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IEEE 802.11 is good enough to build wireless multi-hop networks

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IEEE 802.11 is good enough to build wireless multi-hop networks

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Content IEEE 802.11 IS GOOD ENOUGH TO BUILD WIRELESS MULTI-HOP NETWORKS by Apoorva Jindal A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2009 Copyright 2009 Apoorva Jindal Dedication to my Family and to Divya ii Acknowledgments I would like to express my gratitude to my advisor Professor Konstantinos Psounis, who believed in me and taught me how to perform research. His optimism, his uncanny ability to identify challenging yet solvable problems, and his ideas on how to present thoughts in a well-organized and structured manner, are some of the most important arts that he has passed onto me. I feel lucky to have colloborated with some other very nice and talented people. I am very grateful to Professor Ramesh Govindan, with whom I had the honor of col- loborating towards the end of my PhD. Vision in driving a project towards a logical completion is a skill which I acquired from him during this colloboration. I am also grateful to Sumit Rangwala and Ki-Young Jang for helping me with the design and implementation of distrbuted rate control algorithms. I thank Ravi Jain and Samir Goel who gave me the opportunity to spend a couple of fantastic summers at Google. I am grateful to Chris for his help during my internship at Google. I also want to express my gratitude to Professor Ali Zahid, for whom I was a teaching assistant for several of his classes. Further, I also thank Professor Bhaskar Krishanamachari, who has provided me with valuable guidance and feedback throughout my stay at USC. Finally, I would like to thank Prof. Leana Golubchik and Prof. Cauligi Raghavendra who served in my qualifying exam committee. iii I am grateful to the love of my life Divya, who is a constant source of encourage- ment, support and strength for me. Getting to know her has been the most rewarding experience of my life, and has helped me evolve as a person. Arriving at USC, I was lucky enough to find a great group of friends, who have watched out for me, and helped me in every way possible throughout my stay. I thank Rahul, Avinash, Nupur, Sundeep and Vivek for their friendship and support, and also for making my life lot more interesting than I expected. I also thank my office mates Fragkiskos, Wei, Shyam, Kiran and Vlad for all the interesting discussions we had. Further, I thank Kartikeya and Himanshu, my undergraduate classmates and friends, for the numerous phone conversations on topics involving both research and life in general. However, above all, my deepest respect and gratitude goes to my family, and espe- cially to my parents, who have made me the person I am today. iv Table of Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures ix Abstract xii Chapter 1: Introduction 1 1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Characterizing the Achievable Rate Region for IEEE 802.11- Scheduled Multi-hop Networks . . . . . . . . . . . . . . . . . 3 1.1.2 Worst Case Peformance Bounds on IEEE 802.11-Scheduled Multi- hop Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Distributed Rate Control Protocol for IEEE 802.11-Scheduled Multi-hop Networks . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Related Work 7 2.1 Performance Analysis of IEEE 802.11 in Multi-Hop Networks . . . . . 7 2.2 Distributed Rate Control in IEEE 802.11-Scheduled Multi-Hop Networks 9 Chapter 3: Preliminaries 11 3.1 A Brief Description of IEEE 802.11 MAC Protocol . . . . . . . . . . . 11 3.2 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 4: Achievable Rate Region of IEEE 802.11-Scheduled Multi-Hop Net- works 18 4.1 Characterizing the Achievable Edge-Rate Region . . . . . . . . . . . . 19 v 4.1.1 Expected Service of An Edge . . . . . . . . . . . . . . . . . . 20 4.1.2 Derivation of Collision and Idle Probabilities for Two-Edge Topolo- gies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.3 Determining the Achievable Edge-Rate Region in any Multi- hop Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.4 Network Solution . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Achievable Flow Rate Region . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.1 Two-edge topologies . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 Common Topologies . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.3 Square Topology: Which Route . . . . . . . . . . . . . . . . . 50 4.3.4 A Real Topology: Houston Neighborhood . . . . . . . . . . . . 51 4.3.5 Random Topology . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.6 Different Network Parameters . . . . . . . . . . . . . . . . . . 53 4.3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Network Solution Without The Iterative Procedure . . . . . . . . . . . 55 4.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5.1 Different Transmission Rates and Packet Sizes . . . . . . . . . 57 4.5.2 More Detailed Physical Layer Model . . . . . . . . . . . . . . 57 Chapter 5: Worst Case Bounds on IEEE 802.11 60 5.1 Topology Characterization . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Worst Case Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Imposing Practical Constraints . . . . . . . . . . . . . . . . . . . . . . 74 5.3.1 Non-Interfering Neighbors . . . . . . . . . . . . . . . . . . . . 75 5.3.2 Interference between Edges Belonging to Different Maximal Independent Sets . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3.4 Characterizing the Worst Case Topology . . . . . . . . . . . . . 78 5.3.5 Comparison with Related Work . . . . . . . . . . . . . . . . . 80 5.4 Typical Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4.1 Average Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4.2 A Real Topology: Houston Neighborhood . . . . . . . . . . . . 82 5.5 End-to-End Rate Allocations in Multi-hop Topologies . . . . . . . . . . 82 5.5.1 Throughput Ratio and Number of Hops . . . . . . . . . . . . . 83 5.5.2 Throughput Ratio and Topology Characteristics of the Congested Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5.3 Throughput Ratio and Number of Congested Neighborhoods . . 85 5.6 Discussion: Effect of Our Assumptions . . . . . . . . . . . . . . . . . 85 5.6.1 Assumptions on the Physical Layer . . . . . . . . . . . . . . . 85 vi 5.6.2 Multiple Outgoing Edges from a Node . . . . . . . . . . . . . . 89 Chapter 6: WCP-CAP: Distributed Rate Allocation Protocol for IEEE 802.11- Scheduled Multi-Hop Networks 91 6.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Evaluation of WCP-CAP . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.2 Performance of WCP-CAP . . . . . . . . . . . . . . . . . . . . 97 Chapter 7: Conclusions and Future Work 102 References 104 vii List of Tables 3.1 A brief description of the notation used in the analysis. (Please refer to the text for precise definitions.) . . . . . . . . . . . . . . . . . . . . . . 14 3.2 System parameters used to obtain numerical results. . . . . . . . . . . . 17 5.1 Table defining the second constraint. . . . . . . . . . . . . . . . . . . . 77 6.1 Simulation Reults for the Flow in the Middle Topology . . . . . . . . . 97 6.2 Simulation Reults for the Diamond Topology . . . . . . . . . . . . . . 98 6.3 Simulation Reults for the Half-Diamond Topology . . . . . . . . . . . 99 6.4 Simulation Reults for the Chain-cross Topology . . . . . . . . . . . . . 99 6.5 Delay results for WCP-CAP. . . . . . . . . . . . . . . . . . . . . . . . 100 viii List of Figures 4.1 The Markov chain representing the evolution of a transmitter’s state. . . 21 4.2 Different two-edge topologies: (a) Coordinated stations, (b) Near hid- den edges, (c) Asymmetric topology, (d) Far hidden edges. . . . . . . . 25 4.3 Multiple RTS exchanges at e 1 can collide with the same DATA trans- mission one 2 for the asymmetric topology. . . . . . . . . . . . . . . . 30 4.4 A possible realization of the sequence of events which follow event E e 1 ,FH 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Capacity Regions for different two-edge topologies. The packet loss rate for a 1024 byte packet is equal to 0.2 at e 1 , 0.3 at e 2 and 0.5 at all the interference links. (All the rates are in Mbps.) (a) Coordinated stations. (b) Near hidden edges. (c) Asymmetric topology. (d) Far hidden edges. (The error in the maximum rate achieved at e 1 after fixing the rate ate 2 is less than10.1% for all the four plots.) . . . . . . . . . . . . . . . . . 47 4.6 (a) The Flow in the Middle topology. (b) Achievable rate region for the Flow in the Middle topology. . . . . . . . . . . . . . . . . . . . . . . 48 4.7 (a) Chain topology. (b) Achievable rate region for the Chain topology. . 49 4.8 (a) Square topology. (b) Achievable rate region for the Square topology. 50 4.9 Topology from the deployment in a Houston neighborhood. Arrows show the routing paths and the numerals on top of an arrow is the prob- ability of loss of a1024 byte packet on that link. Dashed lines represent the interference links. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.10 (a) Achievable Rate Region for the Flow in the Middle topology for100 byte packets and 1 Mbps data rate. (b) Achievable Rate Region for the Flow in the Middle topology for 1024 byte packets and 11 Mbps data rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ix 4.11 (a) Achievable rate region for the Flow in the Middle topology with the approximations of Section 4.4. Error between simulations and analysis is less than 20%. (b) Achievable rate region for the Chain topology with the approximations of Section 4.4. Error between simulations and analysis is less than12%. . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1 The solid lines connect nodes which interfere with each other. There are four flows in this topology: 1→ 8, 3→ 9, 3→ 10 and 8→ 11. Congested queues are indicated with a symbol depicting a queue. . . . 63 5.2 Worst case and average throughput ratios for IEEE 802.11 against opti- mal scheduling for different neighborhood sizes. . . . . . . . . . . . . . 72 5.3 (a) Worst case throughput achieved ate c as a function of the interference factor for|N ec | = 10. (b) Worst case throughput ratio achieved at e c as a function of the interference factor for|N ec | = 10. . . . . . . . . . . . 72 5.4 Worst case topology for|N ec | = 8. (a) Worst Case with no assump- tions on the physical layer. (b) Worst case assuming disk model and transmission range equal to interference range. . . . . . . . . . . . . . 74 5.5 e 1 can interfere with all the other four edges belonging to N ec . . . . . . 75 5.6 Characteristics of the worst case topology with disk model. (a) Interfer- ence Ratio. (b) Scheduling Set Ratio. (c) Number of edges inN ec coll . . . . 78 5.7 (a)-(c) Multi-hop topologies which we use to demonstrate that end-to- end rate allocations in a multi-hop topology depend only on the topology characteristics of the congested neighborhoods. Congested queues are indicated with a symbol depicting a queue. . . . . . . . . . . . . . . . . 83 5.8 (a) Worst case and average throughput ratio when the interference range is twice the transmission range. (b) The achievable rate region of the topology of Figure 4.2(c) with a binary and a non-binary interference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.1 Pseudo-code for rate controller at edge e . . . . . . . . . . . . . . . . . 94 6.2 (a) The Diamond topology. (b) The Half-Diamond topology. There are three flows in both these topologies: 1→ 3,4→ 6 and7→ 9. . . . . . 98 6.3 The Chain-cross topology. There are five flows in this topology: 1→ 2, 1→ 7,6→ 7,8→ 9 and10→ 11. . . . . . . . . . . . . . . . . . . . 99 x 6.4 Average End-to-End Delay as a function ofU for the flow 4→ 6 in the Flow in the Middle topology. . . . . . . . . . . . . . . . . . . . . . . . 101 xi Abstract This work formally establishes that IEEE 802.11 yields exceptionally good performance in the context of wireless multi-hop networks. A common misconception is that existing CSMA-CA random access schemes like IEEE 802.11 yield unfair and inefficient rates in wireless multi-hop networks. This misconception is based on works which study IEEE 802.11-scheduled multi-hop networks with TCP or in saturation conditions both of which grossly underutilize the available capacity that IEEE 802.11 provides, or use topologies which cannot occur in practice due to physical layer limitations. To formally establish our thesis, we will derive worst case performance bounds on IEEE 802.11 in multi-hop networks. We first characterize the achievable rate region for any IEEE 802.11-scheduled multi-hop network. To do so, we first characterize the achievable edge-rate region, that is, the set of edge rates that are achievable on a given topology. This requires a careful consideration of the inter-dependence among edges, since neighboring edges collide with and affect the idle time perceived by the edge under study. We approach this problem in two steps. First, we consider two-edge topologies and study the fundamental ways by which they interact. Then, we consider arbitrary multi-hop topologies, compute the effect that each neighboring edge has on the edge under study in isolation, and combine to get the aggregate effect. We then use the characterization of the set of feasible rates to compare the max-min rate allocation achieved by IEEE 802.11 and optimal, and find that: (i) IEEE 802.11 is xii never worse than 16% of the optimal when ignoring physical layer constraints, (ii) in any realistic topology with geometric constraints due to the physical layer, IEEE 802.11 is never worse than 30% of the optimal, and (iii) in typical topologies IEEE 802.11 attains more than 55% of the optimal throughput. Considering that the state-of-the-art distributed approximations to optimal scheduling achieve lower worst case bounds than the above, IEEE 802.11 is surprisingly efficient. To ensure that this good performance is achievable with a distributed rate controller, we propose WCP-CAP. It provides explicit and precise rate feedback to sources while exchanging control information only amongst the neighbors. WCP-CAP achieves max- min rates within15% of the optimal for all the topologies considered in this paper. xiii Chapter 1: Introduction Wireless multi-hop networks are networks in which all nodes wishing to exchange data are not within the radio range of each other. These out-of-range nodes communicate via multi-hop paths consisting of intermediate forwarding nodes. Note that all the cur- rently deployed wireless networks are based on single-hop technologies. Recently there has been significant research activity on building wireless multi-hop networks as they allow building cheap, easily deployable and scalable networks with smaller energy cost. Hence, multi-hop networks are expected to lead the next wave of wireless technology deployments. Enviosioned applications include last mile Internet access [10,20,41], dis- tributed sensing [34], community networking [6], disaster relief operations [30], extend- ing cellular architecture [47] etc. One of the fundamental open research question in the design of wireless multi- hop networks is how to efficiently schedule transmissions in a distributed manner. Researchers have explored the following two approaches to schedule transmissions. (i) First is to find distributed approximation algorithms to implement optimal scheduling. (ii) Second is to use existing CSMA-CA based random access scheduling schemes like IEEE 802.11. Distributed Approximations to Optimal Scheduling: Finding the optimal schedule involves solving a max-weight problem with secondary interference constraints. The queue-size at each edge is set to be its weight. Solving this problem is NP-hard [73]. 1 So, different researchers have proposed distributed approximation algorithms to imple- ment optimal scheduling [16,40,55,71]. Neglecting the overhead, the best scheme [40] amongst these achieves between 1/3 and 1/6 of the optimal for geometric random graphs. Distributed protocols for the network and transport layers can be easily built if one can implement optimal scheduling as backpressure-based techniques can be used to jointly optimize scheduling, routing and rate control in wireless multi-hop networks [27, 50, 57, 74, 79]. However, its not clear if these distributed approximations of optimal scheduling can be implemented with a low overhead. Moreover, implementing them requires the design of new hardware, which makes deployments expensive. Existing CSMA-CA based Random Access Schemes: The initial motivation to use existing CSMA-CA based scheduling schemes was that off-the shelf radios (like IEEE 802.11 radios) can be used to easily and cheaply deploy a multi-hop network. As a result, existing testbeds are built with such radios [10, 11, 26, 69]. However, questions about the suitability of these schemes arise because of the unfair and inefficient through- puts achieved by TCP over IEEE 802.11-scheduled multi-hop networks [56, 66, 77, 82]. These results raise the following question: Is poor throughput performance an artifact of using TCP over IEEE 802.11, or is it a fundamental characteristic of CSMA-CA based random access schemes when used in multi-hop networks. In other words, is it possible to achieve fair and efficient throughputs in multi-hop networks with such scheduling algorithms. The results in this work show that it is possible to achieve the same degree of perfor- mance with existing CSMA-CA based random access scheduling schemes as distributed approximations of optimal scheduling. Hence, the simplicity and low overhead of imple- menting CSMA-CA based random access coupled with its good performance motivates making these scheduling schemes the de-facto standard for multi-hop networks. 2 1.1 Contributions To show that existing CSMA-CA based random access scheduling schemes can achieve fair and efficient rates in multi-hop networks, we derive performance bounds on IEEE 802.11 and show that these performance bounds can be achieved with distributed rate control. We solve this problem in the following three parts. 