialize V =Vertex-set(1-lc,q). 2: while V fc 0 do 3: Pick any v EV, and let I= {v }. 4: for all v 1 E V /I do 5 if {There is no edge between v 1 and I } n { {µ( v 1 ), 17( v 1 )} = {µ( v ), 17( v)} } then 6: I=IU{v 1 }. 7: end if 8: end for 9: Color all the vertices of the resulting set I by an unused color. 10: Let V +- V \I. 11: end while Notice that {µ( v ), 17( v)} denotes the (unordered) set of the users either requesting or caching the packet corresponding to vertex v. Notice also that each set I produced by Algorithm 6.1 is an independent set containing vertices with the same set of users either requesting or caching the corresponding packets. In fact, starting from a "root" node v among those not yet selected by the algorithm, the corresponding set I is formed by all the independent vertices v 1 such that {µ(v 1 ),17(v 1 )} = {µ(v),17(v)}. It is also worthwhile to notice that GCC 2 is nothing else than "naive multicasting", that we have included here, for the sake of completeness, in a form symmetric to that of 128 Algorithm 6.2 GCC2 1: Initialize V =Vertex-set(1-lc,q). 2: while V fc 0 do 3: Pick any v EV, and let I= {v }. 4: for all v 1 E V /I do 5 if p(v 1 ) = p(v) then 6: I=IU{v 1 }. 7: end if 8: end for 9: Color all the vertices of the resulting set I by an unused color. 10: Let V +- V \I. 11: end while GCC 1 . In fact, it produces a set I (and a color) for each requested packet, which is then transmitted (uncoded) and simultaneously received by all requesting users. We can see that both the outer "while-loop" starting at line 2 and the inner "for-loop" starting at line 4 of Algorithm 6.1 iterate at most nB times, respectively. The operation in line 5 of Algorithm 6.1 costs at most complexity n. Therefore, the complexity of Algorithm 6.1 is O(n 3 B 2 ) (polynomial in n and B). Also, it is easy to see that this complexity dominates that of Algorithm 6.2. Therefore, the overall complexity of GCC 6.2.3 Achievable Rate As anticipated before, we shall consider the concentration of the (random) rate of the scheme described above, where the randomness follows from the fact that the caching functions, and therefore the conflict graph, are random. It is clear from the delivery phase construction that the output length of CIC or GCC does not depend on {Wt : f E F} but only on C and Q (see (6.5)), since the conflict graph is determined by the realization of the caches and of the demands. Therefore, without loss of generality, we can treat the files as fixed arbitrary binary vectors and disregard the sup over {Wt : f E F} in the 129 rate definition (see (6.1)). Given n, m, M, the demand distribution q and the caching distribution p, we define Rc1c( M q p)"' JE[x(1-lc,o)ICJ n,m, , , B (6.6) to be the conditional rate, given the cache configuration C, achieved by CIC. Similarly, we let RGCC(n,m,M,q,p) denote the conditional rate achieved by GCC, defined by (6.6) after replacing the chromatic number with the number of colors produced by GCC. The same definition applies to RGCC, ( n, m, M, q, p) and RGCC 2 ( n, m, M, q, p). In the following, it is intended that we consider the limit of the CIC and GCC schemes for F, B -+ co with fixed packet size F / B -+ constant. The performance of the proposed caching schemes is given by the following result. Theorem 6.1 For the shared link network with n users, library size m, cache capacity M, and demand distribution q, fix a caching distribution p. Then, for all e > 0, Jim I!" (RGCC(n, m, M, q, p) S min{,P(q, p), m} + e) = 1, F-+oo (6.7) where m in "' L (1 - (1 - q1 rl, (6.8) J~l and where (6.9) with (6.10) where D is a random set of£ elements selected in an i. i. d. manner from F (with replace- ment). Proof See Appendix C.1. 130 Remarks: 1. Since by construction RCIC( n, m, M, q, p) S RGCC ( n, m, M, q, p) for any realization of C, Q, then also RGCC ( n, m, M, q, p) stochastically dominates RCIC ( n, m, M, q, p). Therefore, Theorem 6.1 immediately implies limp-+ 00 1F (RCIC(n,m,M,q,p) S min{,P(q,p),m} + e) = 1. 2. As mentioned earlier, both RCIC(n,m,M,q,p) and RGCC(n,m,M,q,p) are func tions of the random caching placement C. Hence, not only IE[RGCC ( n, m, M, q, p)] S min{,P( q, p ), m }, but also the (conditional) rate RGCC(n, m, M, q, p) concentrates its probability mass all to the left of the bound min{,P( q, p ), m }, in the limit of F, B -+ co and F / B -+ constant. This means that in the large file limit, choosing a configuration of the caches that "misbehaves", i.e., that yields a rate larger than the bound of Theorem 6.1, is an event of vanishing probability. 3. The achievable rate in Theorem 6.1 is given by the minimum between two terms. The first term, ,P( q, p ), follows from the analysis of GCC 1 given in Appendix C.1. In particular, we show that limp-+ 00 1F (IRGCC,(n,m,M,q,p)-1/J(q,p)I Se)= 1, which means that RGCC, ( n, m, M, q, p) concentrates its probability mass at ,P( q, p ), as F, B -+ co and F / B -+ constant. The second term, m, is simply the average number of distinct requested files, which is a natural upper bound of RGCC 2 , the average number (normalized by B) of distinct requested (uncached) packets. As will be shown later, after careful design of the caching distribution p, the only case in which m < ,P( q, p) is in regimes of very small M, in which caching is shown to provide no order gains with respect to non-caching approaches such as conventional unicasting or naive multicasting of all requested files. Morover, in this regime, m becomes a tight upper bound of RGCC 2 • Accordingly, we disregard e and the fact that (6.7) involves a limit for F -+ co and identify the rate achieved by GCC directly as RGCC(n, m, M, q, p) = min{,P(q, p), m }. 131 4. The events underlying the probabilities Pfl, defined in the statement of Theorem 6.1, can be illustrated as follows. Let D be a random vector obtained by selecting in an i.i.d. fashion £ elements from F with probability q. Notice that D may contain repeated entries. By construction, IF(D = (!1, ... , Jc)) = f1f~ 1 qk Then, Pf/ is the probability that the element in D which maximizes the quantity (pjM)c- 1 (1- PjM)n-c+l is f. 5. For the sake of the numerical evaluation of 1/;( q, p ), it is worthwhile to note that the probabilities Pf/ can be easily computed as follows. Let J, Ji, ... , Jc denote £ + 1 i.i.d. random variables distributed over F with same pmf q, and define (for simplicity of notation) gc(j) "' (pjM)C- 1 (1 - pjM)n-C+l. Since gc(J1), · · · ,gc(Jc) are i.i.d., the CDF of Yi"' max{gc(J 1 ), ·· ,gc(Jc)} is given by IF(Yi Sy)= (IF(gc(J) S y))c = ( L qj)c jE:Fge(j)~y Hence, it follows that Pf/= IF(Yi = gc(f)) = ( L qj) c jE:Fge(j)~ge(f) which can be easily computed by sorting the values {gc(j) : j E F}. 6.2.4 Random Caching Optimization (6.11) (6.12) Driven by Theorem 6.1, we propose to use as caching distribution the one that minimizes the rate RGCC(n, m, M, q, p), i.e., p* = argmin min{1/;(q,p),m}, (6.13) p PJ9/M,'E;f PJ~l where 1/;(p, q) is given by (6.9) with PJ,f in (6.12) and mis given by (6.8). In the following, we refer to random caching according to the distribution p* as RAndom Popularity-based 132 (RAP) caching placement. Consequently, the caching schemes with RAP placement and CIC or GCC delivery will be referred to as RAP-CIC and RAP-GCC, respectively. The distribution p* resulting from (6.13) may not have an analytically tractable ex pression in general. This makes a direct analysis of the performance of RAP-CIC and RAP-GCC difficult, if not impossible. To this end, in the following we also consider a simplified caching placement according to the truncated uniform distribution p defined by: 1 Pf==, f S iii m Pt = 0, f 2 iii + 1 (6.14) where the cut-off index iii 2 M is a function of the system parameters. The form of pin (6.14) is intuitive: each user caches the same fraction of (randomly selected) packets from each of the most iii popular files and does not cache any packet from the remaining m - iii least popular files. If iii = M, this caching placement coincides with the least frequently used (LFU) caching policy [LNM+Ol]. If iii = m, p corresponds to caching packets at random, independently across users, with uniform distribution across all files in the library [MAN14a]. For this reason, we refer to this caching placement as Random LFU (RLFU), and the corresponding caching schemes as RLFU-CIC and RLFU-GCC. We will refer to RLFU with iii = m also as uniform placement (UP) and to UP-GCC as the scheme with UP as caching placement and GCC as delivery scheme. Similarly, we can define LFU-GCC. For later analysis purposes, we shall use a simplified upper bound on the rate of RLFU-GCC given by the following corollary of Theorem 6.1: Lemma 6.1 For any e > 0, the rate achieved by RLFU-GCC satisfies (6.15) 133 where (6.16) with Gm"' LT~i qf, and where m is defined in (6.8). Proof See Appendix C.2. For convenience, we disregard e and the fact that (6.15) involves a limit for F-+ co and refer directly to the achievable rate upper bound as Ru 1 (n, m, M, q, m) "'min { '7(q, m), m}, (6.17) where it is understood that the upper bound holds with high probability, as F -+ co. While RLFU-GCC is generally inferior to RAP-GCC, we shall show in Section 6.4 that RLFU-GCC is sufficient to achieve order-optimal rate when q is a Zipf distribution. In order to further shed light on the relative merits of the various approaches, in Section 6.5 we shall compare them in terms of actual rates (not just scaling laws), obtained by simulation. 6.3 Rate Lower Bound In order to analyze the order-optimality of RLFU-GCC, we shall compare Ru 1 (n, m, M, q, m) with a rate lower bound on the optimal achievable rate R* ( n, m, M, q). This is given by: 134 Theorem 6.2 The rate R(n, m, M, q) of any admissible scheme for the shared link net- work with n users, library size m, cache capacity M, and demand distribution q must satisfy R(n, m, M, q) 2 R 1 b(n, m, M, q) "'ma,ic{P1(£,r)P2(£,r,ZJ max~ z(l-M/l£/zj)l{z,r;:> 1}, C,r,z zE{l, ,rmm{z,r)lJ P1(£,r)P2(£,l,ZJ(l-M/£)l{zE (0,1)}} (6.18) where£ E {1, .. ., m}, r E lR+ with r S n£qc and z E lR+ with z S min { r, £ ( 1 - ( 1 -1 ri}, and where p (£ ) "' 1 _ ( (n£qc-r) 2 ) 1 , r exp 2 , , ncqc (6.19) and " ( (e(1-(1-irl-zl 2 ) P2(£,r,ZJ=l-exp - ( ( 1 )r) · 2£ 1- 1-y (6.20) Proof See Appendix C.3. 6.4 Order-optimality The focus of this section is to prove the order-optimality of the RLFU-GCC scheme in- traduced in Section 6.2.4, when q is a Zipf distribution. Using Lemma 6.1 and Theorem 6. 2, we shall consider the ratio Rub ( n, m, M, q, iii)/ R 1 b ( n, m, M, q) for n, m -+ co, where m -+ co and n, M are functions of m as in Definition 6.2. 5 According to Definition 6.2, RLFU-GCC is order-optimal if the ratio Rub( n, m, M, q, iii)/ R 1 b ( n, m, M, q) is uniformly bounded for all sufficiently large m. Obviously, order-optimality of RLFU-GCC im- plies order-optimality of all "better" schemes, em ploying the optimized RAP distribution and/or CIC coded delivery. 5 When M is not explicitly given in terms of m, it means that the corresponding scaling law holds for M equal to any arbitrary functions of m, including M constant, as a particular case. 135 We shall also compare the (order optimal) rate achieved by RLFU-GCC with the rate achieved by other possibly suboptimal schemes, such as conventional LFU caching with naive multicasting, 6 and the scheme designed for arbitrary demands, achieving the order-optimal min-max rate [MAN14b, MAN14a]. We shall say that a scheme A has an order gain with respect to another scheme B if the rate achieved by A is o(-) of the rate achieved by B. We shall say that a scheme A has a constant gain with respect to another scheme B if the rate of A is 8(-) of the rate of B, and their ratio converges to some r; < 1 as m -+ co. In addition, we shall say that some scheme A exhibits a multiplicative caching gain if its rate is inversely proportional to an increasing function of M. Specifically, we say that the multiplicative caching gain is sub-linear, linear, or super-linear if such function is sub-linear, linear, or super-linear in M, respectively. We notice that the behavior of the Zipf distribution is fundamentally different in the two regions of the Zipf parameter 0 SI< 1 and I> 1. 7 In fact, for I< 1, as m-+ co, the probability mass is "all in the tail", i.e., the probability LT~i qf of the most probable iii files vanishes, for any finite iii. In contrast, for I > 1, the probability mass is "all in the head", i.e., for sufficiently large (finite) iii, the set of most probable iii files contain almost all the probability mass, irrespectively of how large the library size m is. In the following, we consider the two cases separately. 6.4.1 Case 0 S / < 1 In this case, we have: 6 Recall that with conventional LFU every user caches the M most popular files and hence there are no coded multicast opportunities. In fact, it is straight forward to show that if we fix the placement scheme to (conventional) LFU in the shared link network, the best delivery scheme is naive multicasting. 7 The regime I' = 1 requires not more difficult but somehow different analysis because of the bounding of the Zipf distribution (see Lemma 1 in [JCM13b]). For the sake of brevity, given the fact that the analysis is already quite heavy, also motivated by the fact that most experimental data on content demands show I'-/::- 1 [BCF+99], in this work, we omit this case. 136 Theorem 6.3 For the shared link network with n users, library size m, cache capacity M, and random requests following a Zipf distribution q with parameterO SI< 1, RLFU GCC with iii = m yields order-optimal rate. The corresponding (order-optimal) achievable rate upper bound is given by Ru 1 (n, m, M, q, m) =min { (; - 1) ( 1 - (1 -1;;'.f), m}. Proof See Appendix C.4. Notice that, in short, Theorem 6.3 says that UP-GCC is order optimal for the heavy- tail regime 0 S I < 1. This also implies that, in this regime, the rate under random demands behaves, in terms of the order-optimal scaling, as the min-max rate under deterministic demands [MAN14a]. Intuitively, this result is due to the heavy tail property of the Zipf distribution with 0 S I < 1 such that, in this case, the random demands are approximately uniform over the whole library and, from Lemma C.1 in Appendix C.3, we know that the average rate under uniform random demands is order-equivalent to the min-max rate under arbitrary demands. Nevertheless, making use of the knowledge of the Zipf parameter may yield fairly large constant rate gains, especially for I close to 1. In particular, we can optimize the parameter iii as follows. Define H ( /, x, y) "' '£'f~x i-'Y and consider the bounds on the tail of the Zipf distribution given by the following lemma, proved in [JCM13b]: Lemma 6.2 If/ fc 1, then 1 1 11 11 11 1 --(y + 1) -"! - --x -"! S H(r, x, y) S --y -"! - --x -"! + -. 1 - / 1 - / 1 - / 1 - / x'Y (6.21) D 137 Notice that for the Zipf distribution with parameter /, the term Gm in Lemma 6.1 is written explicitly as Gm= zt~:i::i. Then, using Lemma B.1 in Lemma 6.1, we can write Ru 1 (n, m, M, q, iii) < ,P(q, iii) --1 1- 1-- +n 1--=-~ ( iii ) ( ( M)nm~:::) ( H(r, 1,iii)) M iii H(r, l,m) ~ (: - 1 + ( 1 - (:) l-~) n) , (6.22) where (a) follows from the fact that and H(l,1,iii) H(l,1,m) > m,ih-+oo ---+ (6.23) ( 1- ~1~~-~~(iii_+_1)_1_-~~--_-1_~-~~) _l_ml-~ __ 1_ + 1 1-~ 1-~ (1-(:) 1 -~)n (6.24) Minimizing the upper bound given by (6.22) with respect to iii, subject to MS iii S m, and treating m as a continuous variable, we obtain (6.25) Fig. 6.2, shows the significant gains that can be achieved by using RLFU-GCC with optimized iii (as given by (6.25)) compared to UP-GCC for a network with m = 50000, n = 50 and I = 0.9. For example, given a target rate of 20, UP-GCC requires a cache capacity M "' 2000, whereas RLFU-GCC with optimized iii requires only M "' 800. 138 \ \ 45 \ \ \ 40 ...... \ \ 35 Q) «l a: 30 " Q) &l 25 0.. ~ 20 15 10 \ \ ' \ ...... ·:· "'\ .. 1000 ' ' ·--,- 2000 - - - UP-GCC -- RLFU-GCC 3000 4000 5000 M Figure 6.2: Rate versus cache size for UP-GCC and RLFU-GCC with optimized m (see (6.25)), for n = 50, m = 50000, and Zipf parameter 'Y = 0.9. 6.4.2 Case / > 1 This case is more intricate and we need to consider different sub-cases depending on how the number of users scales with the library size: namely, we distinguish the cases of n = w (m'), n = e (m'), and n = 0 (m'). 6.4.2.1 Regime of "very large" number of users n = w (m' ) Theorem 6.4 For the shared link network with library size m, cache capacity M, random requests following a Zipf distribution q with parameter 'Y > 1, if m--+ oo and the number of users scales as n = w (m'), UP-CCC (i.e., m = m) achieves order-optimal rate. Proof Theorem 6.4 can be proved by following the steps of the proof of Theorem 6.3 in Appendix C.4. This is omitted for brevity. 139 6.4.2.2 Regime of "large" number of users n = 8 (m'I) Theorem 6.5 For the shared link network with library size m, cache capacity M, random requests following a Zipf distribution q with parameter I> 1, if m-+ co and the number of users scales as n = 8 (m'I), RLFU-GCC achieves order-optimal rate with the values of iii given in Table 6.1, for different sub-cases of the system parameters. The corresponding (order-optimal) achievable rate upper bound Ru 1 (n, m, M, q, iii) is also provided in Table 6.1. Proof Theorem 6.5 can be proved by following the same machinery of the proof of Theorem 6.6, provided in Appendix C.5. Since the regime n = 8(m'1) is a somehow restricted special case of the range of possible scaling laws of n and m, we do not provide the proof of Theorem 6.5 explicitly in this work. Interested readers can find the details in the extended report [JTLC15]. M ~ Ru 1 (n,m,M,q,ffi) m p 2 1 M20 m min{:-l+a(:))m} l 1 O<;M<l pam 2p'"m-l+o(m) O<p<l 1 l<;M<- p 1 1 p'"M'"m 2pim ( m ) ---l+o -- Mi-± Ml-1; 1 :-l+a(:) M2- m p Table 6.1: Order-optimal choice of iii and the corresponding achievable rate upper bound Ru 1 (n, m, M, q, iii) for RLFU-GCC with I> 1 and n = 8 (m'1). 140 Without loss of generality, we let the leading term of n in terms of m to be pm'! for some p > 0, i.e., n =pm'!+ o(m'I) form-+ co. Hence, we distinguish the following cases: • For p > 1, the network behaves similarly to the case of n = w(m'Y), and UP-GCC achieves order-optimal rate, which scales as 8 ( M) for M 2 1. • For 0 < p < 1, we distinguish three regimes of M, namely, 0 S M < 1, 1 S M < ~' and M 2 ~· The corresponding order-optimal value of iii varies from 1 1 1 p'Ym, via p'i M'im tom. This corresponds to the order-optimal caching placement varying from RLFU to UP. Correspondingly, the scaling law of the rate varies from e (p*m ), via e (;!;)toe(;), where the multiplicative caching gain of RLFU-GCC (with order optimal iii) varies from sub-linear to linear. 6.4.2.3 Regime of "small-to-moderate" number of users n = o (m'I) Theorem 6.6 For the shared link network with library size m, cache capacity M, random requests following a Zipf distribution q with parameter I> 1, if m -+ co and the number of users scales as n = o ( m '1), RLFU-G CC achieves order-optimal rate with the values of iii given in Table 6.2, for different sub-cases of the system parameters. The corresponding (order-optimal) achievable rate upper bound Ru 1 (n, m, M, q, iii) is also provided in the Table 6.2. Proof See Appendix C.5. Note that, for brevity, Table 6.2 excludes the small sub-region {1 S M < ~ 7 } n { M = 1 1 8(n?- 1 ) = tm?- 1 }, where r; > 0 is a positive constant. The order-optimal results and analysis of order-optimality of this region is considerably involved and, at the same time, this region captures just a small special case of the whole parameter space. Therefore, we have not included these results here. The full detailed results and analysis are shown in [JTLC15]. Since Theorem 6.6 contains several regimes, it is useful to discuss separately some noteworthy behaviors. We start by consider the case of n = o(m'f-l ), for which there are 141 M - Rub(n,m,M,q,iii) m OSM<l -' 2n±+o(n±) na M = o(m) M±n± 1 <; M = o ( n°',) ---l+o -- 2n± ( n± ) Mi--f; Mi--f; M = 8(m) = "1m m :-l+o(:) ma l<;M<- n M = o(m) M --+o -- n ( n ) Ma-1 Ma-1 w ( n°',) = M < ma n 11;1 = e(m) = tc1rr. M ----- +a 1 ( n n ) Ma-1 mu-1 ( ) ma min{:-l+o(:) )m} M2- m n Table 6.2: Order-optimal choice of iii and the corresponding achievable rate upper bound Rub(n, m, M, q, iii) for RLFU-GCC with I > 1 and n = o (m'1). Here, 0 < "1 < 1 indicates a fixed positive constant. two relevant regimes of M (see Table 6.2), namely, 0 <; M < 1 and 1 <; M < In particular: • If 0 <; M < 1, the achievable rate upper bound is 2n 1 h. This rate scaling can also be achieved by using naive multicasting for all the requested files from the file set 1 {1, · · · , n 7} and conventional unicasting for the requested files from the remaining 1 set { n 7+1, · · · , m }. It is not difficult to show (details are omitted) that the average 1 1 1 number of distinct files requested from the set { n 7+1, · · · , m} is n 7 +o( n 7). Hence, both the naive multicasting and the conventional unicasting of the requested files from the respective sets require rate equal to n 1 h in the leading order, such that the concatenation of the two delivery schemes requires rate 2n 1 h. In order to 142 achieve this (order-optimal) rate scaling, caching is not needed at all. We conclude that, in this regime of "small storage capacity" (M < 1), caching does not achieve any significant gain over the simple non-caching strategy described above, based on combining naive multicasting for the most popular files and conventional unicasting of the remaining less popular files. • For the case 1 S M < ";, 7 , we notice that the ass um pt ion n = o( m 1'-l) yields ";, 7 = w(m). Hence, the constraint M < ";, 7 is dominated by the obvious condition MS m, which always holds by definition. 8 Considering that n = o(m1'-l) implies that m = w ( n 7~ 1 ) , we distinguish the following three sub-cases: 1 S M = o ( n 7~ 1 ) , M = 8 ( n 7~ 1 ), and w ( n 7~ 1 ) = MS m. We now discuss in more details the two regimes 1 S M = o ( n 7~ 1 ) and w ( n 7~ 1 ) = M S m shown in Table 6.2. In the case 1 SM= o ( n7~') or, equivalently, w(M1'- 1 ) = n < ml'-1, if M = o(m), 1 1 then the order-optimal RLFU parameter is iii = M'Y n 7. In this case, the rate is 8 (M~=+), which exhibits order gain with respect to the rate obtained with UP, given by 8 (min {Ji'.