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Data replication and scheduling for content availability in vehicular networks
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Data replication and scheduling for content availability in vehicular networks
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DATA REPLICATION AND SCHEDULING FOR CONTENT AVAILABILITY IN VEHICULAR NETWORKS by Shyam N Kapadia A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) January 2007 Copyright 2007 Shyam N Kapadia Dedication This dissertation is dedicated to my mother, father, brother, and grand-mother. ii Acknowledgements First and foremost, I am especially thankful to Bhaskar for the countless nights that we spent together working on some part or another of this thesis. Over the years, he has been a tremendous source of inspiration and strengthened my belief that with hard work and dedication any goal can be achieved. His own work ethic and discipline have inspired me in more ways than one. I would also like to thank my co-advisor, Shahram; we have collaborated for over 5 years now. I have learnt a lot under his guidance as well. Bhaskar pushed me more toward complimenting my research with rigorous mathematical analysis and adopted a more \hands-on" to \hands-o®" approach to advising as I moved toward the¯nalyearsofmyPhD.Shahramontheotherhandwasmorehands-oninhisadvising concentrating on the minutest details in terms of paper submissions, presentations etc. I would also like to thank Dr. Helmy for exposing me to the research in wireless ad-hoc networks, the initial collaboration with him is what motivated me to process toward a PhD. My committee members, namely, Dr. Psounis, and Dr. Zimmerman, also provided valuable advice which helped in improving this thesis. I would also like to thank Dr. Gully Burns, a research assistant professor in the Neurobiology department at USC, currently at ISI, for not only ¯nancially supporting me during my Masters and earlier part of my PhD but also the training and experience I received while collaborating with him was invaluable. I learnt the art of writing high quality code for release to the outside world. He gave me a lot of °exibility in terms of my working hours and gave me challenging projects to work on. I have also had a lot of philosophical discussions with him about life and my progress toward the PhD for which I would be eternally grateful. Both on a personal and professional front I found his attitude extremely warm and helpful. iii I would also like to express my gratitude for the members of the ANRG group and other close friends and researchers whose work either implicitly or explicitly in°uenced my work. Speci¯cally, our alumni: Nara, Gang, and Yang have been extremely helpful in giving me kind advice about how to go about my job search. Marco, who will be graduating along with me, has been doing really good work both in his research and also for the group, he de¯nitely has been an excellent student role-model for others in the group. Numerous interesting conversations both on the research and non-research front have provided valuable take-away lessons. Have also been involved in a variety of discussions with Kiran, especially toward the end of my PhD, he has been extremely helpfulinmorewaysthanone. Avinash,thecode-guruoftheANRGgroup,hasimparted and trained a lot of personnel in our group especially in tiny OS related programming stu®. Sundeep contributed equally for the same, we spent a lot of night outs together in the laboratory working on a collaborative group project, that time was by far the most fun time I have had during my PhD. I had a good time with the interns, Maulik and Xiaofan, that joined our group for the summer of 2006, the experience provided an opportunity to adopt the role of a mentor. Also special thanks to other members of the group: Hua, Joon, Yi, Pai-Han, Peter, and Amitabh. Among other friends, wouldliketo start by thankingKarthik, wehaveshared numer- ous courses together since the beginning of my tenure at USC. Hence, have had a lot of study sessions as well as usual discussions on the professional as well as personal fronts over the past few years. Also, would like to thank Rishi, my ¯rst stint with research was in collaboration with him and Karthik, it was a valuable learning experience that paved the way for my thesis. We still have an un¯nished tech report about the collaboration which some day we hope to ¯nish. Then come the two Ramakrishnas, one called Ram and the other Ramki, the former has been a great help in my job search, the latter in giving lots of advice on a lot of topics related to research and career development. Finally, would like to mention some other friends who I have known for a long time now, Mithu, Puneet, Vidula, Rohyt, Kiran, and Sumit, anytime I needed a break from my PhD, I always °ew down to the east coast to meet them. And every single time they iv mademefeellikeaVIP,mostofthesevisitswerestressrelieversbecauseofwhichIcould come back and restart my research with renewed enthusiasm. Last but de¯nitely not least, I would like to thank my mother, she has always made me believe that I am the best and given me support, hope and a sense of purpose in everything I do, always encouraging and motivating me whenever I hit a road block in my progress. My brother, Ram, in his own way, has encouraged me to expedite the completion of my thesis. He has always been around for advice whenever I have needed another perspective on things. I would like to acknowledge my sister-in-law, Neha, who enteredourlivesduringthelastcoupleofyearsofmythesis;shehasde¯nitelycontributed alotonapersonalfront. Also,wouldliketoacknowledgemygrandmotherandmyfather who I am sure implicitly helped me cope up with the varied challenges during my PhD. Moreover, my frequent trips back home, almost twice a year, involved interactions with the rest of my family, my uncles and aunties and cousins, they all in their own way have helped me come through this process. I truly believe that a PhD has helped me emerge a more con¯dent, mature, aware, and experienced human being. v Table Of Contents Dedication ii Acknowledgements iii List Of Tables ix List Of Figures x Abstract xiv Chapter 1: Introduction 1 1.1 Overview of Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 PAVAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Static Replication schemes. . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Zebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Common Assumptions and Architectural Framework 7 2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Architectural Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 3: PAVAN 14 3.1 PAVAN variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.1 Spatio-Temporal Lookahead (STL) parameter . . . . . . . . . . . . 17 3.2 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 Utility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 4: Static Replication Schemes 27 4.1 Family of Replication Policies . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Data items with display time one and long client trip duration . . . . . . 29 4.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 vi 4.2.1.1 Sparse Scenario . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1.2 Dense scenario . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2.2.1 Scale-up experiments . . . . . . . . . . . . . . . . . . . . 38 4.2.2.2 Variation in car density . . . . . . . . . . . . . . . . . . . 39 4.2.2.3 Variation in storage per car . . . . . . . . . . . . . . . . . 42 4.2.2.4 Variation in data item repository size . . . . . . . . . . . 42 4.3 Data items with display time one and short trip duration . . . . . . . . . 42 4.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3.1.1 Sparse Approximation . . . . . . . . . . . . . . . . . . . . 45 4.3.1.2 Dense Approximation . . . . . . . . . . . . . . . . . . . . 46 4.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Data items with higher display time and long client trip duration . . . . . 48 4.5 Data items with higher display time and short client trip duration . . . . 50 4.5.1 Aggregate availability latency as a function of car density (N) . . 53 4.6 Evaluation with a real map . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.6.1 Results with replication schemes . . . . . . . . . . . . . . . . . . . 56 4.7 Evaluation with real movement traces . . . . . . . . . . . . . . . . . . . . 57 4.7.1 UMassDieselNet Traces . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7.2 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 5: Zebroids 67 5.1 Overview of Zebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Carrier-based Replacement policies . . . . . . . . . . . . . . . . . . . . . . 71 5.4 Analysis Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4.1 One-instantaneous zebroids . . . . . . . . . . . . . . . . . . . . . . 74 5.4.2 z-relay zebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Simulation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.6.1 Zebroid replacement schemes . . . . . . . . . . . . . . . . . . . . . 79 5.6.2 Zebroids performance improvement . . . . . . . . . . . . . . . . . . 81 5.6.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.6.2.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.6.3 Zebroid overhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.6.4 Zebroids with inaccurate route predictability . . . . . . . . . . . . 88 5.6.4.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.6.4.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6.5 Maximum improvement with zebroids . . . . . . . . . . . . . . . . 89 5.6.5.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6.5.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 90 vii 5.6.6 Zebroid trade-o®s with car density and storage per car . . . . . . . 91 5.6.6.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6.6.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.6.7 Impact of di®erent trip durations and repository sizes . . . . . . . 92 5.6.7.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.6.7.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.7 Evaluation with a real map . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.7.1 Results with zebroids . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Evaluation with Real Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.8.2.1 Requests issued at the start of the day . . . . . . . . . . 101 5.8.2.2 Requests issued at equal inter-arrival times during the day 102 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 6: Related Work 109 6.1 Other Components of an AutoMata Application . . . . . . . . . . . . . . 109 6.2 Replication in MANETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3 Sparse Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4 Intelligent Transportation Systems (ITS) . . . . . . . . . . . . . . . . . . . 115 Chapter 7: Conclusions 116 Bibliography 119 viii List Of Tables 2.1 Terms and their de¯nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 Four variants of PAVAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Three utility models to evaluate alternative variants of PAVAN. . . . . . . 20 4.1 Approximateoptimalreplicationexponentsfordataitemswithhigherdata item display times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.1 Related studies on intermittently connected networks. . . . . . . . . . . . 113 ix List Of Figures 1.1 An example illustration of an AutoMata application. . . . . . . . . . . . . 2 1.2 Components of an AutoMata application . . . . . . . . . . . . . . . . . . 3 2.1 A hierarchical architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 An example 6£6 map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 An overview of PAVAN, its inputs and output. . . . . . . . . . . . . . . . 15 3.2 The numbers in the cells indicate STL value for the shaded cell numbered 1. 18 3.3 A comparison of PAVAN with di®erent inputs for utility models 1 and 2 as a function of the degree of replication of the titles. . . . . . . . . . . . . 20 3.4 Di®erence in the availability latencies as a function of the degree of repli- cation of the titles for di®erent title display times. . . . . . . . . . . . . . 23 3.5 Comparisonofdi®erenceintheavailabilitylatenciesof SS only andPL t for di®erent AutoMata densities in a 10x10 map as a function of the degree of title replication. The title display time under consideration spans 5 cells. . 24 4.1 Sparse analysis ( Equation 4.6) versus simulation obtained average avail- ability latency for a data item as a function of its replicas for a 10£10 torus, when the number of cars is set to 100. . . . . . . . . . . . . . . . . 34 4.2 Figure 4.2(a) shows the validation of the analytical expression in Equa- tion 4.25 for the probability that availability latency is zero. Figure 4.2(b) shows the probability that the availability latency is zero as a function of the replicas for the data item for 5 di®erent car densities f50, 100, 150, 200, 250g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 The complete picture depicting the availability latency for a data item obtained via simulations as compared with its sparse and dense approxi- mation as a function of its replicas for a 10£10 torus, when the number of cars is set to 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Aggregate availability latency for di®erent replication strategies for a 10£ 10 torus when T =100 and N =50. Figures (a), (b), and (c) depict three di®erent storage values per car: f4,10,25g. . . . . . . . . . . . . . . . . . . 37 x 4.5 Scale-up experiments where the total storage to the data item repository size is held constant at S T T = 3000 600 . The number of cars and the storage per car are varied to realize S T =3000. . . . . . . . . . . . . . . . . . . . . . . 40 4.6 Aggregate availability latency for the three replication schemes as a func- tion of the car density when the storage per car is ¯xed at 3. Here T =100. 41 4.7 Aggregate availability latency for the three replication schemes as a func- tion of the storage per car when data item repository size is 50 and car density is 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.8 Aggregate availability latency for the three replication schemes as a func- tion of the data item repository size for a car density of 50 and storage per car of 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.9 Average availability latency for a data item as a function of its replicas for a¯nitetripduration° of10. Thesimulationcurvesareplottedalongwith the sparse and dense approximations for ¯nite trip duration for a 10£10 torus, when the number of cars is set to 50. . . . . . . . . . . . . . . . . . 47 4.10 Aggregate availability latency for di®erent replication strategies for a 10£ 10torusfora¯nitetripdurationof10whenT =100andN =50. Figures (a), (b), and (c) depict three di®erent storage values per car: f4,10,25g. . 48 4.11 Average availability latency for a data item as a function of its replicas for di®erent data item display times for a 10£10 torus. The latency is given by C r ¾ i where the exponent ¾ increases with data item display time. . . . . 49 4.12 Figure4.12(a)shows± agg ofthesqrt,linearandrandomreplicationschemes versus ® for ¢=4 and N =200. Figure 4.12(b) shows the % comparison of the linear and random schemes wrt the sqrt scheme for this scenario. Region I and Region II, respectively, indicate the parameter space where n=1 and n=0:5 perform the best. . . . . . . . . . . . . . . . . . . . . . 51 4.13 AmapoftheSanFranciscoBayAreaobtainedfromhttp://maps.google.com is shown in Figure 4.13(a). Figure 4.13(b) superimposes a 15£15 grid on this map and labels the cells appropriately with the freeway IDs that they overlap with. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.14 Theintersectionbetweenfreeways880and85iscapturedinthe¯gurealong with the equivalent probability transitions in the Markov model based on data obtained from Caltrans regarding the vehicular densities.. . . . . . . 56 4.15 Performance of various replication schemes as a function of car density when T = 25, ® = 2, and ° = 10. Figure 4.15(b) shows the performance wrt the linear scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.16 Performanceof variousreplication schemesas afunctionof storage percar when T =25, N =50, and ° =10. Figure 4.16(b) shows the performance wrt the linear scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 xi 4.17 Performanceofvariousreplicationschemesasafunctionofdataitemrepos- itory size when N = 50, ® = 2, and ° = 10. Figure 4.17(b) shows the performance wrt the linear scheme. . . . . . . . . . . . . . . . . . . . . . . 58 4.18 The number of active buses for each trace representing the bus encounters for each day of a 60-day period. The buses operated from 7am to 5pm. . . 60 4.19 The CDF of the time between encountersaveraged across all the traces for 2 di®erent separation times 0s (Figure 4.19(a)) and 20s (Figure 4.19(b)). . 60 4.20 Aggregate availability latency for satis¯ed requests and the aggregate un- satis¯ed request metric for the random, square-root, and linear replication schemes are shown in Figure 4.20(a) and (b) respectively. The ratio of the storage per car to the data item repository size, ® T is maintained as 1:5. . 62 4.21 Aggregate availability latency for satis¯ed requests (Figure 4.21(a)) and the aggregate unsatis¯ed request metric (Figure 4.21(b)) as a function of the storage per car for a data item repository size of 25. . . . . . . . . . . 63 4.22 Aggregate availability latency for satis¯ed requests (Figure 4.22(a)) and the aggregate unsatis¯ed request metric (Figure 4.22(b)) as a function of the data item repository size when storage per car ® is ¯xed at 3. . . . . . 63 4.23 CDF of the time between encounters from the Markov model for a 25£25 torus with a car density of 15. . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.24 Aggregate availability latency for satis¯ed requests and the aggregate un- satis¯ed request metric as obtained from an equivalent scenario employing the Markov model. The ratio of the storage per car to the data item repository size, ® T is maintained as 1:5. . . . . . . . . . . . . . . . . . . . . 65 5.1 Availability latency when employing one-instantaneous zebroids as a func- tion of (N,®) values, when the total storage in the system is kept ¯xed, S T =200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Latencyperformancewithone-instantaneouszebroidsviasimulationsalong with the analytical approximation for a 10£10 torus with T =10. . . . . 81 5.3 Latencyperformancewithz-relayzebroidsviaanalysisandsimulationsfor a 10£10 torus with T =10. . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Latency performance with both one-instantaneous and z-relay zebroids as a function of the car density when ®=2 and T =25. . . . . . . . . . . . 84 5.5 Latency performance with both one-instantaneous and z-relay zebroids as a function of ® when N =50 and T =25. . . . . . . . . . . . . . . . . . . 85 5.6 Replacement overhead when employing one-instantaneous zebroids as a function of (N,®) values, when the total storage in the system is kept ¯xed, S T =200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 xii 5.7 ReplacementoverheadwithzebroidsforthecaseswhenN isvariedkeeping ®=2 (¯gure 5.7.a) and ® is varied keeping N =50 (¯gure 5.7.b). . . . . 87 5.8 Availability latency, ± agg , for di®erent car densities as a function of the prediction accuracy metric with ®=2 and T =25. . . . . . . . . . . . . . 88 5.9 Improvement in availability latency with one-instantaneous zebroids as a functionof(N,®)values,whenthetotalstorageinthesystemiskept¯xed, S T =200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.10 Improvement in ± agg with one-instantaneous zebroids for di®erent client trip durations in case of 10£10 torus with a ¯xed car density, N = 100. 93 5.11 Shows improvement in availability latency as a function of the car density for di®erent repository sizes with ®=2 and ° =10. . . . . . . . . . . . . 94 5.12 Performancewithzebroidsasafunctionofdi®erent(N,®)valueswhenthe total storage in the system is held constant at S T = 200, ° = 10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.13 PerformancewithzebroidsasafunctionofcardensitywhenT =25,®=2, and ° = 10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. . . . . . . . . . . . . . . . . . . . 97 5.14 Performance with zebroids as a function of storage per car when T = 25, N = 50, and ° = 10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. . . . . . . . . . . . . . . 98 5.15 Performance with zebroids as a function of data item repository size when N =50, ®=2, and° =10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. . . . . . . . . . . . 99 5.16 Aggregate Availability Latency for Linear, Sqrt, and Random Replication Schemes for Zipf values -0.5, -1.0, -1.5, and -2.0 . . . . . . . . . . . . . . . 100 5.17 Aggregate availability latency and normalized unsatis¯ed requests with zebroids for the case when the ratio of T : ® is maintained as 5 : 1 and requests are issued as per a Zipf distribution at equal inter-arrival times. . 104 5.18 Replacement overhead incurred by employing zebroids when the ratio of T :®ismaintainedas5:1andrequestsareissuedasperaZipfdistribution at equal inter-arrival times. . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.19 Performance with zebroids as a function of the storage per car when the data item repository size is held constant at 10. Requests are issued as per a Zipf distribution at equal inter-arrival times. . . . . . . . . . . . . . . . 106 5.20 Performance with zebroids as a function of the data item repository size when the storage per car is held constant at 3. Again, requests are issued as per a Zipf distribution at equal inter-arrival times. . . . . . . . . . . . . 107 xiii Abstract On-demand delivery of audio and video clips in a vehicular network is a growing area of interest. A given repository of such data items, each with an associated popularity, may be available to the passengers of the vehicles. The vehicles themselves are equipped with a `TiVO' like device that has several gigabytes of storage and a wireless interface allowingshortrangecommunicationat10sto100sofMegabitspersecond. Thegoalisto minimize the latency between request issuance and the time till a copy of the requested item is encountered. This latency is termed the availability latency. This thesis explores two generic tools to alleviate availability latency: (a) data replication (b) data delivery scheduling. With the replication study, we propose a general optimization formulation that seeks to minimize average availability latency subject to a storage constraint per vehicle. We explore the e®ects of a family of popularity-based replication schemes on availability latency. When the vehicles follow a 2D random walk based mobility model, via analysis and extensive simulations, we determine the optimal replication scheme that minimizes latency across a wide parameter space with major dimensions being data item size and client trip duration . Once an appropriate static replication scheme has allocated replicas, the vehicles themselves may be employed as data carriers to further improve availability latency. These data carriers are termed zebroids. However, a zebroid's local storage may be completelyexhausted. Hence,toaccommodatethisnewdataitem,itmayneedtoevictan existing one. Various replacement policies such as LFU, LRU, random etc. are examined and their relative performance is studied. Via analysis and extensive simulations we study the behavior of zebroids as a function of large parameter space comprising data xiv item repository size, storage per vehicle, number of vehicles, popularity distribution of the data items, di®erent replacement schemes for zebroids etc. We validate the Markov model based observations with two independent validation phases employing (a) freeway tra±c information on a city map (b) real world traces from a small bus network. xv Chapter 1 Introduction The notion of `entertainment on wheels' is no longer a distant dream. With today's technology, it is possible to present entertainment content in the form of audio and video clips to passengers as they travel in their vehicles in a city. Advances in technology, both in the area of storage and wireless communications, have contributed to the support of on-demand delivery of such content among mobile vehicles. Vehicles may be equipped with devices consisting of several gigabytes of storage, a fast processor, and a wireless interface with bandwidths of several 10s or 100s of Megabits per second. These devices are termed AutoMata [5] (formerly known as a C2P2 [21] for Car-to-Car-Peer-to-Peer) and they collaborate to form a mobile ad-hoc network to deliver the requested data to a client. The radio range of these devices is in the order of a few hundred feet. The content exchanged between the vehicles may vary from tra±c information such as accident noti¯cations and emergency vehicle arrival noti¯cations to multimedia for entertainment such as audio ¯les, cartoons, movies and other video ¯les. Without loss of generality, throughout this thesis, we will use the term data item from now on with the understanding that a data item can be an audio title, video title or any other useful content. In a typical scenario, a client of this application operating over an AutoMata network is provided a list of available data items (see Figure 1.1). Several other components may be involved in realizing such an application. Figure 1.2 depicts a realization of a component diagram that may be used to realize such an application. We provide a very brief overview of the functionality of some of the components. 1 Figure 1.1: An example illustration of an AutoMata application. Once a user initiates display of a data item, an admission control component [18] ensures availability of both resources and the referenced data. Next, a data delivery scheduling technique [20] utilizes resources as a function of time to deliver the data item toarequestingAutoMatadevice. Thiscomponent,mayswitchbetweenseveralcandidate servers containing the referenced data item based on their proximity, current availability ofresources,andnetworkconditions. Thiscomponentistiedcloselytoanad-hocnetwork routing protocol which facilitates delivery of data between AutoMata devices. Example protocolsareDSR[34],DSDV[47],AODV[48]tonameafew. Anothersystemcomponent may monitor whether the system is providing a target AutoMata with the desired QoS and make adjustments as necessary. Besides the above components, there may be others responsibleforaddressingthesecurity[64]andprivacy[16]concernsoftheuserthatmay be mandatory for practical use of the system. Additionally, suitable physical and MAC layer optimizations may be needed to adapt to the wireless nature of the communication medium between the vehicles. 2 Figure 1.2: Components of an AutoMata application While each of these components may warrant a separate thesis in itself, this the- sis speci¯cally explores possible realizations of three components, namely: Discovery (PAVAN), Data replication (Static Replication Schemes), Data delivery scheduling (Ze- broids). Along the process, we explore various trade-o®s that have in°uenced the design decisions for our proposed solution. Before we provide a brief overview of each of the three studies mentioned above, we brie°y introduce the prime metric of interest in our study namely availability latency. The time interval between when a request is issued by a client and a copy of the requested data item is encountered by it is referred to as availability latency (denoted by ±). A data item is available immediately when it resides in the local storage of the AutoMata device serving the request yielding ± as zero. Users prefer lower availability latency associated with data items. Studies [39] with peer-to-peer network systems have indicated that users prefer a larger list of data items (¯les) each associated with a low latency. This latency is a function of a number of parameters: (i) current location of the client (ii) destination and travel path of the client (iii) mobility model of the AutoMata equipped cars (iv) number of replicas constructed for the di®erent data items (v) placement of data item replicas across the AutoMata equipped cars. Without loss of generality and to simplify the discussion, we assume the term car refers to an AutoMata- equipped vehicle. 