1.1.1 Characterizing the Achievable Rate Region for IEEE 802.11- Scheduled Multi-hop Networks Consider the following question: Given an arbitrary multi-hop topology and a collection of source-destination pairs, what is the set of feasible end-to-end rates on this arbitrary multi-hop network. Solution to this question for optimal scheduling with different phys- ical layer models has been derived by prior work [39, 46]. However, how does this set look like if IEEE 802.11 is used at the MAC layer is still unknown. Characterizing the set of feasible rates is not only a first step towards deriving the worst case perfor- mance bound for IEEE 802.11 in multi-hop networks, but an important result in its own right with several applications. The constraints characterizing the achievable rate region can be fed into an optimization problem to understand network management issues like admission control and allocating resources to a new flow. This characterization can also be used to find the residual bandwidth at an edge, which can then be used to design interference-aware routing and congestion control. Finally, it will allow researchers who propose new rate control or routing protocols for multi-hop networks with IEEE 802.11 to compare the performance of their scheme with the optimal value. 3 1.1.2 Worst Case Peformance Bounds on IEEE 802.11-Scheduled Multi-hop Networks We then study the worst case performance of IEEE 802.11 against the optimal in multi- hop networks. In multi-hop topologies which do not have neighborhoods larger than 20 edges, for max-min fair rate allocation, we formally establish the following: (i) IEEE 802.11 achieves more than16% of the optimal throughput, (ii) In any real system where the physical layer model imposes geometrical constraints on the topology, IEEE 802.11 achieves more than30% of the optimal throughput, and (iii) In typical topologies, IEEE 802.11 attains more than 55% of the optimal throughput. In reality, the neighborhoods in multi-hop topologies will be much smaller, hence, the performance of IEEE 802.11 will be even better. In the process of formally deriving bounds for worst case with IEEE 802.11, we also characterize the worst case topology. Surprisingly, we find that fewer the maximal inde- pendent sets required to cover the entire topology (less interference in the topology), the worse is IEEE 802.11’s throughput ratio as compared to optimal. We also make the following observation during this study. Moving from wired to wireless moves the link- centric view to a neighborhood centric view [66]. The intuition in wired networks that the congested links in a multi-hop topology determine the end-to-end throughput perfor- mance [4, 28, 61, 62] translates into the following for wireless: In a multi-hop wireless topology, the congested neighborhoods dictate the end-to-end throughput performance. (See Chapter 5 for a precise definition of congested neighborhood.) Hence, studying the performance in congested neighborhoods is sufficient to understand the end-to-end performance. 4 1.1.3 Distributed Rate Control Protocol for IEEE 802.11-Scheduled Multi-hop Networks Having established that the best possible rate controller achieves fair and efficient rates with IEEE 802.11, its still not clear if such a rate controller can be implemented in a distributed manner. In other words, its not clear if the complexity has been shifted to the transport layer. So, we design and implement a distributed rate control algorithm for IEEE 802.11-scheduled multi-hop networks. We label the proposed algorithm WCP- CAP. WCP-CAP provides explicit and precise feedback-based rate control, similar to XCP [43] and RCP [24] in wired networks. Explicit feedback implies that intermedi- ate nodes explicitly send congestion notification to the sources, and precise feedback implies that intermediate nodes inform the sources of the precise rate to transmit at (and not just a binary congestion notification). WCP-CAP estimates the available capacity within each neighborhood, and apportions this capacity to contending flows. It uses the characterization of the achievable rate region proposed in this work to accurately estimate the available capacity in each neighborhood. And it does so while exchanging information only amongst the neighbors, and hence is distributed. WCP-CAP achieves max-min rates within15% of the optimal in all the topologies studied in this work. 1.1.4 Implications Our results have the following three implications. (i) As stated before, it motivates the use of CSMA-CA based schedulers. Note that IEEE 802.11 was primarily designed for single-hop WLANs. Adding modifications (without changing the hardware) to make it more suited to multi-hop networks will only improve the performance further. (ii) Secondly, it diminishes the need to find distributed approximation algorithms for opti- mal scheduling as it is not clear if one can gain substantially over random access. For 5 example, as stated before, the best known distributed approximation provably achieves between 1/3 and 1/6 of the optimal in geometric random graph topologies, which is comparable to what IEEE 802.11 achieves. (ii) Thirdly, it prompts researchers to inves- tigate the design of practical congestion control and rate allocation protocols, like WCP- CAP, which can realize this good performance over random access schemes. 1.2 Thesis Organization We first summarize prior work in Chapter 2, and then present our notation, assumptions and simulation set up as well as a brief description of IEEE 802.11 MAC protocol in Chapter 3. Chapter 4 describes how to characterize the achievable rate region of IEEE 802.11-scheduled multi-hop networks. Then, Chapter 5 derives the worst case through- put performance bounds for IEEE 802.11 as compared to the optimal. Finally, Chapter 6 presents WCP-CAP before concluding and discussing future work in Chapter 7. 6 Chapter 2: Related Work To show that IEEE 802.11 can achieve the same degree of performance as distributed approximations to optimal scheduling, this work solves three different problems. (i) The first problem is to characterize the achievable rate region of IEEE 802.11-scheduled multi-hop networks. It involves modeling and analyzing the behavior of IEEE 802.11 in multi-hop networks, a problem which prior works have considered. (ii) The second problem is to use this model to derive performance bounds on IEEE 802.11-scheduled multi-hop networks. This is the first work to look at this problem. (iii) The third problem is to propose a distributed rate control mechanism for IEEE 802.11-scheduled multi-hop networks. This problem has also been studied by prior works. In this section, we briefly summarize the contribution of prior works on the first and third problems, and point out how this work enhances the state-of-the-art. 2.1 Performance Analysis of IEEE 802.11 in Multi-Hop Networks There is a large body of interesting work on modeling the behavior of IEEE 802.11 in a multi-hop network. This work can be subdivided into five broad categories. (i) [18, 68] present a detailed analysis for specific topologies under study (like the flow in the mid- dle topology or the chain topology), but their methodology cannot be applied to any 7 arbitrary topology. (ii) [12, 19, 80] propose a methodology independent of the topology at hand, but in order to keep the analysis tractable, they simplify the operation of the IEEE 802.11 protocol. In particular, they ignore lack of coordination problems due to topology asymmetries, and/or certain aspects of the protocol like the binary exponen- tial backoff mechanism. (iii) [15, 42, 70] focus on modeling and analyzing interference at the physical layer. To eliminate MAC issues which complicate the analysis without effecting the physical layer model, they assume that all transmitters are within range of each other, and ignore certain aspects of the IEEE 802.11 protocol like the binary expo- nential backoff mechanism and ACK packets. Our work is complementary to papers of this category. We use a simplified physical layer model but a complete model for IEEE 802.11 MAC layer with no assumption on the topology at hand. We discuss briefly in Section 4.5.2 how the more sophisticated physical layer model proposed by papers of this category can be incorporated with the MAC layer analysis presented in this work. (iv) [31,48,54,83] are perhaps the closest to our work. They present a general methodology without making any simplifications to the IEEE 802.11 protocol. But their methodology cannot be applied to topologies which have nodes with multiple outgoing edges, and hence, cannot be used to study any arbitrary multi-hop topology. Further, these papers do not incorporate all the possible dependencies which can exist between both neighboring and non-neighboring edges which makes them increasingly inaccu- rate as the packet transmission time increases. (v) [63] proposed a complete model to derive the one-hop throughput for IEEE 802.11 in multi-hop topologies. This model is more accurate than the previous ones because it uses a Markov chain to capture the complete network state in each of its states. However, the Markov chain has an expo- nential number of states which precludes the model’s use for any decent sized network. (For example, a typical20 node network will require constructing and solving a Markov chain with more than500000 states.) 8 To summarize, an accurate, general and scalable method to model and analyze the behavior of IEEE 802.11 scheduling in multi-hop networks is still missing. 2.2 Distributed Rate Control in IEEE 802.11-Scheduled Multi-Hop Networks We briefly discuss broad classes of research pertinent to this work while referring inter- ested reader to [52] for a more comprehensive survey of congestion control/rate allo- cation in wireless networks. Two different kind of protocols have been proposed to improve the performance of TCP over IEEE 802.11 in multi-hop networks. The first kind are AIMD-based, like TCP. Some of them modify TCP by either mod- ifying how TCP reacts to packet losses (the congestion indicator in TCP) by proposing mechanisms to let TCP distinsuish between link failure induced losses and losses due to congestion [14, 37, 44, 51, 84]. Others modify the congestion indicator and/or the flows which gets the congestion indication by recognizing that flows passing through neigh- boring edges also contribute to the congestion at an edge, and should also reduce their rates [21, 29, 65, 82]. Finally, EWCCP [77] is a protocol which derives new additive increase and multiplicative decrease parameters for IEEE 802.11-scheduled networks. An alternative to AIMD-based schemes are schemes in which intermediate routers send explicit and precise feedback to the sources. XCP [43] and RCP [24] are examples of such schemes for wired networks. Such schemes cannot be directly extended to multi-hop wireless networks, as the available capacity at a wireless link depends on the link rates at the neighboring edges, and ignoring this dependence will overestimate the available capacity and lead to performance degradation [59] and eventually to instability. Variants of XCP for wireless multi-hop networks, like WXCP [75] and XCP-b [2], use heuristics based on measuring indirect quantities like queue sizes and the number of link 9 layer retransmissions, to reduce the overestimation in the available capacity. The rate control protocol proposed in this work, WCP-CAP, uses the derivation of the achievable rate region of IEEE 802.11-scheduled networks introduced in the first part of this work to directly estimate the exact capacity of a link as a function of the link rates at the neighboring edges. Hence it is more effective than any of previously proposed schemes, and achieves max-min rates within15% of the optimal for the topologies studied in this work. In another related work, Li et al. [48] have explored theoretical methods to set up centralized optimization problems for 802.11-scheduled multi-hop networks to find rate allocations achieving a given objective, like the max-min fair rate allocation. However, they do not discuss how to achieve these allocations through distributed rate control schemes. 10 Chapter 3: Preliminaries This chapter summarizes the IEEE 802.11 MAC protocol, and introduces our notation, assumptions and the simulation setup. 3.1 A Brief Description of IEEE 802.11 MAC Protocol The fundamental access mechanism of IEEE 802.11 MAC is the Distributed Coordi- nation Function (DCF). In addition to the DCF, IEEE 802.11 also defines an optional Point Coordination Function (PCF), which uses a central coordinator for assigning the transmission right to nodes. As expected, DCF is used to build multi-hop networks as PCF requires a central coordinator. Hence, we will focus only on the DCF in this work. The DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) MAC protocol. The DCF defines two access mechanisms for packet transmissions: basic access mechanism, and RTS/CTS access mechanism. In this work, we only con- sider the RTS/CTS access mechanism because its use is suggested by the IEEE 802.11 standard as it achieves a better scheduling efficiency. The RTS/CTS access mechanism uses a four-way handshake. A node that wishes to send a DATA frame first triggers the backoff procedure described later. When the backoff counter expires, the transmitter senses the channel for a DIFS duration. If the 11 channel is determined to be idle, then a RTS frame is sent to the destination. Upon suc- cessful reception of the RTS frame, the destination waits for a SIFS interval, and then sends a CTS frame back to the source. The source can start sending the DATA frame a SIFS interval after the reception of the CTS frame. Upon successful reception of the DATA frame, the destination waits for an SIFS interval, then sends an ACK frame back to the source. A node that hears either the RTS, CTS or DATA frame sets its Network Allocation Vector (NA V) to the expected length of the transmission, as indicated in the Duration/ID field of the DATA frame. This is called the virtual carrier sensing mecha- nism. The channel is considered to be busy if either the physical carrier sensing or the virtual carrier sensing indicates so. This four-way handshake attempts to prevent any DATA-DATA collisions that may occur due to the hidden terminal problem [1]. Now we describe the backoff procedure. It is implemented by means of the backoff counter and backoff stage. Initially, upon receiving a new packet to be transmitted, the node starts in backoff stage 0, with the backoff window set to its minimum value. Following an unsuccessful transmission attempt, the backoff stage is incremented by 1, and the backoff window size is doubled until the maximum size of the backoff window is reached, after which the backoff stage and the backoff window size remains unchanged on subsequent unsuccessful exchanges. The backoff window size as well as the backoff stage are set back to their initial values after a successful transmission attempt, or if the retry count exceeds the retry limit for the packet. (The packet is dropped if the retry count exceeds the retry limit.) At the start of each backoff stage, the backoff counter is set to an integer chosen uniformly at random between 0 and the current backoff window size. The backoff counter is decremented by 1 in every subsequent slot, as long as the channel is sensed idle in that slot. (Here, a slot is an interval of fixed duration specified by the IEEE 802.11 protocol.) If a transmission by some other node is detected, the node freezes it backoff counter, and resumes its count from where it left off after the end 12 of the transmission and an additional DIFS interval. When the backoff counter reaches 0, the node transmits. 3.2 Notation and Assumptions The input topology is defined by the interference graphG = (V,E) where V is the set of all nodes andE is the set of all edges. An edge between two nodes in the interference graph implies that the two nodes interfere with each other (irrespective of whether they can hear each other’s transmission successfully or not). The interference is assumed to be binary, that is, a transmission emanating from one of these interfering nodes will always cause a collision at the other node, and pairwise, that is, interference happens between these node pairs only. This interference model neglects some physical layer issues like the capture effect [15] and the effect of multiple interferers [23]. However, Section 5.6.1 shows that using a non-binary and non-pairwise interference model will improve the throughput ratio. Hence, the model we choose yields conservative results. Also, the analysis to determine the achievable rate region presented in Chapter 4 can be easily extended to a non-binary and non-pairwise model, as we discuss in Section 4.5.2. In the absence of a collision, a transmission may get lost due to physical layer imper- fections like fading, hardware noise etc. Successful reception of the RTS, CTS, DATA and ACK packets transmitted on some edgee∈E in absence of collisions are modeled as Bernoulli random variables with success probability equal top e RTS ,p e CTS ,p e DATA and p e ACK respectively. (Note that if two nodes are within each other’s interference range but outside each other’s transmission range, then these probabilities are equal to 0.) Table 3.1 summarizes the notation introduced in this section as well as the variables introduced in Section 4.1. 13 T e Transmitter ofe R e Receiver ofe λ e Edge rate ate E[S e ] Expected service time ate p e RTS (p e CTS Probability of successful RTS (CTS, DATA, p e DATA ,p e ACK ) ACK) transmission in absence of collisions T s Time taken to complete one packet transmission T c Time wasted in an RTS collision p e,T c,i Probability of successful RTS-CTS exchange when backoff window value atT e isW i p e,T l,i Probability of successful DATA-ACK exchange when backoff window value atT e isW i p e,T idle Probability that channel is idle aroundT e p e w 0 Probability that the backoff counter ate is equal to0 K e,T Expected number of DATA transmissions per packet N e Set of edges which interfere withe Table 3.1: A brief description of the notation used in the analysis. (Please refer to the text for precise definitions.) We assume that the set of flowsF is also given as an input. Each flow f ∈F is represented by a source-destination pair. Let s(f) denote the source and d(f) denote the destination for flow f. We assume that the arrival process for each flow f has i.i.d. (independent and identically distributed) inter-arrival times, and a long term rate equal to r f . We also assume independence between the arrival process for different flows, 1 and denote the edge rate (sum of the flow rates at the edge) induced by these flows on edge e by λ e . A given set of edge rates Λ E ={λ e : e∈ E} is said to be achievable if the input rate at each queue in the network is less than the service rate at that queue. Then, a given set of end-to-end flow rates is said to be achievable if there 1 Since we assume independent inter-arrival times and independence between the arrival process for different flows, the achievable rate regin we derive in Chapter 4 is a lower bound on the capacity region derived without any assumption on the arrival processes. 