}, m, n}). This also shows that the order-optimal average rate in this regime yields an order gain with respect to the min-max order-optimal rate [MAN14b,MAN14a]. We interpret this order gain as the benefit due to caching according to popularity. Intuitively, when I> 1 and the number of users is not very large (n = o(m1'- 1 )), only a limited number of files are requested with non-vanishing probability. Mean while, 1 S M = o ( n7~') and n = o (m1'- 1 ) imply that the cache capacity is M = o(m), i.e., only a sublinear number of files can be cached. Hence, it is crit- ically important to be able to focus on the files that deserve to be cached. We conclude that, in this case, caching according to the knowledge of the demand distribution makes a significant difference (in fact, a difference in the rate scaling order) with respect to UP. 8 If M 2: m, then each user can cache the whole library and the rate is trivially zero. 143 In the other regime, w ( n 7~ 1 ) = M S m, the cache size M can be large. In this case, LFU (obtained by letting iii = M) combined with the naive multicasting of the uncached requested files achieves order-optimal rate, which scales as 8 (M~- 1 ) (see Table 6.2). Again, this rate exhibits an order gain with respect to the min-max order-optimal rate. Intuitively, this is due to the fact that, in this case, users request relatively few files, most of which are the popular ones. Since the storage capacity is large, then LFU caching covers most of the requests and the source node only needs to serve the unpopular requests, which account for a vanishing rate (8 (M~- 1 ) = o(l)). In addition, we observe that the multiplicative caching gain becomes super-linear for I> 2. Then, we examine the case of o( m '1) = n = w ( m 'f-l), 9 where the number of users is relatively large. The relevant regimes of M (see Table 6.2) in this case are 0 S M < 1, 1 S M < ";, 7 , and ";, 7 S M < m. Keeping the order of n in m fixed and increasing 1 1 1 the order of M in m, the order-optimal iii varies from n 7 via M'i n 7 to m, indicating that the caching placement converges to UP instead of LFU, in contrast to the case of n = o (m'l- 1 ) considered before. This shows that in this regime, with the exception of the "small storage capacity" regime M < 1, LFU with naive multicasting fails to achieve order-optimality. In addition, as the order of M in m increases, the scaling law of the rate varies from 8 (n *),via 8 ( n + 1 ) to 8 (l\f). This indicates that the multiplicative Mi-:y caching gain goes from sub-linear to linear. 6.4.3 Remark We finally remark that in this work, for analytical convemence, we let n be a function of m such that n -+ co as m -+ co. However, when m is a constant independent of n, gWe do not discuss the case of n = 8 ( m 1 - 1 ) for the sake of brevity and ease of presentation. The corresponding result can be found in Table 6.2 and [JTLC15]. 144 and n -+ co, the shared caching networks behave similarly as that in the regime of "very large" number of users, namely, n = w (m'I) as shown in Theorem 6.4. In addition, by using the fact that m is finite, it can be shown that the ratio between Rub ( n, m, M, q, m) and R 11 (n, m, M, q) is upper bounded by 12 when n-+ co, which is shown in [JTLC15]. 6.5 Discussions and Simulation Results In Section 6.4, we have seen that, under a Zipf demand distribution, RLFU-GCC with m given in Tables 6.1 and 6.2, achieves order-optimal rate and so do RLFU-CIC, RAP-GCC and RAP-CIC. In all these schemes, once the cache configuration is given, the delivery phase reduces to an index coding problem. Despite the fact that for general index coding no graph coloring scheme is known to be sufficient to guarantee order-optimality [HL12], for the specific problem at hand we have the pleasing result that CIC and even the simpler GCC are sufficient for order optimality. While this result is proved by considering the RLFU-GCC scheme, for the sake of analytical simplicity, one would like to directly use RAP-GCC or, better, RAP-CIC, to achieve some further gain in terms of actual rate, beyond the scaling law. While the minimization of min{,P( q, p ), m} given in (6. 7) with respect top is a non-convex problem without a appealing structure, it is possible to use brute-force search or branch and bound methods [LD60] to search for good choices of the caching distribution p. Fig. 6.3 shows p* obtained by numerical minimization of the bound min{,P( q, p ), m} for a toy case with m = 3, M = 1, n = 3, 5, 10, 15 and demand distribution q = [0.7, 0.21, 0.09]. We observe how the caching distribution p*, which does not necessarily coincide with q, adjusts according to the system parameters to balance the local caching and coded multicasting gains. In particular, p* goes from caching the most popular files (as in LFU) for n = 3 to UP for n = 15. Recall from Theorems 6.4-6.6 that the optimized p follows this same trend, going from LFU (m = M) to UP (m = m) as n increases, while constrained to be a step function. This, perhaps surprising, behavior arises from the fact that even 145 if the "local" demand distribution q is fixed, when the number of users increases, the "aggregate" demand distribution, i.e., the probability that a file gets requested at least by one user, flattens. This effectively uniformizes the "multicast weight" of each file, requiring caching distributions that flatten accordingly. The corresponding achievable rate, Rccc(n, m, M, q, p*) with p* given by (6.13), is shown in Fig. 6.4, confirming the performance improvement provided by RAP-GCC. For comparison, Fig. 6.4 also shows the rates achieved by RLFU-GCC, Rccc(n, m, M, q, p) with m = 1, 2, and 3. <: 0 0.9 0.8 0.7 '5 0.6 ,Q ~ 0.5 O> <: f3 0.4 ro () 0.3 0.2 0.1 0 2 files D n=15 • n=10 • n=5 ..... • n=3 3 Figure 6.3: The optimal caching distribution p* for a network, where m = 3, M = 1 and n = 3, 5, 10, 15 and the demand distribution is q = [0.7, 0.21, 0.09]. As discussed in Theorems 6.4-6.6, the fact that the caching distribution adjusts to changes in all system parameters, and not just the demand distribution q, is a key aspect of our order-optimal schemes and one of the reasons for which previously proposed schemes have failed to provide order-optimal guarantees. In addition, unlike uncoded delivery schemes that transmit each non-cached packet separately, or the scheme suggested in [NMA13], where files are grouped into subsets and 146 (]) -ro a: 2.5 2 . -·-·-·-·- --·-· : : .... -..... . . . ..... : ............ +-.. ; ....... .. .. .. :. : : ,+-/'' . . . , , ,. ~ . . . . . , Jtf . : : : : .• , .... / · 0-·.;.0;.;..9 · .:-• ""'• •+-G-·+--O-+ -O- , f-· ..o-·- ·-- . : : : ... , .sr ~ 1.5 fy ,' . .... , . .. .rr .. , JI' Y "JI , ClJ Cii > <( 'JI . . ,, ,. ," y/' /f ~ ·.,t · I · " 'I' ~'' , ' i' .,/ 0.5 l , ... . : , . , ;,• , 2 4 6 8 10 12 n -•- p* .... ·+ · P=[1/31/31/3] - + - P=[0.5,0.5,0] -'f"- P=[1,0,0] 14 16 18 20 Figure 6.4: Rccc(n, m, M, q, p) for different caching distributions p and for a network with m = 3, M = 1 and n = 3, 5, 10, 15 and demand distribution q = [0.7, 0.21, 0.09]. Note that p = [1/3, 1/ 3, 1/ 3], p = [0.5, 0.5, O], and p = [1, 0, O J correspond to RLFU with m = 3, m = 2, and m = 1, respectively. coding is performed within each subset, another key aspect of the order-optimal schemes presented in this work is the fact that coding is allowed within the entire set of requested packets. When treating different subsets of files separately, missed coding opportunities can significantly degrade efficiency of coded multicasting. For example, in the setting of Fig. 6.4 with M = 1.5 and n = 20, by following the recipe given in [NMA13], 10 each of the m = 3 files becomes a separate group, delivered independently of each other, yielding an expected rate of 1.5, which can also be achieved by conventional LFU with naive multicasting. On the other hand, for this same setting, RLFU-GCC uses a uniform caching distribution and GCC over all request ed packets, yielding a rate of 0.5. 10 The achievable rate for the scheme proposed in [NMA 13] is computed based on a grouping of the files, an optimization of the memory assigned to each group, and a separate coded transmission scheme for each group, as described in [NMA13]. 147 w,-------,~,-------,~,--------,~,------;=>======il ---Q- UCP-GCC ~RS ~LFU-NM --+-- R..FU- GCC 35 5 10 15 20 25 30 35 40 45 w M (a) M (c) (b) (d) Figure 6.5: Simulation results for I = 0.6. a) m = 5000, n = 50. b) m = 5000, n = 500. c) m = 5000, n = 5000. d) m = 500, n = 5000. RLFU in this figure corresponds to the RLFU with optimized iii given by (6.26). In Figs. 6.5 and 6.6, we plot the rate achieved by RLFU-GCC, given by RGCC(n, m, M, q, p) with ~ . Rub( M ~ ) m = argm1n n, m, , q, m , (6.26) which can be computed via simple one-dimensional search. For comparison, Figs. 6.5 and 6.6 also show the rate achieved by: 1) UP-CCC (i.e., letting iii= m); 2) LFU with naive multicasting (LFU-NM), given by Lf=M+l (1- (1- q1r); and 3) the grouping scheme analyzed in [NMA13], which is referred to as "reference scheme" (RS). The expected 148 20 4:l ED ED m m ~ ® ® 200 M M (a) (b) 20 40 ro 80 100 120 14::> HD 1ED 200 M (c) (d) Figure 6.6: Simulation results for I = 1.6. a) m = 5000, n = 50. b) m = 5000, n = 500. c) m = 5000, n = 5000. d) m = 500, n = 5000. RLFU in this figure corresponds to the RLFU with optimized iii given by (6.26). rate is shown as a function of the per-user cache capacity M for n = {50, 500, 5000}, m = {500, 5000}, and I= {0.6, 1.6}. The simulation results agree with the scaling law analysis presented in Section 6.4. In particular, we observe that, for all scenarios simulated in Figs. 6.5 and 6.6, RLFU-GCC is able to significantly outperform both LFU-NM and RS. For example, when I = 1.6, m = 500 and n = 5000, Fig. 6.6(d) shows that for a cache size equal to just 4% of the library (M = 20), the proposed scheme achieves a factor improvement in expected rate of 5x with respect to the reference scheme and 8x with respect to LFU-NM. Interestingly, we notice that the reference scheme (RS) of [NMA13] 149 often yields rate worse than UP-CCC, a scheme that does not exploit the knowledge of the demand distribution. Computing the chromatic number of a general graph is an NP-hard problem and difficult to approximate [Zuc06]. However, for specific graphs (e.g., Erdos-Renyi ran dom graphs G(n,p)), the chromatic number can be approximated or even computed [Bol88,Luc91a,Luc91b,AK97]. In our case, by using the property of the conflict graph resulting from RAP or RLFU, we have shown that the polynomial time ( 0( n 3 B 2 )) greedy constrained coloring (CCC) algorithm can achieve the upper bound of the expected rate given in Theorem 6.1. Finally, we remark that, as recently shown in [SJT+14, JKc+15], especially when operating in finite-length regimes (B finite), one can design improved greedy coloring algorithms that, with the same polynomial-time complexity, further ex- ploit the structure of the conflict graph and the optimized RAP caching distribution t o provide significant rate improvements. This is confirmed by the simulation shown in Fig. 6.7, where in addition to RLFU-CCC, UP-CCC, and LFU-NM, we also plot the rate achieved by RAP-HgC, where HgC is the Hierarchical greedy Coloring algorithm proposed in [JKC+15], for a network with (a) m = n = 5, (b) m = n = 8, 'Y = 0.6, and a finite number of packets per file B = 500. M (a) (b) Figure 6.7: Two examples of the simulated expected rate by using RAP-HgC (B = 500). The comparison includes UP-CCC (B --+ oo), RLFU-CCC (B --+ oo and m given by (6.26)), LFU-NM (B--+ oo). In these simulations, 'Y = 0.6. a) m = n = 5. b) m = n = 8. 150 6.6 Summary In this Chapter, we built on the shared link network with caching, coded delivery, and random demands, firstly considered in [NMA13]. We formally defined the problem in an information theoretic sense, giving an operational meaning to the per-user rate averaged over the random demands. We analyzed achievability schemes based on random fractional (packet level) caching and Chromatic-number Index Coding (CIC) delivery, where the latter is defined on a properly constructed conflict graph that involves all the requested packets. In particular, any suboptimal (e.g., greedy) technique for coloring such conflict graph yields an achievable rate. Our bound (Theorem 6.1) considers a particular delivery scheme that we refer to as Greedy Constrained Coloring (GCC), which is polynomial in the system parameters. The direct optimization of the bound with respect to the caching distribution yields a caching placement scheme that we refer to as RAndom Popularity-based (RAP). For analytical convenience, we also considered a simpler choice of the caching distribution, where caching is performed with uniform probability up to an optimized file index cut-off value iii, and no packets of files with index larger than iii are cached. This placement scheme is referred to as Random Least Ftequencly Used (RLFU), for the obvious resemblance with conventional LFU caching. We also provided a general rate lower bound (Theorem 6.2). Then, by analyzing the achievable rate of RLFU-GCC and comparing it with the general rate lower bound, we could establish the order-optimality of the proposed schemes in the case of Zipf demand distribution, where order-optimality indicates that the ratio between the achievable rate and the best possible rate is upper bounded by a constant as m,n-+ co (with the special case ofm fixed and n-+ co treated apart). Beyond the optimal rate scaling laws, we showed the effectiveness of the general RAP CIC approach with respect to: 1) conventional non-caching approaches such as unicasting or naive multicasting (the default solution in today's wireless networks); 2) local caching policies, such as LFU, with naive multicasting; and 3) a specific embodiment of the 151 general scheme proposed in [NMA13], which consists of splitting the library into subsets of files with approximately the same demand probability (in fact, differing at most by a factor of two) and then applying greedy coloring index coding separately to the different subsets. Our scaling results point out that the relation between the rate scaling and the var ious system parameters, even restricting to the case of Zipf demand distribution, can be very intricate and non-trivial. In particular, we characterized the regimes in which caching is useless (i.e., it provides no order gain with respect to conventional non-caching approaches), as well as the regimes in which caching exhibits multiplicative gains, i.e., the rate decreases (throughput increases) proportionally to a function of the per-user cache size M. Specifically, we identified the regions where the multiplicative caching gain is either linear or non-linear in M, and how it depends on the Zipf parameter /. Finally, for the regimes in which caching can provide multiplicative gains, we characterized 1) the regions in which the order-optimal RAP-CIC converges to conventional LFU with naive multicasting, showing when the additional coding complexity is not required, and 2) the regions in which (cooperative) fractional caching and index coding delivery is required for order-optimality. 152 Chapter 7 Caching and Coded Multicasting: Multiple Groupcast Index Coding In this chapter, we consider the same setting of [MAN14b], where users make multiple requests instead of a single request. Our contribution is two-fold. First, by using the same combinatorial caching phase of [MAN14b], we generalize the coded delivery phase by using the directed (fractional) local chromatic number proposed in [SDL13] to the case of multiple groupcasting, where multiple means that each user makes L 2 1 requests and groupcasting means that one file or packet (see later) may be requested by several users. We show that order gains can be obtained by using the proposed scheme compared to the naive approach of using L times the delivery phase of [MAN14b]. Second, we present an information theoretical lower bound of the rate, and show that the proposed scheme meets this lower bound within a constant factor of at most 18. The chapter is organized as follows. Section section: network model introduces the network model and the precise problem formulation. The achievable scheme and its performance are presented in Section 7.2 and Section 7.3, respectively. The converse and optimality of the achievable scheme is presented in Section 7.4. Finally, we discuss the results in Section 7.5. 153 7.1 Network Model and Problem Formulation We consider a network with n user nodes U = {1, · · · , n} connected through a single bottleneck link to a server. The server has access to the whole content library F = {1, · · · , m} containing m files of same size of B bits. Each user node has a cache of size M files (i.e., MF bits). The bottleneck link is a shared deterministic channel that transmits one file per unit time, such that all the users can decode the same multicast codeword. At each time unit (slot), each user requests an arbitrary set of L files in F. Such requests form a matrix F of size L x n with columns fu = (fu,1, fu,2 ... , fu,L? corresponding to the requests of each user u E U. The caching problem includes two distinct operations: the caching phase and the delivery phase. The caching phase (cache formation) is done a priori, as a function of the files in the library, but does not depend on the request matrix F. Then, at each time slot, given the current request matrix F, the server forms a multicast codeword and transmits it over the bottleneck link such that all users can decode their requested files. Formally, we have: Definition 7.1 (Caching Phase) The caching phase is a map of the file library F onto the user caches. Without loss of generality, we represent files as vectors over the binary field lF2. For each u E U, let <Pu : lF2F -+ lFrF denote the caching function of user u. Then, the cache content of user u is given by Zu ~ ¢u(Wt : f = 1, · · · , m), where Wt E JF~ denotes the f-th file in the library. 0 Definition 7.2 (Delivery Phase) A delivery code of rate R(M, L) is defined by an encoding function 1./J : lF2F xFLxn -+ lF~(M,L)F that generates the codeword XF = 1/;( {Wt : f = 1, · · · , m }, F) transmitted by the server to the users, and decoding functions Au : JF~(M,L)F x JFrF x FLxn-+ JF~F such that each user u EU decodes its requested files as 0 154 The case of arbitrary demands can be formulated as a compound channel, where the delivery phase is designed in order to minimize the rate for the worst-case user demand. The relevant worst-case error probability is defined as Pe= maxmax max IF (wuJu.e fc Wtu.e). F uEU C=l, ... ,L (7.1) The cache-rate pair (M, R(M, L)) is achievable if there exist a sequence of codes { ¢, 1./J, { .\,, : u EU}} for increasing file size F such that limF-+oo Pe = 0. In this context, the system capacity R*(M, L) (best possible achievable rate) is given by the infimum of all R(M, L) such that (M, R(M, L)) is achievable. In the rest of the work, proofs are omitted for the sake of space limitation. 7.2 Achievable Scheme In this section, we present the proposed achievable scheme. Since worst-case demands are considered, we simply use the same packetized caching function defined in [MAN14b], which is optimal (within a fixed factor) for the case L = 1. For the sake of completeness, we describe the caching scheme in the following. Let t = ~n be a positive integer, then we consider the set P of all subsets T (combinations) of distinct users of size t. Each file is divided into GJ packets. For each file, we use T E P to label all the file packets. Then, user u will cache the packets whose label T contains u. We denote by S the set of all implicitly requested packets. Based on this caching scheme, we design a delivery scheme based on linear index coding (i.e.,¢ is linear function over an extension field of IF 2 ). In particular, we focus on encoding functions of the following form: for a request matrix F, the multicast codeword is given by XF = LWsVs = Vw, sES (7.2) 155 where Ws is the binary vector corresponding to packets, represented as a (scalar) symbol of the extension field IF~ with r; = 2F/(~), the v-dimensional vector Vs E IF~ is the coding vector of packet s and where we let V = [v 1 , · .v 151 ] and w = [w 1 , · · · ,w 151 ]T. The number of rows v of V yields the number of packet transmissions. Hence, the coding rate is given by R(M, L) = v / GJ file units. In order to design the coding matrix V for multiple group-casting, we shall use the method based on directed local chromatic number introduced in [SDL13]. The definition of directed local chromatic number is given as follows: Definition 7.3 (Directed Local Chromatic Number (I)) The directed local chro matic number of a directed graph Hd is defined as: (7.3) where C denotes the set of all vertex-colorings of1i, the undirected version of1id obtained by ignoring the directions of edges in Hd, V denotes the vertices of1id, N+(v) is the closed out-neighborhood of vertex v, 1 and c(N+(v)) is the total number of colors in N+(v) for the given coloring v. 0 An equivalent definition of the directed local chromatic number in terms of an opti- mization problem is given by: 1 Closed out-neighborhood of vertex v includes vertex v and all the connected vertices via out-going edges of v. 156 Definition 7.4 (Directed Local Chromatic Number (II)) Let I denote the set of all independent sets of 11, the directed local chromatic number of Hd is given by: Also, we have: minimize k subject to L XJ S k, \Iv EV IN+(v)nifO L XJ 2' 1, \Iv E V I:vEI XJ E {0, 1 }, VJ EI (7.4) (7.5) (7.6) 0 Definition 7.5 (Directed Fractional Local Chromatic Number) The directed frac- tional chromatic number is given by (7.4) when relaxing (7.6) to XJ E [O, 1]. 0 In the following, we describe the proposed scheme when tis a positive integer. When tis not an integer, we can simply use the resource sharing scheme for caching proposed iu [MAN14b] aud achieve convex combiuatious of the cases where t is au integer. Ju order to fiud the coding matrix V we proceed iu three steps [SDL13]: 1) coustruct iug the directed conflict graph Hd; 2) computing the directed local chromatic number xz(Hd) aud the correspoudiug vertex-coloring c; 3) coustructiug V by using the columns of the generator matrix of a (lcl,xz(Hd))-MDS code. 2 The detailed delivery scheme is described iu the following. 