3 The biggest challenge in such an environment is mobility of the vehicles. This causes the network topology to change dynamically, hence traditional solutions proposed for staticenvironmentsmaynotbedirectlyapplicable. Moreover,connectivityofthenetwork at all times cannot be assumed as frequent partitioning may occur due to insu±cient density of vehicles in certain parts of the network. However, knowledge of the mobility model dictating the vehicular movements may enable design of suitable schemes that allow an application to provide better estimates of the availability latency for the data items. In this thesis, we propose to incorporate knowledge of mobility into the design of various mechanisms that help enhance the availability latency to user desired content. We ¯rst present PAVAN, a policy framework that, for a given data item level of repli- cation,describestheprocedurethatoutputsthelistofavailabletitlesandtheirassociated latency for users. One important ¯nding of this study is that the degree of replication of the data items is the key parameter that in°uences availability latency. We then study the replication parameter in detail by proposing a family of static replication techniques and explore their e®ect on availability latency under di®erent parameter settings. Hav- ing identi¯ed the performance improvements with static replication, we then study the e®ect of data carriers, termed zebroids, in providing further improvements in availability latency. 1.1 Overview of Case Studies In this section, we summarize the ¯ndings of each of the three studies that are part of this thesis proposal. 1.1.1 PAVAN In this study, we present PAVAN, a Policy for Availability in Vehicular Ad-hoc Net- works. This policy framework outlines how the list of titles available to a client and their associated availability latency is computed. Here, we observe that when the degree of replication for the data items is below a certain threshold, the PAVAN variant that uses content density information and a predictive mobility model provides the best latency 4 performance. Identifying data replication as the key parameter that a®ects latency we next explore alternate replication strategies and their tradeo®s. 1.1.2 Static Replication schemes In this study, we explore the e®ects of static replication schemes on availability latency. Given a data item repository, a certain vehicle density and a storage constraint per vehicle, we present an optimization formulation to determine the optimal number of data item replicas that minimize an average availability latency metric. We simplify the data placement issue by allocating the replicas to the vehicles uniformly at random, with the constraint that no two replicas of the same data item are placed in a vehicle. We analytically capture the variation in latency as a function of the data item replication levelswhenthevehiclesobeya2DrandomwalkbasedMarkovmobilitymodel. Westudy the performance of a family of replication strategies in such an environment. Various design parameters are considered, such as size/display time of the data items, short/long client trip durations, and di®erent data item repository sizes, and their e®ect on the optimal replication scheme is presented. This is followed by validation of the Markov model based observations with two independent validation phases employing (a) a real map of an urban environment that dictates the mobility transitions of the Markov model and (b) ¯ne-grained mobility traces from a real environment comprising buses moving around a university campus area. 1.1.3 Zebroids Onceastaticreplicationschemehasallocatedreplicastothevehicles,zebroidscanbeused tofurtherimproveavailabilitylatency. Azebroidisavehiclewhosepathrendezvouswith both the client (data item requestor) and the server (vehicle containing that data item). Aidedbythisspatio-temporaloverlap,zebroidscantransportadataitemfromtheserver to the client. However, a zebroid's local storage may be completely exhausted. Hence, to accommodate this new data item, it may need to evict an existing one. Examples of replacement policies that determine what item to evict are LFU, LRU, random among others. Theperformanceimprovementinlatencyobtainedwithzebroidsunderconditions 5 of in¯nite storage and no interference is captured analytically. This is followed by an exhaustive simulation study that presents the behavior of zebroids as a function of large parameterspace. Aswithstaticreplicationschemes,observationswiththeMarkovmodel are validated by employing a Markov model derived from a real city map and also using the bus-based vehicular traces. 1.2 Organization of this thesis Therestofthisthesisisorganizedasfollows. Chapter2providesadescriptionofthe2-tier architecture used in our study and introduces some common terminology and de¯nitions usedinthisthesis. Chapter3describesPAVANframework,resultsofthesimulationstudy to evaluate the performance of the PAVAN variants. Chapter 4 introduces the family of frequency-based replication schemes that a®ect availability latency. presents the results of the experimental study that evaluates the performance of the various schemes under di®erentparametersettings. Chapter5introduceszebroidsasdatacarriers,describesthe various environments used in this study and a classi¯cation of the di®erent carrier-based replacement policies that are deployed in these environments, followed by detailed simu- lation results with zebroids deployed in the various environments with di®erent policies. Chapter 6 gives a brief overview of the related work in the area. Chapter 7 concludes this document by highlighting the major contributions of this dissertation. 6 Chapter 2 Common Assumptions and Architectural Framework In this chapter, we present the elements common to all the studies. In particular, they share a similar set of assumptions, a 2-tier architecture and the simulation model. Below we describe each in turn. 2.1 Assumptions The list of data items comprising the database repository and their frequency of access is given and does not change. Moreover, the data items are not updated. There exists a 2-tier architecture comprising of (a) A high bandwidth data plane made up of the ad-hoc peer to peer network between the vehicles featuring band- widths in the order of 10s to 100s of Mbps (b) A low bandwidth control plane similar to a cellular infrastructure between the vehicles and adjacent base stations which may be connected to the internet (see Section 2.3). Studies[49,62,52]thathaveexploredtheoptimalnumberofneighbors,andoptimal radiorangetoensureconnectivityinmobilewirelessnetworkscomplimentourwork. We assume that vehicles within radio range communicate directly and multi-hop transmissions are supported. We also assume the presence of suitable physical, MAC and routing layers and do not consider various low level wireless channel issues that have been studied 7 Database Parameters T Number of data items. S i Size of data item i ¢ i Display time of data item i. ¯ i Bandwidth requirement of data item i. f i Frequency of access to data item i. Replication Parameters R i Normalized frequency of access to data item i, R i = (f i ) n P T j=1 (f j ) n ; 0·n·1 r i Number of replicas for data item i, r i =min(N;max(1;b R i ¢N¢® S i c)) n Characterizes a particular replication scheme. ± i Average availability latency of data item i ± agg Aggregate availability latency for replication technique using the n th power, 0·n·1, ± agg = P T j=1 ± j ¢f j AutoMata System Parameters N Number of AutoMata devices in the system. ® Storage capacity per AutoMata. ° Trip duration of the client AutoMata. S T Total storage capacity of the AutoMata system, S T =N¢®. G Number of cells in the 2D torus. Table 2.1: Terms and their de¯nitions extensively [14, 1]. Additionally, we also ignore wireless channel and contention issues. 2.2 Preliminaries Here, we introduce some formalism in the notation used throughout this document. Ta- ble 2.1 summarizes the notation for the commonly used parameters. Assume a network of N mobile AutoMata devices, each with storage capacity of ® bytes. The total storage capacity of the system is S T =N¢®. There are T data items in the database, each with a display time of ¢ i seconds and display bandwidth requirement of ¯ i . Hence the size of each data item is given by S i =¢ i ¢¯ i . The frequency of access to data item i is denoted asf i with P T j=1 f j =1. LetthetripdurationoftheclientAutoMataunderconsideration be °. Let r i represent the number of replicas for data item i. 8 The availability latency for a data item i, denoted as ± i , is de¯ned as the time after which a client AutoMata will ¯nd at least one replica of the data item accessible to it, either directly or via multiple hops, for the data item display time (¢ i ). If this condition is not satis¯ed for a given request for data item i, then we set ± i to ° which indicates that data item i will not be available to the client during its journey. Also, if ¢ i exceeds ° for a certain data item i then we set ± i to °. Note ± i is the instantaneous availability latency for a given request for data item i. Weareinterestedintheaverageavailabilitylatencyobservedacrossallthedataitems. Hence, we weigh the average availability latency ± i for every item i with its f i yielding the aggregate availability latency (± agg ) metric de¯ned as follows: ± agg = T X i=1 ± i ¢f i (2.1) The aggregate availability latency is the primary metric which we seek to optimize in our studies. 2.3 Architectural Framework Avehicularad-hocnetwork,suchasAutoMata,maypotentiallycoveralargegeographical area,suchasametropolitancity. Atsuchlargedistances,discoveringavailabledataitems becomes a very challenging problem. It is easy to see that on-demand °ooding/simple query-based approaches to resource discovery within the ad-hoc network will not scale well. Our solution is to adopt a hierarchical architecture that also leverages the existing large scale heterogeneous wired-wireless cellular network infrastructure. This infrastruc- tureaidsinthecollectionoflocalizedaggregateinformationthatcanbeusedtodistribute the decision making. Our two-tiered architecture, shown in Figure 2.1, consists of sepa- rate data (edges labelled 3) and control networks (edges labelled 2). We now provide an overview of the various components of this architecture. Data network: The data network consists of the vehicular ad-hoc network of Au- toMata devices. The system storage is distributed among the various AutoMata devices 9 Figure 2.1: A hierarchical architecture. withinthisnetwork. Ateachinstant, thecommunicationislocalizedsothatitisbetween nodesthataremovingwithinthesamecell. WeassumethateveryAutoMatainthesame cell is network connected. A typical path between two devices in the same cell may be multi-hop. This is because the range of a cellular base station is almost certainly much larger than the range of high bandwidth network devices (e.g., 802.11a [7]) employed by AutoMata devices. The number of hops is expected to be short, on the order of 3 to 4 hops. Controlnetwork: Thecontrolnetworkisalowdataratecellularnetworkinfrastruc- ture, with base stations dividing a large geographical area into localized cells. It provides three key functionalities: (i) monitoring and collection of pertinent content and mobil- ity information from individual car's AutoMata devices to the base station; (ii) regional consolidation and storage of this information into maps, mobility models and content information by nearby base stations and remote servers within the cellular network in- frastructure; and (iii) periodic update of pertinent regional map, mobility, and content information of AutoMata devices within each cell. A base station may perform the last step by broadcasting information. Control messages are typically small and require a low data rate in the order of tens of Kilo bits per second (Kbps). 10 1 8 15 22 29 36 6 11 16 21 26 31 2 3 4 5 7 9 10 12 13 14 17 18 19 20 23 24 25 27 28 30 32 33 34 35 Start cell Figure 2.2: An example 6£6 map. Brie°y, we now examine the di®erent components of the control information being collected and broadcasted in each cell. 1. Regional Maps and Mobility Model: Several cells adjacent to each other can begroupedintoasingleregionalmap. Figure2.2illustratessuchamapforasystemwith square cells. A base station locally monitors information about the number of AutoMata devices in its cell, which cell a device came from, and which cell a device is moving towards. This information from nearby cells is then used to construct a Markov inter-cell mobility transition table over this regional map (see Section 2.4). 2. Data Item Replication Table: Based on regional as well as global input, information is also maintained about the ID and duration of all data items, as well as their replication levels. This table would have T rows, one for each possible data item. While T might be potentially in the order of hundreds or thousands, note that each row is small and in the order of tens of bytes. 3. Regional Lookahead Table: Based on current data, a regional lookahead table is also created that maintains information about data items and AutoMatas within a certain cell's vicinity. 2.4 Simulation Model We now describe the simulation model that is common to all our studies. We assume a repository of homogeneous data items with identical bandwidth requirement, display 11 time, and size (¯ i = ¯, ¢ i = ¢, S i = S). Figure 2.2 shows an example map used in our study. The map is divided into ¯xed size cells. Only AutoMatas within a cell can communicate with each other either directly if they are in radio-range or via other AutoMatas using multi-hop transmissions. In other words, the AutoMatas within a cell form a connected sub-network. AutoMatas in adjacent cells cannot communicate with each other. Without any loss of generality, to reduce the dimensionality of the problem, we express the data item display time, ¢, as the amount of time required by an AutoMata equipped vehicle to travel ¢ cells. We express ® as the number of storage slots per AutoMata. Each storage slot stores a data item fragment equivalent to a single cell worth of data item display time. Moreover, we assume the amount of data displayed in each cell is identical. Now, we represent both the size of a data item and the storage slots in terms of the number of cells. This means that a data item has a display time of ¢ cells and an AutoMata has ® units of cell storage. For example, a data item with display time of 4 cells (¢ = 4) requires 4 storage slots and an AutoMata provides 100 storage slots (®=100). The trip duration (°) is also expressed as the number of cells traversed by the client during its journey. We also de¯ne availability latency (± i ) for data item i in terms of the number of cells. In other words, ± i is the number of cells after which a client AutoMata will encounter a replica of the data item i, either directly or via multiple hops, for the data item display time. Hence, the possible values of the availability latency are between 0and°. Weonlyconsiderscenariosinwhich¢·°. Assumethat° =6. Foradataitem i with ¢ i =6, ± i is either 0 or 6. ± i =0 means that at least one replica of that data item was present in each of the 6 cells along the path of the client. ± i =6 means that at least one cell along the path of the client was missing a replica of the data item. Similarly, for data item j with ¢ j = 5, ± j is either 0, 1 or 6. If ± j = 0, the client encountered at least one replica of data item j along each of the ¯rst 5 cells along its path. If ± j = 1, the client encountered at least one replica of the data item along the last 5 cells of its path, but not even a single replica in the ¯rst cell. Finally, ± j =6 indicates that there were at least 2 cells along the path of the client, in which no replicas of data item j were present. 12 Asmentionedearlier,aMarkovianmobilitymodeldescribesthemovementofthecars which is probabilistic in nature. The vehicles equipped with AutoMata devices perform a 2D random walk on the surface of a torus which constitutes the map. Each cell of the map constitutes a state. A map of size G£G yields G 2 states. These states are self-contained and a transition from one state to another is independent of the previous history of a car in that state. The mobility model is weighted toward the diagonal both from the left to right and vice-versa (indicated by the shaded boxes in Figure 2.2). The aggregate of the transitions from each cell (state) to every other state gives the G£G probability transition matrix Q = [q ij ] where q ij is the probability of transition from state i to state j. Using Markov chains, it is possible to estimate the distribution of the steady-state probabilities of being in the various cells, by solving ¦ = ¦¤Q, where ¦ is thevectorrepresentingthesteady-stateprobabilitiesofbeinginthevariouscells(states). We employ the Markov mobility model to dictate the movements of the vehicles. For the initial part of this thesis, we assume that the vehicles move about as per a random walk mobility model. Toward the latter half, we consider a Markov mobility model derived from an underlying real city map. The transitions of vehicles located in various cells are controlled by the freeway and side-street locations of the underlying city map. We collected real-time tra±c data during di®erent time periods during the day to obtain the percentage of vehicles transitioning between di®erent freeways which in turn dictated the Markov model transition probabilities. 13 Chapter 3 PAVAN In this chapter, we ¯rst de¯ne the availability problem in terms of the list of titles that will be available to a client during its journey. The client is presented the available list of titles with their associated latency. The idea is to present the client with as accurate a list as possible so as to avoid user frustration. We present PAVAN as a policy framework to generate this list, and evaluate how the di®erent variants of PAVAN di®er in their accuracy of the match between the predicted list and actual list. 3.1 PAVAN variants We now present the details of PAVAN and its alternate variants distinguished on the basis of the input information given to them. Taking examples of data items as video and audio clips, an example output produced by PAVAN is shown on the right hand side of Figure 3.1. In the following discussion, we will assume speci¯c examples of data itemsasaudioorvideotitlesnotingthatitdoesnotcompromisethegeneralapplicability of PAVAN. The output of PAVAN is the available title list displayed on an interactive menu to the user showing all titles predicted to be available and their associated latency after which they will be available to the client. The prediction and presentation of the availability latency empowers users to make informed decisions. The accuracy of PAVAN's output depends on its provided information, i.e., its input. AsnotedinSection2.3,therearethreeessentialpiecesofinformationthatcanbeprovided asinputtoPAVAN:thetitlereplicationtable,theregionalmobilitytable,andtheregional 14 Figure 3.1: An overview of PAVAN, its inputs and output. look-ahead table. The global title replication table (shown in Figure 3.1) is provided to all variants to PAVAN. These alternatives are di®erent depending on whether PAVAN is provided with either the mobility table, lookahead table, or both. Intuitively, the \richer" the information input, the closer the output list is to the actual list (the list of titles produced by a oracle aware of all future movements of AutoMatas). There is a trade-o® between obtaining richer information for the policy decision-making against the overhead of having this information broadcast from the base station to AutoMata devices. An additional input to PAVAN is the maximum delay tolerable by a client. This provides an upper bound on the availability latency. The mobility model given to PAVAN may be categorized into two types. It may be either predictive (P) where the transition matrix, Q, is used in each step, or steady- state (SS) where only the equilibrium probabilities are used. Recall, the equilibrium probabilities are obtained by solving the equality ¦ = ¦¢ Q, where ¦ is the vector representing the steady-state probabilities of being in the various cells (states). 15 PAVAN Policy Input information SS only Steady-state mobility model SSL t Steady-state mobility model and density of the contents in the AutoMatas within a pre-speci¯ed lookahead PL t Predictive mobility model and density of the contents in the AutoMatas within a pre-speci¯ed lookahead PL r Predictive mobility model and density of the AutoMatas within a pre-speci¯ed lookahead Table 3.1: Four variants of PAVAN. We now describe the alternative variants of PAVAN. SS only provides PAVAN with bothSS andtheglobalreplicationtable. ThismeansthelocationofallAutoMatasisthe same and is given by the steady-state matrix ¦. Since no information about the contents of the AutoMatas is provided, SS only uses the global replication table and assumes that anAutoMatacontainsthevarioustitlesgovernedbythisglobaldistribution. Aggregating this information for all AutoMatas yields the title location matrix T. Note that T has identical rows for each AutoMata since no information is known about their contents. For each step along the path, the following procedure is applied: Algorithm 3.1.1: proca(Steps;AutoMatas;titles) for stepÃ1 to Steps cell idÃclients current cell Conf Ã0 for iÃ1 to AutoMatas if (AutoMata i located in cell id) for jÃ1 to titles if (AutoMata i contains title j) Conf(j;step)+=¦(i;cell id)¤T(i;j) 16 Hence, for each title, this procedure yields the `Con¯dence' of that particular title for that step along the journey of the client. The higher the con¯dence, the higher the predicted availability of the title at that step. At the end, the con¯dence for each title across all steps is aggregated into a metric that is then mapped into the list of available titles. This is achieved using the following procedure: Algorithm 3.1.2: procb(Steps;titles) for iÃ1 to titles if Conf(i,j)¸m for every step j, 1·j· Steps Agg metric(i)= P Steps j=1 Conf(i;j) else Agg metric(i)=0 If, for title i, Agg metric(i) > 0, then that title will appear in the client's available list. In all our experiments, we choose a value of m = 1. Intuitively, if the title has a Con¯dence < 1 at even one step, on an average less than 1 copy of that title exists at that step. Hence, the client may not ¯nd a copy of that title at that step. Such a title is not shown in the predicted list. It should be noted that values of m less than one result in optimistic predictions, while values of m greater than one result in conservative predictions. In addition to the mobility model and the global replication table, PAVAN can be provided with the AutoMata density information and what content they carry. This can be limited to a speci¯c area de¯ned by a Spatio-Temporal Lookahead (STL) parameter. 3.1.1 Spatio-Temporal Lookahead (STL) parameter The SSL t variant of PAVAN consumes the global replication table, SS, and the contents of those AutoMatas within a ¯xed geographical area de¯ned by STL. A STL 1 value of k encompasses all k adjacent cells. When STL is 0, SSL t is similar to SS only . Figure 3.2 shows an example 5£ 5 map with the client occupying the shaded cell. This ¯gure shows STL values of 1, 2, and 3. As one increases the value of STL, a client obtains 1 We use k to denote the value of STL. 17 1 2 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 Figure 3.2: The numbers in the cells indicate STL value for the shaded cell numbered 1. information about additional cells that are further away. Note that a cell is assumed to have eight adjacent neighbors. SSL t enables an AutoMata to incorporate the content of all AutoMatas in k adjacent cells into T. The remaining AutoMata devices are assumed to contain titles as per the global replication table. The PL r variant of PAVAN considers the predictive mobility model (P), the density of the AutoMatas within the STL and the global replication table to produce the title availability list. As the value of STL increases, the client obtains more information about the number of AutoMatas in the various cells. When STL spans all cells, the client obtains information about the number of AutoMatas in each cell of the entire network. For a given client, PAVAN knows the location of AutoMatas within STL adjacent cells. For all the other AutoMatas, their location is equally likely to be a cell in the map outside those within the STL. This combined information about the initial positions of the AutoMatas yields the initial location matrix L. At each step, we compute product of L i and Q i , where i indicates the step under consideration, Q is the transition probability matrix de¯ned by the mobility model, and the initial value of L i =L. Note that the con- tents of the AutoMatas are not known; hence, the Title matrix T is calculated according to the global replication table. The L i and T matrices are used with procedures PROCA (replacing ¦ by L) and PROCB in order to obtain the predicted list of available titles. Finally, PL t denotes the variant of PAVAN with the following inputs: the global replicationtable, P andL t . Whenk =1, PL t isprovidedwiththecontentofAutoMatas in its current cell, termed start cell. PL t assumes the remaining AutoMatas are equally 18 likely to be in other cells of the network besides the `start cell'. This yields the initial AutoMataLocationmatrixL. Again,wecomputeproductofLandQ i atstepitoobtain location matrix L i at that step. Note that since the list of titles assigned to some of the AutoMatas is known, namely the AutoMatas present in the start cell, we incorporate that information in T. PL t assumes other AutoMatas have the titles distributed as per the global title replication distribution. Hence, in this case, the rows of T need not all be the same. When k > 1, the client obtains precise information about the density and the contents of the AutoMatas in the cells that are reachable within a distance of k at the current instant. Using this information the client obtains the Location matrix L i at each step i using L i ¤Q i where initially L i =L. Similarly, the T matrix is obtained where the information of the contents of all AutoMatas within the STL is known. L i and T can be input to PROCA (again replacing ¦ by L) and PROCB to obtain Agg metric(i), which is then converted into the client's available titles list. 3.2 Simulation Study We ¯rst describe the parameters of the simulation set-up, followed by a brief description of the simulation results. 3.2.1 Experimental Set-up The experimental set-up consists of a 6£ 6 map as shown in Figure 3. The mobility model is weighted toward the diagonal both from left to right and vice-versa (due to gray boxes). Assume that the client starts from cell 1 and travels along the path f1, 8, 15, 22, 29, 36g. Numbers in the bracket indicate the sequence of visited cell IDs. At the start of a client's journey, each variant of PAVAN retrieves its required information from the control network. Subsequently, each variant of PAVAN (see Table 3.1) produces a predicted list of available titles. Initially, all AutoMatas are distributed uniformly across the cells in the map. This is determined by a random initial seed. The distribution of titles across AutoMatas is also chosen to be uniform. At each step, depending on the current AutoMata locations, each 19 Model Weight(w 1 ) Weight(w 2 ) Weight(w 3 ) of a 10 of a 01 of a 11 1 0 0 1 2 0 -5 1 3 -1 0 1 Table 3.2: Three utility models to evaluate alternative variants of PAVAN. 