14 exists a routing (multiple paths per flow are possible) such that the induced set of edge- rates is achievable. The achievable edge-rate and flow-rate regions are then defined as the closures of the corresponding achievable sets of rates. We assume that each node is running IEEE 802.11 with RTS/CTS at the MAC layer. Recall that we assume RTS/CTS because it achieves a better scheduling efficiency. Let W 0 and m denote the initial backoff window and the number of exponential backoff windows respectively. We assume that the basic time unit is equal to one backoff slot time. Let T RTS , T CTS , T DATA and T ACK denote the time taken to transmit one RTS, CTS, DATA and ACK packet respectively. (Note that the DATA packet includes the UDP, IP, MAC and PHY headers along with the payload.) We also assume that all packets are of the same size, so T DATA is a constant. LetT c denote the time wasted in an RTS collision and letT s denote the time it takes to complete one packet transmission. Then, T c = T RTS +DIFS +δ andT s = T RTS +SIFS +δ +T CTS +SIFS +δ + T DATA +SIFS+δ+T ACK +DIFS+δ whereδ is the propagation delay andDIFS and SIFS are IEEE 802.11 parameters. We assume that the packet size and the data transmission rate is fixed, henceT s is a given constant. To ensure that the difference between optimal scheduling and IEEE 802.11 is only due to the scheduling inefficiencies of IEEE 802.11, we make the overhead imposed by control message exchanges and protocol headers to be the same for both. Thus the packet transmission time T s is the same for both the scheduling schemes. Note that this assumption gives an advantage to optimal scheduling as in practice, the overhead to implement optimal scheduling is expected to be much larger. Further, motivated by prior work that has expressed concerns about the ability to achieve fair and efficient rate allocations under IEEE 802.11 [25, 31, 80], we will com- pare the max-min rate allocation under IEEE 802.11 and under an optimal scheduler. The max-min rate allocation point is defined as follows [5]. A feasible allocation of 15 rates~ x is max-min fair if and only if an increase of any rate within the domain of feasi- ble allocations must be at the cost of a decrease of some already smaller rate. Formally, for any other feasible allocation ~ y, if y s > x s , then there must exist some s ′ such that x s ′≤x s andy s ′ <x s ′. Finally, we will be making the following two assumptions throughout the work to simplify the analysis. Assumption 1. First, we assumeT RTS ≪T s andT CTS ≪T s . The protocol description recommends the use of RTS/CTS only when the size of the DATA packet is much larger than the size of the RTS packets. This is in line with the fundamental principle that the load due to control packets should be a small fraction of the total load. Hence, this assumption is satisfied for normal protocol operation. Assumption 2. Second, we assume that W 0 ≫ 1. Default 802.11 parameters satisfy this assumption. In general, choosing a small value for W 0 will not properly regulate random access to the channel, and will cause a lot of collisions and throughput loss even for WLAN’s. Hence, this assumption is also satisfied for normal protocol operation. 3.3 Simulation Setup We use Qualnet 4.0 as the simulation platform in this work, since it has been shown to provide an accurate and realistic simulation environment [76]. All our simulations are conducted using an unmodified 802.11(b) MAC (DCF) with RTS/CTS. We use default parameters of 802.11(b) (summarized in Table 3.2) in Qualnet unless otherwise stated. Auto-rate adaptation at the MAC layer is turned off and the rate is fixed at 1Mbps. Unless explicitly stated, we set the buffer size and maximum retry limit in 802.11 (the number of retransmission attempts after which the packet is dropped) to a very large 16 Packet Payload 1024 MAC Header 34 bytes PHY Header 16 bytes ACK 14 bytes + PHY header RTS 20 bytes + PHY header CTS 14 bytes + PHY header Channel Bit Rate 1 Mbps Propagation Delay 1μs Slot Time 20μs SIFS 10μs DIFS 50μs W 0 31 m 5 Table 3.2: System parameters used to obtain numerical results. value to avoid packet losses. This allows us to generate the achievable rate region with- out having to worry about transport layer retransmissions to recover from these losses. The packet size is fixed to be 1024 bytes. To use simulations to validate the theoret- ically derived achievable rate region region, we simulate all possible combinations of flow rates with each flow rate varying from 0 to1 Mbps in steps of10 Kbps and plot the achieved output rate at the destination. 17 Chapter 4: Achievable Rate Region of IEEE 802.11-Scheduled Multi-Hop Networks A central question in the study of multi-hop networks is the following: Given an arbi- trary multi-hop topology and a collection of source-destination pairs, what is the achiev- able rate region of this arbitrary multi-hop network. Researchers have formulated a multi-commodity flow problem to answer this question [39, 46]. These papers assume optimal scheduling with different interference models at the MAC layer in their formu- lations. However, characterizing the achievable rate region of an arbitrary multi-hop network with IEEE 802.11 scheduling is still an open problem and is the focus of this chapter. Setting up a multi-commodity flow formulation for IEEE 802.11-scheduled multi- hop networks runs into the following problem: What is the achievable edge-rate region of the given multi-hop topology? The achievable edge-rate region is the region charac- terizing the set of edge rates achievable on the given multi-hop topology. For example, for a wireline network, this region is simply characterized by the constraint that the sum of flow rates at each edge is less than the data rate of the edge. For a multi-hop network 18 with optimal scheduling, this region is characterized using independent sets [39]. Char- acterizing this region is the main missing step in the characterization of the achievable rate region for IEEE 802.11-scheduled multi-hop networks. We adopt the following methodology to characterize the achievable edge-rate region. We first find the expected service time at a particular edge in terms of the collision probability at the receiver and the idle time perceived by the transmitter of that edge. The hard part in the procedure is to find these collision probabilities and idle times because their value depends on the edge-rates at other edges in the network. To find the value of these variables, we decompose the local network topology into a number of two-edge topologies, derive the value of these variables for these two-edge topologies and then appropriately combine them. Finding the expected service time at each edge allows us to characterize the achievable edge-rate region. It is important to note that this decompose and combine approach that we follow provides an intuitive precise description of how neighboring nodes of a multi-hop wireless network affect each other under a random scheduler like IEEE 802.11. We then use the characterization of the achievable edge- rate region to characterize the achievable flow-rate region 2 for any multi-hop network and a collection of source-destination pairs. 4.1 Characterizing the Achievable Edge-Rate Region This section characterizes the achievable edge-rate region Λ E for any multi-hop topol- ogy. 2 Achievable flow-rate region is also referred to as the achievable rate region. Both these terms are used interchangeably in this work. 19 4.1.1 Expected Service of An Edge This section finds the expected service time of a particular edge (denoted bye) in a par- ticular topology (denoted by T ) by constructing and solving a Markov chain (MC) for this edge. The states of this MC describe the current backoff window, backoff counter, and time since the last successful/unsuccessful RTS/CTS exchange (see next two para- graphs for details). The transition probabilities of this MC fore depend on the collision probabilities at the receiver of e, which, in turn, depend on the exact state at the other edges in the network. In order to decouple the MCs and reduce the state space, we find the average value of the collision probabilities by averaging over all possible events which can cause a collision at the receiver. (Note that these events are not independent.) The dependence between the different edges, and, consequently, MCs, is captured via these average probabilities. Prior work on the analysis of IEEE 802.11 has also attempted to reduce the state space of a MC describing the backoff window and counter values. For single-hop net- works, the author in [9] assumed node homogeneity and independence, an approach which has been justified rigorously recently [72]. In the context of multi-hop net- works [31, 54] a somewhat similar approach to ours has been used, but not all events leading to collisions have been considered, and these events have been assumed to be independent. Later sections describe how to find the value of these average collision probabilities, here we focus on finding the expected service time assuming these proba- bilities are given. The evolution of the IEEE 802.11 MAC layer state at the transmitter of edgee after receiving a packet from the network layer is represented by the absorbing MC shown in Figure 4.1. The MC starts from the state START (which represents a packet entering the MAC layer to be scheduled for transmission) and ends in the state DONE (which represents the end of a successful packet transmission). The expected service time at 20 W + 1 0 1 START 1 Ts W + 1 1 m W + 1 1 m W + 1 1 m W + 1 1 m (W , W ) 0 0 (0, W ) 0 1 Ts 0 (T , W ) 0 Tc (C , W ) p c,0 e,T c,0 1−p e,T 1−p l,0 e,T p c,m e,T c,m 1−p e,T W + 1 m e,T p l,m−1 1 1 1 1 1 m (T , W ) 1 1 (0, W ) (W , W ) m m m m Tc (C , W ) (C , W ) m (T , W ) m W + 1 m W + 1 0 1 1 1 0 (T , W ) 1 1 0 (C , W ) 1 1 1 DONE W + 1 m e,T W + 1 m e,T p p l,m e,T p l,m−1 1−p l,m e,T l,m Figure 4.1: The Markov chain representing the evolution of a transmitter’s state. e is equal to the expected time it takes for the MC to reach DONE from START. The state (j,W i ),0≤ j≤ W i ,0≤ i≤ m, represents the transmitter state where the back- off window is equal to W i and the backoff counter is equal to j. The backoff counter keeps decrementing till it expires (reaches state (0,W i )) which is then followed by a transmission attempt. The transmitter first attempts an RTS-CTS exchange, which fails with probability p e,T c,i . (Thus, p e,T c,i denotes the probability that the RTS-CTS exchange at edgee in topologyT is unsuccessful given that either the RTS/CTS exchange or the DATA/ACK exchange was unsuccessful in the previous i transmission attempts. Note that Table 3.1 contains a brief summary of the variables which are being rigorously 21 defined in this section.) The states (C k ,W i ),1≤ k ≤ T c represent an unsuccessful RTS/CTS exchange k time-units before, while the states (T k ,W i ),1≤ k≤ T s repre- sent a successful RTS-CTS exchangek time-units before, followed by the DATA-ACK exchange which fails with probability p e,T l,i . (Thus,p e,T l,i denotes the probability that the DATA-ACK exchange is unsuccessful given that the RTS-CTS exchange was success- ful, and either the RTS/CTS exchange or the DATA/ACK exchange was unsuccessful in the previousi transmission attempts.) If the DATA-ACK exchange is successful, the MC moves to the state DONE. If either the RTS/CTS or the DATA/ACK exchange is unsuccessful, the backoff window is set to W i+1 if i < m, and to W m if i = m, and the backoff counter is chosen uniformly at random in between 0 and the new backoff window value and the MC jumps to the corresponding state. Note that p e,T c,i and p e,T l,i depend on i which denotes the number of successive trans- mission failures. Since the probability that there are more thanm+1 successive trans- mission failures is small for the default values of IEEE 802.11, we approximatep e,T c,i and p e,T l,i fori>m byp e,T c,m andp e,T l,m . In case one decides to not use the default parameters of IEEE 802.11 and setm to a smaller value, then one can introduce additional states in the MC till some valuem ′ >m such that the probability ofm ′ +1 successive transmission failures is small. This MC does not capture the duration of time the backoff counter may get frozen due to another transmission within the transmitter’s neighborhood (due to the physi- cal/virtual carrier sensing mechanism of the IEEE 802.11 protocol). To capture this, let p e,T idle denote the proportion of time the channel around the transmitter of edge e is idle conditioned on the event that there is no successful transmission ongoing ate. We now use the MC to derive the expected service time at edgee (denoted byE[S e ]) in Equation (4.1) in terms of the collision and idle probabilities. For ease of presentation, we define the following two additional variables: Let E[T c,e W i ] and E[T l,e W i ] for 1≤ i≤ m denote 22 the additional time required to reach the start of a successful packet transmission given that the backoff window just got incremented to W i due to an unsuccessful RTS-CTS and DATA-ACK exchange respectively. E[T c,e W i ] =T c + W i +1 2p e,T idle +p e,T c,i E[T c,e Wn i ]+ 1−p e,T c,i p e,T l,i E[T l,e Wn i ] E[T l,e W i ] =T s + W i +1 2p e,T idle +p e,T c,i E[T c,e Wn i ]+ 1−p e,T c,i p e,T l,i E[T l,e Wn i ] E[S e ] =T s + W 0 +1 2p e,T idle +p e,T c,0 E[T c,e W 1 ]+ 1−p e,T c,0 p e,T l,0 E[T l,e W 1 ] (4.1) wheren i =    i+1 if1≤i≤m−1 m ifi =m . Note that the Equation (4.1) is derived based on the following rule for finding the mean time to reach an absorbing state in an absorb- ing MC: LetS denote all the states of a MC, let p ij denote the transition probability from statei to statej, letk∈S denote the absorbing state and letT jk denote the mean time to reach statek from statej. ThenT ik =p ik + P j∈S p ij T jk . To derive the value of the expected service time at a particular edgee using Equation (4.1), one has to first find the value of p e,T c,i , p e,T l,i and p e,T idle for that edge. The next two sections describe how to find the value of these variables for any edge in a given multi- hop topology. Note that we have neglected the effect of post-backoff in this MC. (Post-backoff refers to backing off right after the transmission of the last packet in the queue, in anticipation of a future packet for which there will be no backoff if post-backoff has completed in the meantime.) Since we are interested in determining the boundary of the capacity region, this will have a negligible impact on the accuracy. This is because the boundary of the capacity region depends on the service rate of the backlogged edges, such edges are almost always busy and don’t post-backoff, and their dependence on 23 non-backlogged edges is nearly unaffected by the post-backoff taking place in these non-backlogged edges. 4.1.2 Derivation of Collision and Idle Probabilities for Two-Edge Topologies This section finds the collision and idle probabilities for all possible two-edge topolo- gies. A two-edge topology is defined to be one which has two distinct edges not sharing the same transmitter. These two-edge topologies reveal the types of inter-dependence which can exist between two edges in a multi-hop network and an analysis for these topologies will serve as the building block for the analysis of more complex topologies as will be seen in the next section. Garetto et al. [31] identified four different cate- gories of two-edge topologies which can exist in a given multi-hop network and ana- lyzed them to study unfairness in IEEE 802.11 networks. Here we derive the achievable edge-rate region for these topologies. (This list is exhaustive, that is, all possible two- edge topologies belong to one of these four categories.) We use the following notation throughout this section: e 1 and e 2 denote the two edges under consideration, and λ e j , j = 1,2, denote the edge rates (in packets/time unit). Further, let T e j andR e j ,j = 1,2, denote the transmitter and the receiver of the two edges. Finally, let E t,r RTS and E t,r CTS , t,r∈{T e 1 ,T e 2 ,R e 1 ,R e 2 }, denote the event that the RTS and the CTS packet transmit- ted by nodet is not correctly received at noder due to physical layer errors respectively. For example,E Re 1 ,Te 2 CTS denotes the event that the CTS transmitted byR e 1 is not correctly received atT e 2 due to physical layer errors. Coordinated Stations (CoS) A two-edge topology is a coordinated station topology ifT e 1 andT e 2 interfere with each other. Figure 4.2(a) shows an example of a coordinated station topology. Note that there 24 T 1 e R 1 e e T 2 e R 2 e 2 e 1 (a) T 1 e R 1 e e 1 R 2 e e 2 2 T e (b) T 1 e R 1 e e 1 T 2 e R 2 e e 2 (c) T 1 e R 1 e e 1 R 2 e e 2 2 T e (d) Figure 4.2: Different two-edge topologies: (a) Coordinated stations, (b) Near hidden edges, (c) Asymmetric topology, (d) Far hidden edges. are other two-edge topologies also whereT e 1 andT e 2 interfere with each other, but with no interference links betweenT e 1 andR e 2 and/orT e 2 andR e 1 . However, the performance profile and most of the analysis remains the same, hence, all these topologies are referred to as coordinated stations. The minor change introduced by the lack of interference links betweenT e 1 andR e 2 and/orT e 2 andR e 1 is discussed at the end of this section. We first state the value ofp e j ,CoS l,i in the following lemma. Lemma 1. p e j ,CoS l,i = 1− p e j DATA ×p e j ACK ,0≤i≤m,j = 1,2. Proof. For this topology, the RTS-CTS exchange will successfully avoid any DATA collision and the DATA-ACK exchange will be unsuccessful only when the DATA or the ACK packet gets corrupted due to physical layer effects. We next derive the value of p e j ,CoS c,i . Note that the analysis presented in [9] can be directly applied for this topology to derive the value of p e j ,CoS c,i under saturation condi- tions (when transmitters always have a packet to send). The following lemma finds this probability for non-saturation conditions. 