1) Coustructiou of the directed conflict graph Hd. • Consider each packet requested by a user as a distinct vertex iu 1{d. This means that if a certain packet is requested N times, for some N, because it appears iu the 2 According to the classical coding theory notation, an (p, v, d)-MDS code is a code with length p, dimension v, and minimum Hamming distance d = p - v + L An MDS code is able to correct up to p - v erasures. This implies that a linear MDS code has generator matrix of dimensions v x p such that any subset of R :S v columns form a v x R submatrix of rank R. 157 requests of different users, it corresponds to N different vertices in Hd. Hence, each vertex of Hc,w = (V, E) is uniquely identified by the pair v = {p( v), µ( v)} where p( v) indicates the packet identity associated to the vertex and µ( v) represents the user requesting it. Since each user caches G:::i) packets of each file, and each file is partitioned into GJ packets, the number of requested packets of each file for each user is GJ - G:::i) = (n~l) (by Pascal's triangle). Hence, the total number of files requested Ln corresponds to IVI = Ln(n~l) vertices in Hd. • For any pair of vertices v1, v2, we say that vertex (packet) v1 interferes with vertex v2 if the packet associated to the vertex v1, p(v1), is not in the cache of the user associated to vertex v2, namely, µ(v2). In addition, p(v1) and p(v2) do not represent the same packet. Then, draw a directed edge from vertex v2 to vertex v1 if v1 interferes with v2. 2) Color the directed conflict graph 1{d according to the coloring which gives the directed local chromatic number xz(Hd). The total number of colors needed to give the directed local chromatic number is denoted by lei. 3) Construct V 1 be the generator matrix of an MDS code with parameters (lei, xz(Hd)) over IF~· Notice that it is well-known that for sufficiently large r; such MDS code exists. Since we are interested in F-+ co and r; = 2F/(~), for sufficiently large F this code exists. 4) Allocate the same coding vector to all the vertices (packets) with the same color in e. Then, V is obtained from the MDS generator matrix V 1 by replication of the columns, such that each column in V is replicated for all vertices with the same corresponding color in e. Eventually, all packets are encoded using the linear operation in (7.2). The above constructive coding scheme proves the following (achievable) upper bound on the optimal coding length: 158 Theorem 7 .1 Under worst-case demand, the optimal coding length for the multiple group cast problem with integer t = nM /m satisfies R*(M L) < R 1 c(M L) =max xz(Hd) ' - ' F Gl (7.7) where the upper bound is achieved by the caching and linear coded delivery scheme seen above. D We can also consider a delivery scheme based on the directed fractional local chromatic number, as given in Definition 7.5, that achieves a generally better performance. For the optimality of index coding based on local chromatic number, where only the index coding delivery phase is considered for assigned node side information (i.e., without optimizing caching functions), it can be shown that the the gap between the proposed index code and the converse is bounded by the integrality gap of a certain linear programming (see [YN13]). 7.3 Performance Analysis To show the effectiveness of the proposed coded caching and delivery scheme, we first consider the two special points of the achievable rate versus L corresponding to L = 1, considered in [MAN14b], and L = m, i.e., when every user requests the whole library. Theorem 7.2 When each user makes only one request {L = 1), then the achievable rate of the proposed scheme satisfies (7.8) where RMN (M) is the convex lower envelope of min { n ( 1 - ;';;) H ~%n, m - M}, achieved by~e~~0~m4~ D 159 Theorem 7.3 When each user requests the whole library {L = m), then the achievable rate of the proposed scheme satisfies R 1 c(M,m) Sm- M. (7.9) D The rate m - M can also be achieved, with high probability for a large field size, by using random linear coding. Moreover, it can be shown that this is information theoretically optimal. Theorems 7.2 and 7.3 show that in the extreme regimes of L the proposed scheme performs at least as good as the state of the art scheme in literature. A general upper bound of the achievable rate is given by: Theorem 7.4 The achievable rate of the proposed scheme satisfies (7.10) where Rt{;(M,L) is the convex envelope (with respect to M) of D The qualitative performance of the proposed scheme is shown in Fig. 7.l(a), where we call the scheme that repeats the delivery scheme designed for one request (in [MAN14b]) L times as direct scheme. To measure the performance of the proposed scheme quanti tatively, we let L = o/m, where c./ E (0, 1). Then, let M be a constant, as m-+ co, we can see that the rate of the direct scheme is L · RMN, which scales as 8(m 2 ). While by using (7.10), the rate of the proposed scheme scales at most O(m). Thus, we can see the proposed scheme can have an order gain compared to the direct scheme. 160 In order to appreciate the gain achieved by the proposed scheme over the direct scheme for fixed parameters, we consider the case of n = m = 3 and M = 1. In this case, though, the requests can be arbitrary. Fig. 7.l(b) shows the worst-case (over the requests) coding rate versus L. We can observe that, even for small n, m and M, the gain achieved by the proposed scheme with respect to the direct scheme is fairly large. For example, for L = 2, the proposed scheme requires i file units to satisfy any request, while the directed scheme or random linear coding require 2 file units. Ra.te 0 (m) ,If L (a) (b) Figure 7.1: (a). A qualitative illustration of the different delivery schemes (Mis assumed to be not very small). L · RMN (blue curve) represents rate by using the scheme in [MAN14b] L times. Rrand (black curve) is the rate by random linear coding. R~~ (grey curve) is an upper bound of the achievable rate of the proposed scheme. R 1 c (red curve) represents the rate by the proposed scheme based on directed local chromatic number. (b ). An example of the rate by the proposed scheme. In this example, n = m = 3 and M = 1. In this figure, all the symbols have the same meanings as in Fig. 7.l(a). 7.4 Converse and Optimality To show the optimality (up to a constant factor) of the proposed scheme, we prove an information theoretic lower bound on the rate by using the cut-set bound technique: 161 Theorem 7 .5 Under worst-case demand, the optimal coding length for the multiple group cast problem satisfies R*(M,L) 2 R 11 (M,L) = { ( sM ) m-M} max max Ls - --m- , m . s~l, ,min{lyJ,n) ll:;cJj [r;l D Using Theorem 7.11, we show that: Theorem 7.6 The multiplicative gap between the upper and lower bounds satisfies RLC(M,L) 18 Rlb(M,L) S . (7.11) D From simulation, we observed that this multiplicative gap is generally smaller than 5. For example, when m = n = 100 and M = 20, the we found a gap always less than 3.88. 7.5 Discussions For the index coding problem, since the side information is assigned, it is generally difficult to find a constant multiplicative gap between the achievable rate and the converse. For example, in [HL12], the authors consider a random unicast index coding problem, where each user can cache (3m files uniformly and f3 E (0, 1). It is assumed that m = n and all the users request different files. In this case, the best multiplicative gap between the achievable rate and the converse lower bound is 8 ( 1 ~), which goes to infinity as n -+co. However, for the caching problem, due to the fact that we can design the caching functions, it is possible to find coding schemes with a bounded multiplicative gap for the worst-case demands. In the case treated in this work, for example, any scheme (possibly 162 non-linear) would provide at most a factor of 18x gain with respect to the proposed linear scheme. We notice from Section 7.3 that, to achieve the order gain compared with the di rect scheme, we need to code over all the L requested files simultaneously, in contrast with the repeated application of the scheme in [MAN14b] (direct scheme), where the user can decode instantaneously. Therefore, the proposed scheme may be useful in the FemtoCaching application, where each user in our system corresponds to a local server (a small-cell base station) serving L requests to its own local users on a much faster local connection. In this case, there is no natural ordering of the L requests such that there is no interest to decode them instantaneously, however based on the applications we can choose which requests combine together. 163 Chapter 8 Conclusions and Open Problems Motivated by the fact that video is responsible for 66% of the 100 x increase of wireless data traffic in the next few years, one important application of the caching networks is the video-on-demand service, where traditional methods for network capacity increase are expensive and do not exploit the unique features of video traffic, which are (i) video shows a high degree of asynchronous content reuse; (ii) storage is relatively cheap and the fastest-increasing resource in modern hardware. In this thesis, first, we considered a "infrastructure-less" caching network or (one-hop) D2D caching network, where each server (small cell base station, even user) has a lim ited storage capacity, can distributively make and serve requests. We proposed a caching strategy based on either deterministic or random assignment of packets of the library files, and a coded multicasting delivery strategy where the users send linearly coded messages to each other in order to collectively satisfy their demands. The achievable throughput is 8 (min { ~, ~}), which means that the throughput can linearly increase as M grows when nM 2 m, or it is possible that we can turn memory into bandwidth if nM 2 m. Surprisingly, this distributed network structure has negligible loss compared with the centralized network structure with an omniscient server, which shows power of distributed storage and computing in caching networks. In addition, We showed that the proposed approaches could achieve the information theoretic outer bound in terms of traffic load within a constant multiplicative factor. Due to the property of the wireless 164 medium, besides the gain of coded multicasting, it is also possible to exploit the gain of spatial reuse, namely, multiple links can be activated simultaneously if their interferences are manageable. It is natural to ask whether these two gains are cumulative, i.e., if a D2D network with both local communication (spatial reuse) and coded multicasting can provide an improved scaling law. Somewhat counterintuitively, we showed that these gains do not cumulate (in terms of throughput scaling law). In fact, these two issues are in contrast with each other. In addition, we showed that the spatial reuse gain of the D2D network is order-equivalent to the coded multi casting gain of single base station transmission, based on which we designed a practical non-packetized, decentralized and random caching placement and TDMA-based (Time Division Multiple Access) delivery algorithm such that the order-optimal throughput is achievable. Further, we proposed an holistic design of a novel network structure by combining the D2D caching network and the current cellular network, also exploiting the millimeter wave communication if applicable. The performance of this network structure and the proposed algorithms are evaluated by using realistic network deployment and channel conditions, which demon strates a significant gain compared with all the state-of-the-art schemes. In addition, we extended the analysis of the one-hop D2D caching networks to multihop D2D caching networks. Interestingly, the gain of using multihop communications is infinite compared to both D2D caching networks and shared link caching networks when the number of users and/or the number of files goes to infinity. Second, we studied the shared link caching networks, where there are one omnidirec tional sender with access to the entire library and many receivers with caching ability. In particular, when each user make one request, we designed a random packetized con tent placement and a coded multicasting delivery scheme that shows a "Moore's law" for throughput: in a certain regime of sufficiently high content reuse and/or sufficiently high aggregate cache capacity, namely, the storage capacity in the entire network is enough to cache only the popular contents, the per-user throughput increases linearly or even super-linearly with the local cache size, and it is independent of the number of users, 165 despite the fact that these users make independent video file requests. we proved that the proposed schemes are information-theoretic order-optimal, which is firstly shown in literature for this well-known open problem. In addition, we identified the regimes in which the proposed scheme exhibits order gains compared to state-of-the-art approaches. Further, we extended this results to the case when each user make multiple requests. The results showed that by coding across requests, order gains can be obtained compared to the scheme by treating each request independently. As an emerging area, the research of developing caching techniques (distributed stor age, computing and transmission) to significantly improve overall performance of the entire system has attracted considerably attention in recent years. Among the research we are currently pursuing or intend to pursue in the future, the following three directions are most interesting and potentially far-reaching: The study of caching network is just in its infancy. To fully understand the fundamen tal limits and the feasibility of the joint design of caching and delivery, it is necessary to study more general and practical caching network structures, which opens up new oppor tunities for developing real caching technologies for today's networks. There are several possible extensions of previous works (by us and other researchers). We illustrate some of them in the following. First, for wireless caching networks, the channel in previous works was assumed to be deterministic (error-free) rather than random (e.g., Gaussian, fading channel), one natural extension is to incorporate the realistic channel model into the caching networks. Second, the works in literature mainly focused on the fundamental limits of caching networks, it is important to develop practical coding scheme for basic networks (shared link, D2D caching networks) and formulate a optimization framework by using the results from the fundamental limits such that practical algorithms can be designed. The advancement of Big Data and Internet of Things {Io T) make it beneficial is to develop fundamental theories of the unified framework for (distributively) storing, pro cessing and transmitting big data for different networks. It is important to develop a 166 Shannon Theory (fundamental theory) for distributed storage, computing and transmis sion of big data in the network, where some nodes have (constrained) memory, some nodes can compute, some nodes place demands (function of the sources), and some nodes can serve other nodes. Clearly, the caching networks considered in this thesis are examples of such systems. Other examples are: Distributed Message-Passing System, Distributed Storage and Distributed Processing System (e.g., Apache Hadoop). The key aspects for this research are to formulate clean problem definitions, and identify the metrics and parameters of the system, such as storage capacity, repairability, liveness, atomicity, (transmitting, processing) energy, communication cost (prefetching, updating, serving), complexity and bandwidth. The goal is to study the trade-offs between these metrics and parameters. We illustrate several relevant and interesting topics in the following. First, for the distributed caching networks described previously, it is natural to ask whether it is possible to achieve some degree of repairability while keeping the minimum trans mission rate. In Chapter 2 ( [JCM14]), we showed that for some simple examples (see Fig. 2.3), it is possible to achieve the maximum efficiency of repairability and guaran tee the minimum transmission rate simultaneously. However, in general, the trade-off between the repairability and the transmission rate is still open. Second, motivated by (but unlike) the caching networks, where each node wants to recover the original files, instead, each node may want to retrieve a general function of the sources (e.g., average over any two sources). One of such applications is smart home/city system. Third, from the system level point of view, it is important to bring the queueing models (statistics of the requests arrive/service time) into the system (e.g., parallel/distributed systems with the programming model of map-reduce). The statistics of users' behavior plays an important role in our daily life due to the development of social networking services (Facebook), video streaming services (Netflix), etc. In addition, from our analysis of caching networks, it is also essential to have some knowledge about user demands' statistics. For exam pie, with the knowledge of the users' demand distributions (assumed to be independent, identical over users and time), the 167 achievable throughput can have infinite gains as number of users becomes large compared with that designed for the uniform demand distribution. In reality, rather than simpli fied assumptions, it is critically important to predict the real joint distribution/statistical process of users' demands, or characterizing the properties of the statistics for users' be havior (e.g., closeness centrality/capturing the most influential users), possibly by using tools from machine learning (e.g., collaborative filtering), compressed sensing and social networks (e.g., clustering method). Although a number of methods have been developed and studied, there are still many open questions to be addressed. For example, what model should be used to capture users' social behavior for different purposes (e.g., ad vertising, video streaming)? What tools should be used to predict users' joint demand distribution/statistical process based on the users' social behavior? What is the influence of the inaccuracy in the estimated statistics, or what is the key properties of the user behaviors' statistics, to the targeted systems? Exploring the answers to these questions are key research opportunities that are needed to be pursed in the future. 168 Appendix A Proofs of Chapter 2 A.1 Deterministic Caching and Delivery Schemes: General Case, r > \/2 In this section, we generalize the the deterministic caching and coded delivery scheme illustrated in Chapter 2.2.3 through an example to the general case of any m, n and M, such that t "' 11nn ;:> 1 is an positive integer. When tis not an integer, we can use a resource sharing scheme as in the examples at the end of this section (see also [MAN14b,MAN14a]). • Cache Placement: The cache placement scheme is closely related to the scheme in [MAN14b]. Recall that U = {1, 2, · · · , n} denotes the set of user nodes and Wj denotes packet i of file f. We divide each packet of each file into t G) subpackets. Letting T denote a specific combination of t out of n elements, we index each subpacket by the pair (T, j) with j = 1, ... , t, such that the subpackets of Wj are indicated by {Wj·T,j}. Node u caches all the subpackets such that u E T, for all f = 1, ... , m and i = 1, ... , L 1 , such that the cache function Zu is just given by the concatenation of this collection of subpackets. • Delivery and Decoding: For the delivery phase, let k = 1, · · · , L 11 and denote Su + k - 1 as the index of the k-th packet of file fu requested by user u. As a consequence of the caching scheme described above, any nodes subset of size t + 1 in U has the property that the nodes of any of its subsets of size t share t 169 subpackets for every packet of every file. Consider one of these subsets, and consider the remaining (t+ 1)-th node. For any file requested by this node, by construction, there are t subpackets shared by all other t nodes and needed by the ( t + 1 )-th node. Therefore, each node in every subset of size t + 1 has t subpackets, each of which is useful for one of the remaining t nodes. Furthermore, such sets of subpackets are disjoint (empty pairwise intersections). For delivery, for all subsets of t + 1 nodes, each node computes the XOR of its set oft useful subpackets and multicasts it to all other nodes. In this way, for every multicast transmission exactly t nodes will be able to decode a useful packet using "interference cancellation" based on their cache side information. In order to illustrate the caching and delivery scheme described above, we consider a few exam pies. Example 2 Consider a network with n = 2, m = 2 and M = 1, with t = 1. The two files are denoted by W1 and W2. First, we divide each packet into tG) = 2 subpackets. In this case, TE {{1}, {2}}. Hence, the subpacket labeling is W 1 = {(w;·{l},l' w;·{2},l): . _ l LI} d HT _ {('"i,{1},1 0 ui,{2},1) . · _ l LI} Th h · b . i- , ... , an vv2- vv 2 ,vv 2 .i- , ... , . ecac es are given y. Zl {w i,{1},1 0 ui,{l},l . · = 1 LI} 1 ,vv 2 .i , ... , Z 2 {w i,{2),1 0 ui,{2},1 . · = 1 LI} 1 ,vv 2 .i , ... , Assuming that, without loss of generality, user 1 requests packets [s1 : s1 + L 11 - 1] of file W1 and users 2 requests packets [s2 : s2 + L 11 - 1] of file W2, User 1 sends W~,{l},l : i = + L 11 1 t 2 d 2 d wi,{ 2 L 1 · · - + L 11 1 t s2, ... , s2 - o user , an user sen s 1 . i - s1, ... , s1 - o user 1. The transmission rate is R(l) = 2 x ~ = 1 (recall that the rate is expressed in number of equivalent transmissions of blocks of F bits). 