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of replication (%) k=1 k=3 k=5 k=6 Model 1 Utility 0 10 20 30 40 50 −1.5 −1 −0.5 0 0.5 1 Degree of replication (%) k=1 k=3 k=5 k=6 Model 2 Utility 3.3.a) Utility model 1 3.3.b) Utility model 2 Figure 3.3: A comparison of PAVAN with di®erent inputs for utility models 1 and 2 as a function of the degree of replication of the titles. AutoMata moves to one of its adjoining cell as governed by the mobility model. Another seeddeterminesthechoiceofwhichcellanAutoMatamovesto. EachAutoMataperforms six transitions according to the mobility model. The intersection of AutoMatas with the cellsalongtheclient'spathyieldstheactualcon¯dencevaluesforaparticulartitleseenin a particular run of the simulation. For each run, a di®erent random seed is used starting from the same initial position. For each run, at each step, the client obtains the exact distribution of titles in the network and the corresponding con¯dence values for each title. These values are then translated to a list of titles for this particular run (actual list) using the same procedure PROCB (see Section 3.1). For each run, the predicted list is compared with the actual list and the utility models presented later in Section 3.2.2 depict the di®erences. All presented results are averages across 10,000 simulation runs. 20 3.2.2 Utility Models We use three utility models to quantify the quality of lists computed by di®erent variants ofPAVAN.Thesemodelsassignadi®erentweighttotheaveragenumberoffalsenegatives (denoted a 10 ), false positives (denoted a 01 ), and true positives (denoted a 11 ). A false negative is a title present in the actual list but not in the predicted list. A false positive is a title present in the predicted list but not in the actual list. Finally, a true positive is a title present in both the actual and predicted lists. All utility models are represented as: U =w 1 ¢a 10 +w 2 ¢a 01 +w 3 ¢a 11 We implement the alternative utility models by assigning a di®erent weight to a 10 , a 01 , and a 11 (see Table 3.2). These models are as follows. Model 1 depends on those titles thatappearcorrectlyinboththeactualandthepredictedlists. Soitsutilityvalueranges from 0 to 1. Model 2 severely penalizes those titles that appear in the predicted list but not in the actual list. It assumes that a user would be greatly dissatis¯ed by choosing such titles because they are not available. The utility of this model ranges in value from ¡5 to 1. Model 3 penalizes those titles that appear in the actual lists but not in the predicted ones. These available titles cannot be selected by a user because they are not predicated asavailable. Theutilityofthismodelrangesinvaluefrom¡1to1. Notethatthepenalty for these false negatives is not as signi¯cant as false positives. 3.2.3 Results In our experiments, we used 200 AutoMatas, unless stated otherwise, and 16 titles with unique ids 1, 2, ¢¢¢ 16. The percentage degree of replication of a title with id i is given by: Title¡rep(i)= 8 < : 2¢i 1 ·i·10 20+5¢(i%10) 11·i·16 (3.1) This means title 1 has 4 copies, title 2 has 8 copies, and so on until title 10. Title 11 has 25 copies, title 12 has 30 copies, and so on until title 16. Replicas of a title are 21 assigned to AutoMatas randomly. An AutoMata may contain several di®erent titles, but only one copy of a certain title. Figure 3.3 presents a comparison of alternative variants of PAVAN. The graphs rep- resent the utility values as a function of the di®erent degrees of replication of the movie titles. ThepredictedlistsgeneratedbyPAVANinallcases(whereapplicable)werecalcu- lated using the largest STL value (here, k=6, the length of the path of the client). Next, we brie°y describe the main lessons of this study. Lesson1: Asthedegreeofreplicationincreasesbeyondacertainthreshold all the variants of PAVAN start showing similar utilities. The value of the threshold is di®erent for di®erent models. While for model 1, this replication threshold is 20%, itisapproximately50%withmodel2. Twofactorsimpactthisobservation. Firstis thedegreeoftitlereplication. Secondisthepredictivenatureofaspeci¯cPAVANpolicy. The general trend indicates that as the degree of replication increases, the model utilityvaluesalsoincreaseand convergetoward1(maximumutilityvaluefor allmodels). With the increase in the degree of title replication, the global replication table, which is the base-line input to PAVAN, dominates the titles shown in the predicted lists yielding higher true positives (a 11 ). Here, both false-positives and false-negatives contribute an insigni¯cant amount toward the ¯nal observed utility for all models. Model 2, which penalizes those titles that are present in the predicted but not in the actual list, highlights the di®erences between the PAVAN variants. It is seen that, in general, PL t outperforms the others. This is because it uses information about the densityandthecontentsoftheAutoMataswithintheSTL.Sincethisutilitymodelpenal- izes policies that over-predict, we see that SS only performs the worst followed by SSL t . With lower replication levels, in case of model 1, these policies were doing marginally betterthanPL t becausetheirover-predictivenaturealwaysresultedinhigher a 11 values. The performance of PL r , which uses AutoMata density information and the predictive mobility model, lies between the two extremes. The results above indicate that SS only and PL t represent the two extremes. Hence, we eliminate results from the other two variants for the remaining sets of experiments noting that their performance was always in between SS only and PL t . 22 0 10 20 30 40 50 −6 −4 −2 0 2 4 6 Degree of replication (%) Δ=1−cell Δ=3−cell Δ=5−cell Δ=6−cell Difference in Availability Latency 0 10 20 30 40 50 −6 −4 −2 0 2 4 6 Degree of replication (%) Δ=1−cell Δ=3−cell Δ=5−cell Δ=6−cell Difference in Availability Latency 3.4.a) SS only 3.4.b) PL t Figure3.4: Di®erenceintheavailabilitylatenciesasafunctionofthedegreeofreplication of the titles for di®erent title display times. Lesson 2: The accuracy of availability latency estimated by PL t is the best when compared with other alternatives. Figure 3.4.a indicates the average di®erence between the availability latency of SS only and the actual observed latency as a function of the di®erent degrees of title replication for di®erent title display times. The graph shows the behavior for title display times of 1, 3, 5 and 6 cells. Figure 3.4.b shows the same for PL t . Note the lower the di®erence in the availability latency, the better the match between the predicted and the actual lists. We observe that the availability latency of alternative PAVAN policies is signi¯cantly di®erent and impacted by both the display time of a title and trip duration. The main observation is that the peaks in the curves for SS only (over-predictive) are much higher than that for PL t (conservative). With SS only , the predicted list is very accurate when title display times are greater than two and the degree of replication is less than 5%. This is because the average availability latency is close to 6 in both cases. However, as the degree of replication increases beyond 5%, the predicted availability latency drops at a faster rate than what is seen with the actual availability latency. This di®erence is always positive because SS only alwaysover-predictsirrespectiveofthetitledisplaytime. Notethatover-prediction indicates a smaller predicted availability latency as compared to the actual one. Beyond 23 0 10 20 30 40 50 −10 −8 −6 −4 −2 0 2 4 6 8 10 Degree of replication (%) Difference in Availability Latency SS only − 200cars SS only − 300cars PL t − 300cars PL t − 200cars Figure 3.5: Comparison of di®erence in the availability latencies of SS only and PL t for di®erentAutoMatadensitiesina10x10mapasafunctionofthedegreeoftitlereplication. The title display time under consideration spans 5 cells. 25% degree of replication, availability latency of all titles except those with display time 6 converge to zero. As the degree of replication increases, slowly the actual availability latency catches up with the predicted one thereby making the di®erence between them converge to zero. WithPL t ,theavailabilitylatencydi®erencealwaysliesbetween-1and+1.5irrespec- tiveofthetitledisplaytime. Foralldisplaytimes, beyond20%replication, thedi®erence in the availability latency converges to 0. For degree of replication less than or equal to 5%, the di®erence in the availability latency becomes negative. This means that the predicted available latency is higher than the actual observed latency, thereby indicating that for lower degrees of replication PL t is more conservative. The reason that the di®er- ence is not always 0 is due to the statistical variations inherent in the experiments. The mobility model is probabilistic, hence if we consider extremely large scenarios, then even for lower degrees of replication, the di®erence in the availability latency will converge to 0. Lesson3: Theaccuracyoftheavailabilitylatencycalculationsfordi®erent variants of PAVAN is sensitive to AutoMata density. Figures 3.5 indicates the 24 behavior of SS only and PL t with respect to the di®erence in availability latency metric in a 10x10 map for AutoMata densities of 200 and 300, when considering a title with a display time of 5 cells. When the AutoMata density increases, for a given degree of replication, now there are more number of replicas (AutoMatas) for each title. Hence, similar trends are seen as earlier but the curves for all the PAVAN variants move to the left. Moreover, the curves peak at a lower degree of replication. Even though we have presented the results for one map, we also considered other maps in our simulations. Obtained results show the map is an important parameter that impacts the behavior of PAVAN signi¯cantly. Speci¯cally, a lower degree of overlap between the path of an AutoMata equipped vehicle and the gray cells in the map causes thetransitionprobabilitiestorapidlydiminishtoward0. Insuchcases, evenifthedegree of replication of the titles is 100% the variants of PAVAN will always under-predict. This willbethecaseevenifthetotalnumberofgraycellsincreasesbeyondacertainthreshold because then the transition probabilities di®use quickly. So, within a few steps, the movement prediction probabilities will diminish having very little e®ect on the predicted lists even in the case of 100% title replication. This e®ect will also be seen if the map consists of entirely non-gray cells. In such cases, the trends are similar to those seen for the lower replication titles. In conclusion, SS only is appropriate for certain utility models but not all. Also, the degree of title replication has a profound impact on the availability latency metric. PL t demonstrates a competitive performance for all utility models, all clip display times and degrees of replication. 3.3 Summary PAVAN is a novel policy that computes the time when di®erent titles are available in an ad-hoc network of AutoMata devices. It accomplishes this by employing a Markov mobility model that consumes a regional map, a mobility transition table, and a regional look ahead table. This input data is in the order of a few hundred bytes and provided by a base station. Each AutoMata device invokes PAVAN independently. Obtained results 25 demonstrate that one variant of PAVAN, that employs information about the density of the vehicles, the content that they carry and their mobility pattern, provides the most accurate availability latency when compared with other techniques. We quanti¯ed the quality of lists computed by PAVAN policies using di®erent utility models. The accuracy of the PAVAN predictions crucially depends on the transition proba- bilities of the Markov model. Unfortunately, with the UMassDieselNet [11] traces, the locations of the buses were not available. For a data set that has an underlying map and recorded locations of di®erent vehicles with some ¯ne/coarse granularity as they move about constrained by the map, one can create a Markov model for this data set. The MarkovmodelwillbeemployedbythePAVANmoduletopredictthelistofavailabletitles and the accuracy of the predictions will be measured by looking at the actual vehicular traces. This remains a future research direction as and when such a data set becomes publicly available. Synthetic data sets obtained from microscopic vehicular simulators like VISSIM/CORSIM may also be used for the above process, bringing the evaluations a step closer to reality. 26 Chapter 4 Static Replication Schemes In this chapter, the focus is on how the degree of replication per data item a®ects avail- ability latency. We consider a family of frequency-based replication strategies and study theirimpactonavailabilitylatency. First,ageneraloptimizationformulationispresented to determine which replication scheme minimizes the aggregate availability latency sub- ject to a total storage constraint. Subsequently small data items and long client trip durations, we solve the optimization in the case of sparse density of vehicles. Then, we explore the latency performance in high density scenarios via simulations and present an analytical approximation that captures the trends. The results are extended to consider data items of larger size and short client trip durations. Subsequently, some of the re- sults obtained with a 2D random walk model are evaluated on a map of the city of San Francisco with the major freeways being captured by the transition probabilities of the Markov mobility model. Finally, we explore the performance of the replication schemes on a realistic data set comprising of movement traces of buses in a small neighborhood in Amherst. 4.1 Family of Replication Policies Given a data item repository (T), a certain vehicle density (N), and a storage constraint per vehicle (each with storage ®), we present an optimization formulation to determine the optimal number of data item replicas that minimize the average availability latency metric. We simplify the data placement issue by allocating the replicas to the vehicles 27 uniformly at random with the constraint that no two replicas of the same data item are placed in a vehicle. We de¯ne the normalized frequency of access to the data item i, denoted R i , as: R i = (f i ) n P T j=1 (f j ) n ; 0·n·1 (4.1) R i is normalized to a value between 0 and 1. The number of replicas for data item i, denoted as r i , is: r i =min(N;max(1;b R i ¤N¤® S i c)) (4.2) This de¯nes a family of replication schemes that computes the degree of replication of data item i as the n th power of its frequency of access. Hence, the number of replicas for title i, r i , lies between 1 and N. Note that r i includes the original copy of a data item. One may simplify Equation 4.2 by replacing the max function withb R i ¤N¤® S i c. This would allow the value of r i to drop to zero for a data item i. This means that there is no copy of the data item in the network. In this case, a hybrid framework might provide access to the data item i. For example, a base station employing IEEE 802.16 [22] might facilitate access to a wired infrastructure with remote servers containing the data item i. The aggregate availability latency, ± agg , depends on the value chosen for n, since n determines the replicas per data item. Intuitively, the higher the replicas for a data item i, the lower will be the latency, ± i , experienced by a request for that data item. The core problem of interest here is to keep the aggregate availability latency as low as possible by tuning the data item replication levels, in the presence of storage constraints. We assume that the database size is smaller than the total storage capacity of the system, P T i=1 S i ·S T . Otherwise, data items cannot be replicated when at least one replica of a data item must be present in the system. More formally, the optimization problem can be stated as, Minimize ± agg , subject to T X i=1 S i ·S T (4.3) Implicit in this formulation is the design variable, namely, the desired replication for each data item. The value of n in Equation 4.3 determines a r i value for each data item 28 i with the objective to minimize ± agg . This minimization is a challenge when the total size of the database exceeds the storage capacity of a car, P T i=1 S i > ®. Otherwise, the problem is trivial and can be solved by replicating the data item repository on each device. The optimization space that de¯nes what value of n provides the best ± agg is quite large and consists of the following parameters: (i) density of cars, (ii) data item display time, (iii) size of the data item, (iv) display bandwidth per data item, (v) data item repository size, (vi) storage per car, (vii) client trip duration, (viii) frequency of access to thedataitems,and(ix)mobilitymodelforthecars. We¯rstexplorethisparameterspace where the mobility model employed by the vehicles is a 2D random walk on the surface of a torus. Not only does this provide tractability for mathematical analysis, but it turns out that the biased Markov mobility models based on an underlying city map comprising of freeways and side-streets show performance trends similar to those observed with the simple 2D random walk based mobility model (see Section 4.6). 4.2 Data items with display time one and long client trip duration In this section, we consider small data items i.e. items with a display time of one, where the client trip duration is long. We ¯rst present analytical approximations that capture the performance of availability latency for an item as a function of the number of replicas for that item for both a low and high density of replicas. Subsequently, we employ simulations to determine the optimal replication exponent that minimizes the aggregate availability latency. 4.2.1 Analysis In this section, we assume data items with a display time of one cell and for a scenario with a sparse density of data item replicas, derive a closed-form expression for the aggre- gate availability latency. Subsequently, we use this expression to solve the optimization problem to reveal that a square-root replication scheme minimizes this latency. Then, 29 we derive an expression that approximates the aggregate availability latency in case of a high density of data item replicas. 4.2.1.1 Sparse Scenario In this section, we provide a formulation that captures scenarios with a low density of vehicles. One can obtain the relationship between the ± i and r i under a given storage constraint. Ingeneral, therelationshipisafunctionofthemobilitymodelofthevehicles. For illustration, we have considered that vehicles follow a random walk-based mobility model on a 2D-torus. Aldous et al. [2] show that the mean of the hitting time for a symmetric randomwalkon thesurfaceof a 2D-torusis £(GlogG)whereG is the number ofcellsinthetorus. Moreover,themeanofthemeetingtimefor2randomwalksishalfof the mean hitting time. Furthermore, the distribution of the meeting times for an ergodic Markov chain can be approximated by an exponential distribution of the same mean [2]. Hence, P(± i >t)=exp µ ¡t c¢G¢logG ¶ (4.4) where the constant c ' 0:34 for G ¸ 25. Now since there are r i replicas, there are r i potential servers. Hence, the the meeting time, or equivalently the availability latency for the data item i is the time till it encounters any of these r i replicas for the ¯rst time. This can be modelled as a minimum of r i exponentials. Hence, P(± i >t)=exp Ã ¡t c¢ G r i ¢logG ! (4.5) Note, however that this formulation is valid only for the cases when G >> r i , which is the case for sparse scenarios. The expected value of ± i is given by: ± i = c¢G¢logG r i (4.6) For a given 2D-torus, G is constant, hence we have ± i / 1 r i or equivalently, ± i = C r i where C =c¢G¢logG. Hence, we have the following optimization formulation, 30 Min " T X i=1 f i ¢ C r i # (4.7) Subject to: T X i=1 r i =N¢® (4.8) r i ·N ;8 i=1 to T (4.9) r i ¸1 ;8 i=1 to T (4.10) Theorem 1. In case of a sparse density of vehicles, a replication scheme that allocates data item replicas as a function of the square-root of the frequency of access to data items minimizes the aggregate availability latency. r i = 8 > > > > < > > > > : p f i ¢N¢® P T j=1 p f j 1 N¢® · p f i P T j=1 p f j · 1 ® max ³ 1;min( q f i ¢C ° 0 ;N) ´ in the general case where ° 0 is s.t. P T i=1 r i =N¢® (4.11) Proof. We solve the above optimization using the method of Lagrange multipliers. First, we prove part(i) of the theorem. The Lagrangian for the optimization can be written as: H = T X i=1 f i ¢C r i +' " T X i=1 r i ¡N¢® # (4.12) We solve for r i as follows: @H @r i =¡f i ¢ C r 2 i +'=0 (4.13) 31 r i = s C¢f i ' (4.14) Substituting r i in the constraint, we get: '= Ã P T i=1 p C¢f i N¢® ! 2 (4.15) Finally, we get the optimal value of r i as, r i = p f i ¢N¢® P T j=1 p f j (4.16) The constraints are satis¯ed if 1 N¢® · p f i P T j=1 p f j · 1 ® which proves part (i) of the theorem. Withoutthisconditiononf i ,theaboveoptimizationcanbere-writtenasthefollowing Lagrangian taking all the constraints into account as: G= T X i=1 f i ¢C r i +° 0 " T X i=1 r i ¡N¢® # ¡ T X i=1 ° i ¢(r i ¡N¢®)¡ T X i=1 ¯ i ¢(¡r i +1) (4.17) The Kuhn Tucker Conditions for the modi¯ed Lagrangian are: ¡f i ¢ C r 2 i +° 0 ¡° i +¯ i =0;8 i=1 to T (4.18) T X i=1 r i ·N¢®, ° 0 ¸0, and ° 0 " T X i=1 r i ¡N¢® # =0 (4.19) r i ·N, ° i ¸0, and ° i ¢(r i ¡N)=0;8 i=1 to T (4.20) ¡r i ·¡1, ¯ i ¸0, and ¯ i ¢(¡r i +1)=0;8 i=1 to T (4.21) Solving Equation 4.18, we get, 32 r i = s f i ¢C ° 0 ¡° i +¯ i (4.22) Equations 4.20 and 4.21 imply that either ° i =0 or r i =N and also either ¯ i =0 or r i =1 respectively. Therefore, the optimum solution for r i is given by, r i =max Ã 1;min( s f i ¢C ° 0 ;N) ! (4.23) where ° 0 is such that P T i=1 r i =N¢® proving part (ii) of the theorem. Hence, in a sparse network, the optimal replication that minimizes the aggregate availability latency is obtained if the number of replicas for a data item is proportional to the square root of the frequency of access for that data item. Cohen et al. [13] proved thatforunstructuredpeer-to-peernetworkstheexpectedsearchsizeisminimizedusinga square-root replication strategy which is shown to be optimal. The aggregate availability latency metric in wireless mobile ad-hoc networks is analogous to the expected search size used in peer-to-peer networks. Itshouldbenotedthat,ingeneral,theoptimalreplicationdependsonhow± i isrelated to r i i.e. ± i = F(r i ) and F(¢) is the function that will determine the optimal replication strategy. Theabovemethodologycanbeusedtobeobtaintheoptimalnumberofreplicas as long as F(¢) is di®erentiable. Figure4.1showsthetypicaltrendshownby± i fora10£10torus,wherer i isincreased from 1 to N where N = 100. In other words, in a G = 100 cell torus, N = 100 cars are deployed, with r i of them having a replica for the data item. We only consider a single dataitem,arequestforthatitemcanbeissuedatanyvehiclechosenuniformlyatrandom among all the cars. If the item is stored locally, the latency is 0. The ¯gure indicates that when r i is small, (r i · 20) the analytical approximation in Equation 4.6 is valid. Subsequently, latency reduces at a much faster rate when compared to that predicted by the sparse approximation. This is because for a given G, as r i increases, the latency till any one of the r i replicas is encountered can no longer be modeled as the minimum 33 0 20 40 60 80 100 10 −3 10 −2 10 −1 10 0 10 1 10 2 Number of replicas ( r i ) Average availability latency ( δ i ) Sparse−approximation Simulation Figure4.1: Sparseanalysis(Equation4.6)versussimulationobtainedaverageavailability latency for a data item as a function of its replicas for a 10£10 torus, when the number of cars is set to 100. of r i independent exponentials. In the next section, we provide an approximation that captures the high density case. 4.2.1.2 Dense scenario In this section, we provide an analytical formulation that captures the trends shown by the availability latency in the presence of a high density of replicas. Recall that N cars are distributed uniformly at random across G cells, r i o® the N cars carry a copy of the data item of interest. Here, we use the traditional de¯nition of the expected availability latency for title i, namely, ± i = 1 X k=0 k¢P(± i =k) (4.24) We ¯rst determine an expression for the case when the latency is 0. This occurs if the data item is locally stored at a client or a data item replica is located in the same cell as the client at which the request is issued. Hence, the probability that the latency experienced by a client is zero is given by the following expression: P(± i =0)= r i N +(1¡ r i N )¢ µ 1¡ µ 1¡ 1 G ¶ r i ¶ (4.25) 34 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of Replicas (r i ) P( δ i = 0) Analysis Simulation 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of Replicas (r i ) P( δ i = 0) N=50 N=100 N=150 N=200 N=250 (a) (b) Figure4.2: Figure4.2(a)showsthevalidationoftheanalyticalexpressioninEquation4.25 for the probability that availability latency is zero. Figure 4.2(b) shows the probability that the availability latency is zero as a function of the replicas for the data item for 5 di®erent car densitiesf50, 100, 150, 200, 250g. Figure 4.2(a) indicates that the analytical expression above matches the simulation results quite well. For a given car density N, as the density of replicas increases, the probability that the availability latency experienced by a client is zero also increases. Figure4.2(b)showshowthisprobabilityvarieswithincreasingcardensity. Givenatorus comprising G cells, increase in P(± i = 0) shows a decreasing steepness as N increases. This is because r i varies from 1 to N, and only when r i = N, P(± i = 0) = 1 since the data item is locally stored by every car. De¯ne A k as the event that a data item i is encountered by the client for the ¯rst time in the k th cell. Let P(A k ) denote the probability of event A k occurring. Let p k denote the probability of encountering data item i in the k th cell, given that it was not encountered in the previous k¡1 cells. Note that p k is a conditional probability. Also, p 1 =P(± i =0) as de¯ned by Equation 4.25. Then, p k =1¡ µ 1¡ 1 G¡k+1 ¶ r i ; 2·k·G (4.26) 35 0 20 40 60 80 100 10 −3 10 −2 10 −1 10 0 10 1 10 2 Number of replicas ( r i ) Average availability latency ( δ i ) Sparse−approximation Simulation Dense−approximation Figure 4.3: The complete picture depicting the availability latency for a data item ob- tainedviasimulationsascomparedwithitssparseanddenseapproximationasafunction of its replicas for a 10£10 torus, when the number of cars is set to 100. Note that the model assumes that not encountering the data item in the (k¡1) th cell increases the probability of encountering it in the k th cell. Moreover, when k = G, p k =1 no matter what the value of r i , meaning that the maximum latency that a client will encounter will always be no more than G. Although this is true for a high density of replicas, this approximation is not valid for a sparse replica density, where p k may not increase as k increases, especially for the ¯rst few steps of the client. Note that, P(A k ) is a joint probability since encountering a data item for ¯rst time in the k th cell indicates that it was not encountered in any of the previous k¡1 cells. Clearly, p k and p k¡1 are not independent, hence, we use the multiplication rule to obtain the value of P(A k ) as, P(A k )=p k k¡1 Y j=1 (1¡p j ) ; 2·k·G (4.27) Then, the average availability latency (± i ) for data item i is given by, ± i = G X k=1 (k¡1)P(A k ) (4.28) 36 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 45 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 (a) ®=4 (b) ®=10 (c) ®=25 Figure 4.4: Aggregate availability latency for di®erent replication strategies for a 10£10 torus when T = 100 and N = 50. Figures (a), (b), and (c) depict three di®erent storage values per car: f4,10,25g. Figure 4.3 shows that the above equation captures the trend depicted by the average availabilitylatencyforhigherreplicadensitieswherethesparseanddenseapproximations are plotted together with the latency obtained via simulations. 