25 Lemma 2. (i)p e 1 ,CoS c,i = 1− p e 1 RTS ×p e 1 CTS 1−λ e 2 E[S e 2 ]p e 2 w 0 ,0≤i≤m, (ii)p e 2 ,CoS c,i = 1− p e 2 RTS ×p e 2 CTS 1−λ e 1 E[S e 1 ]p e 1 w 0 ,0≤i≤m, where 2 Wm+1 ≤p e w 0 ≤ 2 W 0 +1 is the probability that the backoff counter at edgee is equal to0. Proof. We first look at edge e 1 . The RTS/CTS exchange is unsuccessful if either the RTS or the CTS is lost due to physical layer errors or an RTS collision happens at R e 1 . An RTS collision will occur only if the backoff counter at edge e 2 also expires in the same slot duration resulting in both T e 1 and T e 2 sending an RTS packet. Thus, p e 1 ,CoS c,i =P(e 2 has a packet to send)×p e 2 w 0 . (a)P(e 2 has a packet to send) =λ e 2 E[S e 2 ] as the probability that a queueing system is non empty is equal toλE[S] whereλ is the packet arrival rate into the system andE[S] is the expected service time. (b) As derived in [9],p e 2 w 0 is upper bounded by 2 W 0 +1 and lower bounded by 2 Wm+1 . Putting everything together yields the result. p e 2 ,CoS c,i is derived using the same arguments. Approximating p e w 0 by its upper bound is accurate when there are few colli- sions and data losses at the physical layer, otherwise approximating it with its lower bound will be more accurate. So we make the following approximation, p e w 0 =    2 W 0 +1 ifp e,CoS l,0 ≤p cutoff 2 Wm+1 ifp e,CoS l,0 >p cutoff where p cutoff is the value of the DATA/ACK exchange loss probability which results in the lower and upper bound yielding the same error. (Its value for the default parameters of Table I is equal to 0.8.) This approximation is not introducing significant inaccuracies for the following reason. Assumption 2 implies that the probability of an RTS collision at some edge e due to another edge with which it forms a coordinated stations topology is rather small (since the upper bound is small). On the other hand, the probability of RTS collisions due to edges with whiche forms an asymmetric or far hidden edges topology (Sections 4.1.2 and 4.1.2) is much larger, and dominates the calculation of the overall RTS collision probability. Finally, if there are 26 only coordinated stations in e’s neighborhood, the effect of the backoff counter being frozen due to carrier sensing will dominate over RTS collisions (see Equation (4.1)). Section 4.3 verifies that making this approximation has no significant impact on the accuracy of the analysis. Finally, we derive the value of p e j ,CoS idle in the next lemma. We use the fol- lowing variable in this derivation. Let K e,T denote the expected number of DATA transmissions per packet at edge e in topology T including the extra transmissions due to unsuccessful DATA-ACK exchange. Using elementary probability, K e,T = P m−1 i=1 i 1−p e,T l,i Q i−1 k=1 p e,T l,k + Q m−1 i=1 p e,T l,i m−1+ 1 (1−p e,T l,m ) . Lemma 3. (i)p e 1 ,CoS idle = 1−K e 2 ,CoS λe 2 Ts−λe 1 Ts 1−λe 1 Ts , (ii)p e 2 ,CoS idle = 1−K e 1 ,CoS λe 1 Ts−λe 2 Ts 1−λe 2 Ts . Proof. The backoff counter for edge e 1 is frozen when a transmission at edge e 2 is going on given that no successful transmission is going on at edge e 1 . 3 The net rate at which packets are transmitted at edge e 2 is equal to K e 2 ,CoS λ e 2 and T s is the expected service time of one packet. Hence, the probability that there is a transmission ongoing at edge e 2 is equal to K e 2 ,CoS λ e 2 T s . Notice that this derivation ignores the extra RTS- CTS traffic generated by an unsuccessful RTS-CTS exchange, but this is fully justified by the assumption that T RTS ≪ T s (Assumption 1). Similarly, the probability that a successful packet transmission is going on at e 1 is equal to λ e 1 T s . Putting everything together yields the result. p e 2 ,CoS idle is derived using similar arguments. 3 If the RTS fromT e2 is successfully received atT e1 , the backoff counter atT e1 is frozen due to virtual carrier sensing, else its frozen due to physical carrier sensing. Hence, whenever there is a transmission on edgee 2 , the backoff counter ate 1 is frozen. 27 Note that if there is no interference link between T e 1 andR e 2 in Figure 4.2(a), then the probability of RTS collision ate 2 will be equal to 0 instead ofλ e 1 E[S e 1 ]p e 1 w 0 . Simi- larly, absence of the interference link between T e 2 andR e 1 will result in the probability of RTS collision ate 1 to be equal to0. Near Hidden Edges (NH) Figure 4.2(b) shows the topology belonging to this category. T e 1 andT e 2 do not interfere with each other, however, there is an interference link between T e 1 and R e 2 as well as T e 2 andR e 1 . The values ofp e j ,NH l,i ,p e j ,NH c,i andp e j ,NH idle ,0≤ i≤ m,j = 1,2, are derived in a manner similar to the derivation of the corresponding probabilities for coordinated stations. The only difference is that now T e 1 (T e 2 ) will freeze its backoff counter only when a CTS sent from R e 2 (R e 1 ) is successfully received at T e 1 (T e 2 ). So, the RTS transmitted byT e 1 (T e 2 ) can now collide in the following four scenarios: (i) bothT e 1 and T e 2 start transmitting an RTS in the same slot duration, (ii) T e 1 (T e 2 ) starts transmitting an RTS andR e 2 (R e 1 ) starts transmitting a CTS in the same slot duration, (iii) T e 1 (T e 2 ) starts transmitting an RTS while T e 2 (T e 1 ) is still sending an RTS, and (iv) The CTS fromR e 2 (R e 1 ) is lost due to physical layer errors at T e 1 (T e 2 ). The next three lemmas state the value of p e j ,NH l,i ,p e j ,NH c,i and p e j ,NH idle , 0≤ i≤ m, j = 1,2. Lemma 4. p e j ,NH l,i = 1− p e j DATA ×p e j ACK ,0≤i≤m,j = 1,2. Lemma 5. (i)p e 1 ,NH c,i = 1− p e 1 RTS ×p e 1 CTS 1−2λ e 2 E[S e 2 ]p e 2 w 0 −K e 2 ,NH λ e 2 T RTS 1−P E Re 2 ,Te 1 CTS ,0≤i≤m. (ii)p e 2 ,NH c,i = 1− p e 2 RTS ×p e 2 CTS 1−2λ e 1 E[S e 1 ]p e 1 w 0 −K e 1 ,NH λ e 1 T RTS 1−P E Re 1 ,Te 2 CTS ,0≤i≤m. Lemma 6. (i)p e 1 ,NH idle = 1−K e 2 ,NH λe 2 (Ts−T RTS )−λe 1 Ts 1−λe 1 Ts , (ii)p e 2 ,NH idle = 1−K e 1 ,NH λe 1 (Ts−T RTS )−λe 2 Ts 1−λe 2 Ts . 28 Asymmetric Topology (AS) Figure 4.2(c) shows an example of the topology belonging to this category. T e 1 andT e 2 as well asT e 1 andR e 2 do not interfere each other, butT e 2 andR e 1 are within each other’s interference range. The main characteristic of this topology is that T e 2 is aware of the channel state as it can hear the CTS fromR e 1 , butT e 1 is totally unaware of the channel state as it can hear neither the RTS nor the CTS from the transmission one 2 . We first derive the collision and idle probabilities for edgee 1 . The following lemma derives the value ofp e 1 ,AS l,i . Lemma 7. p e 1 ,AS l,i = 1 − p e 1 DATA × p e 1 ACK 1−p e 2 w 0 λ e 2 E[S e 2 ] 1−P(E Re 1 ,Te 2 CTS )K e 2 ,AS λ e 2 T s ,0≤i≤m. Proof. The DATA packet send byT e 1 will collide if one of following two events happen: (i) If T e 2 starts transmitting an RTS and R e 1 starts transmitting a CTS in the same slot duration. Both the packets will be received successfully at their respective destinations, and will lead to the DATA packet from T e 1 colliding. (ii) The CTS from R e 1 is not recovered at T e 2 due to physical layer errors, and T e 2 starts a transmission as it is not aware of the ongoing transmission ate 1 . We next derive the value of p e 1 ,AS c,i in the following sequence of lemmas. The first lemma directly follows from the following observation: if T e 1 transmits an RTS while a transmission at edge e 2 is going on, it will collide. As before, note that this lemma ignores the extra RTS traffic generated at e 2 by an unsuccessful RTS-CTS exchange, which is not a problem sinceT RTS ≪T s (Assumption 1). Lemma 8. p e 1 ,AS c,0 = 1−(p e 1 RTS ×p e 1 CTS (1−K e 2 ,AS λ e 2 T s )). Now, lets look at what happens after the first RTS collision. The RTS collision will cause the backoff window at T e 1 to increase to W 1 and a new backoff counter is 29 chosen uniformly at random between (0,W 1 ). If the remaining transmission time at edge e 2 is more than the new backoff counter, then the second RTS transmission at e 1 will collide with the same transmission. (Note that multiple RTS exchanges on e 1 can collide with the same DATA transmission on e 2 , see Figure 4.3. Prior works have not incorporated this effect in their analysis, and hence, their accuracy decreases as Ts W 0 increases.) And if the remaining transmission time at edge e 2 is lower than the new backoff counter, then the probability of RTS collision is equal toK e 2 ,AS λ e 2 T s . So, P(RTS/CTS exchange is unsuccessful at the end of second backoff| a collision occurred at the end of the first backoff ) = (1−p 1 0 )+p 1 0 p e 1 ,AS c,0 , where p 1 0 is the probability that the transmission at e 2 which collided with the first RTS transmission by T e 1 (when the backoff window at T e 1 was W 0 ) ends before the second backoff counter at T e 1 expires (when the backoff window at T e 1 is W 1 ). To evaluate p e 1 ,AS c,1 , note that the backoff window also increments if the first RTS-CTS exchange went through but the subsequent DATA or ACK packet was lost, in which case the RTS collision probability after the second backoff counter expires is equal to K e 2 ,AS λ e 2 T s . Putting everything together yields p e 1 ,AS c,1 = 1− p e 1 RTS × p e 1 CTS 1− 1−p e 1 RTS,0 p e 1 ,AS c,0 + p e 1 RTS,0 (1−p 1 0 ) + p 1 0 p e 1 ,AS c,0 , where p e 1 ,AS RTS,0 = K e 2 ,AS λe 2 Ts p e 1 ,AS c,0 +(1−p e 1 ,AS c,0 )p e 1 ,AS l,0 is the probability that an RTS collision occured at the end of the first backoff given that either the RTS/CTS exchange or the DATA/ACK exchange was unsuccessful at the end of the first backoff. ends on e Transmission 2 1 RTS collides at e 1 RTS collides at e Backoff First starts on e 2 Packet enters RTS exchange successful at e Backoff Backoff Third Second 1 t Transmission MAC layer at e 1 Figure 4.3: Multiple RTS exchanges ate 1 can collide with the same DATA transmission one 2 for the asymmetric topology. 30 We now generalize the derivation of p e 1 ,AS c,1 to find the value of p e 1 ,AS c,i , 1≤ i≤ m. We define the following variables for ease of presentation. (a) Let p e 1 ,AS RTS,i , 0≤ i≤ m, denote the probability that an RTS collision occurred at the end of the(i+1) th backoff given that either the RTS/CTS exchange or the DATA/ACK exchange was unsuccessful at the end of the(i+1) th backoff. If there is no RTS collision at the end of the(i+1) th backoff, then the probability of RTS/CTS exchange being unsuccessful at the end of the next backoff ((i+2) th backoff) is equal top e 1 ,AS c,0 . (b) Letp e 1 ,AS RTSnew,i ,0≤i≤m, denote the probability that an RTS collision occurred at the end of the (i+1) th backoff given that (i) the RTS/CTS exchange or the DATA/ACK exchange was unsuccessful at the end of the (i+1) th backoff, and (ii) the collision occurred with a transmission one 2 which started when the backoff window atT e 1 wasW i , that is, the colliding transmission one 2 started while the backoff counter atT e 1 was decrementing during the (i+1) th backoff. This probability indicates the start of a new transmission ate 2 which might collide with the subsequent RTS exchanges. (c) Let E j,i denote the event that an RTS collision occurred ate 1 when the backoff window atT e 1 wasW i , with a transmission one 2 which had started when the backoff window atT e 1 wasW j . This event indicates the start of the ongoing transmission ate 2 . (d) Finally, letp i j denote the probability that a transmission at e 2 , which started when the backoff window at T e 1 was W j , ends when the backoff window at T e 1 is W i given that it had not ended when the backoff window was W i−1 . This probability is used to count the number of RTS exchanges ate 1 which collides with the same transmission one 2 . Lemma 9. p e 1 ,AS c,i = 1− p e 1 RTS ×p e 1 CTS 1− 1−p e 1 ,AS RTS,i−1 p e 1 ,AS c,0 − P i−1 j=0 P (E j,i−1 ) 1− p i j +p i j p e 1 ,AS c,0 ! ,1≤i≤m, where (i)P (E j,i ) =      p e 1 ,AS RTSnew,j Q i u=j+1 (1−p u j ) Q i u=j+1 (p e 1 ,AS c,u +(1−p e 1 ,AS c,u )p e 1 ,AS l,u ) j <i p e 1 ,AS RTSnew,j j =i 31 (ii)p e 1 ,AS RTS,i = 1−p e 1 ,AS RTS,i−1 K e 2 ,AS λ e 2 T s + P i−1 j=0 P (E j,i−1 ) 1−p i j +p i j K e 2 ,AS λ e 2 T s p e 1 ,AS c,i + 1−p e 1 ,AS c,i p e 1 ,AS l,i −1 (iii)p e 1 ,AS RTSnew,i = 1−p e 1 ,AS RTS,i−1 K e 2 ,AS λ e 2 T s + P i−1 j=0 P (E j,i−1 )p i j K e 2 ,AS λ e 2 T s p e 1 ,AS c,i + 1−p e 1 ,AS c,i p e 1 ,AS l,i −1 . Proof. Given event E j,i−1 occurs, the probability that an RTS collision occurs when the backoff window at T e 1 is W i is equal to 1−p i j +p i j p e 1 ,AS c,0 . On the other hand if there is no RTS collision when the backoff window at T e 1 wasW i−1 , the probability of RTS collision when the backoff window at T e 1 is W i is equal to p e 1 ,AS c,0 . Combining everything together using the law of total probability yields the result. The expressions forP (E j,i ),p e 1 ,AS RTS,i andp e 1 ,AS RTSnew,j follow directly from their definitions. To complete the derivation ofp e 1 ,AS c,i , we next derive the value ofp i j ’s by dividing the total number of favorable cases by the total number of possible cases. Lemma 10. p i j =        P W i u i =0 ... P W 1 u 1 =0 P Ts t=1 I(( P i k=1 u k >t)∩( P i−1 k=1 u k ≤t)) PW i−1 u i−1 =0 ... P W 1 u 1 =0 P Ts t=1 I( P i−1 k=1 u k ≤t) j = 0 P W i u i =0 ... PW j u j =0 I(( P i k=j u k >Ts)∩( P i−1 k=j u k ≤Ts)) PW i−1 u i−1 =0 ... PW j u j =0 I( P i−1 k=j u k ≤Ts) j > 0 whereI(u 1 >t) =    1 u 1 >t 0 otherwise . The next lemma states a combinatorial result which is used to evaluate the summa- tions in the previous lemma. Let 0≤ u k ≤ W k ,k = 1,2,...j be j integers and let Z( P j k=1 u k ≤T) denote the size of the setZ T = n (u 1 ,u 2 ,...u j ) : P j k=1 u k ≤T o . 32 Lemma 11. Z( P j k=1 u k ≤T) =V 0 −V 1 +V 2 +...+(−1) j−1 V j , whereV 0 = T+j j , V l = P 1≤r 1 ≤...r l ≤j C T+j−Wr 1 −Wr 2 −...−Wr l −l j andC u l =    u l u≥l 0 u<l . The only remaining variable to be derived for edge e 1 isp e 1 ,AS idle . To derive its value, we use the fact thatT e 1 cannot hear the transmission one 2 , and hence the channel atT e 1 is always idle. Lemma 12. p e 1 ,AS idle = 1. The next lemma states the value of the collision and idle probabilities for edge e 2 . The proof directly follows from the following two observations: (i) no transmission from e 1 can collide atR e 2 , and (ii) a CTS transmission from R e 1 , if successfully received by T e 2 , will freeze the backoff counter atT e 2 due to virtual carrier sensing. Lemma 13. (i)p e 2 ,AS l,i = 1−(p e 2 DATA ×p e 2 ACK ),0≤i≤m, (ii)p e 2 ,AS c,i = 1−(p e 2 RTS ×p e 2 CTS ),0≤i≤m, (iii)p e 2 ,AS idle = “ 1− “ 1−P “ E Re 1 ,Te 2 CTS ”” K e 1 ,AS λe 1 Ts−λe 2 Ts ” 1−λe 2 Ts . Far Hidden Edges (FH) Only R e 1 and R e 2 are within each others’ range in this topology. Figure 4.2(d) shows the topology belonging to this category. For this topology, an RTS sent by a transmitter will not receive a CTS back if a transmission is going on at the other edge because of virtual carrier sensing at the receiver. Thus, p e j ,FH c,i ,0≤ i≤ m,j = 1,2, is derived in a manner similar to the derivation of p e 1 ,AS c,i . The only difference occurs when the CTS fromR e 2 (R e 1 ) is lost atR e 1 (R e 2 ) causingR e 1 (R e 2 ) to be unaware of the channel state ate 2 (e 1 ) and sending a CTS back in response to the RTS from T e 1 (T e 2 ). Hence, the probability of RTS collision is equal to the probability that there is a transmission ongoing at the other edge conditioned on the event that the CTS was correctly received. 33 The probability of the event that the CTS is not correctly received is derived during the derivation ofp e j ,FH l,i . We next derive the value of the probability of DATA collisions. DATA on edge e 1 (e 2 ) will collide if R e 2 (R e 1 ) transmits a CTS or an ACK. R e 2 (R e 1 ) will send back a CTS only if it had not correctly received the CTS exchanged on e 1 (e 2 ). For this topology, DATA packets will not collide with ACK packets as the preceeding RTS/CTS exchange on the other edge will cause the DATA to collide, and hence the receiver will not send back an ACK packet. We now have to determine the events which can cause R e 2 (R e 1 ) to not correctly receive the CTS exchanged one 1 (e 2 ). Lets first consider edge e 1 . Obviously, one of the events which can lead to the CTS getting corrupted is physical layer errors. If either of the CTS fromR e 2 toR e 1 orR e 1 to R e 2 gets corrupted, it will lead to DATA collision on edge e 1 . Thus, the probability of DATA collision on edgee 1 due to the CTS getting corrupted due to physical layer errors is equal top e 1 ,FH l,CTS = 1− 1−P E Re 1 ,Re 2 CTS 1−P E Re 2 ,Re 1 CTS K e 2 ,FH λ e 2 T s . We now describe events which can cause CTS to get corrupted due to collisions. Let E e 1 ,FH 1 (E e 1 ,FH 2 ) denote the union of the following three events. (i) T e 1 and T e 2 start transmitting an RTS in the same slot duration with T e 1 ’s (T e 2 ’s) transmission starting first, (ii) T e 2 (T e 1 ) starts transmitting an RTS while an RTS transmission is going on at e 1 (e 2 ), and (iii) T e 2 (T e 1 ) starts transmitting an RTS in the same slot duration as R e 1 (R e 2 ) starts transmitting a CTS. Neglecting T RTS (easily justified by Assumption 1), P(E e 1 ,FH 1 ) =P(E e 1 ,FH 2 ) =λ e 2 E[S e 2 ]p e 2 w 0 . We now discuss the sequence of events which will follow event E e 1 ,FH 1 (E e 1 ,FH 2 ). (Figure 4.4 shows a possible realization of the sequence of events following event E e 1 ,FH 1 . Note that prior works have not incorporated the effect of the occurence of events E e 1 ,FH 1 and E e 1 ,FH 2 in their analysis, and hence, their accuracy decreases as Ts W 0 increases.) (a) The transmission of RTS one 1 (e 2 ) will succeed andR e 1 (R e 2 ) will send 34 transmission starts succeeds, DATA RTS exchange transmission starts succeeds, DATA RTS exchange End of DATA transmission End of DATA transmission Second Backoff transmission starts succeeds, DATA RTS exchange Backoff Third ends successfully DATA transmission transmission starts succeeds, DATA RTS exchange e 1 2 e t t RTS collides Second Backoff ends successfully DATA transmission DATA collides with CTS on e DATA collides with CTS on e 2 1 (a), (b) (a), (b) (c), (e) (e) (f) (h) (g) (d) Figure 4.4: A possible realization of the sequence of events which follow eventE e 1 ,FH 1 . back a CTS. This CTS will collide with the RTS transmission on e 2 (e 1 ) at R e 2 (R e 1 ). This collision results in R e 2 (R e 1 ) not receiving both the packets. (b) DATA transmis- sion will commence one 1 (e 2 ) whileT e 2 (T e 1 ) backs off. (c) Backoff counter atT e 2 (T e 1 ) expires and an RTS is transmitted one 2 (e 1 ). R e 2 (R e 1 ) responds back with a CTS. (d) If the DATA transmission on one 1 (e 2 ) has not ended, the CTS transmission byR e 2 (R e 1 ) in step (c) will collide with the DATA transmission at R e 1 (R e 2 ). (e) T e 1 (T e 2 ) backs off and DATA transmission commences one 2 (e 1 ). (f) The backoff counter atT e 1 (T e 2 ) expires, it sends an RTS andR e 1 (R e 2 ) sends back a CTS. (g) If the DATA transmission one 2 (e 1 ) has not ended, the CTS transmission byR e 1 (R e 2 ) will collide with the DATA transmission atR e 2 (R e 1 ). (h) This process goes on till at least one of the DATA packets get successfully exchanged. 