0 Example 3 Consider the example of Section 2.2.3 expressed in the general notation. We haven= m = 3 and M = 2, yielding t = 2. Packets are divided into t(~) = 6 subpackets, 170 .th th f ll . l b l. . f f = 1 2 3 l t w = {(Wi,{1,2),1 wi,{1,2),2 wi,{1,3},1 wi,{1,3J,2 wi,{2,3J,1 l wi e o owing a e ing. or , , , e f f , f , f , f , f , i = 1, ... , L 1 }. The caches are given by: z 1 {Wi,{1,2),1 wi,{1,2),2 wi,{1,3},1 wi,{1,3J,2 .. = 1 LI f = 1 2 3 } f ' f ' f ' f .i , ... , ' '' z 2 {Wi,{1,2),1 wi,{1,2),2 wi,{2,3),1 wi,{2,3J,2 .. = 1 LI f = 1 2 3 } f ' f ' f ' f .i , ... , ' '' z 3 {Wi,{1,3},1 wi,{1,3},2 wi,{2,3),1 wi,{2,3J,2 .. = 1 LI f = 1 2 3 } f ' f ' f ' f .i , ... , ' '' . Assuming, without loss of generality, that user u requests packets [su : Su+ L 11 - 1] from file Wu, for u = 1, 2, 3 and some arbitrary segment indices s1, s2, s3, we apply now the delivery scheme according to the general recipe described above. We have a single subset of size t + 1 = 3, namely {1, 2, 3}. Each user u has t subpackets useful for the other two users, and such that the sets of such subpackets are disjoint. The choice of the sets is not unique. For example, the following choice of coded multicast messages is possible: X1,{1,2,3J X2,{1,2,3} x3,{1,2,3J w; 2 +i,{1,3},1 Ell w;s+i,{1,2J,1, i = l, ... , L11 _ l Wt' +i,{2,3},1 Ell w;s+i,{1,2J,2, i = l, ... , L11 _ l wt' +i,{2,3),2 EB w;2+i,{1,3J,2, i = l, ... , LI!_ 1. As already given in Section 2.2.3, the rate in this case is R(M) = 3 x Ji = ~. 0 Example 4 This example illustrates the strategy when t is not an integer. In this case, we use a cache sharing scheme achieving the lower convex envelope of the rates corresponding to the two integer values ltJ and rtl. Consider a network with n = 2, m = 3 and M = 2, yielding t = Mn/m = 4/3, between 1 and 2. Fort= 1, m = 3 and n = 2, we obtain M1 = 3/2. Fort = 2, m = 3 and n = 2, we obtain M2 = 3. Hence, the cache sharing scheme uses a fraction a such that aM1 + (1 - a )M2 = M = 2, yielding a = 2/3. We allocate 3/2 · 2/3 = 1 storage capacity to the caching placement for M1 = 3/2 and 3 · 1/3 = 1 storage capacity to the caching placement for M2 = 3. The details are in the following. 171 We divide each packet of each library file Wf, f = 1, 2, 3 into 2 subpackets with size aF and(l - a)F, respectively. Since a = 2/3, we denote the resulting packets of W.f as W{J,*) and W{J.~). Then, the packets W{J,*) are stored according to the scheme for M1 = 3/2, t = 1. In particular, each W{J,*) is divided into tG) = 2 subpackets with TE { {1}, {2} }. For f = 1, 2, 3, the subpacket labeling is i = 1, · · · ,L 1 • (A.1) Similarly, the packets W{J.~) are stored according to the scheme for M2 = 3, t = 2, with tG) = 2 subpackets and T = {1, 2}. For f = 1, 2, 3, the subpacket labeling is i = 1, · · · ,L 1 • (A.2) As a result, the caches are given by: Z = {wi,{1J,1 wi,{1,21,1 .. = 1 LI f 1 2 3} 1 {!,*}' {!.~) .i ,···, '= '' (A.3) Z = {wi,{2J,1 wi,{1,21,1 .. = 1 LI f 1 2 3} 2 {!,*}' {!.~) .i ,···, '= '' (A4) Assuming that, without loss of generality, user 1 requests packets [s1 : s1 + L 11 - 1] of file W1 and users 2 requests packets [ s2 : s2 + L 11 - 1] of file W2, user 1 sends W{~:~}ll : i = s2, · · · , s2 + L 11 - 1 to user 2 and user 2 sends W{i,{~)),l : i = s1, · · · , s1 + L 11 - 1 to user 1, 3 1, such that the transmission rate is R(2) = 1/3 · 2 = ~- It is also interesting to compute the converse (rate lower bound) for this case. In this case, for the sake of clarity, we use the same notation used in the example of Section 2.2.3 {see Fig. 2.4). In particular, we label the three files as A, B and C, and let Xu,I denote the codeword sent by user u = 1, 2 in the presence of the request vector f. Consider user 2. From the cut that separates (X1,(A,B),X1,(B,C),Xl,(c,A),Z2) and (W2,A, W2,B, W2,c), by using the fact that the sum of the entropies of the received messages and the entropy 172 of the side information (cache symbols) cannot be smaller than the number of requested information bits, we obtain that L' y;rr L ( RL,(A,B) + RL,(B,C) + RL,(C,A)) + MFL 1 2 3FL 1 • s=l (A.5) Similarly, from the cut that separates (X2,(A,B), X2,(B,C), X2,(C,A), Z1) and (W1,A, W1,B, W1,c ), we obtain L' y;rr L ( RL,(A,B) + RL,(B,C) + RL,(C,A)) + MFL 1 2 3FL 1 • s=l By adding (A.5) and (A.6), we have L' y;rr '\' (RT + RT + RT + RT L,, 1,s,(A,B) 2,s,(A,B) 1,s,(B,C) 2,s,(B,C) s=l +RL,(C,A) + RL,(C,A)) + 2MFL 1 2 6FL 1 . (A.6) (A.7) Since we are interested in minimizing the worst-case rate, the sum RT 1 1 + RT 2 1 must ,s, ,s, yields the same min-max value RT for any s and f. Then, (A. 7) becomes 3RT + 2M F L 11 2 6F L 11 , which yields RT> 2FL 11 - 2 FL 11 M. - 3 (A.8) Finally, dividing by F L 11 we have RT 2 R*(M) = - > 2- -M. FL11 - 3 (A.9) For M = 2, we have R*(2) > 2 - ~ · 2 = ~' which shows the optimality of the cache sharing scheme in this case. 0 173 A.2 Proof of Theorem 2.1 and Corollary 2.1 With the caching placement and delivery schemes of Appendix A.1 with integer t = 11nn, any node in any subset of t + 1 nodes can transmit a subpacket that is useful for all the other t nodes of the subset. Any subset oft+ 1 nodes corresponds to t + 1 (coded) transmissions, each of which has block length :(~') bits. In total, we have (t + 1) (,~ 1 ) transmissions. Therefore, the total transmission length is RT= (t + 1) ( n ) t + 1 Using t = 11nn in (A.10), we have FL" tGJ. (A.10) (A.11) Finally, using the definition of rate, we have that the rate of the scheme is given by (A.12) When t is not an integer, it is easy to see that the convex lower envelope of (A.12) is achievable (see Example 4 in Appendix A.1). A.3 Proof of Theorem 2.2 In this section we generalize the cut-set bound method outlined in Section 2.2.3. Consider a two-dimensional augmented network layout of the type of Fig. 2.4, where a "column" of nodes corresponds to a user and a "row" of nodes corresponds to a demand vector f (for example, in Fig. 2.4 the right-most column corresponds to user 1, and the top row corresponds to f = (A, B, C)). Directly by the problem definition, the cache message Zu is connected to all nodes of column u, and the coded (multicast) message Xu,! is connected to all nodes v fc u of row f. In general, such graph has n columns and mn 174 rows. However, it is clear that by applying cut-set bound inequalities to the subgraph including only subset of such rows, i.e., a subset of the possible demand vectors, we obtain a lower bound to the best achievable rate R*(M). In particular, for the bound of Theorem 2.2, we need to consider two types of cuts. The first type includes m requests vectors {fj : j = 1, ... , m} (i.e., m rows of the graph) constructed as follows. Consider the semi-infinite sequence periodic concatenation of the integers [1, 2, ... , m, 1, 2, ... , m, 1, 2, ... ). Then, fj is the vector of length n formed by the components [j : j + n - 1] of such concatenation. For example, in the case m < n, the first few demand vectors in this set are fi {1, 2, 3, · · · , m - 2, m - 1, m, 1, 2, 3, · · · , m - 1, m, · · · }, f2 {2, 3, 4, · · · , m - 1, m, 1, 2, 3, 4, · · · , m, 1, · · · }, while if m 2 n they are f1 {1,2,3, .. ·,n-2,n-l,n}, f2 {2,3,4,··· ,n-1,n,n+ 1}, Using the fact that the sum of the entropies of the received messages and the entropy of the side information (cache symbols) cannot be smaller than the number of requested information bits, for each v E U, the cut-set bound applied to the cut that separates { Zv, { Xu,1* : j = 1, · · · , m} : Vu fc v} and {Wu,t : f = 1, · · · , m} yields L' YJT n m L L LR"E,s,1*+MFL 1 2'mFL 1 • (A.13) s=l u=l,u#v j=l 175 By summing (A.13) over all v EU, we obtain L' n YJT n m LL L LR"E,s,lj + nMFL' 2 nmFL'. (A.14) v=l s=l u=l,u#v j=l Since we are interested in minimizing the worst-case rate, the sum L~=l RJ sf. must ' ' ] yields the same min-max value RT for any sand fj. This yields the bound L' ( ) T ' ' L" n - 1 mR + nM FL 2 nmF L . (A.15) Dividing both sides of (A.15) by (n- l)mFL' and using the definition of rate R(M) = RT /(FL"), we conclude that the best possible achievable rate must satisfy R*(M) 2 _n_ (1 - M), n-1 m (A.16) which is the second term in the max in the right-hand side of (2.8). Notice that this bound provides the tight converse for Exam pies 2, 3 and 4 in Appendix A.1. For the second type of cut, for l = 1, · · · , min { m, n }, we consider the first l users, with l T J requests vectors 1 fj = {l(j - 1) + 1, ,jl}, (A.17) From the cut that separates { { Zv : v = 1, ... , l}, {{ Xu,lj : j = 1, , l 7 J} II u EU}} and { {Wv,(j-l)l+v : j = 1, · · · , l 7 J} : V = 1, · · ·, l}, 1 With some slight abuse of notation, here we focus only on the first l components of the request vectors and yet we indicate these vectors by fj, meaning that the other n-l elements are irrelevant for the bound. 176 we obtain the inequality L' lmJ YJT T n LL LR"!:,s,lj +lMFL' 2 z l 7 j FL'. s=l j=l u=l (A.18) Since we are interested in minimizing the worst-case rate, the sum L~=l RJ sf. must ' ' ] yields the same min-max value RT for any sand fj. This yields the bound which can be written as ~:, l7J RT +lMFL' 2 z l7J FL', R T> (z - - 1 -M) FL" - l 'f'J ' It follows that the optimal achievable rate must satisfy ( l ) R*(M);:> max l---,;;;-M, lE{l,2, ,mrn{m,n)) l T J which is the first term in the max in the right-hand side of (2.8). A.4 Proof of Theorem 2.3 and Corollary 2.3 (A.19) (A.20) (A.21) We let G = J::.(tf)) denote the multiplicative gap between the rate achievable by our scheme and the best possible achievable rate. Upper bounds on Gare obtained by bound- ing the ratio between the achievable rate our the proposed schemes and the converse lower bound of Theorem 2.2. From Theorem 2.1 we have n R(M) < - -1. - lt J (A.22) Also, it is immediately evident that l;J S 2 for all t 2 1. In order to prove Theorem 2.3, we distinguish between the cases n = w(m) and n = O(m). 177 A.4.1 Casen=w(m) In this case, by using (A. 22), we have R(M) n < ---1 l n,;i:r J :-1 +o (:) (A.23) A.4.1.1 When ~SM= o(m) Let l* = l 2 M J, then by using Theorem 2.2, we obtain R*(M) (A.24) so that we can write (A.25) A.4.1.2 When M = 8(m) • If 2 M 2 3, let l* = l 2 M J, then by using Theorem 2.2, we get R*(M) (A.26) 178 which yields m_l+o(m) 2 GS M-"'--l M S 1 MS 6+o(l). 2 M 2 +o(l) 2- m (A.27) • If 2 M < 3, let l* = 1, by using Theorem 2.2, we obtain R*(M);:>l-M. m (A.28) Then, we have GS J?t-/~;;_(J?t) S :+o(l) S6+o(l) (A.29) m A.4.1.3 When M < ~ Let l* = m, by using Theorem 2.2, we have R*(M) 2 m(l - M). (A.30) Then, we obtain m - 1 + o( m) 1 2 GS M m(l - MJ S M(l - M) + o(l) S M + o(l). (A.31) A.4.2 Case n = O(m) A.4.2.1 When ~SM= o(m) By letting l* = l 2 M J, the lower bound of R*(M) is given by (C.52). • If t = n::; = w(l), the upper bound of R(M) is given by (A.23). Then, we obtain (A.32) 179 • If t = n::; = 8(1), by using (A.22), we have n 1lJ 4t GS _,,,_ + (-"'-) = -ltJ + o(l). 4M o 4M A.4.2.2 When M = 8(m) By using (C.55) and (C.57), we obtain GS 6 + o(l). A.4.2.3 When M < ~ • If n S m, by using (A.22), we have R(M) Sn. Let l* = n, by using Theorem 2.2, we obtain If 'ii' 2 2, by using (C.58), we have R*(M) > n(1-~) --1 n > n(1-2A11) n > - 2 If 'ii'< 2, by using (C.58), we have R*(M) > n (1 - M) n > - 2 (A.33) (A.34) (A.35) (A.36) (A.37) (A.38) 180 Thus, by using (A.35), (C.63) and (C.68), we get n G<-=2. - n/2 • If n > m, the lower bound of R*(M) is given by (A.30). Then, we have n G< ITT - m(l- M) t 2 <- - ltJ M (A.39) (A.40) The proof of Corollary 2.3 follows the exact same step as in the proof of Theorem 2.3 with the exception of case n > m and M < ~. In this case, when L 11 = L 1 (i.e., naive multicast is allowed), then by multicasting all the requested subpackets R(M) Sm (A.41) is achievable. Hence, when n > m and M < ~'we let l* = m and, from Theorem 2.2 and (A.41), we have G < m < m < 2. - (1- M)m - ~m - (A.42) A.5 Proof of Theorem 2.4 In each cluster, by using Theorem 2.1, we have the total number of bits needed to be transmitted in each cluster is Ji'.} (1-1;;'.) FL 11 , therefore, by Definition 2.2 and denoting the achievable rate for each cluster as Rc(M), then when t = g~ is an integer, we have Rc(M) = : ( 1 - ~) . (A.43) 181 When t is not an integer, then the convex lower envelope of (A.43) is achievable. Hence, the achievable throughput is given by A.6 Cr 1 T(M) = JC Rc(M) Proof of Theorem 2.5 Cr 1 ---- JC R(M). (A.44) Due to the protocol channel model, users have to be within a radius of r to receive information simultaneously. Hence, the maximum number of users that can receive useful information simultaneously is 7rr 2 n. 2 Similarly, only users within radius of r of each of these 7rr 2 n users can serve them. Therefore, the maximum number of users that can serve these 7rr 2 n users is at most 47rr 2 n. In this proof, we consider a particular group of served users within a radius of r and with cardinality 7rr 2 n. For other users in the network, we assume they can be served by some genies without any cost. We first compute a lower bound of the min-max number of bits RT needed to serve these 7rr 2 n users. Similar as (A.17), we consider the first l users in the network, with l T J requests vectors: f = fj = {l(j - 1) + 1, 'jl}, (A.45) where j = 1, · · · , l T J and l = 1, · · · , min { m, 7rr 2 n }. from the cut that separates { ( Zv, Xu,lj) : v = 1, , z, u = 1, , r 47rr 2 n l, j = 1, , l 9 J} and {wl,f : J = 1, , ll 9 J} and by using the fact that the sum of the entropies of the received messages and the entropy of the side information (cache symbols) cannot be smaller than the number of requested information bits, we obtain that £:, lTJ l4Kr 2 nl LL L R{;,s,lj + lMFL 1 > s=l j=l u=l l mj !!_·RT+ lMFL 1 l £11 (A.46) 2 Since we consider the asymptotic regime n--+ oo, we ignore the non-integer part of 7rr 2 n. 182 Then, we can obtain (A.47) Hence, for these served Jrr 2 n users, R*(M) 2 max l - --M . ( l ) lE{l,2,"- ,min{m,Kr 2 n}} l T J (A.48) We notice that it is possible to have multiple concurrent transmissions to serve these 7rr 2 n users. The argument used here is similar as the one used in [JCM13b]. By using the protocol model, since each node consumes at most the area of a disk with radius (1+6)r, we can see that the total area consumed by all the nodes in a disk with radius r is at most 7r(r + (1 + 6)r)2 = (2 + 6) 2 7rr 2 . Since each transmission consumes at least the area of a disk with radius of ~r (See the proof of Theorem 1 in [JCM13b] for details), we can obtain the maximum number of concurrent transmissions is (A.49) Thus, there are at most r 4 ( 2 tf l 2 1 concurrent transmissions to serve the users in the disk with radius r. Therefore, the achievable throughput is given by T(M) I 4(2 + 6)21 _Er_ I 6 2 R(M) < 14(2+6)21 1 Cr I 62 ----------(----~). maxlE{l,2, ,min{ m,Kr 2 n}} l - l 4 J M (A.50) 183 A.7 Proof of Theorem 2.6 By using Theorem 2.4 and Theorem 2.5, we obtain T*(M) T(M) < 4, t=w(l),~SM=o(m) 4t 7rr 2 n = O(m), t = 8(1), ~SM= o(m) ITT• (a) JC 14(2 ~/")21 x 6, M = 8(m) (A.fjl) < 2 7rr 2 n = w(m), M < ~ M• t 2 n = O(m),7rr 2 n > m,M < ~ ITT M• 2, n = O(m),7rr 2 n S m,M < ~ where (a) is obtained by Theorem 2.3, in which n is replaced by 7rr 2 n. A.8 Proof of Theorem 2.7 We need to determine the value of p such that the network can cache at least K distinct MDS-coded symbols of each packet from each file with high probability as K -+ co by using Algorithm 2.1. For the sake of analysis, we consider a simpler algorithm whose performance is worse than Algorithm 2.1, but it turns out to be good enough to prove our result. In this new algorithm, each user selects MK /m MDS-coded symbols indepen dently with uniform probability (with probability fr) of a packet from each file. Selection is done with replacement, i.e., there is the possibility of choosing the same coded symbol multiple times. Hence, this simplified selection method is certainly not better than the selection in Algorithm 2.1 (selection without replacement). 184 As in Algorithm 2.1, each user caches the same set of MDS-coded symbols of each packet from each file. Hence, in the following, we refer to "MDS-coded symbol" i without specifying which file and which packet in the file it belongs to, since this will be the same for all files and all packets in each file. We denote by Z the number of distinct MDS-coded symbols (same for each packet from each file) obtained by the new algorithm. Notice K that Z = Li~l Ii, where Ii = 1 if MDS-coded symbol i is cached, otherwise Ii = 0. K Similarly, let Z 1 = :>::! 1 Yi be the number of distinct MDS-coded symbols obtained by using Algorithm 2.1, where Yi is an indicator function similarly defined ash Notice that Algorithm 2.1 stochastically dominates the simplified algorithm in terms of the number of distinct MDS-coded symbols of a packet from each file, i.e., for any a > 0 we have IF(Z 1 Sa) S IF(Z Sa). (A.52) In particular, if we show that IF(Z > K) -+ 1 as K-+ co, the stochastic dominance (A.52) immediately implies that also IF(Z 1 > K)-+ 1 as K-+ co. Noticing that we have n users, each of which makes MK /m independent selections with uniform probability over a set of K / p possible MDS-coded symbol indices, we have that Ii is Bernoulli with probability IF(Ii = 0) = (1- fr )nMK/m. This yields Then, we have K ( nMK) IE[Z] = p 1- ( 1-; r·m- . IE [Z] 2 1- exp(-tp) K, p (A.53) (A.54) where t = 11nn > 1. The proof of Theorem 2. 7 is obtained in the following steps. First, we consider the concentration of Z around its mean IE[Z]. Then, we find that fort> 1 it is possible to find p E (0, 1) such that IE[Z] = (1 + ii)K, where ii > 0 is a constant independent of K. Combining these results, we have that, as K -+ co, the number of 185 cached MDS-coded symbols in the network for all packets of all files is larger than K with probability growing to 1. We start by considering the concentration of Z. To this purpose, we recall here the definition of self-bounding function [BLM13]: Definition A.1 A nonnegative function f : xn-+ [O, oo) has the self-bounding property if there exist functions Ji : xn-l -+ JR such that for all x1, · · · , Xn E X and all i = 1, · · ·, n, and also n L (f(x1, · · · , Xn) - fi(x1, · · · , Xi-1, Xi+l, · · · , Xn)) S f(x1, · · · , Xn)· i=l (A.55) (A.56) 0 We observe that Z is a self-bounding function of the Ii's. To see this, let n = K / p, Xi =Ii for i = 1, .. . ,K/p, z =!(Ii, ... ,IK/p) = I:fli Ij and zi = li(Ii, .. . ,IK/p) = LjfJj· Then, Z - Zi = Ii E {O, 1} such that (A.55) holds. Furthermore, L:~['(z - Zi) L:~f Ii = Z such that also (A.56) holds. As a consequence, we have [BLM13]: Lemma A.1 If Z has the self-bounding property, then for every 0 < µ S JE[Z], IF(Z -JE[Z] 2 µ) S exp (-h (JE(z]) lE[Z]), (A.57) and IF(Z - JE[Z] S -µ) S exp (-h (-JE(z]) lE[Z]) , (A.58) where h(u) = (1 + u) log(l + u) - u, u 2 -1. D 186 Next, we are interested in studying the the quantities h (JE[zi) and h (-JE[z 1 ) in the caseµ= o (JE[Z]). We have h ( JE(z1) = ( 1 + JE(z1) log ( 1 + JE(z1) - JE(z1 ( 1 + JE(z1) ( JE(z1+ 0 ( (JE(z1) 2 ) )- JE(z1 lE~J + lEf;12 - JEfz1 + 0 ( ( JEfz1) 2 ) JE~]2 + o ( ( lErZ]) 2) ' (A.59) and (A.60) Using the above results in (A.57) and in (C.29), and applying the union bound, we have JtD(IZ -JE[Z]I 2 µ) s 2exp (-:i~l + o (:r~l)). (A.61) For what said above, Theorem 2.7 is proved if we find p E (0, 1) such that JE[Z] > (l+<l)K for some 6 > 0 independent of K, and findµ such that µ/JE[Z] -+ 0 and µ 2 /JE[Z] -+ co as K-+ co. To this purpose, we have: 187 Lemma A.2 Fort> 1, the equation: x = 1- exp(-tx) has a unique solution p* E (0, 1). Furthermore, l-ex~(-tp) > 1for0 < p < p*. Proof Consider the function f(x) = 1- exp(-xt), with derivatives f(x) = texp(-tx), J11(x) = -t 2 exp(-xt). (A.62) D This is a monotonically increasing concave function, with slope at x = 0 equal to t > 1, and a horizontal asymptote limx-+oo f(x) = 1. Since f(O) = 0 and the slope at the origin is larger than 1, we have that f(x) > x in a right neighborhood of x = 0. Since the slope for large x is smaller than 1, we have that f ( x) < x for sufficiently large x. Hence, since f ( x) is continuous, f ( x) = x must have a strictly positive solution x = p* < 1. Furthermore, given the concavity and monotonicity, this solution must be unique, such that f(x) > x for x E (0, p*) and f(x) < x for x E (p*, +oo). This also implies that the iteration x(C) = f(x(C-l)) for£ = 1, 2, 3, ... yields a monotonically increasing sequence uniformly upper bounded by p* for any initial condition x(O) E (0, p*) and a monotonically decreasing sequence uniformly lower bounded by p* for all initial conditions x(O) E (p*, +oo). It is immediate to see that both these sequences converge to p*, otherwise this would contradict the uniqueness of the strictly positive solution of x = f ( x). Finally, for any 0 < p < p*, f (p) > p implies f (p) / p > 1. Letting p* denote the unique positive solution of (A.62), we choose p such that p=(l-o)p*, (A.63) where o > 0 is small enough such that p > 0. Then, Lemma A.2 and the lower bound (A.54) imply IE[Z] = (1 + 6(o))K 188 for some 6(0) > 0 that does not depend on K. 1 1 01 Lettingµ= (1 + 6(0))2K2+2 for some constant 6 1 > 0 and using (A.61), we obtain ( ((1+6(o))~K~+5,c )2 ( ( K~+5,c )2)) IF(IZ-(1+6(o))Kl2µ) < 2exp - (l+ 6 (o))K +o (l+ 6 (o))K exp (-K 81 + o ( K 81 )) . (A.64) Thus, using the stochastic dominance (A.52) we have immediately that, for K-+ co, IF( Z 1 2 K) 2 1 - exp ( - K 81 + o ( K 81 )) , (A.65) and Theorem 2. 7 is proved. A.9 Proof of Theorem 2.8 and of Lemma 2.1 First, we show the first term in (2.24). By using the caching placement scheme given in Algorithm 2.1, we can see that the probability that each MDS-coded symbol is stored in each node is given by IF(Each MDS-coded symbol is stored in each node) K (~) m (~-K,:1)!(K,:1-1)! ~! Mp m The expected number of MDS-coded symbols of each packet from each file that are cached exclusively at particular s users is given by (A.67) 189 (A.66) and when K goes to infinity, then the actual number of MDS-coded symbol of each packet from each file that are cached exclusively at particular s users is given by (A.68) Then, we have K n (n) 1 (Mp)s-l ( Mp)n-s+l FL 11 -Ls - - 1-- __ p s s-1 m m K s=2 (A.69) K n ( 1 ) (n) (Mp)s-1 ( Mp)n-s+l FL!! p L 1 + s - 1 s --;,:;;: 1 - --;,:;;: K s=2 ~ (; (:) ( ~) s-1 (l-~) n-s+l) F:!! K n 1 (n) (Mp)s-l ( Mp)n-s+l FL 11 + p L s - 1 s --;,:;;: 1 - --;,:;;: K s=2 ~l~o/f (;(:) (~)s (l-~)n-s) F:!! K n 1 (n) (Mp)s-l ( Mp)n-s+l FL 11 +-L-1 - 1 -- -K · p s- s m m s=2 (A. 70) We divide RT in (A.69) by FL 11 and obtain the first term of (2.24). Next, we wish to show the first term of (2.25) in Lemma 2.1. The first term of (A.70) can be computed as = ~ ;:;p (1-~) (1-(1-~)n _ M~n (1- ~)n-l) F: 11 (A71) 190 The second term of (A.70) is given by K n _1 (n) (Mp)s-l (l - Mp)n-s+l FL 11 pLs-ls m m K s=2 = K _rri_ (1- Mp) ~-1 (n) (Mp)s (1- Mp)n-s FL 11 p MA m L... s-1 s m m K s=2 s K _.,,,,_ (1- Mp) t ~ (n) (Mp)s (1- Mp)n-s FL!! p Mp m s + 1 s m m K s=2 where S is a random variable with Binomial distribution with parameters n and p = ~. Then, we can compute IE [ 8 ~ 1 ] as IE[s~1] n " 1 n! i 1 n-i L.., ~+ 1 1( - )Ip ( - p) i i. n i . i=O t (i + 1)7in - i)!Pi(l - Pin-i i=O ~ (n + 1)! 