4.2.2 Simulation results In this section, we present how the aggregate availability latency realized by the di®erent replicationschemesisa®ectedindividuallybythedi®erentparametersintheoptimization space. In all our experiments, we assume that the various data item popularities are dis- tributedaspertheZipf'slaw[65]. Thismeansthatthefrequencyofthe r th populardata item is inversely proportional to its rank i.e. f i = 1 i v P T j=1 1 j v ; 1<=i<=T (4.29) Here, the exponent v controls the skewness in the popularity distribution of the data items. We denote w =¡v as the skewness parameter. A higher value of w indicates that most of the popularity weight is spread across the ¯rst few popular titles. Note that the data item repository size is T and the denominator is simply a normalization constant. Figure 4.4 depicts the latency performance for di®erent replication schemes when storage per car is increased from 4 to 25 slots. The title repository size is T = 100 37 and the car density is N = 50 which implies that the total storage S T is increased from 200 to 1250 slots. As expected the latency decreases as storage is increased. The replication schemes with exponent values 0, 0:5, and 1 have been popularly studied in the literature [13, 42, 17, 60, 58] and are labelled random, square-root, and linear respectively. Below, we describe the main observations from this ¯gure. The random scheme allocates the same number of replicas per data item irrespective of their popularity. Hence, in all cases, it yields the same aggregate availability latency irrespective of the value of w. As the replication exponent increases from 0 to 1 progres- sivelymorereplicasareallocatedforthepopulardataitems. Thisincreaseinthereplicas is accelerated for higher values of w that provide a bias for the popular titles. Hence, we see a sharp decrease in the availability latency from n = 0 to n = 0:3 for w =¡1:5 and w =¡2. However, the maximum number of replicas per data item can never exceed N. Foravalueofw =¡0:5inwhichcasethepopularityweightisspreadmoreevenlyamong all the data items, it almost doesn't matter what the replication scheme is as seen by the °at latency curves for w =¡0:5. When storage per car is low, ®=4, this represents a scenario with a sparse density of dataitemreplicas. Inthiscase,thesquare-rootreplicationschemeprovidestheminimum latency. Also, the range where the replication exponent n varies from 0:4 to 0:6 shows a latency very close to the square-root scheme. This is true even when the data item popularities are skewed. Moreover, the range 0:4· n· 0:6 shows near optimal latency performance even when the storage is increased (see Figures 4.4(b) and (c)). In other words, through the entire spectrum of the replica density, a replication scheme de¯ned by an exponent in this range will provide near optimal performance. For the rest of this chapter, we will consider the square-root (n=0:5) scheme as representative of this range and compare its performance to the two extremes namely, random (n = 0) and linear (n=1). 4.2.2.1 Scale-up experiments In these set of experiments, we maintain a constant ratio of the total storage to the data item repository size (S T : T). Figure 4.5 presents the performance of the three 38 replication schemes when S T =3000 and T =600. Since S T =N¢®, we vary the values of (N,®) as f(50,60), (100,30), (150,20), (200,15), (250,12)g. Since the total storage in the system S T remains the same the number of replicas allocated per data item also remains. As N increases, the number of potential clients increases, this accounts for the slight upward trend in the latency curves for the di®erent replication schemes. With increasingskewness,forthesametotalstorage,thelatencyrealizedbythesquare-rootand linear schemes reduces. This is because replicas assigned to the more popular data items result in lower latency for those items because as the skewness parameter w increases, a higher popularity weight assigned to these data items. The random replication scheme is blind to the popularity of the data items and hence shows similar latency performance independent of the value of w. For all but w =¡0:5, it performs an order of magnitude worse as compared to the square-root scheme. 4.2.2.2 Variation in car density Next, we study the e®ect of car density on the performance of the replication schemes. Figure 4.6 presents the performance of the three replication schemes as a function of the cardensitywhenthestoragepercarisheldconstantat3for T =100. Increaseinthecar density increases the total storage in the system. Hence, more replicas per data item can be allocated resulting in an overall decrease in the aggregate availability latency. This is true for all replication schemes. However, here for w = ¡0:5 and w = ¡1 (beyond N =100), the random scheme shows slightly better performance than the linear scheme. This is because for a lower skew in title popularities, assigning equal number of replicas per data item is better than providing higher replicas for the popular data items which do not have a su±ciently high popularity weight. However, for higher skew in popularity, the behavior of the linear scheme starts paying richer dividends in reducing the overall latency, hence, it outperforms the random scheme. In all cases, the square-root scheme always yields the lowest aggregate availability latency. 39 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 18 20 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 10 4 Number of cars (N) % of wrt sqrt scheme Linear Random (a) w =¡0:5 (b) w =¡0:5 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 18 20 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 10 4 Number of cars (N) % of wrt sqrt scheme Linear Random (c) w =¡1:0 (d) w =¡1:0 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 18 20 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 10 4 Number of cars (N) % of wrt sqrt scheme Linear Random (e) w =¡1:5 (f) w =¡1:5 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 18 20 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 10 4 Number of cars (N) % of wrt sqrt scheme Linear Random (g) w =¡2:0 (h) w =¡2:0 Figure 4.5: Scale-up experiments where the total storage to the data item repository size is held constant at S T T = 3000 600 . The number of cars and the storage per car are varied to realize S T =3000. 40 0 50 100 150 200 250 0 10 20 30 40 50 60 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 Number of cars (N) % wrt square−root scheme Linear Random (a) w =¡0:5 (b) w =¡0:5 0 50 100 150 200 250 0 10 20 30 40 50 60 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 Number of cars (N) % wrt square−root scheme Linear Random (c) w =¡1:0 (d) w =¡1:0 0 50 100 150 200 250 0 10 20 30 40 50 60 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 Number of cars (N) % wrt square−root scheme Linear Random (e) w =¡1:5 (f) w =¡1:5 0 50 100 150 200 250 0 10 20 30 40 50 60 Number of cars (N) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 50 100 150 200 250 10 0 10 1 10 2 10 3 Number of cars (N) % wrt square−root scheme Linear Random (g) w =¡2:0 (h) w =¡2:0 Figure 4.6: Aggregate availability latency for the three replication schemes as a function of the car density when the storage per car is ¯xed at 3. Here T =100. 41 4.2.2.3 Variation in storage per car The total storage in the system can also be increased by keeping the car density constant and increasing the storage per car. Figure 4.7 shows the performance of the three repli- cation schemes as a function of the storage per car when the car density is held constant at 50 for T =50. As expected increasing storage reduces the latency for all the schemes. In case of w =¡0:5, the random and linear scheme show a latency performance within 10¡20% of the square-root scheme. However, with higher w values this di®erence blows up with the square-root scheme providing a much lower latency as compared to the other two. 4.2.2.4 Variation in data item repository size Finally, we consider the e®ect of increasing the data item repository size for a given value of car density and storage per car. Figure 4.8 depicts the latency performance of the replicationschemesasafunctionofT whenN =50and®=15givingS T =750. Forthe same total storage, as the data item repository size increases, lesser replicas are assigned per data item, resulting in an increase in the overall availability latency. With w =¡0:5, alltheschemesshowanalmostlinearincreaseinthelatencyas T increases. Theincrease in the latency becomes less signi¯cant with increasing skewness because enough replicas can still be assigned to the popular data items which have a major contribution to the aggregate availability latency. With w = ¡1:5 and w = ¡2, the random scheme shows a step function like behavior because increase in data item repository size from 100 to 400 ¯rst causes a reduction in the replicas for the popular data items. However, further increase in T from 400 to 800 does not change the number of replicas for the popular data items causing minimal change in the aggregate availability latency values. 4.3 Dataitemswithdisplaytimeoneandshorttripduration Theanalysisandsimulationresultspresentedsofarassumedthatoncearequestisissued at a client, it is willing to wait as long as it takes for its request to be satis¯ed. In other words, the client trip duration was assumed to be unbounded. For the speci¯c mobility 42 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 Storage per car (α) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 5 10 15 20 25 30 10 0 10 1 10 2 10 3 10 4 Storage per car (α) % wrt square−root scheme Linear Random (a) w =¡0:5 (b) w =¡0:5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 Storage per car (α) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 5 10 15 20 25 30 10 0 10 1 10 2 10 3 10 4 Storage per car (α) % wrt square−root scheme Linear Random (c) w =¡1:0 (d) w =¡1:0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 Storage per car (α) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 5 10 15 20 25 30 10 0 10 1 10 2 10 3 10 4 Storage per car (α) % wrt square−root scheme Linear Random (e) w =¡1:5 (f) w =¡1:5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 Storage per car (α) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 5 10 15 20 25 30 10 0 10 1 10 2 10 3 10 4 Storage per car (α) % wrt square−root scheme Linear Random (g) w =¡2:0 (h) w =¡2:0 Figure 4.7: Aggregate availability latency for the three replication schemes as a function of the storage per car when data item repository size is 50 and car density is 50. 43 0 100 200 300 400 500 600 700 0 10 20 30 40 50 60 70 80 Title Repository Size (T) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 100 200 300 400 500 600 700 10 −1 10 0 10 1 10 2 10 3 10 4 Title Repository Size (T) % wrt square−root scheme Linear Random (a) w =¡0:5 (b) w =¡0:5 0 100 200 300 400 500 600 700 0 10 20 30 40 50 60 70 80 Title Repository Size (T) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 100 200 300 400 500 600 700 10 −1 10 0 10 1 10 2 10 3 10 4 Title Repository Size (T) % wrt square−root scheme Linear Random (c) w =¡1:0 (d) w =¡1:0 0 100 200 300 400 500 600 700 0 10 20 30 40 50 60 70 80 Title Repository Size (T) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 100 200 300 400 500 600 700 10 −1 10 0 10 1 10 2 10 3 10 4 Title Repository Size (T) % wrt square−root scheme Linear Random (e) w =¡1:5 (f) w =¡1:5 0 100 200 300 400 500 600 700 0 10 20 30 40 50 60 70 80 Title Repository Size (T) Aggregate Availability Latency (δ agg ) Sqrt Linear Random 0 100 200 300 400 500 600 700 10 −1 10 0 10 1 10 2 10 3 10 4 Title Repository Size (T) % wrt square−root scheme Linear Random (g) w =¡2:0 (h) w =¡2:0 Figure 4.8: Aggregate availability latency for the three replication schemes as a function of the data item repository size for a car density of 50 and storage per car of 15. 44 model under consideration, namely 2D random walk on a torus, the maximum latency experienced by a client is bounded [2] as long as at least one replica of every time is present in the system at all times. However, in more practical scenarios, the client may have a certain maximum time it is willing to wait for request resolution. This is captured by considering a ¯nite trip duration, °, for the client. The availability latency for item i, ± i , can be any value between 0 and °¡1. If the client's request is not satis¯ed, we set ± i =° indicating that the client's request for item i was not satis¯ed 1 . 4.3.1 Analysis As before with Section 4.2.1, here, we present approximations for the average availability latency in the presence of a low and high density of replicas for small data items in the presence of short client trip durations. 4.3.1.1 Sparse Approximation Recall that latency in the case of a 2D-random walk on a torus can be modeled as an exponential distribution as: P(± i >t)=¸¢exp(¡¸¢t) (4.30) where ¸ = r i c¢G¢logG . The average availability latency with ¯nite trip duration ° is then given by, ± i = Z ° 0 x¢¸¢exp(¡¸¢t)dx+ Z 1 ° °¢¸¢exp(¡¸¢t)dx (4.31) Hence, we get ± i = c¢G¢logG r i ¢[1¡exp( ¡°¢r i c¢G¢logG )] (4.32) 1 Another way of handling ¯nite trip durations is to divide the input requests into satis¯ed and unsat- is¯ed respectively, and only calculating the expected latency for the satis¯ed requests. We adopt such an approach while evaluating the performance of di®erent replication schemes using mobility derived from traces of bus movements from a real test-bed, see Section 4.7. 45 4.3.1.2 Dense Approximation Recall that as de¯ned in Section 4.2.1.2, A k is the event that a data item i is encountered by the client for the ¯rst time in the k th cell and P(A k ) is the probability that event A k occurs. Also, p k is the probability of encountering data item i in the k th cell, given that it was not encountered in the previous k¡1 cells. Then, p k =1¡ µ 1¡ 1 G¡k+1 ¶ r i ; 2·k·° (4.33) Also, we rewrite P(A k ) incorporating the ¯nite trip duration constraint as, P(A k )=p k k¡1 Y j=1 (1¡p j ) ; 2·k·° (4.34) Let P(A °+1 ) denote the probability of not encountering the data item i during the entire trip duration °. Hence, P(A °+1 )= ° Y j=1 (1¡p j ) (4.35) Then, the availability latency (± i ) for data item i is given by, ± i = °+1 X k=1 (k¡1)P(A k ) (4.36) Figure 4.9 shows that the above approximations for low and high density of replicas matchesthelatencyobtainedbysimulations. Inthiscase,thedenseapproximationisalso valid for a low density of replicas because the ¯nite trip duration ° limits the maximum value of the availability latency. For a low density of replicas in most cases the latency will be higher than ° and hence it will be bounded by °. For a higher replica density, the value of ° is not as signi¯cant since the latency for that item will be much lower than °. 4.3.2 Simulation Results Figure4.10depictsthelatencyperformancefordi®erentreplicationschemeswhenstorage per car is increased from 4 to 25 slots when the trip duration is set as 10. When storage 46 0 10 20 30 40 50 10 −2 10 −1 10 0 10 1 Number of replicas ( r i ) Average availability latency ( δ i ) Sparse−approximation Simulation Dense−approximation Figure 4.9: Average availability latency for a data item as a function of its replicas for a ¯nite trip duration ° of 10. The simulation curves are plotted along with the sparse and dense approximations for ¯nite trip duration for a 10£10 torus, when the number of cars is set to 50. per car is low, ® = 4, this represents a constrained storage scenario. The linear scheme that allocates more replicas to the popular data items shows superior performance as compared to the square-root scheme. This is because in such scenarios the replicas per data item is small, hence, only data items having a larger number of replicas will provide alatencylessthan°. Sincethepopulardataitemsaretheonesthatrequestedmoreoften allocatingmorereplicasfortheseitemslowerstheaggregateavailabilitylatency. Contrast this scenario with the case of unbounded trip duration where a square-root replication scheme always provided the minimum latency (see Figure 4.4). The optimal scheme here is a super-linear one which allocates most of the replicas to the ¯rst few popular data items after satisfying the constraint that at least one copy of every data item must be present in the network. For a highly skewed scenario, w =¡2, allocatingalltheremainingstorageforthemostpopulardataitemminimizesthelatency. This is because most of the popularity weight is associated with the most popular data item which is requested very often. As the storage per car is increased further the curves start becoming °atter and at ® = 25, see Figure 4.10(c), a replication scheme characterized by an exponent in the 47 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 Replication Exponent (n) Aggregate Availability Latency (δ agg ) w=−0.5 w=−1.0 w=−1.5 w=−2.0 (a) ®=4 (b) ®=10 (c) ®=25 Figure4.10: Aggregateavailabilitylatencyfordi®erentreplicationstrategiesfora10£10 torus for a ¯nite trip duration of 10 when T =100 and N =50. Figures (a), (b), and (c) depict three di®erent storage values per car: f4,10,25g. range, 0:3 · n · 1:0, shows near optimal performance. This is because the storage is abundant enough for all these schemes to allocate a copy of the popular data items to everycarbringingthelatencyfortheseitemsto0. Thedi®erenceinthereplicasallocated for the lesser popular data items has minimal e®ect on the aggregate availability latency on account of their lower request rate. Recall, the frequency of access to the data items follows a Zipf distribution that depicts a heavy-tailed behavior. 4.4 Dataitemswithhigherdisplaytimeandlongclienttrip duration All the results so far considered a homogeneous repository of data items with a display time (¢) of one. In this section, we consider data items with a higher ¢. In such cases, the latency encountered by a client request is given by the earliest time when a contigu- ous block of ¢ cells containing at least one replica of the request item is encountered. Figure 4.11 depicts the average availability latency for a data item with a higher display time (¢=f2;3;4;5g) as a function of the replicas for that item. Here, again we consider a sparse density of replicas. For a given data item replica density, the latency increases with the display time. As expected the latency reduces with increase in the replicas. A simplecurve-¯tonthelatencycurvesyieldsaclose-matchwiththeexpressionoftheform 48 0 2 4 6 8 10 10 1 10 2 10 3 10 4 Number of replicas (r i ) Average Availability Latency ( δ i ) Δ=2 Δ=3 Δ=4 Δ=5 σ=2.4843 σ=2.0848 σ=1.7007 σ=1.3414 Figure 4.11: Average availability latency for a data item as a function of its replicas for di®erent data item display times for a 10£10 torus. The latency is given by C r ¾ i where the exponent ¾ increases with data item display time. ± i = C r ¾ i (4.37) where¾ representstheexponentforagivendataitemdisplaytimeandC isaconstant that is a function of the size of the torus. Note that the value of ¾ for data items with a display time of one is one (see Equation 4.6). The values of ¾ increases with data item display time. This indicates that an increase in the replicas provides a larger drop in the latency for a data item with a higher display time. Intuitively, encountering a replica in a contiguous block of ¢ cells becomes more and more di±cult as ¢ increases. Hence, an increase in the replica density provides a faster reduction in the latency for the higher ¢ items. This is captured by the increasing value of ¾ with ¢. The speci¯c formulation of Equation 4.37 has special signi¯cance. Equation 4.37 can be plugged in directly into the optimization formulation in Section 4.2.1.1 to determine the optimum replication scheme that minimizes the availability latency in case of data items with higher display times. Following a similar procedure as stated in Theorem 1, we get the following result. 49 ¢ ¾ n= 1 ¾+1 2 1:3414 0.4271 3 1:7007 0.3703 4 2:0848 0.3242 5 2:4843 0.287 Table 4.1: Approximate optimal replication exponents for data items with higher data item display times. Corollary 1. In case of a sparse density of vehicles, with a repository of data items with higher display times (¢ > 1), replication exponent n, such that n < 0:5, minimize the aggregate availability latency metric. Proof. Following a similar procedure as the proof listed in Theorem 1, we obtain the optimal replication exponents for the ¾ values capturing higher data item display times in Figure 4.11. Table 4.1 lists the display times and the corresponding approximate optimal exponent values. 4.5 Data items with higher display time and short client trip duration In this section, we consider scenarios with short client trip durations with data items havinghigherdisplaytimes. Thisimpliesthattheclientisonlywillingtowaitforashort period foritsrequesttobesatis¯ed(denotedby °). Otherwisethe requestis taggedwith a latency equal to °. Figure2.2showsa6£6gridusedasthemapforourexperimentalstudy. Assumethat the client starts from cell 1 and travels along the pathf1, 8, 15, 22, 29, 36g. Numbers in the bracket indicate the sequence of visited cell IDs. Hence, ° =6. For our experiments, we chose N = 200 and T = 100. We simulated a skewed distribution of access to the T data items using a Zipf distribution with a mean of 0:27. The distribution is shown to correspond to sale of movie theater tickets in the United States [15]. Initially, all cars are distributed across the cells of the map as per the steady state distribution which is determined by a random number generator initialized with a seed. 50 0 20 40 60 80 100 0 1 2 3 4 5 6 Storage slots per car (α) sqrt linear random Aggregate Availability Latency (δ agg ) 0 50 100 150 200 250 300 350 −100 −50 0 50 100 150 200 Storage slots per car (α) Latency wrt sqrt scheme (%) linear (n=1) random (n=0) Region I Region II (a) ± agg (b) % Comparison wrt sqrt scheme (¡) Figure 4.12: Figure 4.12(a) shows ± agg of the sqrt, linear and random replication schemes versus ® for ¢ = 4 and N = 200. Figure 4.12(b) shows the % comparison of the linear and random schemes wrt the sqrt scheme for this scenario. Region I and Region II, respectively, indicate the parameter space where n=1 and n=0:5 perform the best. Depending on the particular replication technique, the replicas for each data item are calculated using Equation 4.2 and then distributed across the car. A car only contains a maximumofonereplicaforaparticulardataitem. Thedistributionofdataitemreplicas across the cars is uniform. At each step, depending on the current car location, it moves to one of its adjoining cell (including itself) as governed by the mobility model. Another seeddeterminesthechoiceofwhichcellacarmovesto. Since° =6,eachcarperformssix transitions according to the mobility model. We performed the comparisons for several di®erent data item distribution seeds starting from the same initial car positions. Next, we varied the initial car positions by changing the initial seed. Speci¯cally, we chose 50 di®erent initial seeds and for each of these we used 50 seeds that decide the distribution of the data item replicas among the cars. Thus, each point in all the presented results is an average of 2500 simulations. Below is an overview of the key lessons learnt from these experiments with higher data item display times and a short trip duration: The optimal value of n varies as a function of the scarcity of the network storage 51 When storage is scarce, the optimal aggregate availability latency is realized by using a higher value of n. Evenarandomschemewithn=0isgoodenoughwhenstorageisabundantrelative to the repository size. When storage is extremely scarce, with larger data item sizes (¢ >1), linear (n=1) scheme provides the best performance. This is because it allocates more replicas for the popular data items at the cost of assigning very few for the remaining data items. In this case, the contribution to ± agg is a function of the ± for the more popular data items since for the less popular data items there will be insu±cient replicas to reduce their ±. On the other hand, since the random scheme is blind to the data item access frequencies, on an average, it assigns equal number of replicas for each data item thereby providing the worst performance. The square root (n = 0:5) scheme assigns fewer replicas for the popular data items than the linear scheme. As we increase the amount of storage, there is a cut-o® point alongthestorageaxis, whereallocatingmorereplicasforthepopulardataitemsprovides negligible improvement in ± agg . It is beyond this point that the square root scheme starts outperformingthelinearscheme. Thisisbecausethesquarerootschemecanusetheextra storage savings for allocating replicas for the less popular data items, thereby reducing their ±. To illustrate, Figure 4.12 shows the variation of ± agg as a function of ® for ¢ = 4. Since ± agg is a function of the value of n, hence, here we denote it as ± agg (n = i). For Figure 4.12(b), the y-axis represents the percentage comparison of the linear (n=1) and the random (n=0) schemes with respect to the square root (n=0:5) scheme calculated as, ¡= µ ± agg (n=i)¡± agg (n=0:5) ± agg (n=0:5) ¶ £100; where i=f0;1g (4.38) Figure4.12(b)showstwodistinctregionsinwhichtheschemeswithn=0:5andn=1 perform well under certain parameter settings within the design space. For ®<=20, the linear scheme (n = 1) performs the best. For 20 <= ® <= 360, the square root scheme 52 (n = 0:5) performs the best. Beyond this value even a random scheme (n = 0) provides a competitive latency performance. With ¢ = x and T = y, the value of ® needed to replicate the entire database on each car is ® db = x¢y. At a certain storage threshold (earlier than ® db ), the random scheme assigns enough replicas to the popular data items to bring their ± down. In this case, all the data items have the same number of replicas, thereby producing a low ± for every data item. Hence, from this point onward, even a random scheme provides a good performance. However, this point requires su±cient storage per car and hence a random scheme may be appropriate only for over-provisioned scenarios. As illustrated in Figure 4.12(b) with N = 200, T = 100, and ¢ = 4, the storage threshold is around 360 slotspercar. For¢=5,and6,thisthresholdisapproximately450and540, respectively. These are loose upper bounds. 