4 4 Note that the loss of one of the RTS exchanges in this sequence due to physical layer effects will change the probability of DATA collision. Ignoring this event is easily justifiable using Assumptions 1 and 2. By Assumption 1, the probability of the DATA packet getting corrupted by physical layer errors will be much larger than the same probability for the RTS packet as the DATA packets are much larger than the RTS packets. Andp e1,FH l,i ,0≤ i≤ m will be dominated byp e1 DATA asP(E e1,FH 1 ) andP(E e1,FH 2 ) are much smaller (by Assumption 2). Hence, for the network conditions for which P(E e1,FH 1 ) and P(E e1,FH 2 ) matter, ignoring the loss of RTS exchanges will introduce negligible error. 35 p e 2 ,FH l,CTS ,E e 2 ,FH 1 and E e 2 ,FH 2 are similarly defined for edge e 2 . The value of p e j ,FH l,i ,0≤ i≤ m, is stated in the next lemma, whose proof follows directly from the discussion above. We define the following additional variables for ease of presentation. (a) Let p e j ,FH D,i denote the probability that a DATA collision occurs on e j due to events E e j ,FH 1 orE e j ,FH 2 having occurred during previous exchanges, given the current backoff window at T e j is W i and either the RTS/CTS or the DATA/ACK exchange was unsuc- cessful when the backoff window value at T e j was W 0 ,...W i−1 . If the DATA/ACK loss does not occur due to events E e j ,FH 1 or E e j ,FH 2 having occurred during previous exchanges, the probability of DATA collision after the next backoff is equal to p e j ,FH l,0 . (b) Letp e j ,FH D E 1 ,i (p e j ,FH D E 2 ,i ) denote the probability that eventE e j ,FH 1 (E e j ,FH 2 ) occurs during the current data exchange given that the current backoff window atT e j isW i and either the RTS/CTS or the DATA/ACK exchange was unsuccessful when the backoff window value atT e j wasW 0 ,...W i−1 . EventE e j ,FH 1 (E e j ,FH 2 ) may be followed with a sequence of DATA collisions. Lemma 14. Forj = 1,2, (i)p e j ,FH l,0 = 1− p e j DATA ×p e j ACK 1−p e j ,FH l,CTS 1−P E e j ,FH 1 . (ii)p e j ,FH l,i = 1− p e j DATA ×p e j ACK 1− 1−p e j ,FH D,i−1 −p e j ,FH D E 1 ,i−1 −p e j ,FH D E 2 ,i−1 p e j ,FH l,0 − P i−1 k=0 p e j ,FH D E 1 ,k Q i−1 u=k+1 p k,u (E 1 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” p k,i (E 1 )+p c k,i (E 1 )p e j ,FH l,0 − P i−1 k=0 p e j ,FH D E 2 ,k p k,i (E 2 )+p c k,i (E 2 )p e j ,FH l,0 Q i−1 u=k+1 p k,u (E 2 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” ! , 1≤i≤m. We next state the values of p e j ,FH D,i and p e j ,FH D E l ,i . The expressions for these variables follow directly from their definition. Forj = 1,2,l = 1,2 and0≤i≤m, (i)p e j ,FH D,i = P i−1 k=0 p e j ,FH D E 1 ,k Q i−1 u=k+1 p k,u (E 1 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” p k,i (E 1 )+ P i−1 k=0 p e j ,FH D E 2 ,k 36 Q i−1 u=k+1 p k,u (E 2 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” p k,i (E 2 ) h p e j ,FH c,i + 1−p e j ,FH c,i p e j ,FH l,i i −1 , (ii)p e j ,FH D E l ,i = h 1−p e j ,FH D,i−1 −p e j ,FH D E 1 ,i−1 −p e j ,FH D E 2 ,i−1 P E e j ,FH l + P i−1 k=0 p e j ,FH D E 1 ,k p c k,i (E 1 ) Q i−1 u=k+1 p k,u (E 1 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” P E e j ,FH l + P i−1 k=0 p e j ,FH D E 2 ,k Q i−1 u=k+1 p k,u (E 2 ) Q i−1 u=k+1 “ p e j ,FH c,u + “ 1−p e j ,FH c,u ” p e j ,FH l,u ” p c k,i (E 2 )P E e j ,FH l ih p e j ,FH c,i + 1−p e j ,FH c,i p e j ,FH l,i i −1 . We next derive the expressions forp j,i (E 1 ),p c j,i (E 1 ),p j,i (E 2 ) andp c j,i (E 2 ) by finding the total number of favorable cases and dividing by the total number of cases. Let x i ∼ U(0,W i ) and y i ∼ U(0,W i ). For notational convenience, define the following events: (i)S 1 j,i = P i k=j x k < P i k=j y k , and (ii)S 2 j,i = P i k=j x k +T s > P i+1 k=j y k . Let ¯ S 1 j,i and ¯ S 2 j,i denote the complement of these events. Lemma 15. (i)p j,i (E 1 ) = Pr[∩ i k=j+1 (S 1 j+1,k ∩S 2 j+1,k )] Pr[∩ i−1 k=j+1 (S 1 j+1,k ∩S 2 j+1,k )] , (ii)p c j,i (E 1 ) = Pr[(∩ i−1 k=j+1 (S 1 j+1,k ∩S 2 j+1,k ))∩ ¯ S 1 j+1,i ] Pr[∩ i−1 k=j+1 (S 1 j+1,k ∩S 2 j+1,k )] , (iii)p j,i (E 2 ) = Pr[(∩ i k=j+2 (S 1 j+1,k ∩S 2 j+1,k−1 ))∩S 1 j+1,j+1 ] Pr[(∩ i−1 k=j+2 (S 1 j+1,k ∩S 2 j+1,k−1 ))∩S 1 j+1,j+1 ] , (iv)p c j,i (E 2 ) = Pr[(∩ i−1 k=j+2 (S 1 j+1,k ∩S 2 j+1,k−1 ))∩S 1 j+1,j+1 ∩ ¯ S 2 j+1,i−1 ] Pr[(∩ i−1 k=j+2 (S 1 j+1,k ∩S 2 j+1,k−1 ))∩S 1 j+1,j+1 ] . The only remaining variable to be derived isp e j ,FH idle ,j = 1,2. To derive its value, we use the fact that both the transmitters cannot overhear each other. Lemma 16. p e j ,FH idle = 1,j = 1,2. 4.1.3 Determining the Achievable Edge-Rate Region in any Multi- hop Topology To determine the edge-rate region for a given multi-hop topologyT , recall that we first have to determine the expected service time at each edge which in turn requires the val- ues ofp e,T c,i ,p e,T l,i andp e,T idle for each edgee. To derive these probabilities for an edge, we will decompose the local topology around the edge into a number of two-edge topolo- gies, then find these probabilities for each two-edge topology, and finally find the net 37 probability by appropriately combining the individual probabilities from each two-edge topology. We will use the Flow in the Middle topology (Figure 4.6(a)) as an example throughout the section. Decomposition of the local topology arounde is easily achieved by evaluating how each edge ine’s neighborhood interferes with e, based on the definitions stated in Sec- tion 4.1.2. For example, the local topology around edge 4→ 5 can be decomposed into the following two-edge topologies: (i) Coordinated Stations: 5→ 6, (ii) Near Hidden Edges: None, (iii) Asymmetric Topology: 2→ 3 and8→ 9, and (iv) Far Hidden Edges: 1→ 2 and 7→ 8. The previous section discussed how to find the collision and idle probabilities for each individual two-edge topology. This section focusses on how to combine the probabilities obtained from each individual two-edge topology. Combining these probabilities must account for possible dependencies between the neighboring edges. For example, the transmitters of edges 1→ 2 and2→ 3 in the Flow in the Middle topology, which are both interfering with edge4→ 5, can hear each other. Hence, DATA transmission on these two edges will not occur simultaneously. Thus, the collision probabilities due to these two edges cannot be combined independently to find the aggregate collision probabilities at 4→ 5. We first present the scenarios where probabilities can be independently combined, and then discuss the scenarios where the dependencies have to carefully accounted for. The RTS and DATA collision probabilities can be independently combined if they are caused by two (or more) transmitters / receivers starting transmission in the same slot duration. For example, the RTS collision probability due to coordinated stations, and the DATA collision probability due to asymmetric topologies (if the CTS is received correctly at the other edge) can be independently combined. (For a complete list of events which can be independently combined, see the discussion following Lemmas 18 and 19.) 38 When the computation of any probability (either collision or idle probabilities) depends on the probability of the event that there is no ongoing transmission among a set of edges,N , dependencies have to carefully accounted for and combining proba- bilities is more involved. For example, the computation of the RTS collision probability due to far hidden edges and asymmetric topologies, and the computation of the DATA collision probability due to asymmetric topologies (if the CTS is not received correctly at the other edge) belong to this category. Also, the computation of the idle probability for coordinated stations, near hidden edges and asymmetric topologies belongs to this category. To understand how to compute the probability that there is no ongoing trans- mission among edges belonging toN , it is helpful to distinguish between two type of dependencies which can exist between these edges. Consider edge 4→ 5 in the Flow in the Middle topology (Figure 4.6(a)). In this topology, edges 1→ 2 and 8→ 9 interfere with edge 4→ 5 but do not interfere with each other, whereas 1→ 2 and 2→ 3 interfere with both 4→ 5 and each other. Gen- eralizing, (i) if two edges interfere with each other, then they will not be simultaneously scheduled (ignoring the extra RTS traffic due to the event that a colliding RTS trans- mission is taking place on both the edges, which is easily justified by Assumption 1), and (ii) if two edges do not interfere with each other, then they can be independently scheduled given that none of the edges which interfere with both are transmitting. For example, edges 2→ 3 and 8→ 9 will be independently scheduled given there is no transmission ongoing at edges4→ 5 and5→ 6. Note that prior works do not incorpo- rate the impact of these two dependencies ((i) and (ii)) in the evaluation of the collision and idle probabilities. We now state a lemma which finds the probability that there is an ongoing transmission on at least one of the edges in the given setN . The lemma follows directly from basic probability. In what follows, let X e denote the event that there is a transmission going on at edgee and note thatP(X e ) =K e,T λ e T s . 39 Lemma 17. P (∪ en∈N X en ) = X e i ∈N P(X e i )− X e i ,e j ∈N P(X e i ∩X e j )+...+ (−1) |N|−1 P(∩ e i ∈N X e i ), (4.2) where for N s ⊆ N , if S Ns denotes the set of edges in E which interfere with all the edges in N s P(∩ e i ∈Ns X e i ) =    0, if any two edges inN s interfere with each other Q e i ∈Ns P(X e i ) / 1−P ∪ e k ∈S Ns X e k |Ns|−1 , otherwise . Based on the previous discussion, we can derive the collision and idle probability for each edge in a given multi-hop network. For completeness, we state the value of each probability in the next three lemmas. The individual expressions are large because we combine the effect of each two-edge topology. However, each term in the expression can be traced to a term derived for one of the two-edge topologies. We first define the notation used in these lemmas. Denote byN e the set of edges which interfere with the edge under study e. Any edgee n ∈ E\e which either forms a coordinated station or asymmetric topology or near hidden edge or far hidden edge with e belongs to this set. We subdivide the edges inN e into subsets corresponding to the four two-edge topologies, and the coordinated station topologies and asymmetric topologies are further subdivided into two, giving us the following six sets: (i)N e 1 : edges which form a coordinated station with e and interfere with the receiver of edge e, (ii)N e 2 : edges which form a coordinated station with e and do not interfere with the receiver of edgee, (iii)N e 3 : edges which form a near hidden edge withe, (iv)N e 4 : edges which form an asymmetric topology with e being the edge with an incomplete view of the channel state, (v)N e 5 : edges which form an asymmetric topology with e being the edge which has the complete view of the channel state, and (vi)N e 6 : edges which form 40 a far hidden edge withe. Edges in the setN e 1 ,N e 3 ,N e 4 andN e 6 effect the RTS collision probabilities, edges in the setN e 4 andN e 6 effect the DATA collision probability and edges in the setN e 1 ,N e 2 ,N e 3 andN e 5 effect the proportion of idle time at the transmitter ofe. We first state the value of the DATA collision probability. We reuse the nota- tions used in Lemmas 7 and 14. In a multi-hop topology, P(E e,T 1 ) = P(E e,T 2 ) = 1− Q en∈N e 6 1−λ en E[S en ]p en w 0 . p e,T D,i ,p e,T D E 1 ,i and p e,T D E 2 ,i are defined and derived similarly to the corresponding variables in Section 4.1.2. Also, based on the discussion in Section 4.1.2, we setp en w 0 =    2 W 0 +1 ifN en 4 ∪N en 6 =φ andp en,T l,0 ≤p cutoff 2 Wm+1 otherwise . Lemma 18. (i)p e,T l,0 = 1− p e DATA ×p e ACK 1−P ∪ en∈N e 4 X en ∩E Re,Ten CTS ∪ ∪ en∈N e 6 X en ∩ E Re,Ren CTS ∪ E Ren ,Re CTS Q en∈N e 4 1 − λ en E S en p en w 0 1 − P(E e,T 1 ) ! , (ii) p e,T l,i = 1 − p e DATA × p e ACK 1 − 1 − p e,T D,i−1 − p e,T D E 1 ,i−1 − p e,T D E 2 ,i−1 p e,T l,0 − P i−1 k=0 p e,T D E 1 ,k Q i−1 u=k+1 p k,u (E 1 ) Q i−1 u=k+1 (p e,T c,u +(1−p e,T c,u)p e,T l,u ) p k,i (E 1 ) + p c k,i (E 1 )p e,T l,0 − P i−1 k=0 p e,T D E 2 ,k p k,i (E 2 )+p c k,i (E 2 )p e,T l,0 Q i−1 u=k+1 p k,u (E 2 ) Q i−1 u=k+1 (p e,T c,u +(1−p e,T c,u)p e,T l,u ) ! ,1≤i≤m. In the expression of p e,T l,0 , the first term within square brackets corresponds to the situation where a DATA collision is either caused due to asymmetric topologies due to a CTS loss on the edge between the receiver of the edge under study, R e , and the transmitter of a neighboring edge e n , T en , or far hidden edges due to CTS loss on the edge betweenR e andR en . And the second term within square brackets corresponds to a DATA collision due to asymmetric topologies whenT e andR en start transmitting a CTS the same time. The third term within square brackets denotes DATA collision following 41 eventE e,T 1 . In the expression of p e,T l,i , the two terms within square brackets correspond to the events where the previous exchange was not lost or lost due to DATA collisions following the eventsE e,T 1 orE e,T 2 . Note that the eventsX en ,∀e n ∈E and the CTS getting lost on an edge are indepen- dent, hence Lemma 17 is sufficient to derivep e,T l,i . We next state the value of the RTS collision probability. We reuse the notation used in Lemma 9. Additionally, we define the event X e,T = ∪ en∈N e 3 X en ∩E Ren ,Te CTS ∪ ∪ en∈N e 4 X en ∪ ∪ en∈N e 6 X en ∩E Ren ,Re CTS \ E e,T 1 ∪E e,T 2 which denotes that there is at least one ongoing transmission which will cause an RTS collision ate. Lemma 19. (i)p e,T c,0 = 1− p e RTS ×p e CTS h Q en∈N e 1 1−λ en E[S en ]p en w 0 i h Q en∈N e 3 1−2λ en E[S en ]p en w 0 ih 1−P X e,T ∪E e,T 2 i ! , (ii)p e,T c,i = 1− p e RTS ×p e CTS 1− 1−p e,T RTS,i−1 p e,T c,0 − P i−1 j=0 P (E j,i−1 ) 1−p i j +p i j p e,T c,0 !! ,1≤i≤m. In the expression for p e,T c,0 , the first term within square brackets corresponds to RTS collisions due to coordinated stations, while the second term corresponds to RTS colli- sions due to near hidden edges when the CTS sent byR en is successfully received atT e . Finally, the third term corresponds to an RTS collision due to eventX e,T . In the expres- sion for p e,T c,i , the two terms within square brackets correspond to the events where the previous exchange was not lost or lost due to the eventX e,T respectively. To complete the derivation of p e,T c,i , we next state the expressions for p e,T RTS,i and p e,T RTSnew,i . The expressions for these variables follow directly from their definition. For1≤i≤m, (i)p e,T RTS,i = h 1−p e,T RTS,i−1 P (X e,T )+ P i−1 j=0 P (E j,i−1 ) 1−p i j +p i j P (X e,T ) ih p e 1 ,AS c,i + 1−p e 1 ,AS c,i p e 1 ,AS l,i i −1 , 42 (ii) p e,T RTSnew,i = h 1−p e,T RTS,i−1 P (X e,T ) + P i−1 j=0 P (E j,i−1 )p i j P (X e,T ) ih p e 1 ,AS c,i + 1−p e 1 ,AS c,i p e 1 ,AS l,i i −1 . The next lemma states the value of p e,T idle . This lemma follows directly from the observation that any transmission on an edge belonging toN e 1 ∪N e 2 will freeze the backoff counter one, and any transmission on an edge belonging toN e 3 ∪N e 5 will freeze the backoff counter one only if the corresponding CTS is correctly received at T e . Lemma 20. p e,T idle = 1−P ““ ∪ en∈N e 1 ∪N e 2 Xen ” ∪ “ ∪ en∈N e 3 ∪N e 5 “ Xen ∩ ¯ E Ren ,Te CTS ””” −λeTs 1−λeTs , where ¯ E Ren ,Te CTS denotes the complement of eventE Ren ,Te CTS . Equation (4.1) along with the expressions derived in this section enable the deriva- tion of the expected service time at any edge in any multi-hop topology. Thus, these equations along with the following constraint characterizes the achievable rate region Λ E . X e∈Ov λ e E[S e ]< 1,∀v∈V, (4.3) where O v represents the set of outgoing edges from a node v. We sum over all outgo- ing edges from a node because the network queue for all outgoing edges at a node is the same. (Note that unlike prior works, the proposed methodology can be applied to topologies with nodes having multiple outgoing edges.) Finally, we now comment on the computational complexity of setting up the equa- tions for each edge. The complexity of the algorithm to decompose the local topology around an edgee into its constituent two-edge topologies is polynomial in|N e |. Com- puting the collision and idle probability for each two-edge topology takes constant time. Finally, the complexity of the algorithm to combine the individual collision and idle probabilities is equal to the number of non-zero terms in Equation (4.2). Each non-zero term in this equation corresponds to a distinct set of non-interfering edges inN e . So, the number of non zero terms taking an intersection over 1≤j≤|N e | edges is equal to 43 the number of distinct sets ofj non-interfering edges which isO(|N e | j ). However, the maximum number of non-interfering edges inN e is bounded by a constant in practical topologies [16]. Hence, the number of non-zero terms in Equation (4.2) is polynomial in|N e |. So, the overall computational complexity of setting up equations for an edgee is polynomial in|N e |. 