1 i(l r-i 6i'(i+l)!(n+l-i-l)!n+lp -p ~ (n+ 1)_1 i(l- )n-i L... i+l n+lp p i=O 1 ~ (n + 1) i+l(l - )n+l-i-1 (n+ l)pt:o' i+ 1 p p 1 (1-(1-pr+ll (n + l)p 1 ( ( Mp)n+l) (n+l)~ 1- 1---;;;- . (A. 73) 191 Thus, plugging (A.73) into (A.72), we have K n _1 (n) (Mp)s-l (l - Mp)n-s+l FL 11 pLs-ls m m K s=2 = 3K _!"__ (l - Mp) (IE [-1 ] - (l - Mp)n _ ~ Mpn (l - Mp)n-l) FL 11 p Mp m s + 1 m 2 m m K = 3K _!"__ (l - Mp) p Mp m . ( 1 (1- (1- Mp)n+l)- (1- Mp)n - ~ Mpn (1- Mp)n-l) FL 11 • (n + 1)1\'if m m 2 m m K (A.74) Then, using (A.71) and (A.74) into (A.70), we obtain 192 Therefore, dividing both sides of (A.75) by FL 11 , we have R(M) RT FL11 < __"'_ (1- Mp) Mp 2 m 1 + 1 - 1 - _I' - 4 1 - _I' --~ 1 - _I' . ( 3 ( ( M )n+l) ( M )n 5M ( M )n-1) (n + 1)~ m m 2 m m The second term of (2.24) and (2.25) is obtained by counting the needed codewords (i.e., blocks of linear hashed symbols) such that all the users can successfully decode. It can be seen that each user need (1 - ~K)Ff{ codewords to decode. Hence, the total number of codewords needed to be transmitted is (1- ~K)Ff{ · n. Therefore, we have RT (1- MK)FL". n R(M) - - - m K = n - t. - FL 11 - FL 11 (A. 77) A.10 Proof of Theorem 2.9 A.10.1 Caset=w(l) By using Theorem 2.8, we obtain R(M) __"'_ (1 - Mp) Mp 2 m (A.76) 1 + 1 - 1 - _I' - 4 1 - _I' --~ 1 - _I' ( 3 ( ( M )n+l) ( M )n 5M ( M )n-1) (n + 1)~ m m 2 m m --"'--- (1- Mp) (1 + :3_ (1- e-pt) - 4e-pt - ~pte-pt + o(l)) Mp 2 m pt 2 -- 1 - - 1 + - - e-P - + 4 +-pt + o(l) m ( Mp) ( 3 t ( 3 5 ) ) Mp 2 m pt pt 2 m ( Mp) - 1-- (l+f (t)) Mp 2 m P m ( M) 1- Mp 1 M 1- m 1- M p2 (1 + fp(t))' m (A. 78) 193 where fp(t) = f, - e-pt (ft+ 4 +~pt) is a function of t. 3 If t-+ co, by using (A.78), let p = (1 - o)p* and MS 1 ';' 6 , where o is an arbitrary small positive number and p* is given by Theorem 2.7, we obtain fp(t) -+ 0, p-+ 1 - o and R(M) Thus, by using (2.9), we obtain R(M) < 2 x R*(M) - (1 - 0)2 A.10.2 Case t = 8(1) 4, t = w(l), ~SM= o(m) 6, 2 M' M = 8(m) n =w(m),M < ~ (A. 79) (A.SO) In this case, let n, m -+ co, we have ~ -+ 0. By using Theorem 2.8, we can obtain R(M) S n - t S n. Then by using (A.33), we can obtain By using (A. 78), we have 3 Notice that p is a function oft. R(M) < 4t R*(M) - . (A.81) (A.82) (A.83) 194 By using (2.9), we obtain R(M) 1 R*(M) S p 2 (1 + fp(t)) Denote fg(t)"' ~ (1 + fp(t)) p Then, we have 4t ITT' t 2 ITT M' n = O(m),t = 8(1),~ SM =o(m) n = O(m),n > m,M < ~ 2, n = O(m),n S m,M < ~ 4t ITT' t 2 ITT M' n = O(m), t = 8(1), ~SM= o(m) n=O(m),n>m,M<~ 2, n=O(m),nSm,M<~ R(M) . R*(M) S mm{4t, fg(t) }. (A.84) (A.85) (A.86) Since p = (1 - o)p*, where o is an arbitrary small positive number and p* is given by Theorem 2.7, then we obtain that if t-+ co, p-+ 1- o and if t fc 1 and tis finite, then p is finite, therefore, we can conclude that if t fc 1, then ; 2 (1 + fp(t)) is finite. 195 Appendix B Proofs of Chapter 3 B.1 Proof of Theorems 3.1 and 3.2 We first provide an outline of the proof and then dig into the details. 1. We define T~um = :>::~~l Tu and let (T~*um (p ), p) be the solution of max1m1ze Tsum subject to Po s; p, (B.1) where the maximization is with respect to the cache placement and transmission policies Ile, IT,. As for T*(p), also T~'um(P) is non-decreasing in p. Furthermore, the inequality T* (p) S ~ T~'um (p) follows immediately from the definition of T~'um (p) and T*(p). 2. We parameterize problem (B.1) with respect to the number of nodes in a disk of radius r, referred to (for brevity) as "disk size" and indicated by 9r(m), where r denotes the one-hop transmission range of the protocol model. For any value 9r(m) = g, let T;'urr,(g) denote the largest achievable sum throughput with disk size g, and let p~(g) denote the corresponding outage probability. While obtaining exact expressions for T;'urr,(g) and for p~(g) is difficult, we shall obtain an upper bound ~J:u(g) ;:> T;'urr,(g) and a lower boundp 11 (g) S p~(g). By the monotonicity property 196 said above, it follows that (7i',!,(g),p 11 (g)) dominates cr;'um(g),p~(g)) and, as a consequence, (~7i',!,(g),p 11 (g)) dominates (T*(p),p) for p = p~(g). Also, we have that the set of outage probability values p~(g) obtained by varying g includes the feasibility domain IPo,min, 1] of the original problem ( 4.1). This implies that the set of points (~~i;,(g),p 11 (g)), obtained by varying g, dominates the Pareto boundary of the throughput-outage region T. 3. Finally, we shall consider separately the different regimes of the outer bound, by "eliminating" the parameter 9r(m). Conceptually, this can be obtained by letting p = p 11 (g ), solving for gas a function of p and replacing the result into ~i;,(g ). The resulting outer bound shall be denoted simply by (T" 1 (p),p), given by Theorems 3.1 and 3.2. We focus first on Theorem 3.1, where limn-+oo "';," = 0, and consider in details step 2) of the above outline. In the following, we shall implicitly ignore the non-integer effects when they are irrelevant for the scaling laws. For example, recalling that the network has node density n (we haven nodes in the unit square), the disk size is given (up to integer rounding) by (B.2) For given disk size g, a lower bound on p 0 can be obtained by observing that 1 - p 0 is upper bounded by the maximum over the users u = 1, · · · , n, of the probability that user u can be served by the D2D network. A necessary condition for this to happen is that the message fu is found in the cache of some node inside a disk of size g centered at node u. We denote such event by F~. 1 If g ;:> m/M, then the outage probability lower bound 1 Notice: events are defined in the probability space of the triple (f, G,A) "--' f17= 1 qf,IIc(G)IIt(Alf, G), of requests, cache placements and transmission scheduling decisions. 197 is zero, since we can arrange the files in the caches such that at least one node u finds all files in the library within a radius r. Hence, assuming g < m/M, we have 1 - Po < max IF (Pg) u (a) Mg Mg f-'Y < ~qt=~H(1,l,m) H(r, 1,Mg) H(1, l,m) ' (B.3) where (a) follows by caching all most popular Mg files within a disk of radius r form a given user. In order to estimate the value of H(-, ·,·),we have the following lemma: Lemma B.1 For I fc 1, then For I= 1, then 1 log(y + 1)- log(x) S H(1,x,y) S log(y) -log(x) + -. Proof See Appendix B.5. From (B.3) and Lemma B.1, we have the lower bound x for g 2 Ji'.} for g <Ji'.} (B.5) (B.6) Next, we seek an upper bound on T;'um(g) as a function of the disk size g. According to the protocol model (see Section 7.1), the throughput T~urn is given by T~um = C IE [L], (B.7) 198 where L is the number of active links over any strategy with transmission radius r. Letting (i, j) and ( k, Q denote two distinct transmitter-receiver pairs, using the triangle inequality and the protocol model constraints, we have d(j, l) ;::: d(k, j) - d(k, Q ;::: (1 + L:.)r - d(k, Q ;::: (1 + L:.)r - r = ll.r. (88) Hence, any two receivers must be separated by distance not smaller than ll.r. Equiva lently, disks of radius tr around any receiver must be disjoint. Since there is at least a fraction 1/4 of the area of such disks inside the unit square containing our network, the number of such disjoint disks in the unit square is upper bounded by f ,,ffi.ol. Figure B.1: Illustration of the fact that the number of small disks intersecting the union of the big disks centered at the active receivers is nece'>sari\y an upper bound to the number of active receivers. We wish to upper bound the number of simultaneously active receivers L. In order to do so, consider the situation in Fig. B.1, where the potentially active receivers (thase that can receive according to the protocol rrodel) are at the centers of mutually exclusive 199 disks ofradius ~r. Now, any of these receivers u can effectively receive only if Pg occurs. From (B.6), we have IF(F~) S l -p 11 (g). Now, consider a disk of radius (1+6)r around each active receiver (shown as filled dots in Fig. B.1), and let U(r, 6, L) denote the union of all such disks. It is clear that the number of active receivers L is less than or equal to the number of small disks of radius ~r with non-empty intersection with U(r, 6, L). Since, as argued before, there are at most I /Tl~r 2 l such disks, we can write I 1u~gr2 l LS L l{diskinU(r,6,L)}. (B.9) i=l Taking expectation of both sides of (B.9), and denoting the disks of radius ~r centered around the receivers simply as "disk", we can write 16n -;;::rg IE[L] < LIF(diskinU(r,6,L)) < i=l 16n IF(Any diskn U(r,6, L)). 62g Then we introduce the following lemma. Lemma B.2 (B.10) IF(Any diskn U(r,6, L)) S IF (Ci an active receiver in a disk of radius (1+ 3 f) r). (B.11) Proof See Appendix B.6. Using Lemma B.2 in (B.10), we obtain IE[L] < ~~ (;) ·IF (Ci an active receiver in a disk of radius (1 + 3 f) r) ~ ~~(;) (l-(plb(g))(1+ 3 ~) 2 g), (B.12) 200 where (a) follows from the fact that, recalling (B.2), the number of users in a disk of radius ( 1 + 3 :1) r is given by ( 36) 2 ( 36) 2 mr 1 + T r 2 = g 1 + 2 , (B.13) and the probability that no users in such disk find their requested content within the transmission range r can be lower bounded as Using (B.12) in (B.7), we obtain the sought upper bound T:'Jin(g) on 7;'um(g) as en ( ) < rrub ( ) "' 16C . I ~urn g I ~urn g 6 2 (B.14) In order to discuss the different regimes of the outer bound, we start by considering the maximum throughput regime and the corresponding outage lower bound. This is obtained by maximizing I;;;i;,(g) in (B.14) with respect tog, and is given by the following result: [( (i+"')2g) l Lemma B.3 As m -+ co, the maximum of the quantity 1 - (p 11 (g)) 2 ~ is given by }. (1- e-((p*)) ,;:-", where p* is the solution of (3.8) with C,(p) given by (3.9), and where the optimal g takes on the form g* = p*m", with a given in (3.6). Proof See Appendix B. 7. Using Lemma B.3, the resulting maximum (with respect tog) of the sum throughput upper bound is given by: fub(p*) n°', m (B.15) 201 where fub(P) is defined in (3.10). By replacing g = g* into (B.6), the corresponding value of the outage probability lower bound is given by For large m, we have _1_ (Mp*m")l-"f - _1_ + 1 1 - _1_-~'Y-~----~l-_"(~_ _l_ml-i __ 1_ 1-"( 1-"( where we used the identity (1 - 1)(1 - a)= a. (B.16) (B.17) At this point, we have essentially captured the throughput-outage tradeoff outer bound in the third line in expression (3.7) of Theorem 3.1. There is one small tech nical point that needs to be settled in order to obtain the desired result from (B.15) and (B.17), namely, we have to show that by introducing a perturbation of size o(m-") in the outage probability lower bound p 1 b(g*), the corresponding perturbation of the throughput upper bound is no(m-"). This fact follows from the continuity of T;;"Jin(g) and p 1 b(g) in g, and it is proved in Appendix B.10. After this perturbation argument, the throughput-outage point corresponding to the maximization of T;;',!, (g) with respect tog shall be denoted by ((Tub)*, (p 1 b)*), with coordinates (p 1 b)* = 1 - (Mp*) 1 -'Ym-a and (Tub)* = fub(p*)m-a + o(m-"). The point ((Tub)*, (p 1 b)*) dominates the achievable throughput-outage tradeoff boundary (T*(p),p), for all p ;:> (p 1 b)*, yielding the third line in expression (3.7) of Theorem 2.1 throughput-outage trade-off. Next, we characterize the other regimes of the outer bound on the throughput-outage tradeoff region by using (B.6) and (B.14), for different regimes of the disk size 9r(m). It is clear from (B.6) that by increasing 9r(m) beyond g;(m) = g* given in Lemma B.3, the outage probability lower bound decreases. We consider two cases: 1) 9r(m) = 8 (m") withgr(m) > p*m"; 2) w(m") =gr(m) S min{M,n}. 202 Case 1) In this case, we let g = 9r(m) = p 1 m°' with p1 > p*. Letting m-+ co in (B.6) and in (B.14), we obtain and (B.18) With a derivation similar to what done in Appendix B.10, and not included in the paper for the sake of brevity, this yields part of second line in expression (3.7) of Theorem 3.1 (one of the two terms of the minimum). Case 2) When w (m") = 9r(m) S min{ M' n }, we use (B.6) and the probability bound as in (B.12) and write IF (Ci an active receiver in a disk of radius ( 1 + 3 :1) r) S 1- (Plb(g))(Hsi')2g = 1 - 1 - 1-'( 1-'( ( _l_(Mg)l-1' - _1_ + 1) (Hsi')2g _l_ml- 1 __ 1_ 1-'( 1-'( Sl-o(l), (B.19) (B.20) where the last line follows from the fact that, writing the second term in (B.19) as ( 3£,)2 [( 1-'( )g] 1+2 1 - M 1 -1' (~) (1 + o(l)) , (B.21) we see that the condition for (B.21) to be non-vanishing in the limit for g, m-+ co is that 203 or, equivalently, that g = 8(m°'), where a= ~::~ is the familiar quantity defined in (3.6). Hence, in the case g = w(m°'), the disk size g grows rapidly and the limit of (B.21) vanishes. By using (B.20) into (B.10) with g = 9r(m) we eventually obtain b 16C ( n ) ( n ) ~urn= 6 2 9r(m) + 0 9r(m) . Moreover, from (B.6) we have _1_(Mg (m))l-1' - _1_ + 1 1 - 1-1' r 1-1' _l_ml- 1 __ 1_ 1-'( 1-'( Expressing 9r(m) as a function of p = p 11 , we find m 1 9r(m) = -(1- p) 1 -7 M . (B.22) (B.23) Using this into (B.22) and following a perturbation argument similar to Appendix B.10, we find the desired form ~urn(g) = T~um(P) = n 1 + o 1 ' b ub ( 16CM ( 1 )) 62m(l - p) 1-7 m(l - p) 1-7 (B.24) which yields the first line and the second term in the minimum of the second line in expression (3.7) of Theorem 3.1. By following into the same footsteps, Theorem 3.2 can be proved along the same lines with the only difference that, when there exists a positive constant ~ such that ~ s; limn--+oo ~Ci s; ~~~*'the case 9r(m) = w (ma) does not exist. 204 B.2 Proof of Theorem 3.3 In the case limn-+oo ";," > 1 Xt~ 2 , an obvious upper bound of the sum throughput T;'urr,(p) is provided by T~um C · IE[L] n Mn ( ) ~ CL L qf = Cn H /, 1, Mn u~l J~l H(r, 1, m) (b) - 1 -(Mn) 1 -1' - - 1 - + 1 < Cn 1-1' 1-1' _l_(m + 1)1-1' __ 1_ 1-'( 1-'( < n (cMl-1' nl-1' + o ( nl-1' )) ml-1 ml-1 ' (B.25) where (a) is because we use a deterministic caching scheme (see Appendix B.7) which makes the network store the most n popular messages, and (b) follows from Lemma B.1. Dividing by n, we obtain the upper bound (B.26) Moreover, as n goes to infinity, the outage probability in this case can be computed as > 1- H(1, 1, Mn) > 1- Ml-1' nl-1' + o ( nl-1'). Po - H( l ) - 1-1' 1-1' r, ,m m m (B.27) Again, following a perturbation argument similar to Appendix B.10), for p ;:> 1-M 1 -1' ;;,',-:_~, we have T*(p) S Tu 1 (p) in (B.26). Otherwise, the problem is infeasible. 205 B.3 Proof of Theorem 3.4 As mentioned in Section 3.3, we divide the network into clusters, each of which contains 9c(m) nodes. In this case, let F;;;(m) denote the event that user u can find the requested message inside its cluster of size 9c(m). Letting lu = l{F;;;(m)}, we define (B.28) Our goal here is to find the caching distribution P;(f) that maximizes pf,. With indepen dent random caching, the probability that a user u finds its request fu = f in its cluster is given by IF(F;;;(m)lfu = f) = 1- (1-Pc(f))M(g,(m)-l) (notice that we do not consider requests to files in the user own cache, since these do not generate any traffic). By the law of total probability, we can write m pf,= Lqf ((1- (1-Pc(f))Mg,(m)-M). (B.29) J~l Letting 9c(m) = g for simplicity of notation, and assuming g > 2, we have the convex optimization problem m1n1m1ze subject to m Lqt(l -Pc(f))Mg-M J~l m L_Pc(f) = 1, Pc(!) 2 0 VJ J~l The Lagrangian function for the problem is (B.30) (B.31) 206 Taking the partial derivative with respect to Pc(!) and using the KKT conditions [BV04] we obtain _ r _ ( ( ) 1/(M(g-1)-1)1 + Pc(!) - ll qJM(g - 1) j (B.32) It is immediate to see that the minimum is obtained when the sum probability con- straint holds with equality. In order to solve for the Lagrangian multiplier that imposes the constraint with equality, it is convenient to re-parameterize the problem by defining 1 1 (M(~-l)) M(g 1 1 1 = v and z 1 = qj 19 11 1 where the coefficients z 1 are non-increasing since qf is non-increasing by assumption. Hence, we wish to solve The unique solution must be found among the following conditions: v 1- - = 1 with v -21 z1 z2 v v 2----=1 with v -21 z1 z2 Z3 v v v 3------=1 with v -21 z1 z2 Z3 Z4 (B.33) which can be rewritten compactly as finding the unique index m* for which the equation v (t zl) = m* - 1 J~l f (B.34) has a solution in the interval for v 2 Zm*+l and v S Zm*. Since we are guaranteed that such m* exists, we can write m* -1 v(m*) = m* 1 Lj~l Zj (B.35) 207 From the conditions v(m*) 2 Zm*+l and v(m*) S Zm•, we find that m* is explicitly given as the unique integer in { 1, 2, · · · , m} such that m* * ~ 1 m ~ 1 + Zm*+l L -, J~l ZJ (B.36) and m* * ~ 1 m s; 1 + Zm* L -. J~l ZJ (B.37) Next, we wish to determine m* as a function of g = 9c(m) in the assumption that 9c ( m) -+ co as m -+ co. In order to do so, we shall evaluate the terms in the right-hand side of (B.36) and (B.37). Recalling the expression of ZJ in terms of qf = j 7 (recall that we assume a Zipf distribution for the demands, with exponent IE (0, 1)), we have m* l m* j a 1 Zm*+l L-;- = L (m* + 1) ' J~l f J~l (B.38) and (B.39) where we let a 1 = 1' for brevity. We use the following integral lower and upper M(g-1)-1 bounds 1 1 lm* 1 m* ( j )" 1 x" dx < < (m* + 1)" 1 + (m* + 1)" 1 1 - L m* + 1 J~l 1 rm*+l a' (m* + 1)"' J1 x dx, (B40) and 1 1 lm* a' m* ( J )"' 1 lm*+l a' --- --- x dx < - < --- x dx (m*)" 1 + (m*)" 1 1 - L m* - (m*)" 1 1 · J~l (B.41) 208 Solving the integrals, we obtain the lower bound (LB 1) and the upper bound (UB 1) in (B.40) in the form LB 1 I ( )a' a 1 m* m* a 1 + 1(m*+1)"' + a 1 + 1 m* + 1 m* + 1 1 1 UB 1 ------~-~~ a 1 +1 a 1 +l(m*+l)" 1 ' and we obtain the lower bound (LB 2) and the upper bound (UB 2) in (B.41) in the form LB 2 UB 2 a 1 1 m* -----+- a1 + 1 (m*)"' a 1 + 1 m* + 1 (m* + 1)"' __ 1 __ 1_ a 1 + 1 m* ( m* )" 1 a 1 + 1 · We let m* = c/a 1 for some constant c, and notice that a 1 + 0 as 9c(m) -+ co and that lima'.(.o(l + c/a 1 )" 1 = 1, lima'.(.o(l + a 1 /c)" 1 = 1 and lima'.(.o(c/a 1 )" 1 = 1. Hence, in the limit of a 1 + 0 we can write LB 1 UB 1 and LB 2 UB 2 c/a 1 1 1 - 1 - 1 (1-61(a )) + 62(a) a+ c/a 1 + 1 6 ( 1) a1+1 -1+ 3a' c/ a1 ' ( i) --+u4 a a 1 +1 c/a 1 +1( ( 1 )) ( 1 ) 1 1 + <ls a - 1 + 65 a , a+ 1 where Si, i = 1, · · · , 6 tend to zero from above as a+ 0. It follows that c/a 1 1 1 ~ 1 c/a 1 + 1 1 - 1 -(l - 61(a )) + 62(a) S Zm*+l L, - S 1 - 1+63(a ), a + 1 ZJ a + 1 J~l 209 and c/a 1 1 Lm* 1 c/a 1 + 1 1 1 - 1 - 1 +64(a) S Zm• - S 1 l (1+6s(a )) - 1+65(a ), a + ZJ a+ J~l as m* = c/a 1 and a+ 0. Replacing the common leading term in the LB 1, UB 1, LB 2 and UB 2 above into (B.36) and (B.37), we obtain and which yields _c_~l a 1 +1 Therefore, we obtain c = 1, which yields m* = _1_ = M(gc(m) - 1) - 1 + O(l) a1 I i.e., m* = ~ 9c(m) to the leading order. Clearly, if~ 9c(m) > m, then m* = m. B.4 Proof of Theorem 3.5 Recall that Theorem 3.5 deals with the small library regime limn-+oo ";," = 0. We define the probability (B.42) 1.e., the probability that both user u and user u 1 can find the requested files in the corresponding cluster. We let g,(m) W= L lu, (B.43) u=l denote the number of potential links in a cluster. 210 Given the random and independent caching placement IT~ and the random (or round robin) transmission policy ITi as given at the beginning of Section 3.3, we let T(p) denote achievable values of T min subject to the outage constraint p 0 S p. Also, we define Tsurn = L~=l Tu· We provide first an outline of the proof and then dig into the details. 1. Under policies P; (!) and ITi, we notice that both T min and p 0 are uniquely deter mined by the cluster size 9c(m). Hence, the maximum throughput is obtained by solving: max1m1ze T min subject to 0 < 9c(m) Sn. (B.44) Since the exact solution of (B.44) is difficult to obtain, we instead compute a lower bound and, for the maximizing 9c(m) = g~(m), the corresponding value of the outage probability p~. 2. It follows immediately form the definition that T(p) = T(p~) for all p 2 p~ is achievable, by keeping 9c(m) = g~(m) and using the same caching and scheduling policies. This yields a lower bound on the achievable throughput-outage tradeoff when p 2 p~. 3. In order to obtain a tradeoff for p < p~, we increase 9c(m) above g~(m). By letting 9c(m) grow and calculating the corresponding value of p 0 and (a lower bound on) T min, we obtain a lower bound T(p) for p = p 0 on the achievable throughput-outage tradeoff. B.4.1 Achievable T(p) when p 2 p~ We first compute a lower bound on T~um and the corresponding outage probability p 0 for the caching and transmission policies IT~ with ITi, with cluster size 9c(m). Since the 211 resulting system is symmetric with respect to any user, it follows that each user has exactly the same average throughput, such that T min= ~T~um· Then, we shall maximize the resulting (lower bound on) T min with respect to 9c(m) in order to find g~(m), p~ and T(p) for p 2 p~. For simplicity of notations, in the following we ignore some of the smaller order terms as m and n goes to infinity. The main tool to obtain a lower bound on T~um is the Paley-Zygmund Inequality (see [OzglO] and references therein). Letting again L denote the number of active links, we have T~um C IE[L] C · IE[number of active clusters], (B.45) where (a) is because that in rr;, only one transmission is allowed in each cluster. Moreover, IE[number of active clusters] 1 2 KIE[number of good clusters] = ~ (total number of clusters in the network· W(W > 0)), (B.46) where K is the TDMA reuse factor and we use the fact that a cluster is good if W > 0. From (B.46), we have that a lower bound of T~um can b obtained by lower bounding W(W > 0). The distribution of W is not obvious since the random variables lu and lu' are dependent when u and u 1 are in the same cluster and u fc u 1 • Nevertheless, it is possible to compute the first and second moments of W. Then, with the help of the Paley-Zygmund Inequality, we can obtain a lower bound on W(W > 0) which is good enough for our purposes. For completeness, the Paley-Zygmund Inequality is provided in the following lemma: 212 Lemma B.4 Let X be a non-negative random variable such that JE[X 2 ] < co. Then for any t 2 0 such that t < JE[X], we have IF(X > t) > (JE[X] - t)2 - JE[X2] By using Lemma B.4 with t = 0 and X = W, we get JE[W]2 IF(W > 0) 2 JE[W 2 ]. (B.4 7) D (B48) Therefore, our goal is to find a lower bound for JE[W] and an upper bound JE[W 2 ] under the optimal caching distribution P;, given by Theorem 3.4. First, we focus on JE[W]. Using the expression JE[W] = :>::;'.