4.5.1 Aggregate availability latency as a function of car density (N) Car density, which in turn a®ects the available storage in the system, has a major impact on the performance of ± agg for all the schemes. With the decrease in the car density to N = 100, the number of replicas allocated by the schemes is reduced thereby giving comparatively larger values of ± agg across the same storage axis. As ® is increased, the dropinthe± agg curvesforalltheschemesisnotassteepasseeninthecasewithN =200. Again, this is because the number of replicas is not increasing at such a high rate. The storage is reduced by an order of 2, hence a higher value of ® is needed to produce the same drop in ± agg as was seen in the case with N =200 cars. This is observed across all values of ¢. Forallexperiments,wealsocalculatedthestandarddeviations(SD)andthestandard error of the mean (SEM). The 95% con¯dence intervals determined as 1:96¤SEM are quitesmallandaccordinglythecurvesarequitesmooth. However,thestandarddeviation is quite large, especially for the cases when ± agg is low for high values of ¢ and ®. This is because a low latency requires the data item to be present in every cell along the journey depending on the value of ¢. As ¢ increases, it becomes increasingly di±cult to meet 53 thisconditiontherebyshowingahighvariancein± agg . ThelargeSDvalueisanempirical observation about the nature of the random process. Here, we summarize the main results obtained so far with the 2D random walk based mobility model. For data items with a display time of one and long client trip duration, we have the following: in case of sparse density of data item replicas, we analytically solvedstorageconstrainedoptimizationtominimizetheavailabilitylatency,yieldingthat a replication scheme that allocates replicas for a data item as per the square-root of its frequency of access provides the optimal aggregate availability latency. With a higher density of replicas, a scheme that allocates replicas as the n th root of the data item access frequencies where 0:4 · n · 0:6 provides near optimal latency. In all cases, we presentedtherelativeperformanceofthreewell-knownreplicationschemes,namely,linear (allocates replicas in direct proportion to the data items' frequencies), square-root, and random (allocates equal number of replicas per data item). Fordataitemswithadisplaytimeofoneandshortclienttripduration,wefoundthat a linear replication scheme provides a superior performance. In such cases, a super-linear replication scheme that allocates a large number of replicas to the popular data items provides the optimal latency. With data items with larger display time and long client trip duration, we found that higher n th root replication schemes where n < 0:5, show optimal latency performance. Finally, with a short client trip duration, we found that in extremelylowstoragescenarios,alinearschemeshowssuperiorperformance,inmoderate to high storage scenarios a square-root replication scheme provides superior performance while in abundant storage scenarios even a random replication scheme is good enough to provide a low latency. 4.6 Evaluation with a real map In this section, we describe the performance of the various replication schemes where the vehicular movements are dictated by an underlying map of the San Francisco Bay Area. Figure 4.13(a) depicts a section of the San Francisco Bay area with the major freeways and their intersections. We superimpose a 2D-grid on this map and the individual cells are labelled with the respective freeway id that they cover as shown in Figure 4.13 (b). 54 880680 101 680 101 680 101/ 85 237237237237 680 85 101/ 237 101 680 237/ 85 101101101 880680 85 101 880/ 101 680 280 85 880 101680 280 280/ 85 280280 880 280 680/ 101 85 280280 880/ 280 280280 101 85 880 101 85 880 101 85 880 101 85 880/ 85 85 85 85 85 101 880 85 85 85/ 101 880 880/ 237 880 880 (a) (b) Figure4.13: AmapoftheSanFranciscoBayAreaobtainedfromhttp://maps.google.com is shown in Figure 4.13(a). Figure 4.13(b) superimposes a 15£15 grid on this map and labels the cells appropriately with the freeway IDs that they overlap with. This 2D-grid serves to capture the underlying map at a coarse granularity. Most of the probability mass is concentrated on the cells that represent the major freeways. The non-labelled cells have equal transition probabilities to each of its neighboring eight cells. The outgoing transition probabilities at a cell that represents an intersection between two freeways are calculated as follows. As an example, consider the intersection of the freeways 880 and 85 as shown in Figure 4.14. We obtained the tra±c density seen on the freeways before and after the intersection from Caltrans data provided by the California department of transportation [44]. The website allowed real time gathering of vehicle tra±c data. We considered a time window between 7-8 pm for a particular week and averaged the vehicular density seen during this period. The day-to-day statistics were quite similar, here, we show an example of how the actual data was converted into the probabilitytransitionvaluesthatformedthebasisoftheMarkovmobilitymodel. Similar calculations were employed to populate the entire transition probability matrix. Finally, toeliminate¯nite edgee®ects, weconvertedthe15£15 grid intoatorusbyallowingcars at the boundaries to appear at the opposite ends with equal transition probabilities. 55 Figure 4.14: The intersectionbetween freeways880 and 85is captured in the¯gure along with the equivalent probability transitions in the Markov model based on data obtained from Caltrans regarding the vehicular densities. The transition matrix was used to generate the car movements. We provide a notion ofdirectionalitytothecarmovementsbyensuringthatthenextstepforacarsmovement takes into account both the current cell as well as the previous cell which a car traversed. This is done by storing both the cell IDs as part of the state of the Markov chain. Consequently, the °ip-°op movements of the cars is avoided thereby ensuring that car movements are constrained by the underlying freeway structure of the map and are not entirely random. We used these car movements to investigate the relative performance of the various replication schemes under such a scenario. 4.6.1 Results with replication schemes In this section, we present some representative results for the various replication schemes obtained by employing the Markov mobility model previously derived from a map of the San Francisco Bay area. Below we describe the three sets of experiments used in our evaluation for comparison of the linear, square-root, and random replication schemes. As before, requests are issued, one at a time at each time-step at vehicles in a round-robin manner, as per a Zipf distribution with a mean of 0:27. The three sets of experiments were: For data item repository size T set as 25, client trip duration, °, set as 10, storage per car, ®, set as 2, the latency performance with the various replication schemes is studied as a function of increasing car density N (see Figure 4.15). 56 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 9 Number of cars Aggregate Availability Latency linear sqrt random 0 50 100 150 200 250 0 5 10 15 20 25 Number of cars Latency performance wrt linear sqrt random (a) (b) Figure4.15: Performanceofvariousreplicationschemesasafunctionofcardensitywhen T =25, ®=2, and ° =10. Figure 4.15(b) shows the performance wrt the linear scheme. For data item repository size T set as 25, client trip duration, °, set as 10, car density, N, set as 50, the latency performance with the various replication schemes is studied as a function of increasing storage per car ® (see Figure 4.16). For car density, N, set as 50, client trip duration, °, set as 10, storage per car, ®, set as 2, the latency performance with the various replication schemes is studied as a function of increasing data item repository size T (see Figure 4.17). In all cases, the main conclusion is that the linear replication scheme shows superior performance as seen in Section 4.3 for the case with data item size equal to 1 and ¯nite clienttripduration. Thetrendsseenwiththismodelaresimilartothoseseenwithauni- formMarkovmobilitymodelwithequaltransitionprobabilities. Thisresultsuggeststhat the uniform probability transition matrix based Markov model may be a good indicator of the performance that may be seen with a model derived from real maps. 4.7 Evaluation with real movement traces We now evaluate the latency performance of the static replication schemes using traces obtained from a bus-based DTN test-bed called UMassDieselNet [11]. First, we brie°y 57 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 Storage per car Aggregate Availability Latency linear sqrt random 0 1 2 3 4 5 0 5 10 15 20 25 Storage per car Latency performance wrt linear sqrt random (a) (b) Figure 4.16: Performance of various replication schemes as a function of storage per car when T =25, N =50, and ° =10. Figure 4.16(b) shows the performance wrt the linear scheme. 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 Number of titles Aggregate Availability Latency linear sqrt random 0 5 10 15 20 25 30 0 5 10 15 20 25 Number of titles Latency performance wrt linear sqrt random (a) (b) Figure 4.17: Performance of various replication schemes as a function of data item repos- itory size when N = 50, ® = 2, and ° = 10. Figure 4.17(b) shows the performance wrt the linear scheme. 58 describe the test-bed and present some properties of the mobility model followed by the buses. Then,wedescribethedetailsoftheexperimentalset-upandtheresultscomparing thesquare-root,linear,andrandomreplicationschemesunderdi®erentparametersettings using these traces. 4.7.1 UMassDieselNet Traces Here, we brie°y describe the details of the UMassDieselNet test-bed and present some properties of the mobility model that characterizes the movement of the vehicles that are part of the test-bed. The UMassDieselNet network operates daily around the UMass campus and the surrounding county. It comprises of 30 buses equipped with a Linux based computer coupled with a IEEE 802.11b wireless interface that permits ad-hoc communicationbetweenthebuseswhentheyareinradiorange. AnIEEE802.11baccess point is also connected to the brick computer that allows DHCP access to passengers within the bus. The traces are available for a period of 60 days, the logs describe every encounter between every pair of buses that occurred during the day. These traces do not containthelogsforaccessesmadebetweenthepassengersandtheaccesspointwithinthe bus. Theidentityofthebusesinvolvedintheencounter,thetimeofencounterandamount of data transmitted during the encounter are logged in the trace ¯les. Certain buses had long routes while others had short ones. Unfortunately, due to technical di±culties, the GPS device on the buses were unable to provide details about the bus locations during the encounter. Figure 4.18 shows the number of buses that were active on each day of the 60 day period during which the traces were collected. We only considered traces where the number of active buses was greater than 15. This accounted for 52 traces. In general, the traces indicated a sparse density of buses where there was a high degree of locality in the encounters. In other words, if 2 buses encounter each other at the beginning thentheywillcontinuetoencountereachothermorefrequentlythanotherbuses. Thisis capturedinFigure4.19wherewesetaminimumseparationtimebetweentwoconsecutive encountersofthesamepairofbusestoconsideritadi®erentencounter. InFigure4.19(b), the separation time is set as 20 seconds as compared to 0 seconds set for Figure 4.19(a). 59 0 10 20 30 40 50 60 0 5 10 15 20 25 Trace id No of active buses Figure 4.18: The number of active buses for each trace representing the bus encounters for each day of a 60-day period. The buses operated from 7am to 5pm. 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time between encounters CDF 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time between encounters CDF (a) Separation = 0s (b) Separation = 20s Figure 4.19: The CDF of the time between encounters averaged across all the traces for 2 di®erent separation times 0s (Figure 4.19(a)) and 20s (Figure 4.19(b)). 60 4.7.2 Experimental Set-up In this section, we describe the details of the simulation set-up used for evaluation of the replication schemes employing the UMassDieselNet traces. Each trace represents the movements of buses during that particular day. There is no correlation between trace movements across days. Hence, we process each trace one at a time and then average the resultsobservedacrossallthedaysnotingthattheaverageisindicativeoftheperformance seenonmostdays. However,certaindaysdoappearasoutlierssincethenumberofactive buses di®ers from day-to-day. As before we consider a ¯nite data item repository of size T. Each bus is assumed to carry ® storage slots. Replicas for each data item are determined based on a replication scheme and then allocated across the buses uniformly at random. The constraint is that at least one copy of every data item must be present in the network at all times. We consider the three representative replication schemes: random, square-root, and linear and study the relative performance of the schemes. Since the buses only operate for a ¯nite amount of time we consider two separate metrics (i) Average availability latency for satis¯ed requests (ii) Normalized unsatis¯ed request rate. Requests for the T titles are generated as per a Zipf distribution with an exponent w =¡0:73. The duration during which the buses were active during a day is determined apriori and subject to this duration requests are issued at equal inter-arrival times. A generated request is assigned to a bus chosen uniformly at random. A request is assumed to be satis¯ed either if the data item requested is locally stored or another bus carrying the requested item is encountered at some point after the request is issued. Those requests that are not satis¯ed at the end of the day are tagged as unsatis¯ed requests. 4.7.3 Results In this section, we brie°y describe the main results from evaluation of the performance of the replication schemes using the UMassDieselNet traces. For the ¯rst set of experiments we vary the values of (T,®) as f(5,1), (10,2), (15,3), (20,4), (25,5)g (see Figure 4.20). The linear replication scheme provides the lowest average availability latency for satis¯ed 61 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of Titles (T) Availability Latency for Satisfied Requests (secs) Random Sqrt Linear 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 Number of Titles (T) Unsatisfied Request Metric Random Sqrt Linear (a) (b) Figure 4.20: Aggregate availability latency for satis¯ed requests and the aggregate un- satis¯ed request metric for the random, square-root, and linear replication schemes are shown in Figure 4.20(a) and (b) respectively. The ratio of the storage per car to the data item repository size, ® T is maintained as 1:5. requests (about 10¡25% better than the square-root scheme). The linear and square- root scheme show similar performance in terms of the normalized unsatis¯ed requests. The random scheme shows poor performance both in terms of latency as well as the normalized unsatis¯ed requests. Figure 4.21 shows the performance of the replication schemes when the data item repository size is ¯xed at 25 and the storage per bus is increased. Increase in storage leads to increase in the replicas per data item, hence, as expected for all schemes, the latency and the normalized unsatis¯ed requests go down. The linear scheme continues to show superior performance with respect to both metrics. Figure4.22showstheperformanceofthereplicationschemeswhenthestorageperbus is ¯xed at 3 and the data item repository size is increased. As the data item repository size is increased, lesser replicas are allocated per data item resulting in an increase in the latency as well as the unsatis¯ed requests. Initially the increase in latency is linear but slowly becomes sub-linear. Note that the data item repository size cannot be made arbitrarily large since the number of active buses is constrained. As before the linear scheme always shows the best performance. 62 0 2 4 6 8 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Storage per Car (α) Availability Latency for Satisfied Requests (secs) Random Sqrt Linear 0 2 4 6 8 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Storage per Car (α) Availability Latency for Satisfied Requests (secs) Random Sqrt Linear (a) (b) Figure 4.21: Aggregate availability latency for satis¯ed requests (Figure 4.21(a)) and the aggregate unsatis¯ed request metric (Figure 4.21(b)) as a function of the storage per car for a data item repository size of 25. 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of Titles (T) Availability Latency for Satisfied Requests (secs) Random Sqrt Linear 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 Number of Titles (T) Unsatisfied Request Metric Random Sqrt Linear (a) (b) Figure 4.22: Aggregate availability latency for satis¯ed requests (Figure 4.22(a)) and the aggregate unsatis¯ed request metric (Figure 4.22(b)) as a function of the data item repository size when storage per car ® is ¯xed at 3. 63 The mobility model provided by the traces represents an extremely sparse density of buses where inherently there is a limit to the maximum amount of time for request satisfaction(namelythelastencountertimeonthetrace). The¯nitetripdurationincon- junctionwiththe lowdensityand encountermodel favorsa linearschemewhichallocates more replicas for the popular data items. The popular data items are requested more frequently, and within the¯nite time for request satisfaction, havea higher probability of being satis¯ed on account of the larger number of replicas. The square-root scheme tries to allocate replicas less aggressively to the more popular data items in favor of the less popular ones. This hurts its performance since the less popular data items have a very low probability of being satis¯ed. Nevertheless, in such scenarios, a random scheme that allocates replicas equally across the data items shows the worst performance. WenowconsideranequivalentscenariowiththeMarkovmobilityandstudyitsprop- erties in terms of the time between encounters (see Figure 4.23). The aim is to capture a similarscenarioasdepictedbytheUMassDieselNettraces(comparewithFigure4.19(a)). We consider a similar set-up to experimental scenario described in Figure 4.20. Similar to the trends seen with the traces, Figure 4.24 shows that the linear replication scheme outperforms the square-root and the random schemes in terms of the latency for satis- ¯ed requests. The performance in terms of the normalized unsatis¯ed requests is quite similar for the three schemes. These results suggest that the results obtained from the Markov mobility model may be applicable across a vast range of scenarios comprising di®erent mobility models. Adequate adjustment to the transition probabilities of the Markov model may enable this model to suitably capture the mobility trends of other models such as Manhattan, Highway, Random Way-Point etc. 4.8 Summary In this chapter, we have investigated the performance of various replication schemes for a mobile ad hoc network of AutoMata devices. These schemes compute the degree of replication for each data item as a power law function of its popularity, i.e., frequency of access. We propose a general optimization formulation to minimize the average availabil- ity latency subject to a total constraint enforced by the car density and the storage per 64 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time between encounters CDF Figure 4.23: CDF of the time between encounters from the Markov model for a 25£25 torus with a car density of 15. 0 5 10 15 20 25 0 50 100 150 200 250 300 No of Titles Aggregate Availability Latency linear sqrt random 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 No of Titles Aggregate Unsatisfied Request Metric linear sqrt random (a) (b) Figure 4.24: Aggregate availability latency for satis¯ed requests and the aggregate un- satis¯ed request metric as obtained from an equivalent scenario employing the Markov model. The ratio of the storage per car to the data item repository size, ® T is maintained as 1:5. 65 car. While it may be possible to solve this optimization under some scenarios, because of the large parameter design space we solve the problem indirectly using simulations. Obtained results indicate the following key lessons: For a repository with data items having size 1 and with unbounded client trip duration, a square-root replication scheme minimizes availability latency For a repository with data items having size 1 and with ¯nite client trip duration, linear/super-linear replication schemes show superior performance For a repository with data items having size > 1 and with unbounded client trip duration, higher n th order replication schemes minimize the aggregate availability latency where n>2 (see Equation 4.1) Forarepositorywithdataitemshavingsize>1andwith¯niteclienttripduration, alinearschemeshowssuperiorperformanceforhighlyconstrainedstoragescenarios while a square-root scheme shows superior performance over a large storage space. Obtained results are validated with Markov models derived from di®erent maps of cities such as San Francisco that constraint the movements of the vehicles on the basis of the underlying freeway structure. We also investigate the performance of the various replication schemes with traces obtained from a bus-based DTN test-bed called UMass- DieselNet, also providing the equivalence between the trace-based mobility model and the Markov model. A promising future research direction that will complement the above results is to explore heterogeneity. While this study has considered a homogeneous repository of data items, it may be useful to extend our evaluation to a heterogeneous repository of items with di®erent display times and sizes. Moreover, the vehicles themselves may have di®erent quantities of available storage and di®erent trip durations. 66 Chapter 5 Zebroids The choice of a replication scheme has a signi¯cant impact on the availability latency experienced by the client requests. In particular, we saw that a square-root replication scheme provides the minimum aggregate availability latency in case of unbounded trip duration, while a linear replication scheme provides superior performance for ¯nite trip durations. We now try to answer the following question: Can the availability latency be improved further after an appropriate static replication scheme has computed the replicas per data item? Recall that the replication schemes mentioned earlier do not consider placement of the replicas. Hence, a system may dynamically reorganize the replicas across the vehicles on the basis of the currently active requests to make better useoftheavailablesystemstorage. Speci¯cally,appropriatevehiclesmaybescheduledto carry a data item on behalf of a server (vehicle containing the requested item) to a client requesting it, thereby reducing its latency. These data carriers are labeled zebroids. In thischapter, wequantifytheimprovementsinlatencythatcanbeobtainedbyemploying these data carriers along with the incurred overhead. The organization of this chapter is as follows. Section 5.1 formally de¯nes a zebroid and gives a brief overview. Section 5.2 describes how the zebroids may be employed. Section 5.3 brie°y describes the replacement policies that may be employed by a zebroid. Section 5.4 provides details of the analysis methodology employed to capture the per- formance with zebroids. Section 5.5 describes the details of the simulation environment used for evaluation. Section 5.6 enlists the key questions examined in this study and an- swers them via analysis and simulations. Section 5.7 presents some representative results 67 with zebroids when employing a Markov mobility model for the vehicles that adheres to the location of major freeways in the San Francisco Bay area. Section 5.8 provides an evaluation of the performance of zebroids on traces obtained from a real DTN test-bed comprising 30¡40 buses. Finally, Section 5.9 provides brief conclusions of the study. 5.1 Overview of Zebroids In this section, we provide a brief overview of the use of zebroids as data carriers. Selec- tion of zebroids is facilitated by the two-tiered architecture. The control plane enables centralized information gathering at a dispatcher present at a base-station 1 . Some ex- amples of control information are currently active requests, travel path of the clients and their destinations, and paths of the other cars. For each client request, the dispatcher may choose a set of z carriers that collaborate to transfer a data item from a server to a client (z-relay zebroids). Here, z is the number of zebroids such that 0 · z < N, where N is the total number of cars. When z = 0 there are no carriers, requiring a server to deliver the data item directly to the client. Otherwise, the chosen relay team of z zebroids hand over the data item transitively to one another to arrive at the client, thereby reducing availability latency (see Section 5.2 for details). To increase robustness, the dispatcher may employ multiple relay teams of z-carriers for every request. This may be useful in scenarios where the dispatcher has lower prediction accuracy in the informa- tion about the routes of the cars. Finally, storage constraints may require a zebroid to evictexistingdataitemsfromitslocalstoragetoaccommodatetheclientrequesteditem. Hence, suitable replacement policies may be employed by a zebroid. Inthischapter,wequantifythefollowingmainfactorsthata®ectavailabilitylatencyin thepresenceofzebroids: (i)dataitemrepositorysize,(ii)cardensity,(iii)storagecapacity per car, (iv) client trip duration, (v) replacement scheme employed by the zebroids, and (vi) accuracy of the car route predictions. For a signi¯cant subset of these factors, we address some key questions pertaining to use of zebroids both via analysis and extensive simulations. 1 Theremaybedispatchersdeployedatasubsetofthebase-stationsforfault-toleranceandrobustness. Dispatchers between base-stations may communicate via the wired infrastructure. 68 5.2 Solution Approach When a client references a data item missing from its local storage, the dispatcher iden- ti¯es all cars with a copy of the data item as servers. Next, the dispatcher obtains the future routes of all cars for a ¯nite time duration equivalent to the maximum time the client is willing to wait for its request to be serviced. Using this information, the dis- patcher schedules the quickest delivery path from any of the servers to the client using anyothercarsasintermediatecarriersnamelyzebroids. Hence,itdeterminestheoptimal set of forwarding decisions that will enable the data item to be delivered to the client in theminimumamountoftime. Notethatthelatencyalongthequickestdeliverypaththat employs a relay team of z zebroids is similar to that obtained with epidemic routing [59] under the assumptions of in¯nite storage and no interference. A modi¯ed version of the Bellman Ford's shortest path algorithm is employed to determine this path along the encounter graph of the cars. In the following, we give a brief description of this algorithm. A client waits for a maximum of ° time steps for each issued request to be satis¯ed. Hence, we construct graph G = (V;E), the encounter graph of the cars for a given request where V is the set of all cars (jVj = N) and E is the set of edges f(u,v,t) such that 0·t·°g. An edge (u,v,t) exists in G only if cars u and v encounter each other at time step t and u and v are non-servers. For a given data item request, the dispatcher identi¯es the set of cars that have a copy of this item as servers. Now the quickest path from each of these servers to the client is determined individually using the following algorithm listed below: Heresistheclientcar, parent[v]storesthecaridthathandedacopyoftherequested data item to car v, and d[s] stores the length of the quickest path to the client. Finally, the quickest path from all of these is selected as the minimum delay path to the client. Each car along this path except the server and the client represent a z-relay zebroid. The complexity of this modi¯ed Bellman Ford algorithm is O(N 3 ¢°). However, in actual practice the complexity is much lower since on an average not all N cars encounter each other. Moreover, with a random walk like mobility model, cars that meet once may meet quite often due to the locality property. 