4.1.4 Network Solution Determining the expected service time of all edges requires solving a coupled multivari- ate system of equations. We adopt an iterative procedure that uses the values of the idle and collision probabilities computed in the previous iteration for the current iteration. Proving the existence and uniqueness of a fixed point, and convergence of this iterative procedure to this fixed point is out of scope and left as future work. The interested reader is referred to [35, 38] for related fixed-point theory. We now give some insights into the complexity associated with these proofs. The same iterative procedure has been used to solve the multivariate equations arising in both IEEE 802.11-scheduled single-hop [9, 53] and multi-hop networks [31, 54]. Note that single-hop networks are topologically homogeneous, and hence the same fixed point equation governs the collision probability at each node. In contrast, for multi-hop net- works, the fixed point equation governing the collision and idle probabilities are dif- ferent for each node; even the structure of these equations can different for each node. Hence, proving uniqueness and convergence results is significantly more involved for multi-hop networks. Even for the simpler setting of single-hop networks, only a recent work [64] has derived conditions for the uniqueness of a fixed point solution for the most general case where nodes can be parametrically heterogeneous (but topologically homogeneous); while convergence of the iterative procedure is still not well understood. No progress has been made in the context of multi-hop networks yet. 44 In the absence of formal proofs, prior works have relied on extensive simulations to assess the convergence of the iterative procedure. We have adopted the same approach, and performed extensive simulations on different representative topologies. For these topologies, the average number of iterations to converge was 6 and the maximum was 8 irrespective of the initial conditions. For a detailed description of these topologies, please see Section 4.3. 4.2 Achievable Flow Rate Region The achievable flow rate region of a given multi-hop network and a collection of source- destination pairs is characterized by the set of the following constraints: r f ≥ 0 ∀f∈F λ e = X f∈F r e f ∀e∈E g(f)+ X e∈Iv r e f = X e∈Ov r e f ∀f∈F,∀v∈V ~ λ e ∈ Λ E where r e f denotes the flow rate of flow f flowing through edge e, g(f) =          r f ifv =s(f) −r f ifv =d(f) 0 otherwise andI v andO v denote the set of incoming edges into and outgoing edges from the node v respectively. The first constraint ensures non-negativity of flow rates, the second constraint expresses edge rates in terms of flow rates and the third is the standard flow conservation constraint. The final constraint says that the vector of edge rates ~ λ e induced at the edges should lie within the achievable edge-rate region. Note 45 that these constraints can now be fed into an optimization problem to optimize routing and rate allocation for different utility functions. However, building and studying these optimization problems is beyond the scope of this work. 4.3 Model Verification In this section, we verify the accuracy of the analysis by finding the achievable rate region for the four two-edge topologies and five different multi-hop topologies via sim- ulations and comparing it to the theoretically derived achievable rate region. The multi- hop topologies we use are either characteristic representative topologies, real topologies or randomly generated topologies. We also include the achievable rate region of optimal scheduling, derived using the methodology proposed by Jain et al. [39], to shed light on how far from the optimal IEEE 802.11 is. Recall that we assume that the overhead imposed by control message exchange and protocol headers is the same for both IEEE 802.11 and optimal scheduling. Also recall that we will be comparing the max-min fair rate allocation under IEEE 802.11 and under an optimal scheduler. 4.3.1 Two-edge topologies We plot the achievable edge-rate regions derived analytically and via simulations for the four two-edge topologies in Figures 4.5(a)-4.5(d). We make the following observations from these figures. (i) A close match between the analytical and simulation results verifies the accuracy of the analysis. (ii) The asymmetric topology has the smallest achievable rate region amongst the four two-edge topologies, which implies that the loss in throughput with IEEE 802.11 scheduling is largest for this topology. On the other hand, the coordinated station topology has the largest achievable rate region. (iii) In the asymmetric topology, even though IEEE 802.11 is highly unfair toe 1 in saturation 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Edge−rate at e 1 (in Mbps) Edge−rate at e 2 (in Mbps) Simulation Theoretical Optimal TDMA (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Edge−rate at e 1 (in Mbps) Edge−rate at e 2 (in Mbps) Simulation Theoretical Optimal TDMA (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Edge−rate at e 1 (in Mbps) Edge−rate at e 2 (in Mbps) Simulation Theoretical Optimal TDMA Saturation Operating Point (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Edge−rate at e 1 (in Mbps) Edge−rate at e 2 (in Mbps) Simulation Theoretical Optimal TDMA (d) Figure 4.5: Capacity Regions for different two-edge topologies. The packet loss rate for a1024 byte packet is equal to0.2 ate 1 ,0.3 ate 2 and0.5 at all the interference links. (All the rates are in Mbps.) (a) Coordinated stations. (b) Near hidden edges. (c) Asymmetric topology. (d) Far hidden edges. (The error in the maximum rate achieved at e 1 after fixing the rate ate 2 is less than10.1% for all the four plots.) conditions (see arrow on the figure) as also observed in [31, 32], with rate control it is possible to achieve a max-min rate allocation of 0.277 Mbps/edge, which is not that far from the max-min rate allocation of0.332 Mbps/edge achieved by an optimal scheduler. 47 1 2 3 6 5 4 7 8 9 (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Rate of outer flows: 1−>3, 7−>9 (in Mbps) Rate of Middle Flow: 4−>6 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (b) Figure 4.6: (a) The Flow in the Middle topology. (b) Achievable rate region for the Flow in the Middle topology. 4.3.2 Common Topologies The first two multi-hop topologies we consider have been used by prior works to study the performance of IEEE 802.11 in multi-hop networks: (a) Flow in the Middle topology which was used in [18, 80, 82], and (b) Chain topology which was used in [25, 68, 78]. Flow In the Middle Topology Figure 4.6(a) shows the Flow In the Middle topology. All links are assumed to be lossless. There are three flows in this topology: 1→ 3,4→ 6 and7→ 9. Flows1→ 3 and7→ 9 do not interfere with each other, but both of them interfere with flow4→ 6. 5 Since flows1→ 3 and7→ 9 are symmetric, we assume that they have equal rates. We plot the achievable rate of these two flows against the achievable rate for the middle flow (4→ 6) in Figure 4.6(b). We make the following observations from this figure. (i) The analytical and simulation curves are close to each other verifying the accuracy of 5 We say that two flows interfere with each other if any two edges over which they are routed interfere with each other. 48 n 3 2 1 n−2 n−1 (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Rate of flow 1−>15 (in Mbps) Rate of flow 15−>1 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (b) Figure 4.7: (a) Chain topology. (b) Achievable rate region for the Chain topology. the analysis. We compare the error between simulations and analysis for the maximum rate achieved by flow 4→ 6 when the rate of flows 1→ 3 and 7→ 9 is fixed. The error is less than 9%. Note that comparing the achievable flow rate region also verifies the analysis presented in Section 4.1 as the induced edge-rates should lie within the achievable-edge rate region for a set of flow-rates to be achievable (see Section 4.2). (ii) The achievable rate region with IEEE 802.11 scheduling is not convex. This non- convexity can also be seen, perhaps more clearly, in Figure 4.7(b) which shows the achievable rate region of the Chain topology, which is our next example. (iii) The max- min rate allocation for this topology with IEEE 802.11 is 0.194 Mbps/flow and is0.213 Mbps/flow with optimal scheduling. Thus, IEEE 802.11 achieves 91% throughput as compared to optimal scheduling at the max-min rate allocation. Chain Topology Figure 4.7(a) shows the Chain topology. All links are assumed to be lossless. We set n = 15. There are two flows in this topology: 1→ 15 and 15→ 1. We plot the achievable rate region of these two flows in Figure 4.7(b). We make the following observations from this figure. (i) The analytical and simulation curves are close to each 49 1 2 8 7 6 4 5 3 (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Rate of flow 1−>8 (in Mbps) Rate of flow 8−>1 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (b) Figure 4.8: (a) Square topology. (b) Achievable rate region for the Square topology. other verifying the accuracy of the analysis. We compare the error between simulations and analysis for the maximum rate achieved by flow 1→ 15 when the rate of flow 15 → 1 is fixed. The error is less than 12%. (ii) The achievable rate region with IEEE 802.11 scheduling is not convex for this topology also. (iii) The max-min rate allocation for this topology with IEEE 802.11 is0.09 Mbps/flow and is0.14 Mbps/flow with optimal scheduling. Thus, IEEE 802.11 achieves 64.3% throughput as compared to optimal scheduling at the max-min rate allocation. 4.3.3 Square Topology: Which Route The next topology we study is the Square topology of Figure 4.8(a). All links are assumed to be lossless. There are two flows present in this topology: 1 → 8 and 8 → 1. There are two possible routes for each flow: 1 → 2 → 3 → 4 → 8 and 1 → 5 → 6 → 7 → 8 for flow 1 → 8, and 8 → 4 → 3 → 2 → 1 and 8→ 7→ 6→ 5→ 1 for flow 8→ 1. We use his topology to illustrate that our anal- ysis yields the optimal routes as a by product, and show that IEEE 802.11 and optimal scheduling can have different optimal routes. 50 We plot the achievable rate region for this topology in Figure 4.8(b). We make the following observations from this figure. (i) Again, the simulation and analytical curves are close to each other. The error in the maximum rate achieved by flow 8→ 1 when the rate of flow 1→ 8 if fixed is less than 14%. (ii) The maximum throughput with IEEE 802.11, when only one of the flows is on, is equal to0.33 Mbps and is achieved by routing0.165 Mbps along one path and0.165 Mbps along the other path. (iii) When both flows are on, the max-min point with IEEE 802.11 is achieved by single-path routing with non-overlapping routes for the two flows, for example 1→ 8 routed along 1→ 2→ 3→ 4→ 8 and flow 8→ 1 routed along 8→ 7→ 6→ 5→ 1. However, optimal scheduling can achieve the max-min point by both single-path and multi-path routing. Thus, the optimal routing paths for IEEE 802.11 and optimal scheduling can be different. (iv) The max-min rate allocation with IEEE 802.11 is 0.18 Mbps/flow and is 0.213 Mbps/flow with optimal scheduling. Thus, IEEE 802.11 achieves 84.5% throughput as compared to optimal scheduling at the max-min rate allocation. 4.3.4 A Real Topology: Houston Neighborhood The next topology we choose is the real topology of an outdoor residential deployment in a Houston neighborhood [11]. The node locations (shown in Figure 4.9) are derived from the deployment and fed into the simulator. The physical channel that we use in the simulator is a two-ray path loss model with Log-normal shadowing and Rayleigh fading [67]. The ETX routing metric [22] (based on data loss in absence of collisions) is used to set up the routes. Nodes 0 and 1 are connected to the wired world and serve as gateways for this deployment. All other nodes route their packets towards one of these nodes (whichever is closer in terms of the ETX metric). The resulting topology as well as the routing tree is also shown in Figure 4.9. The loss rates at each link are determined from the simulator by letting each node send several broadcast messages one 51 by one and measure the number of packets successfully received at every other node. The topology information and loss rates are fed into the analytical model to find the achievable rate region for this topology. There are 16 flows in this topology. Hence, we only compare the max-min rate allocation from simulations and theory. A very good match is observed: the simulator allocates46 Kbps/flow whereas the theory allocates44 Kbps/flow (error =4.4%). Optimal scheduling allocates67.3 Kbps/flow at the max-min rate allocation. Thus, IEEE 802.11 achieves 65.3% of the throughput as compared to optimal scheduling at the max-min rate allocation. 0 2 3 4 6 7 9 10 11 12 13 15 16 17 1 8 14 0.56 0.23 0.42 0.50 0.29 0.23 0.03 0.57 0.15 0.31 0.39 0.26 0.32 0.03 0.17 5 Figure 4.9: Topology from the deployment in a Houston neighborhood. Arrows show the routing paths and the numerals on top of an arrow is the probability of loss of a1024 byte packet on that link. Dashed lines represent the interference links. 4.3.5 Random Topology We create the final topology by randomly placing 75 nodes in a 1000m×1000 m area. Both transmission and interference range are set equal to 200m. We assume links used for routing packets to be lossless and assumep e RTS = p e CTS = 0.4 on all the other links as links used in routing paths typically are low loss links. We select6 source-destination pairs at random. We compare the max-min rate allocation from simulations and theory. A very good match is observed: the simulator allocates 94 Kbps to five of the flows and 650 Kbps to the sixth flow whereas theory allocates96Kbps to five of the flows and600 52 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Rate of outer flows: 1−>3, 7−>9 (in Mbps) Rate of Middle Flow: 4−>6 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (a) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Rate of outer flows: 1−>3, 7−>9 (in Mbps) Rate of Middle Flow: 4−>6 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (b) Figure 4.10: (a) Achievable Rate Region for the Flow in the Middle topology for 100 byte packets and 1 Mbps data rate. (b) Achievable Rate Region for the Flow in the Middle topology for1024 byte packets and11 Mbps data rate. Kbps to the sixth flow (error = 7.6%). Optimal scheduling allocates 141.7 Kbps to five of the flows and 706 Kbps to the sixth flow at the max-min rate allocation. Thus, at the max-min point, IEEE 802.11 achieves76.35% of the total sum throughput as compared to optimal scheduling. 4.3.6 Different Network Parameters All the previous comparisons were made for a particular set of network parameters. In this section, we investigate the accuracy of the analysis when the network parameters are modified from their default values. We compare the achievable rate region derived via simulations and theory for the Flow in the Middle topology (Figure 4.6(a)) for: (a) 100 byte DATA packets at 1 Mbps data rate in Figure 4.10(a), and (b) 1024 bytes packets at 11 Mbps data rate in Figure 4.10(b). The error between simulations and analysis for the maximum rate achieved by flow 4→ 6 when the rate of flows 1→ 3 and7→ 9 is fixed is less than15% for both scenarios. Note that for both the scenarios, Assumption 1 does not hold, and hence we see a larger error. For smaller DATA packets, the reason why Assumption 1 does not hold is obvious. However, why increasing the 53 data rate to 11 Mbps makes this assumption invalid is not obvious as the DATA packet size is still two orders of magnitude larger than the RTS packet size. In IEEE 802.11, the PHY header contains information used to determine the data rate of the incoming transmission (to allow auto-rate adaptation [1]), and hence is always transmitted at 1 Mbps. And the PHY layer header is exchanged for both control (RTS, CTS and ACK) and DATA packets. For a data rate of 11 Mbps, the transmission time of the 1024 byte DATA packet is comparable to the transmission time of the PHY layer header which is transmitted at 1 Mbps. Hence, the transmission time of a RTS packet is comparable to the transmission time of a DATA packet, which violates Assumption 1. Note that this is a protocol issue which needs to be fixed as this violates the basic premise of protocol design that the load due to control packets should be a small fraction of the total load. From Figures 4.10(a) and 4.10(b), we also observe that IEEE 802.11 achieves more than84% throughput at the max-min rate allocation as compared to optimal scheduling for both the scenarios. Note that in both these examples the overhead is significantly larger than in previous scenarios. 4.3.7 Summary We now summarize the observations made in this section. (i) Under the assumptions we make, our analysis is accurate as we incorporate all the events leading to collisions/busy channel in our proofs. And our assumptions are shown to be accurate via simulations as the analytical results have an average error of 9% and a maximum error of15%. (ii) The achievable rate region with IEEE 802.11 scheduling is non-convex. (iii) IEEE 802.11 achieves more than 64% throughput as compared to optimal scheduling at the max-min rate allocation for all the topologies studied in this chapter. This is an interesting and unexpected observation. We explain this observation in Chapter 5 where we derive the 54 worst case bounds on the performance of IEEE 802.11 as compared to optimal. (iv) The optimal routing paths for IEEE 802.11 and optimal scheduling can be different. 4.4 Network Solution Without The Iterative Procedure As discussed in Section 4.1.4, we need an iterative procedure to solve the coupled multi- variate system of equations derived in Section 4.1. In this section, we discuss if it is pos- sible to decouple the equations to avoid using an iterative procedure by sacrificing some accuracy in the analysis. We look at the following questions: (i) under what network conditions can the equations be decoupled without an unreasonable loss in accuracy, and (ii) what are the approximations to be made to remove the coupling. A careful look at Lemmas 17 and 18 and the expression for K e,T derived in Sec- tion 4.1.2 tells us that the equations cannot be decoupled for networks with a non- negligible probability of RTS/CTS loss on edges without a significant loss in accuracy. For networks with a negligible probability of RTS/CTS loss, one can make the fol- lowing two approximations to decouple the equations. (I) The first approximation is to replace λ e E[S e ] by min λe λsat,ne ,1 in the expressions for the following two prob- abilities: (i) the DATA collision probability (Lemma 18), and (ii) the RTS collision probability (Lemma 19). λ sat,n denotes the saturation throughput of a WLAN with n transmitters transmitting to a single receiver (derived in [9]) and n e =|N e | is the number of edges interfering with e. Note that λ e E[S e ] is upper bounded by 1. Since approximatingλ e E[S e ] by its upper bound is inaccurate whenλ e is small, in these situa- tions we replaceE[S e ] by1/λ sat,ne . (λ sat,n as a function ofn flattens out rather fast [9]. As a result, even if just a few neighboring edges are saturated,1/λ sat,n would be a good lower bound since the topology that minimizes service times is the one where all nodes are within range.) (II) The second approximation is to approximate P(∩ e i ∈Ns X e i ) = 55 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Rate of outer flows: 1−>3, 7−>9 (in Mbps) Rate of Middle Flow: 4−>6 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Rate of flow 1−>15 (in Mbps) Rate of flow 15−>1 (in Mbps) Simulation (802.11) Theoretical (802.11) Optimal Scheduling (b) Figure 4.11: (a) Achievable rate region for the Flow in the Middle topology with the approximations of Section 4.4. Error between simulations and analysis is less than 20%. (b) Achievable rate region for the Chain topology with the approximations of Section 4.4. Error between simulations and analysis is less than12%. Q e i ∈Ns P(X e i ) / 1−P ∪ e k ∈S Ns X e k |Ns|−1 when no two edges inN s interfere with each other in Lemma 17 with Q e i ∈Ns P(X e i ) / 1− P e k ∈S Ns P (X e k ) |Ns|−1 . With the first approximation, the DATA collision probabilities can be derived for each edge independently. Now, given the DATA collision probabilities at each edge, with the second approximation, one can find the RTS collision probabilities and idle probability at each edge independently. Using these approximations will introduce some inaccuracies. However for the topologies studied in this chapter, the inaccuracies are not large. For example, Fig- ures 4.11(a) and 4.11(b) compare the achievable rate region derived with these approx- imations with the simulation results for the Flow in the Middle topology and the Chain topology respectively. With the two approximations, the maximum error is less than 20% for both the topologies. 