~;") p;, = Yc(m)p;, we shall focus on the computation of p;, as follows: _ ~ _ * ~ ( j ) M(gc(m)-1)-1 - L,,qf qm +1 L,, m* + 1 J~l J~l = H(r, 1, m*) _ (m* + 1)(-1') t ( J ) M(gc(m)-1)-1 H(1,l,m) H(r,l,m) m*+l ' J~l (B.49) 213 where (a) is because v 2 Zm*+l (see Theorem 3.4 and its proof in Section B.3). Similarly, we have c m* ( ( V ) M(g,(m)-1)) Pu= Lqf 1- - i=l Zj (a) m* ( (Zm* )M(g,(m)-1)) 2 L_qt 1- --;- i~l f = H(r, 1, m*) _ (m*)(-1') t (L) M(gc(m)-1)-1 H(1,l,m) H(r,l,m) m* ' J~l (B.50) where (a) is because v S Zm* (again, see Theorem 3.4 and its proof in Section B.3). By (B.49), (B.50) and Lemma B.1, we have p~ < < (m + 1) 1 -1' (B.51) 214 where (a) is because m* = Mg;(m). and p~ > > ( m-'Y ) ( I ) M(gc(m)-1)-1 1 . ml-"( -1 Mgc(m) M(g,(m)-1)-1+1 (( Mgc(m) ) M(gc(ml 1) 1 Mgc(m) (Mgc(m) ) M(gc(J) 1) 1 ) . +1 + +1 -1 I I I 1 where (a) is because m* = Mg,(m). "( Therefore, by using (B.51) and (B.52), we obtain (B.52) (B.53) 215 By using (B.53), we have (B.54) Now, since here we deal with an achievability strategy and we can choose the clustering strategy at wish, we choose 9c(m) = c2m". By Theorem 3.4, it follows that m* = cim" with ~: = ~. Clearly, this requires that n 2 9c(m) = c2m" for all sufficiently large n. Then, by using (B. 54), as m -+ co, we have IE[W] /'Ygc(m) ( Mg~m)) l-'f + o (gc(m) ( Mg~m)) l-'f) 1'Yc2m" ( M~m") l-'f + o (c2m" ( M~m") l-'f) 1ci-'Yc2 - o(l). Next, we compute IE[W 2 ]. Since ( g,(m) ) 2J IE L lu u=l 9c(m)p~ + 9c(m)(gc(m) - l)p~u'' then under the optimal caching distribution P;, we need to compute P~u'. (B.55) (B.56) 216 Let Bl be the event that user u requests file f and can find message f in its cluster, such that F;;;(m) = Ut~l B{ Then, we can write P~u' IF({lu=l}n{lu'=l}) IF ( (Ui!1 B~) n ( Uj~1 B;,,)) IF ( Ui!1 uj~1 ( B~ n B;,,)) m m 0'> LLIF ( B~ n B;,,) i=l j=l m m LL IF (B~) IF ( B;,, IE~) i=l j=l m m m L L 1F(B~J1F(B;,,1B~) + LIF(B~)IF(B~,IB~) i=l j=l,j-=f=i i=l m m < L L (qi(l - (l -Pc(i))M(g,(m)-1))) (qj(l - (1-Pc(j))M(g,(m)-1)-1)) i=l j=l,j-=f=i m + L ( qi(l - (1 - Pc(i))M(g,(m)-l))) qi i=l m m L ( qi(l - (1 - Pc(i))M(g,(m)-1))) L ( qj(l - (1 - Pc(j))M(g,(m)-1)-1)) i=l j=l,j-=f=i m + L (qi(l - (1-Pc(i))M(g,(m)-l))) % (B.57) i=l 217 where (a) is because that B{, n B;,, are disjoint for different pairs of (i, j). Replacing Pc(!)= P;(f) in (B.57), we can continue as m* m* P;u' SL (qi(l - (l -P;(i))M(g,(m)-1))) L (qj(l- (1- P;(j))M(g,(m)-1)-1)) i=l j=l,j-=f=i m' + L (qi(l - (l -P;(i))M(g,(m)-1))) qi i=l 2 < (t (qi(l - (1-P;(i))M(g,(m)-l)))) +; (qt(l - (1-P;(i))M(g,(m)-l))) m' (a) p;2 + L (qt(l- (1- P;(i))M(g,(m)-1)))' (B.58) i=l where (a) is because that p; = I:Z:' 1 (qi(l- (l-P;(i))M(g,(m)- 1 ))). The second term in (B.58) can be upper bounded by the following lemma. Lemma B.5 I:;:' 1 (qt(l - (1- P;(i))M(g,(m)-l))) upper bounded by o (p; 2 ). Proof See Appendix B.8. At this point we are ready to obtain a lower bound on W(W > 0) via Lemma B.4 and (B.48). From (B.56) we can write IE[W 2 ] < Yc(m)p; + Yc(m) 2 P;u' < Yc(m)p; + Yc(m) 2 (p; 2 + o (p; 2 )). Then, Lemma B.4, (B.55) and (B.59) yield W(W > 0) > > > IE[W] 2 IE[W 2 ] Yc(m)pf, + Yc(m) 2 (pf, 2 + o (pf, 2 )) (1ci-1'c2)2 (B.59) (B.60) 218 where (a) is because that we pick £1 = M. c2 1' By using (B.46), we obtain JE[number of good clusters] > (B.61) where a= 11'M 1 -1'. Since m* = ~ 9c(m) = c 2 !;1 m" by Theorem 3.4, then as m-+ co, by using (B.51) and (B.52), the corresponding average outage probability is given by Po 1-p~ 1 1-'j' -Q ( -Qi - ac 2 m + o m . (B.62) Therefore, we have C 1-'j' T~um2- +o - . - ac2 n ( n ) K 1 + ac~-1' m" m" (B.63) By the symmetry of the system and of the caching and transmission policies IT~ and Ili, the achievable throughput is lower bounded by - 1- C ac~-1' 1 ( 1 ) Tmin=-Tsurn~- 2 +o - · n K 1 + -1' m" m" ac 2 (B.64) 219 1-7 Next, we wish to find c 2 that maximizes the coefficient ac 2 in the throughput lower l+ac~ 1 bound (B.64). Setting the derivative to zero and looking for a maximum point, we find 1 h . 1 . b ( l-1') 2 - 7 L D - ab 1 - 7 b . (B 64) h t e unique so ut1on c2 = = -a- . et - 1 +ab 2 1 , y using . , we ave T · >--+o - - CD 1 ( 1 ) rmn - K ma ma ' (B.65) with outage probability Po 1-p~ (B.66) Letting p~ = 1 - ab 1 -1'm-l/a, following a perturbation argument similar to Appendix B.10, we have that for all p 2 p~, CD 1 T(p) = -- + o (m-"), Km" (B.67) is achievable. Thus, we have proved the last regime in (3.14) in Theorem 3.5. B.4.2 Achievable T(p) for p < p~ By choosing a throughput-suboptimal value c2 = p2 > b in (B.62) and (B.64), we have that for Po= 1- ap~-l'm-a Sp Sp~, then 1-7 with B = "P 2 2 7 , is achievable. This yields the third regime in Theorem 3.5. l+ap 2 (B.68) Next, we turn our attention to the case of p 0 = 1 - w (m-"). This is obtained by increasing the cluster size in order to decrease the outage probability and correspondingly decrease the throughput. As before, we find expressions for p 0 and lower bounds on T min 220 as a function of 9c(m). We consider two cases for the value of 9c(m). One is when 9c(m) = w (;:;")and 9c(m) S 1m/M. The other is when 9c(m) = p1m/M, where p1 2 f. B.4.2.1 Case 9c(m) = w (,:a) and 9c(m) S 1m/M In this case, the cluster size is so large that W(W > 0) -+ 1 as m -+ co. In order to show this, we shall show that for arbitrary o 1 > 0, with high probability WE [(l-o 1 )1E[W], (1+ 01)1E[W]] with IE[W] -+ co as m -+ co. This will be proved using Chebyshev's Inequality, which requires the computation of IE[W] and Var[W]. By using (B.54), we obtain IE[W]. Sincegc(m) =w(m°'), then limm-+oolE[W] =co. Next, we need to compute Var[W] (B.69) We focus now on the term p~u', which is given by the following lemma. Lemma B.6 P~u' is given by: c 2'( (Mgc(m)) 2 (l-'f) ((Mgc(m)) 2 (l-'f)) Puu' SI + O . m m (B. 70) Proof See Appendix B.9. 221 Therefore, as m -+ co, by using Lemma B.6, we have Var[W] 9c(m)(p~ -p~u') + 9c(m) 2 (P~u' -p~ 2 ) < 9c(m) ()'1' ( Mg~m)) 1-1' -121' ( Mg~m)) 2(1-1')) +gc(m)2 ( r21' ( Mg~m)) 2(1-1') -r21' ( Mg~m)) 2(1-1')) +o (gc(m) ( Mg~m)) 1-1') /l'gc(m) ( Mg~m)) 1-1' + o (gc(m) ( Mg~m)) 1-1'). (B. 71) Thus, by using (B.54) and (B.71) into Chebyshev's Inequality, it is not difficult to show that, for any 01 > 0, IF (IW - IE[W] I s 01IE[W]) ::- 1 - 0 (1). (B. 72) as m --+ CXJ. Since, as observed before, limm-+oo IE[W] = co, we conclude that for any 0 < I< 1, as m-+ co, IF(W > 0) = 1 - o(l). It follows that all clusters are good, such that - C n ( n ) T~um = K 9c(m) + 0 9c(m) . (B. 73) As m-+ co, by using (B.51) and (B.52), the corresponding outage probability is given by Po (B. 7 4) By the usual symmetry argument, we have (B. 75) 222 Finally, letting p =Po, we can solve for 9c(m) = -±M(l -p)' 2 " + o (M(l-p)' 2 ")· ,1-1 7 By using (B.75) and letting A= 1 1 -7, with the similar perturbation argument shown in Appendix B.10, we have that when p = 1- /"I ( Mg;j;m)) l-"f, then T(p) =CA M 1 K m(l-p)'-" ( M ) +o ' m(l-p)1-7 (B. 76) is achievable. This settles the second regime of Theorem 3.1. B.4.2.2 Case 9c(m) = p1m/M, where p1 2 / In this case, by using Theorem 3.4, we have m* = m. We can obtain that W(W > 0) = 1 - o(l) as m-+ co. Thus, we have - lC n CM (M) Tmin=;;;,Kgc(m) = Kp1m +o m . (B. 77) The corresponding outage probability is computed next. Here, we need to find a different bounding technique other than the one we used before. In this case, we directly plug v = ::-1, into p;, and use the integral approximations of summations to obtain the Lj=1 Zf: lower bound of p;,, instead of using that fact that v S Zm* and v 2 Zm*+l as we used before. The reason is that in this case, m* = m as shown by Theorem 3.4, which means that m* _J_ Mg,(m) h > th' k d t d · t. t- 'Y w en Pl /, is ma es Zm* an Zm*+l no goo approx1ma ions anymore. 223 Operating along these lines, pf, can be computed as c Pu m 1 _ VM(g,(m)-1) '\' qf L M(gc(m)-1) f=l qf(gc(m)-1)-1 1 - m - 1 L M(gc(m) 1) 1 ( ) M(g,(m)-1) m 1 L:1 t t=1 qt ( ) M(g,(m)-1) 1 - r:- 1 1 f q; M(gc(~)-1)-1 "\'~ _ M(gc(m) 1) 1 f=l Di=l qi ( ) -(g,(m)-2) 1- (m- l)M(g,(m)-1) f q;gc(,,;)-2 J~l ( 1 )-(M(g,(m)-1)-1) 1 - (m - l)M(g,(m)-1) f ( j-'Y ) M(gc(m)-1)-1 H(1,l,m) J~l (m - l)M(g,(m)-1) 1 1- ~~~~~~~~~~~- H( /, 1, m) ( LJ~l J M(gc(m)-1)-1) M(g,(m)-1)-1. (B. 78) 224 The lower bound of pf, is given by (m - l)M(g,(m)-1) 1 p~ > 1- . ---------~~~~- _l_(m + 1)1-1' __ l_ ( )M(g,(m)-1)-1 1-/ 1-/ 1 + f1m xM(gc(m)-1)-1 dx (m - l)M(g,(m)-1) 1 1 - . ----------------=~~~- _l_(m + 1)1-1' __ l_ ( ) M(g,(m)-1)-1 1-1' 1-1' 1 + H 1 (mM(gc(m)-1)-1 +1 _ 1) M(gc(m)-1)-1 (m _ l)M(g,(m)-1) 1 - (1 - "')~-~- ' (m+1) 1 -1'-l 1 1 mM(gc(m)-1)-1 +l + 1 _ 1 ( ) M(g,(m)-1)-1 l+ M(gc(m)-1)-1 M(gc(m)-1)-1 +l (m - l)M(g,(m)-1) ml-1' 1 1-(1-1)-------------~--- ml-'( (m + 1)1-1' - 1 mM(g,(m)-1)-1+1' mM(g,(m)-1)-1+1' 1 mM(gc(m)-1)-1 +l + 1 _ 1 . ( ) M(g,(m)-1)-1 l+ M(gc(m)-1)-1 M(gc(m)-1)-1 +l (m _ l)M(g,(m)-1) ml-1' 1 - (1 - <v)--~-----~~- ' m 1 -1' (m + 1) 1 -1' - 1 1 1 . mM(g,(m)-1)-1+1' (l 1 ) M(g,(m)-1)-1 + M(gc(m)-1)-1 mM(g,(m)-1)-1+1' ( + l ) M(g,(m)-1)-1 mM(gc(m)-1)-1 + ---'--- M(g,(m)-1)-1 ( l ) M(g,(m)-1) 1 1 - (1 - "') 1 - - I m M(gc(m)-1)-1+1 M(gc(m)-1)-1 ( 1 _ ) I M(gc(m) 1) 1+1 I M(g,(m)-1)-1+1' ml-1' mM(g,(m)-1)-1+1' . (m + 1)1-1' - 1 ( 7 +1 )M(g,(m)-1)-1 mM(gc(m) 1) 1 + I M(g,(m)-1)-1 ( l) M(p1m/M-l) l-(1-1) 1-- m 1 M(gc(m)-1)-1+7 M(gc(m)-1)-1 (1 + o(l)) ( 1 _ ) ! M(gc(m)-1)-1+1 I M(g,(m)-1)-1+1' 1- (1-1) (~)Pl (~\1' (1 + o(l)) 1 - ( 1 - I) (~)Pl -1' ( 1 + o( 1)) . (B. 79) 225 Thus, we have Po 1-p~ < l-(l-(l-1)(~)p 1 -1'(l+o(l))) (1-1) (~)pi-1' (l+o(l)). (B.80) Therefore, letting p = (1 - 1)e1'-P 1 , and following a perturbation argument similar to Appendix B.10, we have that the throughput T(p)=--+o - CM (M) K p1m m (B.81) is achievable. This settles the first regime of Theorem 3.5. B.5 Proof of Lemma B.1 When I fc 1, then, since x\ is an decreasing function, we have (B.82) and y 1 1 y 1 H(r,x,y) =Lil'= xl' + L il' i=x i=x-1 < -dx 1 +- 1 y 1 1 - -l+l x'' xi 1 1- 1 1- 1 = --y 1' - --x 1' + -. 1- 1 1 -1 xl' (B.83) 226 When I = 1, similarly, since ~ is an decreasing function, we have and = log(y + 1) - log(x), 1 y 1 1 S 1 dx 1 + - -1+1 x x 1 = log(y) - log(x) + -. x B.6 Proof of Lemma B.2 (B.84) (B.85) Recall that we denote the disks of radius !/± R centered around the receivers as "disk", our goal is to show that IF(Any diskn U(r,6, L)) S IF (Ci an active receiver in a disk of radius (1 + 3 f) r). (B.86) which is equivalent to show that {Any disk n U(r, 6, L)} c;; {Cl an active receiver in a disk of radius ( 1 + 3 f) r }. (B.87) To see (B.87), we first consider a simple illustration which is easier to explain, but it is not accurate. As shown in Fig. B.2, the network is divided into squarelets whose diagonal are f±r. These squarelets are the analogue to the the sectors with radius of 1/±r (a quarter of disk with radius 1/±r). Now we want to see which events can cause a squarelet 227 to intersect with U(r, !::., L ), which ffi3ans that the area of this squarelet is consumed due to communicating links according to the protocol roodel. From Fig. B. 2, we can see that if the link from user d to u is activated, then the upper bound of the maximum area this link can consume is the area of all the blue squarelets. If we consider sq uarelet A and let user v be a receiver, we can see that A cannot intersect with U(r, !::., L) if there is no any active receiver in a disk centered at v, with radius (1 t 3 t) r. Therefore, if there is at least one active receiver in a disk centered at v, with radius (1 + 3 t) r, then it is possible that A can intersect with U(r,l::.,L). " / u' '\ ~ - \ / 2 .v11 +- l::.) u \.. / Figure B.2: In this figure, u' is a transmitter and u is a receiver. v is another receiver corresponding to another transmitter. The diagonal of each squarelet is %r. The maxi mum area that are consuffi3d by receiver u is the disk centered at u, with radius (1 + l::.)r. The blue squarelets are the maximum activated squarelets that are caused by the active receiver u. A indicates the squarelet containing receiver v. Now we prove this lemma accurately. From the arguments in Appendix 8.1, we know that all the disks with radius of 4r have to be disjoint. Moreover, there is at least a fraction ~of the area of such disks inside the network. Therefore, to obtain an upper bound of the maximum concurrent transmissions, we maximumly pack such sectors 2 2 In th? following. we derote the sector that is ~ of the disk with radius of f-l" as «sector~'. 228 inside the network as shown in Fig B.3 (Of course, Fig B.3 shows an over optimistic way of packing, since we cannot guarantee all the disks with radius ~r are disjoint. However, at least all the sectors are disjoint.). Notice that {Any disk n U(r, L'., L)} c;; {Any sector n U(r, L'., L) }. (B.88) Now we consider each such sector as an analogue of the squarelet considered before. This shows that if the receiver u is activated, then the upper bound of the maximum number of sectors that can intersect with U(r, L'., L) are the blue sectors. Now pick a arbitrary node v, if there is no any active receiver inside a disk centered at v of radius (1 + 3 f) r, then the sector A cannot intersect with U(r, L'., L), which means that if there is at least one active receiver inside a disk centered at v of radius ( 1 + 3 f) r, then the sector A may intersect with U(r, L'., L). Since vis arbitrary, then {Any sectorn U(r,L'., L)} c;; {Cl an active receiver in a disk of radius (1 + 3 f) r }. (B.89) By using (B.88) and (B.89), (B.87) is proved. B.7 Proof of Lemma B.3 Using (B. 6), we are interested in the quantity ( ( 1 ~1'(Mg) 1 -1' - 1 ~1' + 1) (1+ 3 ~ ) 2 g) 1 - 1 - 1 1- 1 ~ .. 90) -m 1'-- g 1-'( 1-'( We consider three regimes for g, namely, g = o(m°'), g = w(m°') and g = 8(m°') =pm°'. 229 /'\ /'\ /'\ /'\ /'\ /'\ /'\ ' ( / / / / / / / / ( / / .J / / .J .I .I ( ·a •A ~ ./ "" .I / r-.. .J .I .I ./ I ~ "' .I A ~ ./ .I .I / ./ .....-:; ..- · a "-~ /\ .J .I ./ .I .I (1+ !:>.)·- ( " v .J .I .I / .I .I ( .I .I ./ .I ./ ./ .I .I \. .I .I ./ .I ./ ./ .I .I Figure B.3: In this figure, u and v are receivers. The radius of each grey sector is %r. Each grey sector is i of each disk with radius ~r. The maximum area that are consumed by receiver u is the disk centered at u, with radius (1 + ll.)r. The blue sectors are the maximum activated sectors that are caused by the active receiver u. A indicates the sector containing receiver v. When g = o(m"), by using (B.90), we have where (a) is because that (1- x) 1 ;::: 1- tx for 0 ~ x ~ 1 and t;::: 1. When g = w(m"), by using (B.90), we obtain (B.92.) 2,30 When g =pm", by using (B.90), we get (1- (plb(g))(1+3~)2g) ~ p:" (1- (1-p1-1'Ml-1'm(a-1)(1-1')) (1+3~)2pm") 0) --"----(1- (1-p1-1'Ml-1'm-°')(1+3~)2pm") pm" n pm" 1 - ( 1 - p1-1' M 1 -1'm-°')p m ( ( -(1-7)M-(1-7) ") (1+3~ )2P2-7M1-7) ® ~ ( 1 _exp (- ( 1 + 3 ~) 2 p2-1' Ml-1')) ~" ( 1+3f'.)2~7 1 2 2 ( 1 + 3&') 2-7 p . ( 1 _exp (- ( ( 1 + 3 ~) 2 ~ 7 p) 2 -1' Ml-1')) ~" (B.93) where (a) follows by using (a-l)(l-1) =-a; (b) follows because limx-+oo(l-x- 1 )" = 2 e-1; (c) is obtained by defining p = (1+ 3 &') 2 -7 p. We conclude that (l-(p 11 (g))(1+ 3 ~) 2 g) ~ = o(,;:a) and, when g =pm", then (1- (plb(g))(1+ 3 n 2 g) ~ = 8 (;:a)· Now we compute the optimal constant p, which is shown in the following lemma. Lemma B.7 The optimal value of p* to maximize (1- (p 11 (g))(1+ 3 ~) 2 g) ~is the solu tion of Moreover, the solution satisfies (5* 2 -1' M 1 -1' >a, and Eq. x =log (1 + (2 -1)x) is a fixed point equation and has a non-negative solution for x > a. 231 Proof From (B.93), we know that to maximize (1- (p 11 (g))(1+ 3 ~) 2 g) ~we need to maximize ?; ( 1 - exp( -p 2 -1' M 1 -1')). Differentiating this expression with respect to p, and equating to zero, we find (B.94) This proves the first part of Lemma B.7. Then, by letting x = p 2 -1' M 1 -1', we get x =log (1 + (2 - 1)x). (B.95) Let f(x) =log (1 + (2 -1)x) - x. We observe that if f(x) = 0, then there are two roots, one is 0 which must b excluded since x = p 2 -1' M 1 -1' > 0 and the other root is greater than 0. Differentiating with respect to x, we find d dxf(x) __ 2~--1~_ - 1 1 + (2 - 1)x (2 -1)(1 - x) - 1 1+(2-1)x (B.96) We observe that d~f(x) < 0 for x >a, d~f(x) > 0 for x <a, and d~f(x) = 0 for x =a. Thus, f(x) achieves its maximum value when x =a. Now we can see that f (a) log(l + (2-1)a)- a log(2 - 1) - a > 0, (B.97) when 0 SI< 1. Thus, the positive root of f(x) = 0 is greater than a. This proves the second part of Lemma B.7. 232 Let ¢(x) =log (1 + (2 - 1)x), then if ¢(x) is a contraction from JR to JR, then we can show that ¢( x) = x is a fixed point equation and can be solved by iterations numerically [Rud76]. Therefore, we need to show when x >a, ¢(x) is a contraction from JR to JR. Let x, y E JR. With out loss of generality we assume x > y. When x > a, we get 1¢(x) - ¢(Y)I (a) < (b) < klx-yl, (B.98) where k = i+(2-1')Y < 1. (a) is because that log(l + x) < x, when x fc 0. (b) is because when y >a, then k = i+(2!1')Y < 1. Thus ¢(x) is a contraction from JR to JR when x >a. Therefore, we conclude that ¢(x) =xis a fixed point equation for x >a. B.8 Proof of Lemma B.5 We have m* L (qt(l - (l -P;(i))M(g,(m)-1))) i=l (a)'\' 2 (qm*+l) M(gc(m) 1) 1 m* ( M(gc(m)-1) ) s L.,qi 1- -- i=l qi 1) 1 ( m* 7 ) = H(21, 1, m; _ (m* + 1)-~ L i-1' ( i ) M(gc(m)-1)-1 ' H(r,l,m) H(r,l,m) i~l m*+l (B.99) 233 where (a) is because v ~ Zm*+l· Now, in order to compute an upper bound on (B.99), we consider the first and the second term separately. In order to upper bound the first term, we have to consider the cases of I fc ~ and I= ~· For I fc ~' by using Lemma B.1, the first term in (B.99) can be upper bounded as: H(21, 1, m*) H(1, 1, m) 2 < < _l_m*1-21' __ l_ + 1 1-2'( 1-2'( ( _l_(m + 1)1-1' - _1_)2 1-'( 1-'( (1 -1)2 m*1- 2 1' - 21 1- 21 ((m + 1)1-1' - 1)2 (1 -1)2 m*1- 2 1' - 21 1 - 21 (ml-1' - 1)2 (1 -1)2 ( Mg;(m) )1-21' - 21 (1 - 21) (m 1 -1' - 1)2 (B.100) For I=~' by using Lemma B.1, the first term in (B.99) can be upper bounded as: H(21, 1, m*) H(1,l,m) 2 By letting 9c(m) = c'J:!'", we have H(21, 1, m*) H(r, 1, m) 2 < < H(l, 1, m*) H(~, 1, m) 2 log(m*) + 1 c~~ (m + 1)1-~ - l~~ r log(Mgc(m)) + log2 + 1 (2(m + 1)~ - 2)2 ~log (m) + log(c1) + 1 (2m 1 -1' - 2) 2 (B.101) (B.102) 234 The second term in (B.99), for any I< 1, can be lower bounded as: > m* (m* + 1)-1' _1~~~- I> ( _l_ml-1' __ 1_ + 1) 2 (m* + l)M(gc(,J)-1)-1 1-'( 1-'( i=l (m* + 1)-1' 1 ( _l_ml-'( _ _J'_) 2 (m* + l)M(gc(m)-1)-1 1-'( 1-'( > rm*+l M(gc(m)-1)-2 .)l x-M(gc(m) 1) 1/dX (m* + 1)-1' 1 ( _l_ml-1' _ _J'_) 2 (m* + l)M(gc(,J)-1)-1 1-'( 1-'( 1 ( l M(gc(m)-1)-2 ) . (m* + 1) - M(gc(m) 1) 11' _ 1 l _ M(g,(m)-1)-2 M(g,(m)-l)-1 I 1 2 ( M(g,(m)-1)-2 ) (l _ ) (m* + 1) - 1' l _ 1- M(g,(m)-l)-1 I I (ml-1' - i)2 l _ M(g,(m)-1)-2 M(g,(m)-l)-1 I M(gc(m)-1)-2 1 (m* + 1) M(gc(m) 1) 11 > M(gc(m)-1)-2 M(gc(m) 1) 1 I (B.103) In order to obtain the scaling behavior of (B.99) we consider the cases of I < ~' I > ~ and I=~· 235 For I< ~'let 9c(m) = c 1 J:F", by using Lemma B.1, (B.100) and (B.103), we have m' L ( qt(l - (1- P;(i))M(g,(m)-1))) i=l 1 2 (1-1)(1-21) < (1 - 1)2 c 1 - 'Im 2 7 - 1 - 21 (m 1 -'I - 1)2 (ml-'! - 1)2 +o ((cim~ + 1)1-2'1) (ml-'! - 1)2 (1 - 1)2 1-2 - 3(1-7) ( 1 ) 2 1-2 - 3(1-7) = c 'Im 2-7 1 + - (1 - 1)c 'Im 2-7 1 - 21 1 m 1 -'I - 1 1 2 1 ((cimi=~ + 1)1-2'1) - +o (ml-'! - 1)2 (ml-'! -1)2 = c 1 m 2-1 + o m 2-1 . 1(1-1) 1-2 _3(1-7) ( _3(1-7)) 1- 21 1 (B.104) For I> ~'let 9c(m) = c 1 r;", by using Lemma B.1, (B.100) and (B.103), we have m' L (qt(l - (l -P;(i))M(g,(m)-1))) i=l (B.105) This settles the scaling behavior of the term :>::z:' 1 (qt(l - (1- P;(i))M(g,(m)-l))) for I ;;f ~ 236 For the case I=~' let 9c(m) =cir;", we use (B.101) and (B.103) to obtain m' L ( qf(l - (1 - P;(i))M(g,(m)-1))) i=l = 1 log (m) + log(c1) + 1 __ 1_ + 0 (_1_) 12 (m~-lr 2m m = _1_ logm + 0 (_1_) . 