69 Algorithm 5.2.1: QuickestPath(V;E;s) for each v in V[G] d[v]=1 parent[v]=NIL prevEncounterTime[v]=¡1 d[s]=0 for iÃ1 to jV[G]j¡1 for each edge (u,v) in E[G] for tÃ1 to ° if (d[v]>t)&(prevEncounterTime[v]·t) d[v]=t parent[v]=u prevEncounterTime[v]=t We use the above algorithm in sparse scenarios when N is small (N <= 100). In other cases, we simulate an epidemic routing kind of dissemination for the trip duration number of steps. Then we determine the earliest time the client car was infected, if at all, this establishes the minimum latency encountered by a client. Since each car stores the parent car that handed over the requested item to it, the sequence of cars involved in carrying the data item from a server to the client along the minimum latency path can be determined as a simple look-up operation. If we restrict the length of the quickest path to a maximum of 2 hops, with the additional restriction that the ¯rst edge (u,v,t) along this path should have t = 1, then the zebroid used becomes a one-instantaneous zebroid with z =1. Recall that u will be a server and t = 1 signi¯es the instantaneous transfer of the data item from u to v, the zebroid. The zebroid then meets the client directly thereby satisfying the request. In some cases, the dispatcher might have inaccurate information about the routes of the cars. Hence, a zebroid scheduled on the basis of this inaccurate information may not rendezvous with its target client. To minimize the likelihood of such scenarios, the 70 dispatcher may schedule multiple zebroids. This may incur additional overhead due to redundant resource utilization to obtain the same latency improvements. The time required to transfer a data item from a server to a zebroid depends on its size and the available link bandwidth. With small data items, it is reasonable to assume that this transfer time is small, especially in the presence of the high bandwidth data plane. Large data items may be divided into smaller chunks enabling the dispatcher to schedule one or more zebroids to deliver each chunk to a client in a timely manner. This remains a future research direction. Initially,numberofreplicasforeachdataitemarecomputedasperastaticreplication scheme (see Chapter 4). This scheme computes the number of data item replicas as a function of their popularity. It is static because number of replicas in the system do not change and no replacements are performed. Hence, this is referred to as the `no- zebroids' environment. We quantify the performance of the various replacement policies with reference to this base-line that does not employ zebroids. One may assume a cold start phase, where initially only one or few copies of every data item exist in the system. Many storage slots of the cars may be unoccupied. When the cars encounter one another they construct new replicas of some selected data items to occupy the empty slots. The selection procedure may be to choose the data items uniformly at random. New replicas are created as long as a car has a certain threshold of its storage unoccupied. Eventually, a majority of the storage capacity of a car will be exhausted. 5.3 Carrier-based Replacement policies In this section, we brie°y describe the di®erent replacement schemes that may be em- ployed by a zebroid. The replacement policies considered here are reactive since a re- placement occurs only in response to a request issued for a certain data item. When the local storage of a zebroid is completely occupied, it needs to replace one of its existing items to carry the client requested data item. For this purpose, the zebroid must select an appropriate candidate for eviction. This decision process is analogous to that encoun- tered in operating system paging where the goal is to maximize the cache hit ratio to 71 prevent disk access delay. We present below a list of carrier-based replacement policies employed in our study which are adapted from di®erent page replacement policies used in operating systems [57]. 1. Least recently used (LRU) LRU-K [45] maintains a sliding window containing the time stamps of the K th most recent references to data items. During eviction, the data item whose K th most recent reference is furthest in the past is evicted. Here, we consider the case with K = 1. Depending on whether the evictions are based on the least recently used data item across all client requests (lru-global) or onlytheindividualclient'srequests(lru-local), weconsiderglobalorlocalvariants of the LRU policy. 2. Least frequently used (LFU) (a) Local (lfu-local): Each AutoMata keeps track of the least frequently used data item within its local repository. During eviction 2 , this is the candidate replica that is replaced. (b) Global (lfu-global): Thedispatchermaintainsthefrequencyofaccesstothedataitemsbasedonrequests from all clients. When a zebroid contacts the dispatcher for a victim data item, the dispatcher chooses the data item with the lowest frequency of access. 3. Random policy (random) In this case, the chosen zebroid evicts a data item replica from its local storage chosen uniformly at random. The replacement policies incur the following overheads. First, the complexity associ- ated with the implementation of a policy. Second, the bandwidth used to transfer a copy of a data item from a server to the zebroid. Third, the average number of replacements incurred by the zebroids. Note that in the no-zebroids case neither overhead is incurred. The metrics considered in this study are aggregate availability latency, ± agg , percent- age improvement in ± agg with zebroids as compared to the no-zebroids case and average number of replacements incurred per client request which is an indicator of the overhead incurred by zebroids. 2 The terms eviction and replacement are used interchangeably. 72 Note that the dispatchers with the help of the control plane may ensure that no data item is lost from the system. In other words, at least one replica of every data item is maintained in the ad-hoc network at all times. In such cases, even though a car may meet a requesting client earlier than other servers, if its local storage contains data items with only a single copy in the system, then such a car is not chosen as a zebroid. 5.4 Analysis Methodology Here, we present the analytical evaluation methodology and some approximations as closed-form equations that capture the improvements in availability latency that can be obtained with both one-instantaneous and z-relay zebroids. First, we present some preliminaries of our analysis methodology. Let N be the number of cars in the network performing a 2D random walk on a p G£ p G torus. An additional car serves as a client yielding a total of N +1 cars. Suchamobilitymodelhasbeenusedwidelyintheliterature[55,53]chie°ybecause it is amenable to analysis and provides a baseline against which performance of othermobilitymodelscanbecompared. Moreover,thisclassofMarkovianmobility models has been used to model the movements of vehicles [6, 50, 66]. We assume that all cars start from the stationary distribution and perform inde- pendent random walks. Although for sparse density scenarios, the independence assumption does hold, it is no longer valid when N approaches G. Let the size of data item repository of interest be T. Also, data item i has r i replicas. This implies r i cars, identi¯ed as servers, have a copy of this data item when the client requests item i. All analysis results presented in this section are obtained assuming that the client is willing to wait as long as it takes for its request to be satis¯ed (unbounded trip duration ° =1). With the random walk mobility model on a 2D-torus, there is a guarantee that as long as there is at least one replica of the requested data item in the network, the 73 client will eventually encounter this replica [2]. Later, we extend our analysis to consider ¯nite trip duration °. Consider a scenario where no zebroids are employed. In this case, the expected avail- ability latency for the data item is the expected meeting time of the random walk under- taken by the client with any of the random walks performed by the servers. Aldous et al. [2] show that the the meeting time of two random walks in such a setting can be mod- elled as an exponential distribution with the mean C = c¢G¢logG, where the constant c ' 0:17 for G ¸ 25. The meeting time, or equivalently the availability latency ± i , for the client requesting data item i is the time till it encounters any of these r i replicas for the ¯rst time. This is also an exponential distribution with the following expected value (note that this formulation is valid only for sparse cases when G>>r i ): ± i = cGlogG r i The aggregate availability latency without employing zebroids is then this expression averaged over all data items, weighted by their frequency of access: ± agg (no¡zeb)= T X i=1 f i ¢c¢G¢logG r i = T X i=1 f i ¢C r i (5.1) 5.4.1 One-instantaneous zebroids Recall that with one-instantaneous zebroids, for a given request, a new replica is created onacarinthevicinityoftheserver, providedthiscarmeetstheclientearlierthananyof the r i servers. Moreover, this replica is spawned at the time step when the client issues the request. Let N c i be the expected total number of nodes that are in the same cell as any of the r i servers. Then, we have N c i =(N¡r i )¢(1¡(1¡ 1 G ) r i ) (5.2) In the analytical model, we assume that N c i new replicas are created, so that the total number of replicas is increased to r i +N c i . The availability latency is reduced since the client is more likely to meet a replica earlier. The aggregated expected availability latency in the case of one-instantaneous zebroids is then given by, 74 ± agg (zeb)= T X i=1 f i ¢c¢G¢logG r i +N c i = T X i=1 f i ¢C r i +N c i (5.3) Note that in obtaining this expression, for ease of analysis, we have assumed that the new replicas start from random locations in the torus (not necessarily from the same cell as the original r i servers). It thus treats all the N c i carriers independently, just like the r i original servers. As we shall show below by comparison with simulations, this approximation provides an upper-bound on the improvements that can be obtained because it results in a lower expected latency at the client. It should be noted that the procedure listed above will yield a similar latency to that employed by a dispatcher employing one-instantaneous zebroids (see Section 5.2). Since the dispatcher is aware of all future car movements it would only transfer the requested data item on a single zebroid, if it determines that the zebroid will meet the client earlier than any other server. This selected zebroid is included in the N c i new replicas. 5.4.2 z-relay zebroids The expected availability latency with z-relay zebroids can be calculated using a coloring problem analog similar to an approach used by Spyropoulos et al. [55]. Consider a data item i requested by the client. Recall that, there are N total cars and r i replicas for data itemi. Assumethateachoftheser i replicasiscoloredred, whiletheothercarsincluding theclientarecoloredblue. Wheneveraredcarencountersabluecar,thelatteriscolored red. The expected number of steps until the client is colored red then gives the average availability latency with z-relay zebroids. If at a given step, there are k red cars (k¸r i ), thentherewillbeN¡k bluecars. Recallthatmeetingtimebetweencarscanbemodelled as an exponential distribution. Hence, by the property of exponential distribution, the average time until any of the k red cars meets any of the N+1¡k blue cars is C k¢(N+1¡k) . Now, the expected time until all the cars are colored red is P N k=r i C k¢(N+1¡k) Notethattheclientmaybecoloredredinanyoneofthesestepswithequalprobability. Consequently, the expected time till the client is colored red is given by, 75 ± i = C N +1¡r i N X m=r i m X k=r i 1 k¢(N +1¡k) (5.4) Evaluating the above expression, we get, ± i = C N +1 ¢ 1 N +1¡r i ¢[N¢log N r i ¡log(N +1¡r i )] (5.5) Now, the aggregate availability latency (± agg ) with z-relay zebroids is obtained by de¯nition, ± agg (zeb) = T X i=1 [f i ¢ C N +1 ¢ 1 N +1¡r i ¢ (N¢log N r i ¡log(N +1¡r i ))] (5.6) 5.5 Simulation Methodology The simulation environment considered in this study comprises of vehicles such as cars that carry a fraction of the data item repository. A prediction accuracy parameter in- herently provides a certain probabilistic guarantee on the con¯dence of the car route predictions known at the dispatcher. A value of 100% implies that the exact routes of all cars are known at all times. A 70% value for this parameter indicates that the routes predicted for the cars will match the actual ones with probability 0:7. Note that this probability is spread across the car routes for the entire trip duration. We now provide the preliminaries of the simulation study and then describe the parameter settings used in our experiments. Similartotheanalysismethodology,themapusedisa2Dtorus. AMarkovmobility modelrepresentingaunbiased2Drandomwalkonthesurfaceofthetorusdescribes the movement of the cars across this torus. Each grid/cell is a unique state of this Markov chain. In each time slot, every car makes a transition from a cell to any of its neighboring 8 cells. The transition is 76 a function of the current location of the car and a probability transition matrix Q = [q ij ] where q ij is the probability of transition from state i to state j. Only AutoMata equipped cars within the same cell may communicate with each other. The parameters °, ± have been discretized and expressed in terms of the number of time slots. An AutoMata device does not maintain more than one replica of a data item. This is because additional replicas occupy storage without providing bene¯ts. Eitherone-instantaneousorz-relayzebroids maybeemployedperclientrequest for latency improvement. Unless otherwise mentioned, the prediction accuracy parameter is assumed to be 100%. This is because this study aims to quantify the e®ect of a large number of parameters individually on availability latency. Here, we set the size of every data item, S i , to be 1. ® represents the number of storage slots per AutoMata. Each storage slot stores one data item. ° represents the duration of the client's journey in terms of the number of time slots. Hence the possible valuesofavailabilitylatencyarebetween0and°. ± isde¯nedasthenumberoftimeslots afterwhichaclientAutoMatadevicewillencounterareplicaofthedataitemforthe¯rst time. If a replica for the data item requested was encountered by the client in the ¯rst cellthenweset± =0. If± >° thenweset± =° indicatingthatnocopyoftherequested data item was encountered by the client during its entire journey. In all our simulations, for illustration we consider a 5£5 2D-torus with ° set to 10. Our experiments indicate that the trends in the results scale to maps of larger size. We simulated a skewed distribution of access to the T data items that obeys Zipf's law [65] with a mean of 0:27. This distribution is shown to correspond to sale of movie theater tickets in the United States [15]. For the sake of completeness, we provide a brief overview of a Zipf distribution. This means that the frequency of the r th popular data item is inversely proportional to its rank i.e. 77 f i = 1 i v P T j=1 1 j v ; 1<=i<=T (5.7) Here, the exponent v controls the skewness in the popularity distribution of the data items. We denote w = ¡v as the skewness parameter. A higher absolute value of w indicates that most of the popularity weight is spread across the ¯rst few popular titles. NotethatthedataitemrepositorysizeisT andthedenominatorissimplyanormalization constant. Weemployareplicationschemethatallocatesreplicasforadataitemasafunctionof thesquare-rootofthefrequencyofaccessofthatitem. Thesquare-rootreplicationscheme is shown to have competitive latency performance over a large parameter space [19]. The data item replicas are distributed uniformly across the AutoMata devices. This serves as the base-line no-zebroids case. The square-root scheme also provides the initial replica distribution when zebroids are employed. Note that the replacements performed by the zebroids will cause changes to the data item replica distribution. Requests generated as pertheZipfdistributionareissuedoneatatime. Theclientcarthatissuestherequestis chosen in a round-robin manner. After a maximum period of °, the latency encountered by this request is recorded. Initially, all cars are distributed across the map as per the steady-state distribution governed by Q. This initial placement of cars across the map is determined by a random numbergeneratorinitializedwithaseed. Allresultspresentedinthissectionareaverages over 10 such seeds each invoking 20,000 requests. Hence, each point in all the presented results is an average of 200,000 requests. The95%con¯denceintervalsaredeterminedforallsetsofresults. Theseintervalsare quite tight for the metrics of latency and replacement overhead, hence, we only present them for the metric that captures the percentage improvement in latency with respect to the no-zebroids case. 78 5.6 Results In this section, we describe our evaluation results where the following key questions are addressed. With a wide choice of replacement schemes available for a zebroid, what is their e®ect on availability latency? A more central question may be: Do zebroids provide signi¯cantimprovementsinavailabilitylatency? Whatistheassociatedoverheadincurred in employing these zebroids? What happens to these improvements in scenarios where a dispatcher may have imperfect information about the car routes? What inherent trade- o®s exist between car density and storage per car with regards to their combined as well as individual e®ect on availability latency in the presence of zebroids? We present both simple analysis and detailed simulations to provide answers to these questions as well as gain insights into design of carrier-based systems. 5.6.1 Zebroid replacement schemes In this section, we describe how replacement policies employed by zebroids impact avail- abilitylatency. Forillustration,wepresent`scale-up'experimentswhereone-instantaneous zebroids are employed (see Figure 5.1). By scale-up, we mean that ® and N are changed proportionally to keep the total system storage, S T , constant. Here, we set T = 50 and S T = 200. We choose the following values of (N,®) = f(20,10), (25,8), (50,4), (100,2)g. The¯gureindicatesthatarandomreplacementschemeshowsacompetitiveperformance. This is because of several reasons. Recall that the initial replica distribution is set as per the square-root replication scheme. The random replacement scheme does not alter this distribution since it makes replacements blind to the popularity of a data item. However, the replacements cause dynamicdatare-organizationsoastobetterservethecurrentlyactiverequest. Moreover, the mobility pattern of the cars is random, hence, the locations from which the requests are issued by clients are also random and not known a priori at the dispatcher. These ¯ndings are signi¯cant because a random replacement policy can be implemented in a simple decentralized manner. 79 0 20 40 60 80 100 1.5 2 2.5 3 3.5 Number of cars Aggregate availability latency (δ agg ) lru_global lfu_global lru_local lfu_local random Figure5.1: Availabilitylatencywhenemployingone-instantaneouszebroidsasafunction of (N,®) values, when the total storage in the system is kept ¯xed, S T =200. The lru-global and lfu-global schemes provide a latency performance that is worse than random. This is because these policies rapidly develop a preference for the more popular data items thereby creating a larger number of replicas for them. During evic- tion, the more popular data items are almost never selected as a replacement candidate. Consequently, there are fewer replicas for the less popular items. Hence, the initial dis- tribution of the data item replicas changes from square-root to that resembling linear replication. The higher number of replicas for the popular data items provide marginal additional bene¯ts, while the lower number of replicas for the other data items hurts the latency performance of these global policies. The lfu-local and lru-local schemes have similar performance to random since they do not have enough history of local data item requests. We speculate that the performance of these local policies will approach that of their global variants for a large enough history of data item requests. On account of the competitive performance shown by a random policy, for the remainder of the paper, we present the performance of zebroids that employ a random replacement policy. As part of our future work, it remains to be seen if there are other more sophisticated replacement schemes that may have a performance better than random. Moreover, the distribution of replicas seems to havea profound impact on the availability latency in the 80 10 1 10 2 10 3 10 −1 10 0 10 1 10 2 Number of cars no−zebroids anal no−zebroids sim one−instantaneous anal one−instantaneous sim Aggregate Availability latency (δ agg ) 10 1 10 2 10 3 0 10 20 30 40 50 60 70 80 90 100 Number of cars % Improvement in δ agg wrt no−zebroids (ω) analytical upper−bound simulation 5.2.a) ± agg 5.2.b) ! Figure 5.2: Latency performance with one-instantaneous zebroids via simulations along with the analytical approximation for a 10£10 torus with T =10. presence of zebroids. Exploring and analyzing the cumulative e®ect of di®erent replica distributions with zebroids presents a promising future research direction. A concrete goal is the investigation of a replacement scheme that over time converges to a replica distribution resembling that provided by a square-root replication scheme. 5.6.2 Zebroids performance improvement We ¯nd that in many scenarios employing zebroids provides substantial improvements in availability latency. 5.6.2.1 Analysis We ¯rst consider the case of one-instantaneous zebroids. Figure 5.2.a shows the variation in ± agg as a function of N for T =10 and ®=1 with a 10£10 torus using Equation 5.3. Both the x and y axes are drawn to a log-scale. Figure 5.2.b show the % improvement in ± agg obtained with one-instantaneous zebroids. In this case, only the x-axis is drawn to a log-scale. For illustration, we assume that the T data items are requested uniformly. Initially,whenthenetworkissparsetheanalyticalapproximationforimprovementsin latencywithzebroids,obtainedfromEquations5.1and5.3,closelymatchesthesimulation 81 results. However, as N increases, the sparseness assumption for which the analysis is valid, namely N << G, is no longer true. Hence, the two curves rapidly diverge. The point at which the two curves move away from each other corresponds to a value of ± agg · 1. Moreover, as mentioned earlier, the analysis provides an upper bound on the latency improvements, as it treats the newly created replicas given by N c i independently. However, these N c i replicas start from the same cell as one of the server replicas r i . Finally, the analysis captures a one-shot scenario where given an initial data item replica distribution, the availability latency is computed. The new replicas created do not a®ect future requests from the client. Next, we consider the case where z-relay zebroids are employed (see Figure 5.3). Sim- ilar observations, like the one-instantaneous zebroid case, apply since the simulation and analysis curves again start diverging when the analysis assumptions are no longer valid. However, the key observation is that the latency improvement with z-relay zebroids is signi¯cantly better than the one-instantaneous zebroids case, especially for lower stor- age scenarios. This is because in sparse scenarios, the transitive hand-o®s between the zebroids creates higher number of replicas for the requested data item, yielding lower availability latency. Moreover, it is also seen that the simulation validation curve for the improvements in ± agg with z-relay zebroids approaches that of the one-instantaneous zebroid case for higher storage (higher N values). This is because one-instantaneous zebroids are a special case of z-relay zebroids. 5.6.2.2 Simulation Weconductsimulationstoexaminetheentirestoragespectrumobtainedbychangingcar density N or storage per car ® in order to also capture scenarios where the sparseness assumptions for which the analysis is valid do not hold. We separate the e®ect of N and ® by capturing the variation of N while keeping ® constant (case 1) and vice-versa (case 2) both with z-relay and one-instantaneous zebroids. Here, we set the repository size as T = 25. Figure 5.4 and 5.5 respectively capture the two cases mentioned above. With Figure 5.4.b, keeping ® constant, initially increasing car density has higher latency bene¯ts because increasing N introduces more zebroids in the system. As N is further 82 10 1 10 2 10 3 10 −1 10 0 10 1 10 2 Number of cars no−zebroids sim no−zebroids anal z−relays sim z−relays anal Aggregate availability latency (δ agg ) 10 1 10 2 10 3 0 10 20 30 40 50 60 70 80 90 100 Number of cars analytical upper−bound simulation % Improvement in δ agg wrt no−zebroids (ω) 5.3.a) ± agg 5.3.b) ! Figure 5.3: Latency performance with z-relay zebroids via analysis and simulations for a 10£10 torus with T =10. increased, ! reduces because the total storage in the system goes up. Consequently, the number of replicas per data item goes up thereby increasing the number of servers. Hence, the replacement policy cannot ¯nd a zebroid as often to transport the requested data item to the client earlier than any of the servers. On the other hand, the increased number of servers bene¯ts the no-zebroids case in bringing ± agg down. The net e®ect results in reduction in ! for larger values of N. Similar trends are seen by keeping N constant and increasing ® (see Figure 5.5.b). The trends mentioned above are similar to that obtained from the analysis. However, somewhat counter-intuitively with relatively higher system storage, z-relay zebroids pro- vide slightly lower improvements in latency as compared to one-instantaneous zebroids. We speculate that this is due to the di®erent data item replica distributions enforced by them. Note that replacements performed by the zebroids cause °uctuations in these replica distributions which may e®ect future client requests. Exploring other suitable choices of parameters that can capture these changing replica distributions may be a useful future research direction. 83 0 50 100 150 200 250 300 350 400 0 1 2 3 4 5 6 Number of cars Aggregate availability latency (δ agg ) no−zebroids one−instantaneous z−relays 0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 Number of cars % Improvement in δ agg wrt no−zebroids (ω) one−instantaneous z−relays 5.4.a 5.4.b Figure 5.4: Latency performance with both one-instantaneous and z-relay zebroids as a function of the car density when ®=2 and T =25. 5.6.3 Zebroid overhead We ¯nd that the improvements in latency with zebroids are obtained at a minimal re- placement overhead (<1 per client request). 5.6.3.1 Analysis Withone-instantaneouszebroids,foreachclientrequestamaximumofonezebroidisem- ployed for latency improvement. Hence, the replacement overhead per client request can amounttoamaximumofone. Recallthattocalculatethelatencywithone-instantaneous zebroids, N c i new replicas are created in the same cell as the servers. Now a replacement is only incurred if one of these N c i newly created replicas meets the client earlier than any of the r i servers. Let X r i and X N c i respectively be random variables that capture the minimum time till any of the r i and N c i replicas meet the client. Since X r i and X N c i are assumed to be independent, by the property of exponentially distributed random variables we have, Overhead=request=1¢P(X N c i <X r i )+0¢P(X r i ·X N c i ) (5.8) 84 0 2 4 6 8 10 0 1 2 3 4 5 6 Storage per car no−zebroids one−instantaneous z−relays Aggregate availability latency (δ agg ) 0 2 4 6 8 10 0 10 20 30 40 50 60 Storage per car % Improvement in δ agg wrt no−zebroids (ω) one−instantaneous z−relays 5.5.a 5.5.b Figure 5.5: Latency performance with both one-instantaneous and z-relay zebroids as a function of ® when N =50 and T =25. Overhead=request= r i C r i C + N c i C = r i r i +N c i (5.9) Recallthatthe number ofreplicasfor dataitem i, r i , is afunction of the totalstorage in the system i.e., r i = k¢N¢® where k satis¯es the constraint 1· r i · N. Using this along with Equation 5.1, we get Overhead=request=1¡ G G+N¢(1¡k¢®) (5.10) Now if we keep the total system storage N ¢ ® constant since G and T are also constant, increasing N increases the replacement overhead. However, if N¢® is constant then increasing N causes ® to go down. This implies that a higher replacement overhead is incurred for higher N and lower ® values. Moreover, when r i = N, this means that every car has a replica of data item i. Hence, no zebroids are employed when this item is requested,yieldinganoverhead/requestforthisitemaszero. Next,wepresentsimulation results that validate our analysis hypothesis for the overhead associated with deployment of one-instantaneous zebroids. 