56 4.5 Extensions We now discuss how to modify the analysis if some of the simplifying assumptions made on the physical layer model and packet sizes do not hold. 4.5.1 Different Transmission Rates and Packet Sizes Different edges in the network can have different average transmission rates due to the automatic rate adaptation employed at the IEEE 802.11 physical layer. Moreover, there can be multiple sized packets flowing through the network. Both these events will result in different transmission times at each edge. To account for these, the pdf of the trans- mission time for each edge would be derived based on the packet pdfs and the automatic rate adaptation algorithm, the expected service time at each edge would be derived as a function of the transmission time at that edge, and then the law of total probability would be used to integrate out this dependence. 4.5.2 More Detailed Physical Layer Model The analysis in Section 4.1 assumed a binary and pairwise interference model. How- ever, recent measurement studies suggest that interference is neither binary [60] nor pairwise [23]. Even though our main objective is to analyze the achievable rate region under IEEE 802.11 MAC, it is important to discuss how the derivation of Section 4.1 gets modified if a more realistic interference model is used. First lets discuss how to remove the binary assumption. [70] proposed a non-binary interference model by associating a capture and a deferral probability to model that a collision might not result in packet loss and the channel might not be sensed busy at a node inspite of the ongoing interfering transmission. For each of the two-edge topologies, incorporating the capture and deferral probabilities will change the collision 57 and idle probabilities. Here we illustrate how to incorporate these probabilities for the coordinated stations only, the analysis for the remaining two-edge topologies will be similarly modified. Lets consider the idle and collision probabilities at edge e 1 . (i) p e 1 ,CoS idle : The backoff counter at e 1 will be frozen only if the ongoing transmission at e 2 causes the channel to be sensed busy at T e 1 (transmitter of e 1 ). (ii) p e 1 ,CoS c,i : The following two modifications will be required. First, the RTS collsion probability will be multiplied with the probability that a simultaneous transmission also causes a packet loss (complement of the capture probability). Second, the event that an ongoing transmission ate 2 does not cause the channel atT e 1 to be sensed busy can also lead to a RTS collision. (iii)p e j ,CoS l,i : Simultaneous RTS transmissions may get captured on both the edges, which will lead to simultaneous CTS transmissions on both the edges. If both these CTS transmissions also get captured, simultaneous DATA transmissions will ensue on both the edges. Now lets discuss how to remove the pairwise assumption. Many simultaneous trans- missions can cause deferral/collision at a node even though each of them individually might not have the same effect. [42, 63] proposed a model for this physical layer effect. For each edge e, there is a deferral and collision probability associated at both the receiver and the transmitter ofe defining its behavior if a setS of edges are transmitting simultaneously. Thus, instead of considering the effect of interfering edges on e, we should consider the effect of interfering sets one. Given that each edge in the setS does not cause any interference individually, the set of edges S can interact in only one of the following two ways: (i) either it causes a deferral at the transmitter, which can be analyzed using techniques developed for analyzing coordinated stations, or (ii) it does not cause a deferral at the transmitter but causes a collision at the receiver, which can be analyzed using techniques developed for analyzing asymmetric topologies. 58 Hence, even with a non-binary and non-pairwise interference model, the essence of the analysis in terms of decomposing a local topology around an edge into a number of interfering sets and then combining them using the results from Section 4.1.3, remains unchanged. So, we believe that the analysis presented in this chapter can be extended to a more realistic interference model. 59 Chapter 5: Worst Case Bounds on IEEE 802.11 The objective of this chapter is to evaluate the worst case performance of IEEE 802.11 scheduling against optimal scheduling in multi-hop networks. To fulfil this objective, we characterize the neighborhood topology which minimizes the ratio of the throughput achieved by IEEE 802.11 over that achieved by the optimal at the edge of interest at the max-min rate allocation. Neighborhood Topology: A neighborhood topology is one where there is a particular edge of interest, and all the other edges in the topology interfere with this edge. The edge of interest is assumed to be the congested edge in the neighborhood topology, and we denote it bye c . The set of neighboring edges is denoted byN ec . For convenience, we adopt the convention thate c / ∈ N ec . (Next paragraph discusses why we choose to look at neighborhood topologies only.) We also assume that no node has multiple outgoing edges. (Section 5.6.2 discusses how our results change by relaxing this assumption.) Finally, we also assume that there are no physical layer losses in absence of collisions. Section 5.6.1 discusses how modifying this assumption changes our results. Note that instead of deriving worst case bounds over all possible multi-hop topolo- gies and traffic matrices, we simplify our problem by studying neighborhood topolo- gies only. (Note that the traffic matrix in neighborhood topologies will obviously have 60 only single-hop flows over each edge.) This simplification does not come at the cost of reducing the scope of the objective which is to compare the worst case performance of IEEE 802.11 against optimal scheduling in multi-hop networks. Studying neighbor- hood topologies suffices because there is a direct one-to-one connection in the through- put performance of neighborhood topologies and multi-hop topologies. This connection exists because the throughput performance of any multi-hop topology is dictated by the throughput performance of its congested neighborhood topologies. The rationale is similar to wired networks where the throughput performance of a multi-hop topology is dictated by its congested links [28,61,62]. For wireless networks, the idea of congested links gets replaced by congested neighborhoods. Now lets try and understand what does the term congested neighborhood imply. In a multi-hop topology, for any rate allocation which operates close to the boundary of the capacity region, each end-to-end flow passes through the neighborhood of at least one edge whose queue is fully utilized. These edges are called congested edges, and their neighborhoods are termed congested neighborhoods. (In our definition of neighborhood topologies, the congested edge is the edge of interest.) Each end-to-end flow can potentially pass through a lot of congested neighborhoods, however the throughput achieved by each end-to-end flow is dictated by the throughput achieved in the most congested neighborhood it passes through. Hence, studying the throughput performance of the worst case neighborhood topology is sufficient to under- stand the worst case performance in multi-hop networks. Please see Section 5.5 for a more detailed exploration of the relationship between the throughput performance of multi-hop flows and congested neighborhoods. Max-Min Rate Allocation: To compare the throughput at the congested edge, we have to pick up a rate allocation point to compare. Amongst the commonly used rate allo- cation points, like proportionally fair rate allocation, maximum sum throughput rate 61 allocation and max-min rate allocation, we choose the max-min rate point because it yields the worst throughput ratio at edgee c among the three. Intuitively, more collisions are observed at the max-min allocation which results in a lower throughput with IEEE 802.11. (We have also verified this statement by brute force for up to 5 link neighbor- hoods.) Note that a similar methodology can be used to compare the throughput of the congested edge at any other rate allocation point. 5.1 Topology Characterization In this section, we define two metrics which will be used to characterize the worst case neighborhood topology. In the following definition, a time slot is equal to the transmis- sion time of one packet. Definition 1 (Interference Factor). The Interference Factor is the minimum number of time slots optimal scheduling will require to schedule all edges inN ec at least once. Lets look at what interference factor is using an example. Consider the neighborhood around edge 2 → 3 in the topology shown in Figure 5.1. N 2→3 has 5 edges. First scheduling edges 1→ 2 and 4→ 5 simultaneously, and then scheduling edges 3→ 9,3→ 10 and3→ 4 one by one schedules each of the5 edges at least once in four time slots. Thus, the interference factor for this topology is equal to 4. Note that the set of edges scheduled simultaneously form a maximal independent set. Thus, the interference factor can also be viewed as the minimum number of maximal independent sets required to cover all the edges inN ec . The throughput achieved by optimal scheduling at e c at the max-min allocation depends on the interference factor. We state the relationship between the two in the next theorem after defining the following two variables. Let k Nec denote the interfer- ence factor of the set of edgesN ec , and letS Nec I denote the set of maximal independent 62 11 9 1 2 3 4 5 6 7 8 10 Figure 5.1: The solid lines connect nodes which interfere with each other. There are four flows in this topology: 1→ 8, 3→ 9, 3→ 10 and 8→ 11. Congested queues are indicated with a symbol depicting a queue. sets scheduled by optimal scheduling to minimize the number of time slots to schedule edges inN ec at least once. Theorem 1. Optimal scheduling yields a throughput of k Nec +1 T s −1 at edgee c at the max-min rate allocation. Proof. e c is the congested edge inN ec ∪e c by assumption. Hence, by definition of the max-min rate allocation, e c has the least rate. Now, we prove that among the edges in N ec , there exists an edge e i which has the same rate at e c . We prove by contradiction. Lets say all edges in N ec have rate higher than e c . LetS Nec I denote the set of maximal independent sets scheduled by optimal scheduling to minimize the number of time slots to schedule edges in N ec at least once. Then, each edge in N ec belongs to more than one maximal independent set inS Nec I . Lets consider I j ∈S Nec I . All edges in I j are contained in some other maximal independent also belonging toS Nec I . Thus, removing I j fromS Nec I still covers all edges in N ec , however, it requires one less set to do so. Hence, a contradiction. Let e i denote the edge in N ec which has the same rate as e c . Increasing the rate at either implies scheduling it once more, which will reduce the rate at the other edge. Thus, the max-min allocation will allocate k Nec +1 T s −1 rate to bothe i ande c . 63 Let I min denote the maximal independent set which has the minimum cardinality amongst the sets inS Nec I . (Equivalently, I min = argmin n |I| :I∈S Nec I o , where|.| denotes the cardinality of a set.) Similarly, letI max denote the maximal independent set which has the maximum cardinality amongst the sets inS Nec I . (Equivalently, I max = argmax n |I| :I∈S Nec I o .) Definition 2 (Scheduling Set Ratio). The Scheduling Set Ratio is defined as the ratio of the cardinality ofI min andI max (=|I min |/|I max |). The scheduling set ratio value lies between 0 and 1, and the more homogeneous is the distribution of the number of edges in the maximal independent sets inS Nec I , the closer to 1 would be its value. For example, the scheduling set ratio of N 2→3 in the topology of Figure 5.1 is equal to0.5. 5.2 Worst Case Neighborhood In this section, we give an O(|N ec | 3 ) algorithm to determine the worst case neighbor- hood topology. We first define the following two variables. Let N e busy denote the set of edges on which any transmission will cause the channel to be sensed busy at T e . Transmissions on these edges contribute to the busy probability at edge e (denoted by p e busy ). Recall that the busy probability is the complement of the idle probability. With the assumption of no physical layer losses in absence of collisions, edges forming coordinated stations, near hidden edges and asymmetric edges wheree is aware of the channel belong to this set (see Section 4.1.2). Let N e coll denote the set of edges on which any transmission will not cause the channel to be sensed busy at T e , but will cause an RTS collision at R e . Transmissions on these edges contribute to the RTS collision probability. With the 64 assumption of no physical layer losses in absence of collisions, edges forming asym- metric edges withe not being aware of the channel and far hidden edges belong to this set (see Section 4.1.2). Recall that the RTS collision probability also depends on the backoff window value atT e . However,p e c,i ,i≥ 1 is a deterministic function ofp e c,0 . So, we adopt the convention that whenever we refer to the RTS collision probability from now on, we implyp e c,0 . The performance of a given neighborhood topology depends on the following two interference characteristics. (i)∀e i ∈ N ec , how e i and e c interfere with each other. In other words whethere i ande c interfere as coordinated stations, or as near hidden edges or asymmetrically or as far hidden edges. (ii)∀e i ,e j ∈ N ec ,i6= j, whether e i and e j interfere with each other or not. Note that whether e i ande j interfere as coordinated stations or asymetrically etc does not impact the throughput one c for either IEEE 802.11 or optimal scheduling (see Section 4.1.3). In practice, physical layer models like the disk model or the distance-based attenua- tion model with or without random shadowing fading will yield geometrical constraints which will disallow the construction of any weird interference characteristics one may come up with. However, in this section, we neglect these geometrical constraints, and find the worst case neighborhood topology without ensuring that it is constructible in practice. One might wonder why even consider topologies which may not occur in real networks. The advantage of this exercise is twofold. Firstly, it allows us to find the absolute worst case, independently of the physical layer model. Secondly, and more importantly, the intuition derived from the characterization of the worst case neighbor- hood topology without any physical layer constraints also holds when we find the worst case topologies after imposing these constraints (as we observe in Section 5.3). 65 5.2.1 The Algorithm Before presenting the algorithm, we first define two new variables which will be used to describe the algorithm. Letk ec busy andk ec coll denote the interference factor for the edges inN ec busy andN ec coll respectively. If none of the edges inN ec busy (N ec coll ) interfere with each other, thenk ec busy = 1 (k ec coll = 1) ; and if all the edges inN ec busy (N ec coll ) interfere with each other, thenk ec busy =|N ec busy | (k ec coll =|N ec coll |). The algorithm has two parts. The first part exhaustively searches over the entire space of the number of edges in N ec coll (which varies from 0 to|N ec |), k ec busy (which varies from 1 to|N ec busy |) and k ec coll (which varies from 1 to|N ec coll |). More precisely, if TR worst denotes the throughput ratio of the worst case neighborhood topology and TR worst (|N ec coll |,k ec busy ,k ec coll ) denotes the worst case throughput ratio given|N ec coll |,k ec busy andk ec coll , then TR worst = min |N ec coll | min k ec busy ,k ec coll TR worst (|N ec coll |,k ec busy ,k ec coll ) (5.1) The second part of the algorithm determines the value of TR worst (|N ec coll |,k ec busy ,k ec coll ). DeterminingTR worst (|N ec coll |,k ec busy ,k ec coll ) To determine TR worst (|N ec coll |,k ec busy ,k ec coll ), we have to derive the two interference char- acteristics stated at the start of the section. We first state a theorem to specify how e c interferes with all the edges inN ec . Theorem 2. The worst case neighborhood topology has the following properties. (a) All edges inN ec busy interfere as near hidden edges withe c . (b) All edges in N ec coll interfere as asymmetric edges with e c . By definition of N ec coll , betweene i ∈N ec coll ande c ,e c is the edge which is not aware of the channel state. 