12 m m (B.106) From (B.53), and by using (B.105) and (B.105), we obtain the desired result. B.9 Proof of Lemma B.6 By using (B.58) and (B.99), we have two cases of I to consider, namely, I fc ~and I=~· When I fc ~'by using (B.51), (B.52), (B.100) and (B.103), we have P~u' (B.107) 237 When I=~' by using (B.51), (B.52), (B.101) and (B.106), we have P~u' ~ (Mgc(m)) + log(Mgc(m)) + log2 + 1 __ 1_ 2 m (2(m+l)~-2r 2m +o ( Mg~m)) < ~(Mg~m))+a(Mg~m)) Thus, (B.107) and (B.108) give the desired result. B.10 Continuity and perturbations In this section, under the condition that and we want to show that (B.108) (B.109) (B.110) (B.111) 238 From calculus, We know that Tub (1 - (M *)1-1' -a) sum P m (B.112) dTs~bm + d~;b x o(m-") + o(o(m-")) ag g=p*m-a dTstbm Thus, the goal is to compute d:lb ----;:Jg g=p*m-a (B.113) dTub d lb which requires to compute J~m and ~g . To obtain d1J~bm, we first need to compute the derivative in the form of F(x) = f(x)g(x), which is given by dF(x) dx F(x) (l(x) log f(x) + ~i:\ J1(x)) f(x)g(x) (l(x)logf(x) + g(x) j1(x)). f(x) (B.114) Then, by denoting 9r(m) as g, we obtain dT 3 ~ = ~_1_6_. ((l- (plb(g))(Hs~) 2 g) ") dg a 9 6 2 9 = ~~ (-~ :g ((plb(g))(H 3 ~ ) 2 g) + ( l - (plb(g))(Hs~ ) 2 g) :g (~)) = _1_6_ (-"(P11(g))(i+s~)2g (1 + 36)2 (1og(p11(g)) + _g_ (8p11(g))) 62 g 2 p1b(g) ag + ( 1- (plb(g))(Hs~)2g) ~:; n) = _1_6_ (-"(P11(g))(i+s~)2g (1 + 36)2 (1og(p11(g)) + _g_ (8p11(g))) 6 2 9 2 p1 1 ( 9 ) a 9 + ( 1- (plb(g))(H 3 ~ ) 2 g) ~~). (B.115) 239 Then, we get 8P11(g) = ~ (l -G(Mg)1-1' _ G + 1) ag ag _1_m1-1' __ 1_ 1-'( 1-'( ( Mg)-1' M (-1-ml-1' __ 1_) _ (-1-(Mg)l-1' __ 1_ + l) m-1'. dm 1-'( 1-'( 1-'( 1-'( dg 1 ml- 1 1 ( ) 2 1-'( - 1-'( _l_ml-i __ 1_ · (B116) 1-'( 1-'( Therefore, we obtain dTs'tibm erg dplb ag 16 (-~(plb(g))(Hlf' )2g (1 + ~ )2 (log(plb(g)) + ~ ( ~))) 62 (Mg) 7M -'-m1-1 __ 1_ 1-1 1-1 (B117) 240 By letting m -+ co, we obtain dTs~bm erg dplb ag g=p*m-a = - 1 + - ---pn e-((p) + o(l) -log(p 11 (g))m"'1m 1 -'I 16 ( 36) 2 1 Af'l- 1 ( * ) n 62 2 p* 1 - I m" - _1_6_ (1 + 36)2 -1:_ (e-((p*) + o(l)) _ri_p*m" 62 2 p* m" p1b(g) + _1_6_ (1- e-((p*) + o(l)) _l _ _ri_ M'f-l p*'im"'Ym 1 -'I 62 p*2 m2" 1 - I 16 ( 36) 2 Af'l- 1 * n 2 = - 1 + - --p*'f-l (e-((p) + o(l))-(-(l -p 11 (g)) + 0 ((l -p 11 (g))) m"'1m 1 -'I 6 2 2 1 - I m" - _1_6_ (1 + 36)2 (e-((p*) + o(l)) _n_ 62 2 plb(g) 16 ( (( *) ) M'f-l -2 n 1 + - 1- e- P + o(l) --pn -m"'Ym -'! ~ l-1 m~ __ _1_6_ ( 36) 2 Af'l- 1 n-l ( -((p*) ( l) - 62 1 + 2 1 - Ip e + o 1 · (M 1 -'1 p*1-'! -"-m"(l-'f)-(l-'f)m"'1m 1 -'I + 0 (-"-m"'1m 1 _'1_l_)) ma ma m2a 16 ( 36) 2 ( -((p*) ) n - 62 1 + 2 e + o(l) plb(g) 16 ( (( *) ) M'f-l -2 n 1 + - 1- e- P + o(l) --pn -m"'Ym -'! ~ l-1 m~ = - ~ 6 2 (1+ 3 ~) 2 1 ~ 1 (e-((p*) +o(l)) (1+ 0 (~")) n - ~~ ( 1+ 3 ~) 2 (e-((p*) + o(l)) n 16 * Af'l- 1 _ 2 + - (1- e-((p) + o(l)) --pn n 62 l-1 = _1_6_ (M'f-l pn- 2 (1 - e-((p*) + o(l)) 62 l-1 - ( 1 + 3 ~) 2 ( e-((p*) + o(l)) - ( 1 + 3 ~) 2 1 ~ 1 ( e-((p*) + o(l)) + 0 ( ~")) n = _1_6_ (M'l-1 n-2 (1 - -((p*)) 62 1 - Ip e ( 36) 2 * ( 36) 2 1 * ) - 1+ 2 e-((p)_ 1+ 2 1 _ 1 e-((p)+o(l) n = O(n). Thus, we obtain dTs'tibm = T;!,(1- (Mp*) 1 -'Ym-a + o(m-")) + d~fb erg (B.119) 242 Appendix C Proofs of Chapter 6 C.1 Proof of Theorem 6.1 Let J(C, Q) denote the (random) number of independent sets found by Algorithm 6.1 applied to the conflict graph Hc,Q defined in Section 6.2, where C is the random cache configuration resulting from the random caching scheme with caching distribution p, and Q is the packet-level demand vector resulting from the random i.i.d. requests with demand distribution q. Recall that we consider the limit for F, B -+ co with fixed packet size F / B. Then, since the term min (6.8) has been already shown to upper bound the average rate due to GCC 2 (see Remark 3 in Section 6.2.3), Theorem 6.1 follows by showing that (C.1) for any arbitrarily small f > 0. By construction, the independent sets I generated by GCC 1 have the same ( un ordered) label of users requesting or caching the packets {p(v): v EI}. We shall refer to such unordered label of users as the user label of the independent set. Hence, we count the independent sets by enumerating all possible user labels, and upperbounding how many independent sets I Algorithm 6 .1 generates for each user label. 243 Consider a user label Uc c;; U of size£, and let Jc,o(Uc) denote the number of indepen dent sets generated by Algorithm 6.1 with label {µ( v ), 17( v)} =Uc. A necessary condition for the existence of an independent set with user label Uc is that, for any user u E Uc, there exist a node v such that: 1) µ( v) = u (user u requests the packet corresponding to v ), and 2) 17( v) =Uc\ { u} (the packet corresponding to vis cached in all users Uc\ { u} and not cached by any other user). Therefore, the following equality holds with probability 1 (pointwise dominance) :Jc,o(Uc)=max L l{'l(v)=Uc\{u}}. uEUe v:p(v)=ifu (C.2) In (C.2), with a slight abuse of notation, we denote the condition that the packet p(v) associated to node v is requested by user u as p( v) 3 fu, indicating that the "file" field in the packet identifier p(v) is equal to the u-th component of the (random) request vector f. The indicator function captures the necessary condition for the existence of an independent set with user label Uc expressed (in words) above, and the maximum over u E Uc is necessary to obtain an upper bound. Notice that summing over u E Uc instead of taking the maximum would overcount the number of independent sets and yield a loose bound. 244 Then, using (C.2) and the definition of J(C, Q), we can write JE[J(C, Q)ICJ lE tu?tu Jc,o(Uc) cj lE t L max L l{'l(v) =Uc\ {u}} cj uEUe £=1 U.e.<;;_U v:p(v)=ifu f~n (gqfi) (tu?tu~t~vp~fu l{'l(v)=Uc\{u}}) (C.3) t (~) L (fr qdi) (u;;1 1 ~X/ L 1 {17(v) = {1, ,£} \ {qr}J) £=1 dEFf i=l v:p(v)3du J t (~) d~' (g qdi) ( L 1 {f = argmax L 1 {'l(v) = {1, ... ,£} \ {u}}} (C.5) JE<l fEF vp(v)3j L l{'l(v)={l, ... ,£}\{u}} (C.6) v p(v)3f where (C.3) follows by writing the conditional expectation with respect to the demand vector explicitly in terms of a sum over all possible files, after recognizing that the indi- cator function 1 {17( v) =Uc \ { u}} is a random variable only function of the cache placement C (in fact, this depends only on whether the £ - 1 users in Uc \ { u} have cached or not the packet associated to node v ), and where (C.4) follows by noticing that the term max L l{'l(v) =Uc\ {u}} uEUe v p(v)3fu depends only on the£ (possibly repeated) indices {Ju : u E Uc}. Therefore, after switching the summation order and marginalizing with respect to all the file indices corresponding to 245 the requests of the users not in Uc, due to the symmetry of the random caching placement and the demand distribution (i.i.d. across the users) we can focus on a generic user label of size £, which without loss of generality can be set to be {1, ... , £}. At this point, the sum with respect to Uc c;; U reduces to enumerating all the subsets of size £ in the user set of size n, yielding the binomial coefficient GJ. Finally, (C.6) follows from replacing the max with a sum over all possible file indices, and multiplying by the indicator function that picks the maximum. At this point, we need to study the behavior of the random variable YC,f= L l{'l(v)={l, ... ,£}\{u}}, (C.7) v p(v)3f where u E {1, ... , £} and where, by construction, the sum extends to the nodes corre- sponding to file f requested by user u, i.e., not present in its cache. By construction of the caching scheme, these nodes are B(l - PJM). Furthermore, the random variable 1 {17( v) = {1, ... , £} \ { u}} takes value 1 with probability (pJM)C-l (1 - PJM)n-C, corre- sponding to the fact that £ - 1 users have cached packet p( v) and n - £ users have not cached it (user u has not cached it by construction, i.e., we are conditioning on this event). However, they are not i.i.d. across different v. By denoting PC,f"' (ptM)C- 1 (1-ptM)n-C, we can see that IE[YC,J] =IE I L 1 {'l(v) = {1, ,£} \ {u}}l = B(l -ptM)PC,f, (C.8) lvp(v)3f J Then, for p(v),p(v 1 ) 3 f, IF(l {'l(v) = {1, ... ,£} \ {u}} = 1,1 {17(v 1 ) = {1, ... ,£} \ {u}} = 1) = PCJIF(l {1J(v 1 ) = {1, ... ,£} \ {u}} = lll{'l(v) = {1, ... ,£} \ {u}} = 1) = PCJ(pjM)C-1(1 - pjMr-c, (C.9) 246 where ( B-2 ) I PJMB-2 PfM - 1 Pf= ( B-1 ) =Pf - B _ l =Pf+ 6(B), PJMB-1 and 6(B) -+ 0 as B -+ co independently with Pf· Let Pf;,f "' (pjM)f- 1 (1 - pjM)n-f, then we obtain Pf;,t (pjM)c-1(1 - pjMr-c ((Pf+ 6(B))M)c- 1 (1- (Pf+ 6(B))Mr-c dPc f I Pu+ d 6(B) + o(6(B)) Pf PJ~Pj Pc,t + ((e - l)(pfM)c- 2 (1 - pfMin-c M - (n - £)(pfM)c- 1 (1 - pfMin-f-l) Ml , · 6(B) PJ=Pf +o(6(B)) Pu+ 6 1 (B), (C.10) where 6 1 ( B) -+ 0 as B -+ co independently with Pu. Then, we have IE ( L 1 {'l(v) = {1, ... ,£} \ {u}}) 2 1 vp(v)3f J L Pc,t + L L PuPiJ vp(v)3f v:p(v)?? f v 1 :p(v 1 )3 f,v 1 f-v L Pc,t+ L vp(v)3f vp(v)3fv 1 p(v 1 )3f,v 1 fv ( L Pu+ L L PuPu) + L L Pu6 1 (B) vp(v)3f vp(v)3fv 1 p(v 1 )3f,v 1 fv vp(v)3fv 1 p(v 1 )3f,v 1 fv ( L Pu+ L L PuPu) + s;B) L L PuPu vp(v)3f vp(v)3fv 1 p(v 1 )3f,v 1 fv f,f vp(v)3fv 1 p(v 1 )3f,v 1 fv B(l -pfM)Pu(l - Pu)+ (B(l -ptM)) 2 Pl,f + 0 (B 2 ) ((l -pfM)) 2 Pl,f, (C.11) 247 Therefore, by using the fact that Var(Yu) = lE[YJ,t] - JE[Yu] 2 and from Chebyshev's inequality 1 , we obtain we have that 2 y CJ !'+ ( M)C- l ( 1 - M)n-c for B -+ co. B(l - PfM) Pf Pf , Equivalently, we can write (C.12) for any arbitrarily small f > 0, where we define the function (already introduced m Remark 5 in Section 6.2.3), (C.13) It follows that we can replace the last line of (C.6) by the bound (holding with high probability) B(gc(f) + E). In order to handle the indicator function in (C.5), we need to consider the concentration (C.12) of Y~f around the values gc(f). Sorting the values {gc(f) : f E F} in increasing order, we obtain a grid of at most m discrete values. The limit in probability (C.12) states that the random variables Yc,j/B concentrate around their corresponding values gc(j), for any j E F. Taking f sufficiently small, the intervals [gc(j) - E, gc(j) + E] for different j are mutually disjoint for different values of j, unless there are some j fc j1 such that gc(j) = gc(l). For the moment we assume that all these values are distinct, and we handle the case of non-distinct values at the end (we will see 1 Here we need only convergence in probability. 2 As usual, .:£+indicates limit in probability [GSOl]. 248 that this does not cause any problem). Now, we re-write the indicator function in (C.5) as 1 {1 = argmax L 1 {'l(v) = {1, ... ,£} \ {u}}} = 1 {1 = argmaxYc,j/B} (C.14) ;Ed ;Ed v p(v)3j and compare it with the indicator function 1 {1 = argmaxgc(j)}. JE<l (C.15) If l ~ d then both indicator functions are equal to 0. If l E d, suppose that l = argmaxjE<l gc(j) such that (C.15) is equal to 1. Then, (C.14) is equal to 0 only if for some j Ed: j fc l, Yc,j/B > Yu/B. Since Yc,j/B E [gc(j) - E,gc(j) + E] and Yu/BE [gc(f) - E, gc(f) + E] with high probability, and, by construction, gc(j) + f < gc(f) - E, it follows that this event has vanishing probability as B -+ co. Similarly, suppose that l E d and that l fc argmaxjE<l gc(j) such that (C.15) is equal to 0. Then, (C.14) is equal to 1 only if Yu/B > Yc,jmax/B, where jmax = argmaxjEd9c(j). Again, since Yu/ B E [gc(f) - E, gc(f) + E] and Yc,jmax/ B E [gc(jmax) - E, gc(jmax) + E] with high probability, and, by construction, gc(f) + f < gc(jmax) - E, it follows that this event has vanishing probability as B -+ co. We conclude that Jim IF (11 {1 = argmaxYc,j/ B}- 1 {1 = argmaxgc(j)} IS f) = 1. B---+oo ;Ed ;Ed (C.16) Since convergence in probability implies convergence in the r-th mean for uniformly ab- solutely bounded random variables [GSOl] and indicator functions are obviously bounded by 1, we conclude that 249 as B -+ co, where D is a random subset of£ elements sampled i.i.d. (with replacement) from F with probability mass function q. Now, replacing the last line of (C.6) with the deterministic bound B(gc(f) + E) (which holds with high probability as explained before) and taking expectation of the indicator function using the convergence of the mean said above, we can continue the chain of inequalities after (C.6) and show that the bound IE[J(C, Q)ICJ St (n) L IF(!= argmaxgc(j))(gc(f) + E) B e JED C~l fEF (C.17) holds with high probability for B -+ co, for any arbitrary f > 0. In the case where for some distinct j, j1 the corresponding values of gc(j) and gc(l) coincide, we notice that outcome of the indicator functions (C.14) and (C.15) are irrelevant to the value of the bound, as long as they pick different indices which yield the same maximum value of the function gc(} Hence, the argument can be extended to this case by defining "equivalent classes" of indices which yields the same value in the bound. Theorem 6.1 now follows by using (C.13) and by noticing that the probabilities IF(!= argmaxjEV9c(j)) coincide with the terms Pf/ defined in (6.10). C.2 Proof of Lemma 6.1 Applying Theorem 6.1 to the case p = p, we have that, for all f > 0, Jim IF(RGCC(n,m,M,q,p)Smin{,P(q,p),m}+E) =1. F-+oo (C.18) 250 Then, we can write ,P( q, p) (a) < ~ (n) ~ (l _ M)n-Hl( M)c-1 L, £ L,Pfl Pf Pf c~1 f~1 (:-1) (1- (1- ~)E[Nm]) + n f~+l qf (:-1) (1- (1-~) nGm) +n(l-Gm) 1/J(q, iii), (C.19) where Nm is the (random) number of users requesting files with index less than or equal to iii, (a) follows from Jensen's Inequality, and (b) because E[Nm] = n LT~i qf = nGm. C.3 Proof of Theorem 6.2 First, notice that since the users decoder AuO operate independently, the rate of the optimal scheme R*(n, m, M, q) is non-increasing inn. In fact, an admissible scheme for n users is also admissible for any n1 < n users. 3 The first step of the proof consists of lower bounding the rate of any admissible scheme with the optimal rate of a genie-aided system that eliminates some users. By construction of the genie, we can lower bound the optimal rate of the genie-aided system by the optimal rate over an ensemble of reduced systems with binomially distributed number of users, reduced library size, and uniform demand distribution. Finally, we lower bound such ensemble average rate with a lower bound on the optimal rate in the case of arbitrary 3 To see this, simply add n - n 1 virtual users to the reduced system with n 1 users, generate the corresponding random i.i.d. demands according to q, and use the code for the system of n users, to achieve the same rate, which is clearly larger or equal to the optimal rate for the system with n 1 users. 251 (non-random) demands, by using a result proven m [NMA13], which we state here for convenience, expressed in our notation, as Lemma C.1. Fix£ E {1, · · · , m} and consider the following genie-aided system: given the request vector f, all users u EU such that fu > £ are served by a genie at no transmission cost. For each u EU such that fu S £, the genie flips an independent biased coin and serves user u at no transmission cost with probability 1 - 'IL, while the system has to serve qfu user u by transmission on the shared link with probability 'IL. We let N denote the qfu number of users that require service from the system (i.e., not handled by the genie). It is immediate to see that N ~ Binomial(n, £qc). In fact, any user u has probability of requiring service from the system with probability IF( u requires service) m L IF(u requires servicelfu = f)qf J~l f L IF(u requires servicelfu = f)qf J~l f L qc qf = £qc, J~l qf (C.20) (C.21) where (C.20) follows from the fact that, by construction, IF(u requires servicelfu = f) = 0 for f > £. Notice also that £qc S 1, since we have assumed a monotonically non-increasing demand distribution q, and if £qc > 1 then qf 2 1/ e for all 1 s f s e, such that :>::~~l qf > 1, which is impossible by the definition of probability mass function. Now, notice that the optimal achievable rate for the genie-aided scheme provides a lower-bound to the optimal achievable rate R*(n, m, M, q) of the original system. In fact, as argued before, the genie eliminates a random subset of users (which depends on the realization of the request vector and on the outcome of the independent coins flipped by the genie). We let R;enie ( n, m, M, q) denote the optimal rate of the genie-aided scheme. Furthermore, we notice that in the genie-aided system the only requests that are handled 252 by the system are made with uniform independent probability over the reduced library {1, ... , £}. In fact, we have IF(fu =flu requires service) IF(fu = f, u requires service) IF( u requires service) IF( u requires servicelfu = f)qJ £qc 1 i; qi - 1 { } O Cqe - C for f E 1, ... , £ for f > e (C.22) It follows that, for a given set of users requiring service, the optimal rate of a system restricted to those users, with library size equal to £, and uniform demand distribution, is not larger than the optimal rate of the genie-aided original system. Moreover, by the symmetry of the system with respect to the users, this optimal rate does not depend on the specific set of users requesting service, but only on its size, which is given by N, as defined before. Consistently with the notation introduced in Section 7.1, for any N = N this optimal rate is denoted by R*(N, £, M, (1/£, ... , 1/£) ). Then, we can write: R*(n, m, M, q) > RZeme(n, m, M, q) > IE[R*(N,£,M, (1/£, .. , 1/£))] n L R*(N,£,M, (1/£, ... , 1/£))IF(N = N) N~l n > LR*(N,£,M,(1/£, ,1/£))IF(N =N) (C.23) N=r > R* ( r, £, M, (1/ £, ... , 1/ £))IF (N 2 r) (C.24) where (C.23) holds for any 1 Sr Sn, since the summation contains non-negative terms, and where (C.24) follows again by the fact that the optimal rate is non-increasing in the number of users. 253 A lower bound on R* ( r, e, M, (1 I e, ... ' 1/ £)) can be given in terms of the lower bound (converse) result on the optimum rate for a shared link network with arbitrary demands (see Lemma 3 in [NMA13]). This is given by the following: Lemma C.1 Any admissible scheme achieving rate R(r, £, M, {1/£, ... , 1/£}) for the shared link network with r users, library size£, cache capacity M, and uniform demand distri- bution {1/£, ... , 1/£} must satisfy R(r,£,M, {1/£, ... ,1/£}) 2 z (1- w) IF(Z 2 z), (C.25) for any z = {1, ... , £}, where Z is a random variable indicating the number of distinct files requested when the random demand vector is i.i.d. ~ Uniform{l, ... , £}. D Using Lemma C.1 in (C.24) we have R*(n, m, M, q) 2 zIF (N 2 r) IF(Z 2 z) ( 1- w) · (C.26) Next, we further lower bound the two probabilities IF (N 2 r) and IF(Zc 2 z) and find the range of the corresponding parameters. To this purpose, we recall the definition of self-bounding function: Definition C.1 Let X c;; JR and consider a nonnegative v-variate function g : xv -+ [O, oo). We say that g has the self-bounding property if there exist functions Yi : xv-l -+ JR such that, for all (x1, ... , xv) E xv and all i = 1, ... , v, and 0 ~ g( Xl, · · · , Xv) - 9i( Xl, · · · , Xi-1 1 Xi+l, · · · , Xv) ~ 1, v L(g(x1,··· ,xv)-gi(x1,··· ,Xi-1,Xi+i,··· ,xv)) Sg(x1,··· ,xv)· i=l (C.27) (C.28) 0 254 The following lemma [BLM13] yields a concentration property of random variables expressed as self-bounding functions of random vectors. Lemma C.2 Consider X c;; JR and the random vector X = (X1, ... , Xv) E xv. Let Y = g(X) where g(-) has the self-bounding property of Definition C.1. Then, for any 0 < µ S IE[Y], IF(Y - IE[Y] S -µ) S exp (- 2 ;~]) . (C.29) D Next, we observe that g( x1, ... , xv) = :>:::r~i Xi is self-bounding when its argument is a binary vector (i.e., X = {O, 1}). Hence, N satisfies Lemma C.2 and we can write IF(N 2 IE[N] - µ) 2 1 - exp ( - 2 ;~N]) , (C.30) with 0 < µ S IE[N]. Since N ~ Binomial(n, £qc) we have IE[N] = n£qc. Hence, letting µ = IE[N] - r in (C.30) we obtain IF ( N 2 r) 2 1 - exp ( (n£qc - r) 2 ) ~ p (£ ) 2 fl 1 , r , ncqc (C.31) for 0 < r S n£qc. The variable Z defined in Lemma C.1 can be written as c Z = L 1 {Cl u requesting file f} . (C.32) J~l Although the binary random variables 1 {Cl u requesting file f} are not mutually inde- pendent, nevertheless Z is given as the sum of the components of a binary vector and therefore Lemma C.2 applies. In particular, we have IE[Z] = £ ( 1 - ( 1 - ~) r) 255 such that, operating as before, we arrive at the lower bound ( (IE[Z]-z) 2 ) c, IF(Z;:>z);:>l-exp - 2 1E[Z] =P2(£,r,z) (C.33) for 0 < z S IE[Z]. Then, for some£, r, by maximizing the obtained lower bound (C.26) with respect to the free parameter z, we obtain R*(n,m,M, q) ;:> IF(N 2 r) max IF(Z 2 z)(z - zM/l£/zj) zE{ 1, , r min{IE[Z],r} l} (C.34) Let z E (0, IE[Z]], we consider two cases: z 2 1 and 0 < z < 1, 1. if z;:> 1, let r 2 1, then R*(n, m, M, q) > IF(N 2 r) max IF(Z 2 z)(z - zM/l£/zj) zE{ 1, , r min{IE[Z],r} l} (a) > IF(N;:>r) max~ IF(Z;:>z)(z-zM/l£/zj) zE{l, ,rmrn{z,r}l} (b) > IF(N ;:>r)IF(Z;:>Z) max~ (z-zM/l£/zj), (C.35) zE{l, ,rmrn{z,r}l} where (a) is because z S IE[Z] and (b) is because that if z S z then IF(Z 2 z) 2 IF(Z 2 Z). By using the lower bounds (C.31) and (C.33) in (C.26), we have R*(n,m,M,q) 2 P1(£,r)P2(£,r,Z) max~ (z- zM/l£/zj). zE{l, ,rmrn{z,r}l} (C.36) 256 2. if 0 < z < 1, then let r = 1, z = 1 and, using (C.31) and (C.33) into (C.26), we have R*(n, m, M, q) > IF(N 2 l)IF(Z 2 1)(1- M/£) 0) IF(N 2 l)IF(Z 2 Z)(l - M/£) > IF(N 2l)P2(£,1,Z)(l-M/£) (b) P1 (£, r)P2(£, 1, Z)(l - M /£), (C.37) > where (a) follows from observing that Z is an integer, so that IF(Z 2 1) = IF(Z 2 Z) when z E (0, 1). Similarly, (b) holds also for r > 1, since in this case IF(N 2 1) 2 IF(N 2 r). Therefore, taking the maximum of (C.36) and (C.37), we obtain R*(n, m, M, q) 2 max{Pi(£,r)P2(£,r,z) max~ (z-zM/l£/zj)l{z,r;:> 1}, zE{l, , rmm{ z,r )l) P1 (£, r )P2(£, 1, Z)(l - M /£)1 {z E (0, 1)}} . (C.38) Maximizing over£, r, and z, we obtain (6.18) in Theorem 6.2. C.4 Proof of Theorem 6.3 Letting iii = m and using Lemma 6.1 we obtain Ru 1 (n,m,M,q,m) min{G';-1) (1-(1-~)n),m} ~ min { c; - 1) ( 1 - ( 1 - n::)) ':- 1, m} < min { n ( 1- ~), :- 1, m} < min {: - 1, m, n} , (C.39) 257 where in (a) is because that (1- xr 2 1 - nx for x S 1. The proof of Theorem 6.3 follows by showing that min { M - 1, m, n} is order-optimal. In the following, we will evaluate the converse shown in Theorem 6.2, and compute the gap between R 1 b ( n, m, M, q), given in ( 6 .18 ), and Rub ( n, m, M, q, m) to show the order optimality of RLFU-GCC by appropriately choosing the parameters£, r, z, z. Specifically we choose: C=m, r = 6(1 -1)n, z= !7m (1-exp (-6(1-1):))' (C.40) (C.41) (C.42) where 6 E (0, 1) and 17 E (0, 1) are positive constant independent of the system parameters m, n, M, and determined in the following while z will be determined later according to the different value of m, n, M. Note that <7m (1 - exp (-6(1 -1){.);)) S 6(1 -1)n hence by definition z S r. We now compute each term in (6.18) individually. 4 To this end, using (C.40) and (C.41) we first find an expression for n£qc and£ (1- (1- ~rJ in terms of 5, er, m, n and f· Specifically, by using Lemma B.1, we can write n£qc > > m-'Y nm-~-~ H(r,l,m) nml-1 _l_ml-"f __ 1_ + 1 1-"( 1-"( (1 -1)n + o(n), (C.43) 4 In evaluating (6.18), anytime that the value of z or IZl diverges as m--+ oo, we ignore the non-integer effect without mentioning. 258 and n£qc from which it follows that < nml-1 _l_(m + 1)1-1' __ 1_ 1-'( 1-'( (1-1)n + o(n). n£qc = (1 -1)n + o(n). Furthermore, using (C.41), we have ( ( 1 ) 8(1-'()n) m 1- 1- m (C.44) (C.45) m ( 1 - exp (-6(1 -1) :) ) + o ( m ( 1 - exp (-6(1 -1l;l9J)l Then, by using (C.40)-(C.42), (C.46) and (C.45) in (6.19) and (6.20), we obtain and 1- exp ( 1- exp ( 1- exp ( 1 - o(l), P2(£, r, Z) = 1- exp ( (a) 2 1 - o(l) (n£qc - 6(1 - 1)n)2) 2n£qc (((l -1)n + o(n)) - 6(l -1)n)2) 2((1-1)n + o(n)) ((1- 6)(l -1)n + o(n))2) 2(6(1-1)n + o(n)) ( e ( 1 - ( 1 - i ri - z) 2) 2e(1-(1-iri ( C.4 7) (C.48) 259 where (a) follows from the fact that (e (l - (l - ir/-; z) 2 = e (m (1- exp (-6(1-ilii_))) 2£(1-(1- 7 )) m Thus, by using Theorem 6.2, we obtain R 11 (n, m, M, q) (a) 2 Pi(£,r)P2(£,r,Z) max (z-zM/l£/zj) zE{l, , rZIJ 2 (1- o(l))(l - o(l)) max (z - zM/l£/zj) zE{l, ,rzi} 2(1-o(l)) 2 max (z-zM/l£/zj), zE{l, ,rZ!J where (a) is because z Sr. (C.49) In the following, we consider two cases, namely n = w(m) and n = O(m). For each of these two cases, we treat separately the sub-regions of M illustrated in Fig. C.1 and Fig. C.2, respectively. n = w(m) 1 2 " SM= o(m) 1 M<- 2a M=8(m) {:CC -<3 2M Figure C.1: The sub-cases of the regimes of M when n = w(m). 