85 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of cars one−instantaneous zebroids Average number of replacements per request (N=20,α=10) (N=25,α=8) (N=50,α=4) (N=100,α=2) Figure 5.6: Replacement overhead when employing one-instantaneous zebroids as a func- tion of (N,®) values, when the total storage in the system is kept ¯xed, S T =200. 5.6.3.2 Simulation Figure 5.6 shows the replacement overhead with one-instantaneous zebroids when (N,®) are varied while keeping the total system storage constant. The trends shown by the simulation are in agreement with those predicted by the analysis above. However, the total system storage can be changed either by varying car density (N) or storage per car (®). Figures 5.7.a and Figure 5.7.b respectively indicate the replacement overhead incurred with both one-instantaneous and z-relay zebroids when ® is kept constant and N is varied and vice-versa. We present an intuitive argument for the behavior of the per-request replacement overhead curves. When the storage is extremely scarce so that only one replica per data item exists in the AutoMata network, the number of replacements performed by the zebroids is zero since any replacement will cause a data item to be lost from the system. The dispatcher ensures that no data item is lost from the system. At the other end of the spectrum, if storage becomes so abundant that ® = T then the entire data item repository can be replicated on every car. The number of replacements is again zero since each request can be satis¯ed locally. A similar scenario occurs if N is increased 86 0 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of cars z−relays one−instantaneous Average number of replacements per request 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Storage per car Average number of replacements per request z−relays one−instantaneous 5.7.a 5.7.b Figure 5.7: Replacement overhead with zebroids for the cases when N is varied keeping ®=2 (¯gure 5.7.a) and ® is varied keeping N =50 (¯gure 5.7.b). to such a large value that another car with the requested data item is always available in the vicinity of the client. However, there is a storage spectrum in the middle where replacements by the scheduled zebroids result in improvements in ± agg (see Figures 5.4.b and 5.5.b). Moreover, we observe that for sparse storage scenarios, the higher improvements with z-relay zebroids are obtained at the cost of a higher replacement overhead when com- pared to the one-instantaneous zebroids case. This is because in the former case, each of these z zebroids selected along the lowest latency path to the client needs to perform a replacement. However, the replacement overhead is still less than 1 indicating that on an average less than one replacement per client request is needed even when z-relay zebroids are employed. Note that the average replacement per request metric does not explicitly capture the bandwidthoverheadassociatedwiththetransferofitemstothezebroids. Thisbandwidth overhead may be signi¯cant in the case of multiple simultaneous active requests. We intend to explicitly incorporate these bandwidth considerations in our model as part of our future research. 87 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 4 Prediction percentage no−zebroids (N=50) one−instantaneous (N=50) z−relays (N=50) no−zebroids (N=200) one−instantaneous (N=200) z−relays (N=200) Aggregate Availability Latency (δ agg ) Figure 5.8: Availability latency, ± agg , for di®erent car densities as a function of the prediction accuracy metric with ®=2 and T =25. 5.6.4 Zebroids with inaccurate route predictability We ¯nd that zebroids continue to provide improvements in availability latency even with lower accuracy in the car route predictions. We use a single parameter p to quantify the accuracy of the car route predictions. This parameter inherently provides a certain probabilistic guarantee on the con¯dence of the car route predictions for the entire trip duration. 5.6.4.1 Analysis Since p represents the probability that a car route predicted by the dispatcher matches the actual one, hence, the latency with zebroids can be approximated by, ± err agg =p¢± agg (zeb)+(1¡p)¢± agg (no¡zeb) (5.11) ± err agg =p¢± agg (zeb)+(1¡p)¢ C r i (5.12) 88 Expressions for ± agg (zeb) can be obtained from Equations 5.3 (one-instantaneous) or 5.6 (z-relay zebroids). 5.6.4.2 Simulation Figure 5.8 shows the variation in ± agg as a function of this route prediction accuracy metric. We observe a smooth reduction in the improvement in ± agg as the prediction accuracy metric reduces. For zebroids that are scheduled but fail to rendezvous with the client due to the prediction error, we tag any such replacements made by the zebroids as failed. It is seen that failed replacements gradually increase as the prediction accuracy reduces. In this study, we have considered a metric that probabilistically governs errors in the car route predictions. Another possible choice for the metric is similar to that used by Jun et al. [36] where the car routes are assumed to follow a Gaussian distribution de¯ned by a mean and a variance. The estimates about the mean and the variance can be built at the dispatcher based on the history of the individual car movements. Exploring such alternate choices of prediction control metrics presents a promising future research direction. 5.6.5 Maximum improvement with zebroids Surprisingly, we ¯nd that the improvements in latency obtained with one-instantaneous zebroids are independent of the input distribution of the popularity of the data items. 5.6.5.1 Analysis The fractional di®erence (labelled !) in the latency between the no-zebroids and one- instantaneous zebroids is obtained from equations 5.1, 5.2, and 5.3 as ! = P T i=1 f i ¢C r i ¡ P T i=1 f i ¢C r i +(N¡r i )¢(1¡(1¡ 1 G ) r i ) P T i=1 f i ¢C r i (5.13) Here C = c¢G¢logG. This captures the fractional improvement in the availability latency obtained by employing one-instantaneous zebroids. Let ® = 1, making the total 89 storage in the system S T = N. Assuming the initial replica distribution is as per the square-root replication scheme, we have, r i = p f i ¢N P T j=1 p f j . Hence, we get f i = K 2 ¢r 2 i N 2 , where K = P T j=1 p f j . Using this, along with the approximation (1¡x) n '1¡n¢x for small x, we simplify the above equation to get, ! =1¡ P T i=1 r i 1+ N¡r i G P T i=1 r i (5.14) Inordertodeterminewhenthegainswithone-instantaneouszebroidsaremaximized, we can frame an optimization problem as follows: Maximize !, subject to T X i=1 r i =S T (5.15) Theorem 2. With a square-root replication scheme, improvements obtained with one- instantaneous zebroids are maximized with a uniform input popularity distribution of the data items. Proof. Using the Lagrangian multipliers method, the optimization can be expressed as: Max ( 1¡ T X i=1 G¢r i N¢(G+N¡r i ) +¸¢ " T X i=1 r i ¡N #) (5.16) We solve for r i to obtain: r i =G+N¡G¢ r G+N G¢N¢¸ (5.17) Note that in Equation 5.17, while r i is independent of i, it is the same for all titles i and is given by N T . It can be veri¯ed that the maximum ! occurs at this value of r i since @ 2 ! @r 2 i <0. This implies that f i = 1 T , in other words, all the data items must have the same popularity. 5.6.5.2 Simulation We perform simulations with two di®erent frequency distribution of data items: Uni- form and Zipf (with mean=0.27). Similar latency improvements with one-instantaneous 90 0 20 40 60 80 100 0 10 20 30 40 50 60 Number of cars % Improvement in δ agg wrt no−zebroids (ω) one−instantaneous zebroids (N=20,α=10) (N=25,α=8) (N=50,α=4) (N=100,α=2) Figure 5.9: Improvement in availability latency with one-instantaneous zebroids as a function of (N,®) values, when the total storage in the system is kept ¯xed, S T =200. zebroids are obtained in both cases. This result has important implications. In cases with biased popularity toward certain data items, the aggregate improvements in latency acrossalldataitemrequestsstillremainthesame. Eveninscenarioswherethefrequency of access to the data items changes dynamically, zebroids will continue to provide similar latency improvements. 5.6.6 Zebroid trade-o®s with car density and storage per car Our ¯ndings indicate that higher latency improvements can be obtained with zebroids when there are more cars with lower storage than fewer ones with higher storage. 5.6.6.1 Analysis Consider the case of one-instantaneous zebroids. The fractional di®erence (labelled !) in ± agg between the no-zebroids and one-instantaneous zebroids cases is obtained in Equa- tion 5.13. Using the approximation (1¡x) n '1¡n¢x for small x, we simplify the above equation to get, 91 ! =1¡ P T i=1 G¢f i r i ¢(G+N¡r i ) P T i=1 f i r i (5.18) Recallthatthe number ofreplicasfor dataitem i, r i , is afunction of the totalstorage inthesystemi.e.,r i =k¢N¢®wherekhastosatisfytheconstraintthat1·r i ·N. Given a total system storage N¢®, we ¯nd that except for N all other terms in Equation 5.18 are constant. Also, with increasing N, ! increases. However, for a constant N¢®, if N increases,®hastoreduce. Thisindicatesforagivensystemstorage,higherimprovements in latency with zebroids are obtained with higher car density and lower storage per car. 5.6.6.2 Simulation Figure 5.9 validates the insight obtained from the analysis in that the improvements in latency go up with higher N and lower ® values when N ¢® = 200. The increase in N increases the zebroid densityenabling the dispatcherto almost always¯nd a zebroid that can deliver the requested data item to the client earlier than any of the potential servers. This trade-o® between the two system parameters of number of cars and storage per car may have important implications in the design of carrier-based networks that improve availability latency. Although we have assumed a constant storage per car for all cars, in practical scenar- ios,theremaybevariablestoragepercar. PartoftheAutoMatadevicestoragespacemay be reserved by a user for his preferred titles which the user may not seek to erase/evict. Theseconsiderationsmaycreateamoreheterogeneousenvironmentwithvariablestorage per car. In addition, zebroids may need to take into account user preferences prior to makingthesereplacements. Incorporatingalltheseconsiderationsintothemodelislikely to pay richer dividends in estimating latency in real deployments. 5.6.7 Impact of di®erent trip durations and repository sizes In this section, we describe the impact of di®erent trip durations and repository sizes on availability latency in the presence of zebroids. 92 0 20 40 60 80 100 0 5 10 15 20 25 30 35 40 45 Replicas per data item % Improvement in δ agg wrt no−zebroids (ω) γ=5 γ=20 γ=10 0 20 40 60 80 100 0 5 10 15 20 25 30 35 40 45 Replicas per data item % Improvement in δ agg wrt no−zebroids (ω) γ=5 γ=10 γ=20 5.10.a) Analysis 5.10.b) Simulation Figure 5.10: Improvement in ± agg with one-instantaneous zebroids for di®erent client trip durations in case of 10£10 torus with a ¯xed car density, N = 100. 5.6.7.1 Analysis In most practical scenarios, the client will wait for a maximum duration within which it expectsitsissuedrequesttobesatis¯ed. Here,weconsiderthecasewheretheclienthasa ¯nite trip duration °, similar to that considered in the simulation environment (° =10). Theavailabilitylatency, ± i ,canbeanyvaluebetween0and°¡1. Iftheclient'srequestis not satis¯ed, weset ± i =° indicatingthat the client'srequest for item i wasnot satis¯ed. Recall that latency in the case of a 2D-random walk on a torus can be modelled as an exponential distribution as: P(± i >t)=¸¢exp(¡¸¢t) (5.19) where ¸ = r i c¢G¢logG . The average availability latency with ¯nite trip duration ° is then given by, ± i = Z ° 0 x¢¸¢exp(¡¸¢t)dx+ Z 1 ° °¢¸¢exp(¡¸¢t)dx (5.20) Hence, we get 93 0 50 100 150 200 0 10 20 30 40 50 60 70 Number of cars % Improvement in δ agg wrt no−zebroids (ω) T=5 T=10 T=20 T=25 T=50 Figure 5.11: Shows improvement in availability latency as a function of the car density for di®erent repository sizes with ®=2 and ° =10. ± i = c¢G¢logG r i ¢[1¡exp( ¡°¢r i c¢G¢logG )] (5.21) The aggregate availability latency with ¯nite trip duration is then given by, ± agg (no¡zeb)= T X i=1 f i ¢ c¢G¢logG r i ¢[1¡exp( ¡°¢r i c¢G¢logG )] (5.22) In the presence of one-instantaneous zebroids, the aggregate availability latency can be obtained using a procedure similar to that used in Section 5.4.1, giving ± agg (zeb)= T X i=1 f i ¢ c¢G¢logG (r i +N c i ) ¢[1¡exp( ¡°¢(r i +N c i ) c¢G¢logG )] (5.23) The above equations yield the improvements in latency with ¯nite trip duration. We consider a 10£10 torus with N = 100 cars each with one storage slot (® = 1). The distribution of data item replicas is assumed to be uniform. Hence, as we increase the size of the data item repository (T) the number of replicas per data item decreases. Speci¯cally, the value of T is varied as f1;2;4;10;20;25;50;100g. Then, the number of replicas per data item changes as f100;50;25;10;5;4;2;1g. Figure 5.10.a and 5.10.b 94 capture the latency performance obtained via analysis and simulations respectively for di®erent trip durations. We now describe the behavior of the curves for a given ¯nite trip duration °. When T = 1, every car has a copy of the item, hence, no car can serve as a zebroid. As T increases, replicas per item go down. Hence, some cars can potentially serve as zebroids, providing some improvements in availability latency. As T is further increased, a peak is reached beyond which the improvements start diminishing. This is because if T is large then the number of server replicas per item becomes small. Hence, the likelihood of ¯nding another car in the vicinity of such a server that will meet the client earlier also reduces. Moreover, as ° increases, peak improvements in latency are obtained with higher T. 5.6.7.2 Simulation Here, we present simulation results that capture the e®ect of di®erent repository sizes on the availability latency when the trip duration ° = 10 (see Figure 5.11). For a ¯xed storagepercar, su±cientcardensityisneededtoprovidehigherimprovementsinlatency for a given repository size. This implies that from a system designer's point of view, if an estimate of the total car density is known, then su±cient gains in latency with zebroids can be realized by adjusting the repository size of titles presented to the users. While a homogeneous repository of data items has been assumed throughout this study, sizes of data items such as audio clips are typically smaller than video clips. One way in which our model can be extended to consider such a heterogenous repository is to assume that every data item can be divided into a set of constant-sized blocks. Di®erent blocks of an item may be stored across di®erent cars. During data delivery, zebroids will be scheduled to ensure timely delivery of the various blocks of a requested data item to a client. Next, we validate a subset of the observations with zebroids with traces collected from a small vehicular test-bed. 95 5.7 Evaluation with a real map In this section, we describe the performance improvements obtained with zebroids in a scenario where the vehicle movements are dictated by an underlying map of the San Francisco Bay Area. The details about how the Markov model was derived from the underlying map are described in Chapter 4, Section 4.6. Below we describe brie°y the experimentalset-upandcorrespondingresultsobtainedwithzebroidswhensuchaMarkov model is employed. 5.7.1 Results with zebroids In this section, we present some representative results for the improvements in latency that are obtained with both one-instantaneous and z-relay zebroids when employing the Markov mobility model previously derived from a map of the San Francisco Bayarea. As before, requests are issued, one at a time at each time-step at vehicles in a round-robin manner, as per a Zipf distribution with a mean of 0:27. At a time only one request is active, each request is active for a maximum of client trip duration number of steps (set as ° =10). In the following, we describe brie°y the experiment set-up followed by a brief description of the main observations. The total storage in the system is held constant as S T = 200. The values of (N,®) arevariedasf(20,10),(25,8),(50,4),(100,2),(200,1)gtorealizethisvalueofS T . For data item repository size T set as 25, client trip duration, °, set as 10, the latency performance with zebroids is studied as a function of the di®erent (N,®) values (see Figure 5.12). Similar trends are obtained as were seen in Section 5.6.6. In other words, having more cars with lower storage provides higher latency improvements as opposed to having fewer cars with higher storage. For data item repository size T set as 25, client trip duration, °, set as 10, storage per car, ®, set as 2, the latency performance with zebroids is studied as a function of increasing car density N (see Figure 5.13). 96 0 50 100 150 200 0 1 2 3 4 5 6 7 8 9 Number of cars Aggregate Availability Latency (δ agg ) no−zebroids one−instantaneous z−relays 0 50 100 150 200 −5 0 5 10 15 20 25 30 35 40 45 50 Number of cars % Improvement in δ agg wrt no−zebroids one−instantaneous z−relays (a) (b) Figure 5.12: Performance with zebroids as a function of di®erent (N,®) values when the total storage in the system is held constant at S T = 200, ° = 10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. 0 100 200 300 400 500 0 1 2 3 4 5 6 7 8 9 Number of cars Aggregate Availability Latency (δ agg ) no−zebroids one−instantaneous z−relays 0 100 200 300 400 500 0 5 10 15 20 25 30 35 40 45 50 Number of cars % Improvement in δ agg wrt no−zebroids one−instantaneous z−relays (a) (b) Figure 5.13: Performance with zebroids as a function of car density when T =25, ®=2, and ° =10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. 97 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 Storage per car Aggregate Availability Latency (δ agg ) no−zebroids one−instantaneous z−relays 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 20 Storage per car % Improvement in δ agg wrt no−zebroids one−instantaneous z−relays (a) (b) Figure 5.14: Performance with zebroids as a function of storage per car when T = 25, N = 50, and ° = 10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. For data item repository size T set as 25, client trip duration, °, set as 10, car density, N, setas50, thelatencyperformancewithzebroidsisstudiedasafunction ofincreasingstoragepercar®(seeFigure5.14). AsseeninSection5.6.2,weobserve that there exists a storage density range that provides the highest improvements with zebroids. For car density, N, set as 50, client trip duration, °, set as 10, storage per car, ®, setas2,thelatencyperformancewithzebroidsisstudiedasafunctionofincreasing data item repository size T (see Figure 5.15). Again, the trends seen are similar to that observed in Section 5.6.7. For a small data item repository size, employing zebroids has almost no bene¯ts over simply using the square-root static replication scheme. However, as the data item repository size increases, for the same total storage in the system (S T = N ¢®), the replicas per data item reduces, thereby increasing the utility of zebroids. However, if the data item repository is too large then the utility of zebroids is reduced because of the constraint that at least one replica of every data item must be present in the system at all times. Moreover, 98 0 10 20 30 40 50 0 1 2 3 4 5 6 7 8 9 Number of titles Aggregate Availability Latency (δ agg ) no−zebroids one−instantaneous z−relays 0 10 20 30 40 50 0 2 4 6 8 10 12 14 16 18 20 Number of titles % Improvement in δ agg wrt no−zebroids one−instantaneous z−relays (a) (b) Figure 5.15: Performance with zebroids as a function of data item repository size when N =50, ®=2, and ° =10. Figure (b) shows the percentage improvement with zebroids when compared to the no-zebroids case. the replicas per data item are further reduced thereby reducing the probability of ¯nding a lower latency path from any server to the requesting client. In all cases, it is seen that z-relay zebroids provide a higher latency improvement as compared to one-instantaneous zebroids albeit at a higher replacement overhead. Again, these results suggest that the uniform probability transition matrix based Markov model may be a good indicator of the performance that may be seen with a model derived from real maps. 5.8 Evaluation with Real Traces In this section, we evaluate the performance of employing zebroids using traces obtained from the UMassDieselNet bus-based DTN test-bed [11]. The details of the test-bed and the properties of the traces have been explained in Chapter Section. Next, we describe the simulation set-up for the evaluation of zebroids followed by a brief explanation of the main results. 99 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Title Repository Size Aggregate Availability Latency Zipf=−0.500000 linear−no−carriers sqrt−no−carriers random−no−carriers linear−with−carriers sqrt−with−carriers random−with−carriers 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Title Repository Size Aggregate Availability Latency Zipf=−1.000000 linear−no−carriers sqrt−no−carriers random−no−carriers linear−with−carriers sqrt−with−carriers random−with−carriers 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Title Repository Size Aggregate Availability Latency Zipf=−1.500000 linear−no−carriers sqrt−no−carriers random−no−carriers linear−with−carriers sqrt−with−carriers random−with−carriers 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 7000 Title Repository Size Aggregate Availability Latency Zipf=−2.000000 linear−no−carriers sqrt−no−carriers random−no−carriers linear−with−carriers sqrt−with−carriers random−with−carriers Figure 5.16: Aggregate Availability Latency for Linear, Sqrt, and Random Replication Schemes for Zipf values -0.5, -1.0, -1.5, and -2.0 100 5.8.1 Experimental Set-up The UMassDieselNet traces represent the movements of buses during a particular day. There is no correlation between trace movements across days. Hence, we process each trace at a time and then average the results observed across all the days noting that the average is indicative of the performance seen on most days. However, certain days do appear as outliers since the number of active buses di®ers from day-to-day. As before we consider a ¯nite data item repository of size T. Each bus is assumed to carry ® storage slots. Initially, replicas for each data item are determined based on a square-root replicationschemethen allocatedacrossthe busesuniformlyatrandom. The constraint is that at least one copy of every data item must be present in the network at all times. Requests for data items are generated as per a Zipf distribution. Weconsider 2 di®erentscenarios,(a)requestsareissuedforeachdataitemateachbusatthestartofthe day (see Section 5.8.2.1) (b) requests for the data items are issued at equal inter-arrival times (see Section 5.8.2.2). 5.8.2 Results In this section, we brie°y describe the main results from evaluation of the performance of zebroids using the UMassDieselNet traces. 5.8.2.1 Requests issued at the start of the day As a base-line result for comparison, we ¯rst consider an optimistic scenario to evaluate the maximum improvements that can obtained with zebroids. Requests for each data item are issued at each bus at the start of the day. For each request, we consider the best possible latency that can be obtained by employing zebroids. The scheduled relay teams of zebroids for the di®erent requests do not interfere with each other. In other words, if there is a lower latency path from a server (bus with the requested data item) to the requesting client via other buses then storage constraints within the buses will not prevent the relay team of z zebroids to deliver the item to the client. We consider three di®erent environments for the case without zebroids where the replicas for the data item are allocated as per the linear, square-root, and random replication schemes. Starting 101 from these three initial distributions, we employ zebroids to study the improvement in the aggregate availability latency. If a request cannot be satis¯ed then it is tagged with a maximum trip duration. This maximum trip duration is calculated on the basis of the duration of the traces across the 52 days. Hence, we only consider one metric namely the aggregate availability latency across all the requests. Figure 5.16 shows the latency performance with the di®erent replication schemes compared with the respective cases when zebroids are employed for 4 di®erent data item popularity distributions. The values of (T,®) are varied as f(5,1), (10,2), (15,3), (20,4), (25,5) g. In all cases, signi¯cant improvements in latency are obtained by employing zebroids. The improvements with the random replication scheme with and without ze- broidsareindependentofthespeci¯c(T,®)value. Thisisbecausetherandomreplication scheme allocates replicas blind to the popularity of the data items. The linear replication scheme provides the lowest latency, because of the ¯nite trip duration (see Chapter), the improvements are more pronounced with skewed popularity distributions and larger data item repositories. 5.8.2.2 Requests issued at equal inter-arrival times during the day Here, requests are generated as per a Zipf distribution with an exponent w = ¡0:73. The duration during which the buses were active during a day is determined apriori and subject to this duration requests are issued at equal inter-arrival times. A generated request is assigned to a bus chosen uniformly at random. A request is assumed to be satis¯ed either if the data item requested is locally stored or another bus carrying the requested item is encountered at some point after the request is issued. Those requests that are not satis¯ed at the end of the day are tagged as unsatis¯ed requests. Hence, we consider two separate metrics (i) Average availability latency for satis¯ed requests (ii) Normalized unsatis¯ed request rate. For each request, a relay-team of z buses may be employed to improve its availability latency. The z-relay zebroids are scheduled on the basis of the state of the network at the time that the request is issued. Note that this relay team of buses are scheduled across space and time. As per the schedule, the time when one of the buses is supposed 102 to hand over a copy of the requested item to another one, the recipient may not have any free slots to perform the transfer. This is because the slots may all be occupied by items reserved for previously scheduled requests. When the z-relay team of zebroids is scheduled, the dispatcher only takes into account the spatio-temporal rendezvous of the buses with the client without regards to the content that the buses carry. This is because the content of the buses will change as other requests are issued into the system. Hence,noteveryscheduledz-relaytransfermaybesuccessful. Ifeveryzebroidinaz-relay schedule is able to carry the requested item then the item will be successfully delivered to the client yielding lower availability latency. Hence, we consider two other metrics: numberofscheduledzebroidsandnumberofzebroidsthatwereactuallyabletocarrythe data item. As before the metrics obtained with zebroids are compared with a scenario without zebroids (no-zebroids case). For the ¯rst set of experiments we vary the values of (T,®) as f(5,1), (10,2), (15,3), (20,4), (25,5), (30,6)g, see Figure 5.17. The latency for satis¯ed requests with zebroids is about 15¡20% lower than the no-zebroids scheme while the normalized unsatis¯ed requests with and without zebroids are the same. The replacement overhead for zebroids is depicted in Figure 5.18. While the zebroids scheduled for all the (T,®) values is the same, a higher percentage of these scheduled zebroids are employed for higher data item repository sizes. Figure 5.19 shows the performance with zebroids when the data item repository size is ¯xed at 10 and the storage per bus is increased. Increase in storage leads to increase in the replicas per data item, hence, as expected the latency with the both cases, with and without zebroids, reduces. Similarly, the normalized unsatis¯ed requests go down. Initially, the latency with zebroids is lower than the no-zebroids case. But as the storage is increased, the no-zebroids case starts outperforming the case with zebroids. This is becauseemployingzebroidsresultsinchangesinthedistributionofthedataitemreplicas. Because of the ¯nite duration of the traces, a steady-state for this distribution is never reached. So the changes in the number of data item replicas caused by replacements for the earlier requests have a detrimental e®ect on the latency for the future requests. With the no-zebroids case, the data item replicas are allocated as per the square-root 103 0 5 10 15 20 25 30 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Title Repository Size (T) Availability Latency for Satisfied Requests (secs) no−zebroids z−relays 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 Title Repository Size (T) Unsatisfied Request Metric no−zebroids z−relays Figure 5.17: Aggregate availability latency and normalized unsatis¯ed requests with ze- broids for the case when the ratio of T :® is maintained as 5 : 1 and requests are issued as per a Zipf distribution at equal inter-arrival times. 