260 1 2<>(1 - c') <; M ~ o(m) ~8(1- o) 2 1 m { 2~ 23 M~8(m) m 2M < 3 n~O(m) M< 1 2"(1-c') { <> 8 (/- o): <; M ~ o(m) ~0(1 - a) < 1 M = 8(m) {-!a-8-(l_m __ o)-n 2 3 1 m M<--- aO(l-a) n m ' < 3 2 a0(1 - a)n Figure C.2: The sub-cases of the regimes of M when n = O(m). C.4.1 Region of n = w(m) In this regime, using the fact that ~ = w(l), the expression of z, given in (C.42), reduces to: z=!7m(l-o(l)). (C.50) Consequently, (C.49) can be rewritten as: R 11 (n,m,M, q) ;:> (1- o(l)) 2 max (z- zM/l£/zj). zE{l, ,r<>m(l-o(l))l} (C.51) 261 C.4.1.1 When 2 c>(l~o(l)) SM= o(m) Letting z = l 2 'Af J, from (C.51) we obtain R 1 b(n,m,M,q) ;:> (1-o(l)) 2 (z-l~JM) (1 - o(l)) 2 ( l-"'-J - li\Y j Ml 2M l m J l2% J (1 - o(l)) 2 ( 4 ~ + o ( 4 ~)), (C.52) from which using (C.39) we have: Rub(n,m,M,q,m) J?T-l+o(J?T) =4 (1) lb( ) S 2(m (m)) +o · R n,m,M,q (1-o(l)) 4 M+o 4 M (C.53) C.4.1.2 When M = 8(m) • If 2 'Af 2 3, letting z = l 2 'Af J, from (C.51) we get R 1 b(n, m, M, q) > (1-o(l)) 2 (l-"'-J- li\YJ Ml 2M l m J l 2% J > (1-0(1)) 2 (2~-1) (1-2:+0(1)) > (1- o(l)) 2 ( i\? 2 - l + o(l)), (C.54) from which, using (C.39), we obtain Rub(n,m,M,q,m) J?T-l+o(JIT) 6 (1) (C.55) lb S ( m ) S +o . R (n,m,M,q) (1-o(l))2 2M 2 - 1 +o(l) 262 • If 2 'Af < 3, letting z = 1, from (C.51), we obtain R 1 b(n, m, M, q) 2 (1- o(l)) 2 ( 1- ~), (C.56) from which, using (C.39), we have Rub(n,m,M,q,m)< J?,T-l+o(J?,T) <m+o(l)<6+o(l). (C.57) Rlb(n,m,M, q) - (1- o(l))2 (1- ;',;) - M - C.4.1.3 When M < 2 a(l~o(l)) letting z = crm (1- o(l)), from (C.51), we get lb 2 ( M) R (n,m,M,q)2(1-o(l))crm 1-l~J , (C.58) from which, using (C.39), we obtain Ru 1 (n,m,M,q,iii) m Rlb(n, m, M, q) S (1- o(l))2crm ( 1- lrJ) 1 S 1 + o(l). (1 _ a(l-0(1))) er 2l~J (C.59) C.4.2 Region of n = O(m) C.4.2.1 When ~6(1- 1) 2 1 z, given in (C.42), boils down to: z crm(l-exp(-6(1-1):)) > crm(l-e- 1 ). (C.60) •If 2 (a(l~e '))SM= o(m), by letting z = l 2 'Af J, R 1 b(n,m,M,q) is given by (C.52) and consequently using (C.39), we have Rub(n,m,M,q,m) J?,T-l+o(J?,T) =4 (1) lb( ) S 2(m (m)) +o · R n,m,M, q (1- o(l)) 4 M + o 4 M (C.61) 263 • If M = 8(m), by letting z = l 2 'Af J when 2 'Af 2 3 and letting z = 1 when 2 'Af < 3, and by using (C.55) and (C.57), respectively, we conclude that Ru 1 (n,m,M,q,m) 6 (l) lb( ) < + 0 . R n,m,M,q - (C.62) • If M < 2 (c>(l~e '))'letting z = crm (1- e- 1 ), by using (C.40)-(C.42) and (C.49), we have from which using (C.39), we obtain Ru 1 (n, m, M, q,m) < R 11 (n, m, M, q) C.4.2.2 When ;';::6(1- 1) < 1 z boils down to: z crm (1- exp (-: 6(l-1l)) m <;} crm (: 6(1-1)- ~ (: 6(1-1))2) 1 n2 2 2 cr6(1-1)n - 2cr m 6 (1 - 1) (b) 1 1 2 2 > cr6(1-1)n - 2 cr 6 (l - 1 / (1 -1) n 1 cr6(1 -1)n - 2 cr6(1 -1)n 1 2 cr6(1 - 1)n, (C.63) (C.64) (C.65) 264 where (a) follows from 1- e-x 2 x - x 2 2 , (b) is due to the fact that ;';:;<l(l -1) < 1. •If ifsd-1') ';:SM= o(m), by letting z = l 2 'Af J, R 1 b(n,m,M,q) is given by (C.52) and finally, using (C.61), we obtain Rub(n,m,M,q,m) 4 (l) lb( ) < + 0 . R n,m,M,q - (C.66) • If M = 8(m), letting z = l 2 'Af J when 2 'Af 2 3 and letting z = 1 when 2 'Af < 3, and by using (C.55) and (C.57), respectively, we conclude that Rub(n,m,M,q,m) 6 (l) lb( ) < + 0 . R n,m,M,q - (C.67) • If M < ifsd-1') ';:,letting z = ~<T6(l -1)n, and using (C.40)-(C.42) and (C.49), we have R 1 b(n,m,M, q) 2 (1- o(l)) 2 ~<T6(1-1)n (1- l ~ j) . ~a8(l-1)n (C.68) 265 Recalling that by assumption :;;;s(l-1) < 1ander<1, we have that ~ 08 /;'_'!)n 2 2. We hence consider two cases. For ~ 08 (';'-'!)n 2 3, using (C.39) and (C.68), we obtain Ru 1 (n, m, M, q,m) R 11 (n, m, M, q) < < < n (1 - o(l))2~cr6(1 -1)n (1 - l ~ j) ~a0(1-1)n 2 (1 - o(l))2cr6(1 -1) ( 1 - 1 m ) ~n ~a0(1-1)n l 2 (1 - o(l))2cr6(1 -1) ( 1 - 2 a5(1,,;; 7 )n) 2 (1 - o(l))2cr6(1 -1) ( 1 - 2 ~~) 8 (1 - o(l))2cr6(1 -1)' while for 2 S ~ 08 (';'-'!)n < 3, using (C.39) and (C.68), we obtain Ru 1 (n, m, M, q,m) R 11 (n, m, M, q) < < < n (1 - o(l))2~cr6(1 -1)n (1 - l ~ j) ~a0(1-1)n 2 (1 - o(l))2cr6(1 -1) ( 1 - ~'ie) 2 (1- o(l))2cr6(l -1) (1- ~) 8 (1 - o(l))2cr6(1 -1) C.5 Proof of Table 6.2 in Theorem 6.6 (C.69) (C.70) In this section, we provide the proof of Table 6.2 in Theorem 6.6, where we assume n, m-+ oo and n = o (m'Y). Table 6.2 considers the regions 0 SM< 1, 1 SM< ";,",and 1 M 2 ";, 7 , excluding the sub-region {1 S M < ";, 7 }n{ M = tm 7- 1 } whose order-optimality 266 is analyzed in [JTLC15] and whose corresponding order-optimal results are provided in [JTLC15]. Except for the region 0 S M < 1, we further consider the subregions illustrated in Figs. C.3 and C.4, treated separately in the following proofs. ma l<M<- - n { M~o(m) 1 <; M ~ o ( n°',) _l_ > 3 { ~, ' = n,n c;;=-r M~ 8(m) =K1m __!_<3 ~, - see [51] w ( n°':_1) = M < m°' __!__ > 2 { M~o(m) n {~' M~ 8(m) 1 = K1m _ < 2 ~, - Figure C.3: The sub-cases of the regimes of M when 1 S M < ~ 7 , where r;, r; 1 are some constants which will be given later. C.5.1 Region of 0 SM< 1 1 In this case, we want to prove that RLFU-GCC with iii = n 7 is order optimal. From Lemma 6.1, we have that: Ru 1 (n, m, M, q, iii) To this end, using (C.71) and Lemma B.1 , we can write the rate for RLFU-GCC 1 with iii= n-::Y as: (C.72) 267 n -- 1 = w(l) ma- n -- =8(1) ma-1 M = o(m) { _1_>3 M ~ 8(m) "1 =tqm 1 -<3 "1 - " 1 <ro(a - 1) n a: ma-1 <ro(a - 1) n a: mu-1 > 1 Figure C.4: The sub-cases of the regimes of M when M 2 ~ 7 , where 1< 1 , O", 6 are some constants which will be given later. Next, we evaluate the converse replacing in Theorem 6.2, the following parameters: S(r-1) ~ r = n1, I - 0"6(1- 1) ~ z = n1, I z = l zJ, 268 with 0 < 6 < 1, and 0 < er < 1 positive constants determined in the following and such that z S r. After some algebraic manipulations, we obtain Ru 1 (n, m, M, q,m) R 11 (n, m, M, q) < 1 2n7 (1- o(l))2" 8 (1'-l) (1- 1 ) n* 1' l a5(~-1) J 2 (C.73) which shows the order-optimality of the achievable expected rate. Eq. (C.73) proves that in this regime (0 S M < 1 small enough), caching cannot provide large gain, or the gain of caching is at most additive so that M cannot affect the order of the expected rate. C.5.2 Region of 1 SM<':" In this regime of M, we have three cases to consider, which are 1 < M = o ( n "~'), M = 8 ( n"~') and w ( n"~') = M < ':" (see Fig. C.3). C.5.2.1 When 1 SM= o ( n7~ 1 ) This case further splits in two scenarios: M = o(m) and M = 8(m) (see Fig. C.3). 1 1 •If M = o(m), letting m = M'Yn'Y, usmg (C.71) and Lemma B.1, after some algebraic manipulations, we obtain < 1 1 ( 1 1) 2M'Yn'Y M'Yn'Y M +o M . (C.74) 269 Next, we prove the order-optimality of the expected rate achieved by the proposed scheme. Following the similar steps as before, we use Theorem 6.2 to compute the converse. The parameters required in Theorem 6.2 are summarized in the following. 1 1 £=iii= M7n7, (C.75) (C.76) (C. 77) z = lzJ, (C.78) with 0 < 6 < 1, and 0 < er< 1 positive constants determined in the following. Note that by definition z < r < iii. Next we compute each term in (6.18) individually. To this end, using (C.75) and (C.76), we first find an expression for n£qc and £ (1 - (1 -1 rJ in terms of 6, er, m, n and/. Specifically, using (C.75) and Lemma B.1, we have n£qc nml-1 > _l_ml-'f __ 1_ + 1 1-'( 1-'( (C.79) 270 and n£qc < from which n·m· (iii)-'( H(r,l,m) nml-1 _l_(m + 1)1-'f __ 1_ 1-'( 1-'( 1 m/-1 1 m ( ) '(-1 1-l - 1-l m+l 1-=J' 1 ( 1-=J' 1 ) (r - 1 )M " n 7 + o M " n 7 , Next using (C.75) and (C.76), we have Then, by using (6.19), (C.76), and (C.81), we obtain ( ( 8( 1) 1-7 1) 2) n£qc - 'Y; M-cr n 7 1- exp 2n£qc (C.80) (C.81) (C.82) 271 while using (6.20), (C.77) and (C.82), we have P2(£, r, Z) =1-ex (-(e(1-(1-irl-zl2) p 2e(1-(1-irl 0;) 1 - o(l), (C.84) where (a) follows from 1n1<{, - -- < 1n1 6(1- l)Ml=i ~ , (l (l l)r) 6(r- l)Ml=i ~ 21 £ / (C.85) with (C.85) derived from (C.82) usmg 1 - exp(-x) > x - x 2 2 for x > 0, and 1 - exp( -x) S x for x > 0. Finally, replacing Eqs. (C.75)-(C.78), (C.83), and (C.84) in Theorem 6.2 and using footnote 4, we obtain 272 R 11 (n, m, M, q) 2P1(£,r)P2(£,r,Z) max (z-zM/l£/zj) zE{l, ,rZIJ 2 (1- o(l))(l - o(l)) max (z - zM/l£/zj) zE{l, Jil} 2 (1-o(l)) 2 z(l- l~J) M 1-~------ l M*n* J from which, using (C.74), we obtain Ru 1 (n, m, M, q,m) R 11 (n, m, M, q) < < 2M-;jn'7 + 0 M~n"Y 1 1 ( 1 1) (1- o(l))2" 8 ~1'-l) (1- _d__ 1 ) M7n* 00(1-1) 2+o(l) ( 1 - (1))2 08(1'-l) (1 - 1 ) ' 0 2, -----1J:'__ _ 1 00(1-1) (C.86) (C.87) where <Y, 6 E (0, 1) can be chosen accordingly (possibly a function of 1) such that ---~ 2 ---- is a positive constant, which shows the order-optimality of the 05(7-1) (i 1 ) 21 ~-1 00(1-1) expected rate. 273 • If M = 8(m) = t<1m + o(m), where 0 < "1 < 1 is a given constant such that 1 1 SM< ~ 7 and 1SM=o(n7- 1 ), using (C.71) and letting iii= m, we obtain < :-1 +o (:) (C.88) Following the similar steps as before, we use Theorem 6.2 to compute the converse. The parameters appeared in Theorem 6.2 are summarized in the following. £=iii =m, <l(r-1) n r= r m1-l' _ <Y<l(r-1) n z=---- 1 m'f-1 z = max { l 2~ j , 1} , (C.89) (C.90) (C.91) (C.92) with 0 < 6 < 1, and 0 < <Y < 1 positive constants determined in the following. Note that by definition z < r. As before, we compute each term in (6.18) individually. To this end, using (C.89) and (C.90), we first find an expression for n£qc and £ (1 - (1 - J rJ in terms of 6, "' m, n and/. Specifically, using (C.89) and Lemma B.1, following similar steps as in (C.43) and (C.44), we obtain (r- 1) n ( n ) n ( n ) 1 +o -- 1 S n£qc S (r-1)-- 1 +o -- 1 , r ml- ml- ml- ml- (C.93) 274 from which, using (6.19) and (C.90), we obtain nr_,qc - -1'-m1-1 ( ( " 8(1'-1) n ) 2 ) 1 - exp -~-----~- 2n£qc ( ( 1'-1 n ( n ) 8(1'-1) n ) 2 ) -;y m?-1 + O m?-1 - -1' P2 m?-1 > 1- exp -~--~-----~~~-~ 2 (r- l)m7-1 +o (m7-1)) 1 - o(l). (C.94) On the other hand, using (C.75) and (C.76), via Taylor expansion, we obtain = 6(r- 1) n + 0 (-n-), I m'f-1 m'f-1 (C.95) from which, using (6.20) and (C.91), after some algebra, we have P2(£, r, Z) =1-ex (-(e(1-(1-irl-zl2) P 2e(1-(1-1ri 0;) 1 - o(l), (C.96) where (a) follows from the fact that (e(1-(1-1rJ--z) 2 =-(-n-) ( ( 1) r) e 1 . 2£ 1 - 1 - I ml'- In the following, we distinguish between two cases: m = _L > 3 and m = _L < 3 M K:1 M K:1 - (see Fig. C.3). 275 - If M = ~ 1 > 3, from (C.92), we have: z = l2: j' (C.97) from which replacing Eqs. (C.89)-(C.91), (C.94), (C.96) and (C.97) in (6.18), we have R 11 (n, m, M, q) Thus, we obtain Ru 1 (n, m, M, q,m) < R 11 (n, m, M, q) m M (1- o(l))4~ ( 2 'Af - 1) 2 1 M 2- m 2 < -1-1=12, 2-3 proving the order-optimality of the achievable expected rate. (C.98) (C.99) 276 - If M = ~ 1 S 3, from (C.92), we have: z = 1, (C.100) from which replacing Eqs. (C.89)-(C.91), (C.94), (C.96) and (C.100) in (6.18), we have R 11 (n, m, M, q) 2P1(£,r)P2(£,r,Z) max (z-zM/l£/zj) zE{l, ,rZIJ 2 (1 - o(l)) 2 ( 1 - ~) . Thus, we obtain Ru 1 (n, m, M, q,m) < R 11 (n, m, M, q) Jlf-1 (1 - o(l))3 (1 - ;',;) l\f (1- ;',;) (1 - o(l))3 (1 - ;',;) m MS3. 1 C.5.2.2 When w(n7-1) = M < ";," (see Fig. C.3 for a reminder) In this case, letting m = M, we have Ru 1 (n,m,M,q,m) < (1-G;yc)n. Next we consider two regimes: M = o(m) and M = 8(m). • If M = o(m), using (C.103) and Lemma B.1, we have R ub( M -) Ml-1' ( Ml-1') n, m, , q, m ~ n + o n . (C.101) (C.102) (C.103) (C.104) 277 To compute the converse, in this case we use the second term of (6.18), with the parameters given by: £=cM, 6(r- l)c 1 -1' 1-1' r= nM , I z =er, (C.105) (C.106) (C.107) and c > 1, 0 < 6 < 1, and 0 < er< 1 positive constants determined in the following. Next we compute each term in (6.18) individually. To this end, using (C.105) and ( C .106), and recalling that iii = M we first find an expression for n£qc and £ (1- (1- irJ in terms of 6,cr,m,n and/. Specifically, using (C.75) and Lemma B.1, we have n£qc > and n£qc < from which n · cM · (cM)-1' H(r, l,m) n(cM) 1 -1' _l_ml-1' __ 1_ + 1 1-'( 1-'( (r- l)cl-1' nMl-1' + o(nMl-1'), I n · cM · (cM)-1' H(r, l,m) n(cM) 1 -1' (C.108) (C.109) 278 Using (6.19), (C.106), and (C.110), we obtain ( n£qc - 8(1'-~c1-7 nMl-1') 2) 2n£qc ( (1'-l~c1-7 nMl-1' + o ( nMl-1') - 8(1'-~c1-7 nMl-1') 2) 2 ((r- l)c 1 -1'nM1-1' + o (nMl-1')) (1 - 6)2(r- l)cl-1' Ml-1' ( Ml-1') 2 212 n +on , (C.111) while using (6.20) and (C.107) we have ( (1- o-)2) P 2 (£,1,Z) = 1-exp 2 · (C.112) Then, replacing (C.105)-(C.107), (C.111), and (C.112) in the second term of (6.18), we obtain R 11 (n, m, M, q) 2 Pi(£, r)P2(£, 1, Z)(l - M/£) ;:> (1-o(l)) cl-'5) 2 ~~;- l)cl-1' nM 1 -1' + o(nM 1 -1')) (1-exp cl ~0-) 2 )) (i- c:) ( (( l-a-) 2 )) ( 1) (1-6)2("'-l)cl-1' 2 (1 - o(l)) 1 - exp 2 1 - ;:: 2 ~ 2 nM 1 -1' + o(nM 1 -1'), (C.113) from which using (C.104), we obtain Ru 1 (n, m, M, q, m) < R 11 (n, m, M, q) < (1 - o(l)) ( 1 - exp ( (l- 2 "l 2 )) (1 - ~) (l- 8 ) 2 ~~;l)c'- 7 nMl-1' + o(nMl-1') 1 ------------- + o(l) (C.114) ( 1 _ ( (l-c>) 2 )) (l _ l) (1-8)2(1'-l)c'-7 ' exp 2 c 2,2 279 which shows the order-optimality of the achievable expected rate for RLFU and RAP. • If M = 8(m) = tqm + o(m), with 0 < "1 < 1, using (C.103) and Lemma B.1, we have Ru 1 (n,m,M,q,m) < (,.;-1' -1) nm 1 -1' +o(l) ( 1-"r 1 ) nM 1 -1' + o(l). (C.115) To compute the converse, similar as before, we use the second term of (6.18). The values of the parameters£, r, and zin (6.18) will be different based on the fact that M m = _1_ > 2 and Mm = _1_ < 2 (see Fig. C.3). K:1 K:1 - - When M = ~ 1 > 2, the parameters£, r, and zin (6.18) are given as in (C.105) (C.107) with the additional constraint that 1 < c < 2 to guarantee that£ Sm. Following the same steps as in the case of M = o(m), we obtain (C.113), from which, using (C.115) we obtain: Ru 1 (n, m, M, q, m) R 11 (n, m, M, q) < ---~--1 ----- -!E<O(l11)9) ( l _exp ( (l- 2 c>) 2 )) (l - ~) (1-8)2(;;1)c7-1 which shows the order-optimality of the achievable expected rate for RLFU and RAP. - When M = ~ 1 S 2, the parameters£, r, and z in (6.18) are given by: C=m, _ 1-l 1'-1, Ml-1' r - --tt: 1 un , I z =er, (C.117) (C.118) (C.119) 280 with 0 < 6 < 1, and 0 < er < 1 positive constants determined in the following. Next we compute each term in (6.18) individually. Specifically, using (C.117) and Lemma B.1, recalling that iii = M we have n£qc nml-1 > (C.120) and n£qc < (C.121) Thus, by using (C.120) and (C.121), we obtain from which, using (C.118) we have: ( n£qc - 7,.r 1 snM 1 -'( f) 2n£qc 1 ( (1 - 6)2(r- l)t<r 1 M1-'Y) ;:- - exp 212 n > (1 - 6)2(r - l)t<r1 nMl-'f - ~ ( (1 - 6)2(r - l)t<r1 nM1-'!) 2 2,2 2 2,2 (C.123) 281 Furthermore using using (6.20) and (C.119), we have ( (1- <T)2) P 2 (£, 1,Z) = 1- exp 2 , (C.124) from which replacing (C.117)-(C.119), (C.123) and (C.124) in the second term of (6.18) we obtain R 11 (n, m, M, q) Thus, by using (C.115) and (C.125), we obtain Ru 1 (n, m, M, q, m) lb < R (n,m,M,q) - ( M)'f-1 < 1 1- ;;;;M +o(l). (1-810) 2 (1'-1) 1 (1- ((l-c>10) 2 )) 1- - Note that 21 2 21 1 exp 2 m 1 _Jcp-1 { 1, r S 2 m < 1-M - m 1- l, I> 2 (C.125) (C.127a) (C.127b) 282 (C.126) from which using (C.126), and (C.127), we obtain Ru 1 (n, m, M, q, m) max{l, 1- 1} lb < -----~~-~~-~~ R (n, m, M, q) - (1-8) 2 (1'-1) 1 (l - ((l-c>) 2 ))' 2,2 21 i exp 2 (C.128) which shows the order-optimality of the achievable expected rate for RLFU and RAP. C.5.3 Region of M 2 ':" In this case, we want to prove that RLFU-GCC with m = m is order optimal. To this end, using (C.71), we can write the rate for RLFU-GCC with m =mas: Ru 1 (n,m,M,q,m) < (:-1) (1-(1-~)nG;n)+(l-G;n)n < :-1 +o (:) (C.129) At this point, we distinguish between two cases: m7_ 1 = w(l) and m7_ 1 = 8(1) (see Fig. C.3). C.5.3.1 When m7-i = w(l) To compute the converse, again, we evaluate the first term of (6.18) in Theorem 6.2, with the parameters defined as: C=m, <l(r-1) n r= r m1-l' <Y6(1- 1) n z= m1-l' I 1' M ' 1 l c>8(1'-1) m j z= max{l 2 'Afj,1}, M = o(m) M = 8(m) (C.130) (C.131) (C.132) (C.133) 283 with 0 < 6 < 1, and 0 < er < 1 positive constants determined in the following. Note that by definition z < r. Next we compute each term in (6.18) individually. To this end, using (C.130) and (C.131), and recalling that iii = m we first find an expression for n£qc and £ (1- (1 - :l' rJ in terms of 6, er, m, n and/. Specifically, using (C.130) and Lemma B.1, following similar steps as in (C.43) and (C.44), we have 1-l n ( n ) n ( n ) ----+o -- S n£qc S (r-1)--+o -- , r mf-l m/-1 m/-1 m/-1 (C.134) from which, replacing (C.131) and (C.134) in (6.19), we have: ( ( l::r=-11 _n _ fu=-11 _n ) 2 ) I m1-1 I m1-1 > 1- exp 2((r-l)m;.'-1 +o(m;.'-1)) (a) 1 - o(l). (C.135) Furthermore, using (C.130) and (C.131), via Taylor expansion, we obtain 1 -7-mf-1 m 1-(1-m) ( 5(7-1) n ) 6(r- 1) n ( n ) +o -- ' I m'f-1 m'f-1 (C.136) from which, replacing (C.132), and (C.136) in (6.20), we have: P2(£, r, Z) ( ( 8('!-l)_n +o(-n )-"8('!-1) n ) 2 ) I m1-1 m1-1 1 ml 1 > 1 - exp -~--~--------~-~ - 2 (8('1-1) _n_ + 0 (-n-J) I m1-1 m1-1 =1-o(l). (C.137) 284 Thus, replacing (C.135) and (C.137) in the second term of (6.18), we obtain R 11 (n, m, M, q) 2P1(£,r)P2(£,r,Z) max (z-zM/l£/zj) zE{l, ,rm 2 (1- o(l))(l - o(l)) max (z - zM/l£/zj) zE{l, ,rzi} 2 (1- o(l)) 2 max z(l - M/l£/zj). zE{l, ,rZ!J (C.138) Next, we find an explicit expression for maxzE{l, .rm z (1 - M /l £/ z J ). Using (C.133), after some algebraic manipulations, we have <T<l(rl-l) (1- <T<l(rl-l)) :+o(:), M=o(m) max z 1--- > ( M) zE{l, ,rm l£/zj (1-:-), M=8(m) and :s3 ~( 2 ~-l)+o(l), M=8(m) and :>3, (C.139a) (C.139b) (C.139c) where (C.139a) follows from footnote 4 and from the fact that M 2 ~ 7 , while (C.139c) follows from the fact that z = l 2 'Af J > 2 'Af - 1 and lm/zj > ~ - 1. 2M Replacing C.139 in (C.138) and using (C.129), after simple algebraic manipulations we have: 1 --~---~+o(l) M=o(m) c>8(~-1) ( 1 - c>8(~-1)) ' Rub(n,m,M,q,iii) < m m Rib(n,m,M,q) M + o(l) S 3 + o(l), M = 8(m) and M S 3 2 2 m (1-o(l)) 1 M +o(l)<12+o(l), M=8(m)andM>3 (2 - m) 285 C.5.3.2 When m:/'-i = 8(1) (see Fig. C.3) Since M 2 ':", we have that 2 Ji'.} and M = 8(m). To compute the converse, similar as before, we use (6.18) in Theorem 6.2, with all the parameters given by: C=m, 6(r- 1) n r= m1-l' I z= \ O', c>8(1'-1) 1' 1, z= 1, l2~ J' c>8(1'-1) n s 1 1' m7 1 n c>8(1'-1) n ml 11 1' m7 1 > 1 c>8(1'-1) n < 1 I ml 1 - c>8(1'-1) n > l c>8(1'-1) m < 2 I ml 1 ' I M c>8(1'-1) n > l c>8(1'-1) m > 2 I ml 1 ' I M - (C.141) (C.142) (C.143) (C.144) with 0 < 6 < 1, and 0 < O" < 1 positive constants determined in the following. Next, we compute each term in (6.18) individually. In doing this, we consider to further regions " 8 (1'-l) n < 1 and " 8 (1'-l) n > 1 (see Fig. C.3). I ml 1 - I ml 1 • If " 8 (1'-l) n S 1, usmg (C.141) and Lemma B.1, following similar steps as in 1' m7 (C.43) and (C.44), we have 1-l n ( n ) ----+o -- / m'f-1 m'f-1 S n£qc S (r- 1) m~-l + o (m~-1)' (C.145) 286 from which using (6.19) and (C.142), we obtain ( " 8(1'-1) n ) 2 ) nr_,qc - -1'-rn:FT 2n£qc ( (1'-l)_n __ 8(1'-l)_n_ + 0 (-n-)) 2 ) I m1-1 I m1-1 m1-1 2 ((r- l)m7-1 + O (m7-1)) ((1-6)¥~)2) 2 ((r- l)m 7 _ 1 ) (l - 5 )2(1- l)) (C.146) 2,2 ' where (a) follows from the fact that 8(1) = m7_ 1 2 Ji'.} 2 1. Furthermore, using using (6.20) and (C.143), we have (C.147) Replacing (C.141)-(C.143), (C.146) and (C.147) in the second term of (6.18), we obtain R 11 (n, m, M, q) 2 Pi(£, r)P2(£, 1, Z)(l - M / £) ( ( (1-6)2(1-1))) ( M) > 1-exp 1-- - 21 2 m ' (C.148) 287 from which using (C.129), we have Ru 1 (n, m, M, q,m) R 11 (n, m, M, q) < (a) < li'.f-l+o(l) ( 1 - exp ( (1-8~~~'f-1))) (1- ;',;) 1 m (l) ( 1 - exp ( (1-8~~~'f-1))) M + o c>8('f-1) ( 1 - exp ( ~1-8~~~'f-1))) + o(l), where (a) is because that M S m7-1 S " 8 (~-l). (C.149) • If " 8 (~-l) m7 1 > 1, using (C.141) and (C.142), and following the same procedure as (C.85), via Taylor expansion, we obtain 1 -7-mf-1 m 1-(1-m) ( 5(7-1) n ) 0) 6(1-l) n (l) 1 + 0 ' I m't- (C.150) where (a) is due to 8 ('f;l) m7_ 1 = 8(1). Hence, replacing (C.141), (C.142), and (C.145) in (6.19), we obtain: 1- exp ( 1- exp ( ~ 1-exp( 1- exp ( (C.151) 288 where (a) is because that m7_ 1 2 " 8 (~-l), while replacing (C.143) and (C.150) in (6.20), we obtain: (C.152) Replacing (C.141)-(C.144), (C.151), and (C.152) in the first term of (6.18), and noticing, from (C.143), that, when " 8 (~-l) m7 1 > 1, by definition, z < r, we obtain R 11 (n, m, M, q) 2Pi(£,r)P2(£,r,Z) max (z-zM/l£/zj) zE{l, ,rm ;:> (1-exp( (l-6] 2 )) (1-exp( (l-<T)2)) max z(l-M/l£/zj). 2/<T 2<T zE{l, , rm (C.153) Replacing (C.141) and (C.144) in (C.153), we have (C.154a) (C.154b) 289 from which replacing (C.154) in (C.153) and using (C.129) we obtain Rub(n,m,M,q,iii) < R 1 b(n,m,M,q) - 2, (1) ( ( (1 8)2)) ( ( (1 <')2)) + 0 ' ml(1- 1) 1 - exp - 2 ~" 8 1 - exp -~ (1-exp(-~)) (1-e~p(-¥)) (~-" 8 ~'1-l)) +o(l), (C.155a) (C.155b) where (C.155a) holds for " 8 (~-l) Ji'.} < 2, while (C.155b) holds for " 8 (~-l) Ji'.} ;:> 2. 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Asset Metadata
Creator
Ji, Mingyue
(author)
Core Title
Fundamental limits of caching networks: turning memory into bandwidth
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
07/29/2015
Defense Date
03/31/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
algorithms,caching networks,coding theory,device-to-device networks,fundamental limits,index coding,information theory,OAI-PMH Harvest,shared link networks,wireless communications
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English
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Caire, Giuseppe (
committee chair
), Goldstein, Larry (
committee member
), Molisch, Andreas F. (
committee member
)
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davidjmy@gmail.com,mingyuej@usc.edu
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https://doi.org/10.25549/usctheses-c3-614118
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Dissertation
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Ji, Mingyue
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
algorithms
caching networks
coding theory
device-to-device networks
fundamental limits
index coding
information theory
shared link networks
wireless communications