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Title Repository Size (T) Normalized Replacement Overhead Scheduled Used Figure 5.18: Replacement overhead incurred by employing zebroids when the ratio of T : ® is maintained as 5 : 1 and requests are issued as per a Zipf distribution at equal inter-arrival times. 104 replication scheme and this distribution does not change. This provides a better latency performance on an average. Similar trends are seen in the unsatis¯ed requests metric beyond a storage per bus value of 2. As more data item replicas are introduced into the system, lesser number of requests canbene¯tfromemployingzebroids. ThisiscapturedinFigure5.19(d)wheretheaverage number of zebroids scheduled per request reduces with increase in the storage per bus. Recall that for a given request, a relay team of z zebroids is only scheduled if it provides a lower latency than that provided by the no-zebroids case. Figure 5.20 shows the performance with zebroids when the storage per bus is ¯xed at 3 and the data item repository size is increased. As the data item repository size is increased, lesser replicas are allocated per data item resulting in an increase in the latencyaswellastheunsatis¯edrequests. Thisisseenbothinthecaseswithandwithout zebroids. Moreover,theincreaseinlatencyforsatis¯edrequestswithoutzebroidsismuch sharper, signi¯cant latency improvements can be obtained by employing zebroids with larger repositories (see 5.20(b)). This can be seen by the higher number of zebroids scheduled with a larger repository in Figure 5.20(d). However, if the repository size is verylargethenbecauseoftheconstraintthatatleastonecopyofeverydataitemmustbe present in the network at all times, very few buses can be employed as zebroids, yielding a similar latency for the cases with and without zebroids. 5.9 Summary In this chapter, we examined the improvements in latency that can be obtained in the presence of data carriers, termed zebroids, that deliver a data item from a server to a client. We quanti¯ed the variation in availability latency as a function of a rich set of pa- rameters such as car density, storage per car, data item repository size, and replacement policies employed by zebroids. Our key ¯ndings are as follows. A naive random replace- ment policy employed by the zebroids exhibits competitive latency bene¯ts at a minimal replacement overhead. Zebroids continue to provide improvements even in the presence of lower accuracy in the predictions of the car routes. Improvements in latency obtained 105 0 1 2 3 4 5 6 0 2000 4000 6000 8000 10000 12000 Storage per bus (α) Availability Latency for Satisfied Requests (secs) no−zebroids z−relays 0 1 2 3 4 5 6 −25 −20 −15 −10 −5 0 5 10 15 20 25 Storage per bus (α) % wrt no−zebroids (a) Latency for satis¯ed requests (b) Percentage wrt no-zebroids 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 Storage per bus (α) Unsatisfied Request Metric no−zebroids z−relays 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Storage per bus (α) Normalized Replacement Overhead Scheduled Used (c) Unsatis¯ed requests (d) Replacement overhead Figure5.19: Performancewithzebroidsasafunctionofthestoragepercarwhenthedata item repository size is held constant at 10. Requests are issued as per a Zipf distribution at equal inter-arrival times. 106 0 5 10 15 20 25 30 0 2000 4000 6000 8000 10000 12000 Title Repository Size (T) Availability Latency for Satisfied Requests (secs) no−zebroids z−relays 0 5 10 15 20 25 30 −20 −15 −10 −5 0 5 10 15 20 25 30 Title Repository Size (T) % wrt no−zebroids (a) Latency for satis¯ed requests (b) Percentage wrt no-zebroids 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Title Repository Size (T) Unsatisfied Request Metric no−zebroids z−relays 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Title Repository Size (T) Normalized Replacement Overhead Scheduled Used (c) Unsatis¯ed requests (d) Replacement overhead Figure 5.20: Performance with zebroids as a function of the data item repository size when the storage per car is held constant at 3. Again, requests are issued as per a Zipf distribution at equal inter-arrival times. 107 withone-instantaneouszebroidsaremaximizedwithauniforminputpopularitydistribu- tion of the data items. Also, for a given total system storage, presence of more cars with a low storage capacity yields higher improvements in latency when compared with fewer cars with a high storage capacity. The conclusions seem to be robust to the fact that the Markov mobility model may be derived from a underlying real city map. Finally, similar observations as obtained with the Markov model are realized from employing zebroids on traces collected from small vehicular test-bed (UMassDieselNet). Extensions to zebroids that consider data items of larger size remains to be inves- tigated. The larger data items may be divided into chunks and multiple relay team of zebroidsmaybescheduledtoensurethateachchunkreachestheclientattheappropriate time. Incorporation of explicit bandwidth constraints into the Markov model so that it can be better equipped for scenarios where multiple simultaneous requests are active in thesystemalsoremainsasubjectoffutureinvestigation. Also, zebroidsmayalsobeused for delivery of data items that carry delay sensitive information with a certain expiry. Extensions to zebroids that satisfy such application requirements present an interesting future research direction. 108 Chapter 6 Related Work Inthischapter,weprovideabriefoverviewoftherelatedworkinthearea. We¯rstpresent adescriptionofvarioussystemcomponentsthatwillbeapartofanoperationalAutoMata application. Next, weprovideadescriptionofthevariousstudiesonreplicationinmobile ad-hoc networks. Subsequently, we present related works in the area of delay tolerant networks where data carriers like zebroids have been employed. Finally, we conclude with a concise account of some related studies in the area of Intelligent Transportation Systems (ITS) where our results may be directly applicable. 6.1 Other Components of an AutoMata Application The graphical user interface that displays the list of available titles with their associated latency may be complimented by a number of other components that are needed for realizing an AutoMata application. Below, we describe some typical components that may be part of such a system. Note that the related works described in this subsection are beyond the scope of this thesis, we present them here for providing a uni¯ed picture. Even though these components have been described in the introduction, we relist them here for completeness. Once a user initiates display of a data item, an admission control component [18] ensures availability of both resources and the referenced data. Next, a data delivery scheduling technique [20] utilizes resources as a function of time to deliver the data item toarequestingAutoMatadevice. Thiscomponent,mayswitchbetweenseveralcandidate 109 servers containing the referenced data item based on their proximity, current availability ofresources,andnetworkconditions. Thiscomponentistiedcloselytoanad-hocnetwork routing protocol which facilitates delivery of data between AutoMata devices. Example protocolsareDSR[34],DSDV[47],AODV[48]tonameafew. Anothersystemcomponent may monitor whether the system is providing a target AutoMata with the desired QoS and make adjustments as necessary. Besides the above components, there may be others responsibleforaddressingthesecurity[64]andprivacy[16]concernsoftheuserthatmay be mandatory for practical use of the system. Additionally, suitable physical and MAC layer optimizations may be needed to adapt to the wireless nature of the communication medium between the vehicles. 6.2 Replication in MANETs Replication in MANETs has been explored in a wide variety of contexts. Several studies predict partitions in the network and pre-emptively replicate data to ensure continuous data availability. Li et al. [40, 38] use a group mobility model to predict the movement of the nodes and replicate pre-emptively to ensure existence of at least one server per group for continuous data availability. Our study is di®erent in that not only does it use a di®erent mobility model which is Markovian in nature, but the metric we seek to optimize is the latency until the ¯rst encounter of a replica of a requested data item. Our mobility model is more °exible because by suitably adjusting the transition probabilities between the various cells in the map it can approximate other mobility models such as highway, manhattan, random walk mobility model [4] and others. Moreover, we consider storage constraints per node and how available storage can be best utilized to improve availability latency. Storage aspects have not been considered in the studies mentioned above. Hara [23] proposes three replica allocation methods, one that allocates replicas to nodes only on the basis of their local preference to data items, another that additionally considers the contents of the connected neighbors while performing the allocation to re- move some redundancy and ¯nally, one that discovers bi-connected components in the networktopologyandallocatesreplicasaccordingly. Thefrequencyofaccessofthenodes 110 to data items is known in advance and does not change. Moreover, the replica alloca- tion takes place periodically in a speci¯c period termed the relocation period. Several extensions to this work have been proposed where replica allocation methods have been extendedtoconsiderdataitemswithperiodic[25,27]andaperiodic[30,26]updates. Fur- ther extensions to the proposed replica allocation methods consider the stability of radio links [32], topology changes [33] and location history of the data item access log [28, 29]. In [31], the authors consider data items that are correlated by virtue of being requested simultaneously and present the performance of the replica allocation methods proposed earlier. All the above studies are based on simulations of the proposed replica allocation methods. Our study di®ers from the above studies based on the initial proposals by Hara et al. [23] in the following ways. The above studies use a heuristic approach where the methods are evaluated via simulations. On the other hand, in our study, we propose an optimization formulation for the optimal replication strategy that minimizes aggregate availability latency. We analytically compute the optimal replication for sparse density scenarios and employ extensive simulations to study dense cases. Secondly, the above studies consider the number of requests satis¯ed as a fraction of the total requests as the prime metric of interest without regards to the latency per request. In our studies, we notonlystrivetoincreasethenumberofsatis¯edrequestsbutalsostrivetominimizethe availabilitylatencyperrequest. Thirdly, wedonotassumethepresenceofanyrelocation period during which replica allocation takes place. Related studies [51, 61, 24] have appeared in the context of cooperative caching in mobile ad-hoc networks where the objective is to use caching to reduce the mobile node's latency in accessing data items. The proposed techniques take into account the request pattern of the nodes and the topology of the network as the nodes move while making a caching decision. In our studies, nodes (cars) do not perform any caching. We simply calculate the number of replicas per data item that will minimize availability latency. These replicas are then placed across the cars uniformly at random, hence, car's local storage is fully utilized. 111 However, after the static replication schemes have allocated the replicas across the cars, we have experimented with zebroids that perform dynamic data reorganization to better equip the system storage to the active user requests. Since the zebroids are cars whose local storage is fully utilized, we use di®erent cache replacement policies that yield di®erent replica distributions over time. We are not aware of any other work that employsdynamicdatareorganization,acrossadistributedstorageenvironmentcomprised by mobile vehicles, to improve user latency. 6.3 Sparse Network Architectures Recently, several novel and important studies such as ZebraNet [35], DakNet [46], Data Mules [53], Message Ferries [63], SWIM [54], and Seek and Focus [55] have analyzed factors impacting intermittently connected networks consisting of data carriers similar in spirit to zebroids. Table 6.1 provides an overview of these studies. Factors considered by each study are dictated by their assumed environment and target application. A novelcharacteristicof our study is the impact on availabilitylatency for a givendatabase repository of items. In future work, it may be useful to integrate these diverse studies along with our work under a comprehensive general model/framework that incorporates all possible factors, environmental characteristics, and application requirements. The various studies can be characterized into reactive and proactive on the basis of whether the mobilityof thenodesiscontrolled. In the reactive case, nodemovementsare dictated by a speci¯c mobility model and applications rely on the movement inherent in these nodes for data delivery. In the proactive case, the node movement can be adapted pro-actively to deliver data and also in some cases reduce data delivery latency. We ¯rst summarize the studies that use proactive schemes followed by a brief description of ones that employ reactive schemes. Li and Rus [41] introduce a scheme that computes the optimal trajectory for relaying a message among nodes. However, for larger networks, supporting multiple simultaneous message transmissions using this algorithm is di±cult because the scheduling problem becomes intractable. Along the same lines, in [63], special nodes called message ferries that follow non-random movement paths allow data delivery between other nodes whose 112 Study Potentially Mobility Model Delivery Energy Delay How many Storage Mobile copies Nodes created? ZebraNet [35] All Controlled + species Many to X Many X dependent One DakNet [46] One Controlled One to One One Message All Random + One to X X One Ferries [63] Controlled One SWIM [54] Most Random without Many to X Many predictions Some Data Mules [53] Some Random without Many to One X predictions One Seek and All Random without One to X X One Focus [55] predictions One Our Work All Random with Any to X One or X predictions One More Table 6.1: Related studies on intermittently connected networks. movements are governed by a random mobility model. The movement of message ferries can be controlled in order to obtain data from nearby source nodes and deliver it to the appropriate sink nodes when invicinity. Message transmissions onlytakeplace via direct transmissions. In a follow-up work [37], latency-energy tradeo®s are demonstrated by the authors where a single ferry follows a known trajectory and the other nodes schedule their sleep/wakeup schedule in accordance with when the ferry will be in the vicinity. Comparison with reactive protocol like DSR indicate signi¯cant energy savings while su®ering only a small drop in delay performance. In the ZebraNet project [35], sensors are attached to zebras to study the wild life behavior. The sparse nature of the network prevents formation of a connected network. Hence, data is obtained from the sensors when humans drive by in a car or some other vehicle. Similarly, in the DakNet project [46], vehicles are used to transfer data between villages and cities using a store and forward mechanism. Our work with zebroids di®ers from all the above studies that can categorized in the proactive realm, in that, it is reactiveandemploysarandomwalkmobilitymodel. Themovementofthevariousnodes (vehicles in our case) is dictated by this model and cannot be controlled explicitly as was utilized in the above studies. 113 Nodes that obey a random mobility model have been widely studied in the context of reactiveschemes. Therehavebeenalargenumberofstudiesontheanalysisofproperties of random walks [2] on a torus. Some recent studies like [55, 53] have used the 2D random walk mobility model in the context of routing in intermittently connected mobile networks. Our mobility model is similar to that used in these studies. In [53], DataMules are used to route data from static sensors to the base-stations in a sparse sensor network. In our studies, all nodes are mobile and have the same capabilities. In [55], using latency or meeting time and number of transmissions as the metrics of interest, the authors present comparisons of a randomized, utility based and an hybrid (seek and focus) approach with an optimal scheme via analysis as well as simulations. The metric of meeting time used in this study is analogous to our notion of availability latency. In a followup study [56], the authors introduce a new multi-copy routing algorithm where the source \sprays" a certain number of copies into the network and then waits until one of those copies meets the destination. Through analysis and simulations, this spray and wait algorithm is shown to have the lowest average delay and low transmission cost in terms of the number of packets when compared with alternate schemes. However, our study di®ers from the work by Spyropoulos et al. [55], in that, their study does not consider a storage constraint per node as is the case with zebroids. Thestorageconstraintrequireszebroidstomakedecisionsaboutwhichdataitemreplicas should be kept in local storage to minimize overall availability latency. Moreover, we do not expect energy to be a very constrained resource in a vehicular ad-hoc network, hence we have not consider number of transmissions as a metric in our studies. Finally, their studydoesnotemployaninterferencemodelthatwillconstraintheavailablebandwidths for data transfer in a vehicular ad-hoc network. Some studies in the mobile infostation context have relied on replication and caching techniques to reduce data delivery latency. Small and Haas [54] propose the Shared Wireless Infostation Model (SWIM) where the infostation architecture is combined with thead-hocnetworkingmodel. Here,theinfostationsactasdatasinks. Byreplicatingand hencespreadingdataacrossthemobilenodesdatadeliverylatencyattheinfostationscan begreatlyreduced. Thisstudyusesadi®erentialequationmodeltoanalyticallydetermine 114 thetimeuntilthedataisspreadtoallthenodes. Presenceofstaticinfostations,adi®erent mobility model, an absence of a storage constraint per node and a di®erent architectural framework and application makes this study signi¯cantly di®erent from ours. 6.4 Intelligent Transportation Systems (ITS) In the ITS framework, the vehicular network is viewed as a MANET and messages are forwarded from one vehicle to another realizing several applications like vehicle accident noti¯cationbroadcast,pre-emptiveemergencyvehiclearrivalinformationetc[9,10]. Car- Net [43] is a scalable ad hoc network system of cars using a grid location service and geographic routing to achieve scalability for applications such as tra±c congestion and °eet tracking. Along similar lines, several other projects such as FleetNet [8], VGrid [3], DRIVE [12] have been proposed. Our work compliments these studies in that the lessons about replication under storage constraints and minimization of availability latency in vehicular networks are directly applicable to these systems. This is because our frame- work is fairly general, in that, examples of data items across which we seek to minimize latency may be highway repair noti¯cation messages etc. 115 Chapter 7 Conclusions We brie°y summarize the major contributions of this dissertation. The proposed Au- toMata application for delivery of content such as audio or video clips to passengers in theirvehicleswhichistheprimarymotivationofthisthesisisnovel. Theproblemstackled in this dissertation examine complementary aspects such as data discovery (chapter 3), data replication (chapter 4), and data delivery scheduling (chapter 5) in such an environ- ment. Moreover, the metrics of interest in such an application (for example availability latency, overhead etc.), the questions we propose, our solution approach, and our pro- posed methodology for evaluation in itself serve as a guideline to future research when data from live vehicular test-beds will be available for testing out the behavior in real deployments. We have also presented a feasibility analysis to show that employing the cellularinfrastructureasacontrolplaneandvehicularad-hocpeer-to-peernetworkasthe data plane has su±cient bandwidth to realize such an application. Next, we summarize the key contributions of each of the three studies that are part of this thesis. In the PAVAN work presented in Chapter 3, the key contributions are as follows: We introduce PAVAN as a novel policy framework for addressing the availability problem to compute when di®erent data items are available in an ad-hoc network of AutoMata devices We propose novel utility models to evaluate the e®ectiveness of the PAVAN predic- tions with regards to the false-positives, false-negatives, and true-positives. 116 In the static replication schemes study presented in Chapter 4, the key contributions are as follows: Given a data item repository, proposed an optimization formulation to minimize the average availability latency subject to a storage constraint per vehicle Analyticallysolvedtheoptimizationinthecaseofasparsedensityofreplicasyield- ing the square-root replication scheme as the optimum Obtained a mathematical expression that captures the behavior of availability la- tency as a function of the number of data item replicas Examinedtherelativeperformanceof3popularreplicationschemes: linear,square- root, and random for a large number of parameter settings: { Data item size =1 and unbounded client trip duration { Data item size =1 and ¯nite client trip duration { Data item size >1 and unbounded client trip duration { Data item size >1 and ¯nite client trip duration Validated main results employing vehicular movements dictated by an underlying map of the San Francisco Bay Area Evaluated the latency performance of the replication schemes using traces obtained from a bus-based DTN test-bed called UMassDieselNet [11]. Provided an equiva- lence between the Markov model and the mobility model represented by the traces Finally,inthezebroidsstudypresentedinChapter5,thekeycontributionsareasfollows: Introduced the concept of zebroids as data carriers and proposed a modi¯ed Bell- man Ford's algorithm to yield the optimum delivery schedule for the relay team of zebroids that minimizes latency for a given client request Quanti¯ed the variation in availability latency with zebroids as a function of a rich set of parameters such as car density, storage per car, data item repository size, 117 popularity of data items, client trip duration, number of zebroids employed, and replacement policies employed by zebroids. For a sparse density of data item replicas, proposed and validated a mathematical formulation to capture the improvements in latency with zebroids when compared to the latency provided by static replication schemes Evaluation of the performance of di®erent replacement policies at zebroids that yielded that a random replacement scheme provides superior performance Changes in popularity of the data items do not impact the latency gains obtained with one-instantaneous zebroids. Validated main results with zebroids using a map of the San Francisco Bay area to dictateandconstrainthemovementsofthevehiclesinaccordancewiththelocation of major freeways captured by the equivalent Markov model Evaluated the performance employing zebroids using traces obtained from the bus- based UMassDieselNet test-bed. We acknowledge that realistic validation of the Markov model is far from complete. In that regards, much additional work remains to demonstrate the practical applicability of the Markovian approach to realize an AutoMata based application. This represents a concrete direction for future doctoral dissertations. However, this dissertation does present a concrete step toward understanding the dif- ferent nuances involved in realizing an application in a vehicular ad-hoc network. The huge parameter space involved makes it extremely challenging to come up with compre- hensive mathematical models that can capture system performance. Starting from ¯rst principles and tying the di®erent pieces together as we develop enough understanding of thesamepresentsapromisingapproachinthisdirection. Inmanyways, thisthesisalong with other missing pieces of the AutoMata application will serve as a proof of concept that it is now realistic to start thinking about deployment of on-demand delivery appli- cations for passengers in their vehicles. 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Abstract (if available)
Linked assets
University of Southern California Dissertations and Theses
Asset Metadata
Creator
Kapadia, Shyam N.
(author)
Core Title
Data replication and scheduling for content availability in vehicular networks
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Computer Science
Publication Date
02/18/2007
Defense Date
12/11/2006
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
availability latency,data carriers,linear,Markov chains,Markov mobility,OAI-PMH Harvest,random replication,square-root,static replication,trace-based validation,vehicular networks
Language
English
Advisor
Krishnamachari, Bhaskar (
committee chair
), Psounis, Konstantinos (
committee member
), Zimmermann, Roger (
committee member
)
Creator Email
shyamkapadia@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m278
Unique identifier
UC1303569
Identifier
etd-Kapadia-20070218 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-163567 (legacy record id),usctheses-m278 (legacy record id)
Legacy Identifier
etd-Kapadia-20070218.pdf
Dmrecord
163567
Document Type
Dissertation
Rights
Kapadia, Shyam N.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
availability latency
data carriers
linear
Markov chains
Markov mobility
random replication
square-root
static replication
trace-based validation
vehicular networks