66 Proof. Among the kind of two-edge topologies belonging toN ec busy andN ec coll , near hid- den edges and asymmetric topology cause the maximum throughput loss at e c respec- tively (see Section 4.3). Hence the worst case topology will contain only these two kind of two-edge topologies. Since, the number of edges inN ec coll andN ec busy are known (|N ec busy | =|N ec |−|N ec coll |), Theorem 2 completely specifies howe c interferes with edges inN ec . We next determine which of the e i ,e j ∈ N ec ,i6= j interfere with each other in the worst case neighborhood topology. We break this task into the following three steps: (i) We first determine how many edge pairs with one edge lying inN ec busy and the other lying inN ec coll interfere with each other. (ii) We next determine which of thee i ,e j ∈N ec busy ,i6= j interfere with each other. (iii) Finally, we determine which of thee i ,e j ∈ N ec coll ,i6= j interfere with each other. We first state a theorem to determine how many edge pairs with one edge lying in N ec busy and the other lying inN ec coll interfere with each other. Theorem 3. The worst case neighborhood topology has the following properties. (a) No edge inN ec busy interferes with any edge inN ec coll . (b) The throughput achieved by optimal scheduling is equal to max k ec busy ,k ec coll +1 T s −1 . Proof. Whether an edge inN ec busy interferes with any edge inN ec coll or not, does not effect the throughput ate c of IEEE 802.11. However, it may reduce (depending on which edge pairs interfere) the throughput at e c of optimal scheduling. So, to construct the worst case topology, we maximize the throughput achieved ate c by optimal scheduling which will be obtained when no edge inN ec busy interferes with any edge inN ec coll . 67 When no edge in N ec busy interferes with any edge in N ec coll , the interference factor of N ec is equal to max k ec busy ,k ec coll . Thus, by Theorem 1, the throughput achieved ate c by optimal scheduling at the max-min rate allocation is equal to 1 (max (k ec busy ,k ec coll )+1)Ts . We next state a theorem which determines how many edge pairs in N ec busy interfere with each other. We use the following notation in the theorem. Let the set containing the k ec busy maximal independent sets covering all the edges in N ec busy be denoted byS N ec busy I . Note that all the edges belonging to a particular maximal independent set I j ∈S N ec busy I do not interfere with each other. Theorem 4. The worst case neighborhood topology has the following properties. (a) Two edges belonging to different maximal independent sets I j ,I k ∈S N ec busy I ,j6= k, interfere with each other. (b) An edge inN ec busy will be contained in only one maximal independent set inS N ec busy I . (c)|I j | =|N ec busy |/k ec busy ,∀I j ∈S N ec busy I . (d) p ec busy ≤ k ec busy 1−(1−λ ec T s ) |N ec busy |/k ec busy where λ ec is the packet arrival rate at edgee c . Proof. Whether edges in N ec busy interfere with each other or not, effects only the value of p ec busy (the busy probability at e c ) because of the following two reasons. (i) Edges in N ec busy do not cause DATA collision at e c . (ii) The RTS collision probability at e c from all the edges inN ec busy will be approximately the same irrespective of how they interfere with each other (see Section 4.1.2). Note that the throughput achieved by optimal scheduling has already been fixed in Theorem 3, hence minimizing the throughput ratio ate c implies minimizing the through- put achieved by IEEE 802.11 ate c which in turn implies maximizingp ec busy . We first prove subcase(a). If two edges in N ec busy interfere with each other, then the busy probability at e c is higher than when they do not interfere with each other. By 68 definition, the edges belonging to aI∈S N ec busy I do not interfere with each other. Since there is no such restriction on distinct edges belonging to different maximal independent sets, they will interfere with each other in the worst case neighborhood. We next prove subcase (b). If an edge belongs to two maximal independent sets I j ,I k ∈S N ec busy I , removing it from one of them will increase the busy probability at e c without changing the value of k ec busy , and hence, the throughput achieved by optimal scheduling. So, the worst case topology will not have any edge contained in two or more different maximal independent sets inS N ec busy I . We prove subcases (c) and (d) using a sequence of lemmas. Lemma 21. Increasing the rate at any of the edges inN ec busy reduces the rate ate c . Proof. The busy probability at e c is a monotonically increasing function of the rate at e i ,∀e i ∈N ec busy (see Lemma 17). Hence, increasing the rate at any of the edges inN ec busy will increasep ec busy , which will in turn reduce the rate achieved ate c . Lemma 22. For a rate allocation assigning the maximum possible equal rate to all edges in N ec busy ∪e c , there always exists a neighborhood topology with the worst case throughput ratio ate c withe c being the congested edge. Proof. Lets say that edge e i ∈ N ec busy is the congested edge at the maximum possible equal rate allocation and note c . LetN e i denote the set of edges inN ec busy which interfere with e i . Now, change the neighborhood of e c so that the edges in N ec interfere with e c in exactly the same fashion as the edges in N e i interfere with e i . Now, e c becomes the congested edge. SinceN e i ⊆ N ec , the performance of optimal scheduling can only improve, and hence, the throughput ratio of this new topology is either worse or the same as the previous one. Hence, we have a topology with the worst case throughput ratio withe c as the congested edge. 69 This lemma shows that looking only at topologies wheree c is the congested edge is sufficient to find the worst case neighborhood topology for the rate allocation assigining the maximum possible equal rate to all edges inN ec busy ∪e c . Lemma 23. IEEE 802.11 allocates equal rates to all the edges in the worst case neigh- borhood topology at the max-min allocation. Proof. The rate allocation which assigns the maximum possible equal rate to all the edges inN ec busy ∪e c has the following two properties: (i) Increasing the rate at any of the edges inN ec busy reduces the rate ate c . (ii)e c is the congested edge. These two properties imply that the rate at none of the edges can be increased without either reducing the rate at e c or making the system unstable. Thus, for a neighborhood topology, the rate allocation which assigns the maximum possible equal rate to all the edges is also the max-min rate allocation. Now, we prove the last two subcases of the theorem. At the max-min allocation, all edges in N ec busy have a rate equal to λ ec . Since all edges belonging to distinct max- imal independent sets inS N ec busy I interfere with each other, and edges belonging to the same independent set do not interfere with each other, by Equation (4.2), p ec busy = P I j ∈S N ec busy I 1−(1−λ ec T s ) |I j | . Maximizingp ec busy under the constraint P I j ∈S N ec busy I |I j | = |N ec busy | yields|I j | =|N ec busy |/k ec busy ,∀I j ∈S N ec busy I . Thus, the maximum value ofp ec busy is equal tok ec busy 1−(1−λ ec T s ) |N ec busy |/k ec busy . The theorem implies that distributing the|N ec busy | edges uniformly amongst thek ec busy independent sets (maximizing the value of scheduling set ratio) minimizes the through- put ratio. Theorem 4 defines the topology precisely only when |N ec busy |/k ec busy is an inte- ger. Otherwise, Theorem 4(d) can be used to derive a lower bound on the value of TR worst (|N ec coll |,k ec busy ,k ec coll ). 70 Finally, we now state a theorem to determine how many edge pairs inN ec coll interfere with each other. Let the set containing the k ec coll maximal independent sets covering all the edges inN ec coll be denoted byS N ec coll I . Theorem 5. The worst case neighborhood topology has the following properties. (a) Two edges belonging to different maximal independent sets I j ,I k ∈S N ec coll I ,j6= k, interfere with each other. (b) An edge inN ec coll will be contained in only one maximal independent set inS N ec coll I . (c)|I j | =|N ec coll |/k ec coll ,∀I j ∈S N ec coll I . (d)p ec c,0 ≤ k ec coll 1−(1−λ ec T s ) |N ec coll |/k ec coll whereλ ec is the packet arrival rate at edge e c . The proof of Theorem 5 follows along similar lines as the proof of Theorem 4. The only difference is as follows. Whether edges in N ec coll interfere with each other or not, effects only the value ofp ec c,0 because of the following two reasons. (i) Edges inN ec coll do not cause the channel to be sensed busy at T ec . (ii) The DATA collision probability at e c from all the edges in N ec coll will be approximately the same irrespective of how they interfere with each other (see Section 4.1.2). Also, similar to Theorem 4, Theorem 5 is exact only when |N ec coll |/k ec coll is an integer, otherwise Theorem 5(d) can be used to derive a lower bound on TR worst (|N ec coll |,k ec busy ,k ec coll ). Using the upper bounds on p ec busy and p ec c,0 from Theorems 4 and 5, we derive a lower bound on TR worst (|N ec coll |,k ec busy ,k ec coll ). Applying these bounds to evaluate Equa- tion (5.1), we find that the lower bound onTR worst always occurred when|N ec busy |/k ec busy and|N ec coll |/k ec coll were integers. Recall that the bound is exact for these conditions. 71 0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 |N e c | Throughput Ratio (IEEE 802.11/Optimal) Worst Case Worst Case assuming Disk Model Average Ratio Figure 5.2: Worst case and average throughput ratios for IEEE 802.11 against optimal scheduling for different neighborhood sizes. 1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Interference Factor Throughput Achieved at e c (in Mbps) IEEE 802.11 Optimal Scheduling (a) 1 2 3 4 5 6 7 8 9 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Interference Factor Throughput Ratio (IEEE 802.11/Optimal) (b) Figure 5.3: (a) Worst case throughput achieved at e c as a function of the interference factor for|N ec | = 10. (b) Worst case throughput ratio achieved ate c as a function of the interference factor for|N ec | = 10. Computational Complexity Now we comment on the computational complexity of the algorithm to findTR worst . In Equation (5.1), the iteration over|N ec coll | requires|N ec |+1 steps, the iteration overk ec busy andk ec coll requires|N ec coll |(|N ec |−|N ec coll |) steps. Each step requires evaluating the value of TR worst (|N ec coll |,k ec busy ,k ec coll ) which takes constant time. Thus, the overall algorithm has a computational complexity ofO(|N ec | 3 ). 72 5.2.2 Numerical Results The solid line in Figure 5.2 plots the worst case throughput ratio as a function of the number of neighboring edges (|N ec |) for the system parameters summarized in Table 3.2. Note that we use the default parameters of IEEE 802.11, and do not optimize on the parameters to improve the throughput. (Unless explicitly stated, all the numerical results presented in this chapter also assume the system parameters of Table 3.2.) Note that in typical topologies, one expects that the number of neighboring edges per neighborhood will be less than 20. Hence, we display the plot till N ec = 20. The ratio keeps on decreasing as|N ec | increases, and it becomes16% for|N ec | = 20. Now, we characterize the worst case topology. |N ec coll | for the worst case topology is always equal to|N ec |. This is not surprising as collisions cause exponential backoffs reducing the throughput drastically. The interference factor for the worst case topology is always equal to 1 irrespective of the value of|N ec |. This is surprising, as a smaller interference factor implies fewer edges interfering with each other which should actu- ally improve the throughput performance of IEEE 802.11. However, note that optimal scheduling also has a better throughput performance for smaller interference factors. Thus, increasing the interference factor deteriorates the performance of both schedul- ing schemes. However, the deterioration is more significant for optimal scheduling than IEEE 802.11. Figures 5.3(a) and 5.3(b) show this by plotting the worst case throughput achieved ate c for IEEE 802.11 and optimal scheduling and the corresponding ratio for different values of the interference factor. As an example of how the worst case topology looks like, we plot this topology for |N ec | = 8 in Figure 5.4(a). It contains8 non-interfering asymmetric edges. 73 e c (a) e c (b) Figure 5.4: Worst case topology for|N ec | = 8. (a) Worst Case with no assumptions on the physical layer. (b) Worst case assuming disk model and transmission range equal to interference range. 5.3 Imposing Practical Constraints The worst case topology derived in the previous section might not be constructible in practice due to the geometrical constraints imposed by the physical layer. For exam- ple, the number of edges which do not interfere with each other but interfere with e c is bounded in practice [16]. Hence, the worst case performance of IEEE 802.11 for practical topologies should be better than the one derived in the previous section. In this section, we assume disk model at the physical layer. We also assume that the transmission range is equal to the interference range. Interference is still assumed to be binary and pairwise. With these assumptions, we study the worst case performance of IEEE 802.11. Note that these are idealized assumptions on the physical layer, and in practice, the interference range will be more than the transmission range, as well as physical layer effects like capture will effect performance. Section 5.6.1 shows that incorporating real world physical layer effects will only further improve the throughput 74 ratio. Thus, the results presented in this section are a lower bound on throughput ratios which will be observed in real systems. Our assumptions on the physical layer impose two constraints on how edges inN ec can interfere with each other. 5.3.1 Non-Interfering Neighbors e c e 1 Figure 5.5: e 1 can interfere with all the other four edges belonging to N ec . The first constraint bounds the maximum number of non-interfering neighboring edges of e c . Using a derivation similar to [16], we derive the following two rules: (i) The maximum number of nodes which interfere with R ec but notT ec , as well as do not interfere with each other is equal to 4. This rule bounds the maximum number of non- interfering edges inN ec coll to be4. Thus, the worst-case topology derived in the previous section is not feasible for|N ec |> 4. (ii) The maximum number of nodes which interfere with eitherT ec orR ec but do not interfere with each other is equal to8. This rule implies that any neighborhood with more than 8 edges cannot have an interference factor equal to1. 75 5.3.2 Interference between Edges Belonging to Different Maximal Independent Sets Let I 1 ,I 2 ∈S Nec I . Let C j ,j = 1,2 denote the set of edges in I j which also belong to N ec coll . How many edge pairs with one edge in C 1 and the other edge in C 2 interfere with each other effects the throughput ratio. Theorem 5(a) proves that all such edge pairs interfere with each other in the worst case neighborhood topology derived without incorporating physical layer constraints. The second constraint bounds the maximum number of interfering edge pairs with edges belonging to different maximal independent sets. Let e 1 ∈ C 1 be a neighbor of e c . The maximum number of edges in C 2 which e 1 can interfere with depends on the cardinality ofC 1 andC 2 . Lets understand why using an example. Let there be four edges inC 2 , and let there be only one edge inC 1 (edgee 1 ). As shown in Figure 5.5, place the transmitter of e 1 very close to R ec and place the transmitters of edges in C 2 along the circumference of the circle with a radius equal to the transmission range. Then,e 1 will interfere with all the four edges inC 2 . Now lets assume that the number of edges inC 1 is equal to2, while the number of edges inC 2 still remain4. Then, there is no geometrical placement such that the two edges in C 1 can interfere with more than 2 edges in C 2 . This can be easily proved by writing all the geometrical constraints imposed on the transmitter and receiver of these edges, and checking if a solution exists. Table 5.1 states the maximum number of edges inC 2 an edge inC 1 can interfere with for different cardinalities of C 1 and C 2 . These values are also derived by writing the geometrical constraints imposed on the transmitter and receiver of these edges, and checking if a solution exists. Now, letB j ,j = 1,2 denote the set of edges inI j which also belong toN ec busy . In a similar fashion, we derive that the maximum number of edges in B 2 which an edge in B 1 can interfere with is governed by Table 5.1 also. 76 Cardinality of Cardinality Number of edges inC 2 ofC 1 ofC 2 which an edge inC 1 can interfere with |C 1 | = 1 |C 2 |≥ 1 |C 2 | |C 1 |> 1 |C 2 | = 1 1 |C 1 |> 1 |C 2 |> 1 2 Table 5.1: Table defining the second constraint. 5.3.3 Numerical Results We now look at the worst case throughput ratio with the disk model at the physical layer. We construct the worst case topology using brute force by constructing all possible topologies allowed after imposing these two constraints, evaluating the performance of IEEE 802.11 and optimal scheduling for each of these topologies and finding the one which has the worst throughput ratio. Note that having an analytical model allows us to quickly evaluate the performance of IEEE 802.11 for each topology. If one had to evaluate the performance using ns-2 or Qualnet simulations, this approach will become prohibitively expensive. For example, on a 3.06 GHz Linux box, finding the worst case neighborhood topology for|N ec | = 4 using Qualnet simulations will approximately take 819 hrs, and this time will exponentially increase as the number of neighboring edges increases. The dashed line in Figure 5.2 plots the worst case throughput ratio for the disk model. The worst case performance never goes below 30% with the constraints imposed by the disk model. We also see jumps at multiples of8 because the maximum number of non- interfering neighbors (which can be scheduled simultaneously)e c can have is equal to8; hence, after a multiple of 8, optimal scheduling takes one extra slot to schedule, which deteriorates its throughput performance. Note that the performance of IEEE 802.11 also deteriorates, but not as much as optimal scheduling, hence the observed jump at multiples of8. 77 0 5 10 15 20 0 1 2 3 4 | N e c | Interference Factor Worst Topology with Disk Model Minimum Possible (a) 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 | N e c | Scheduling Set Ratio Worst Topology with Disk Model Maximum Possible (b) 0 5 10 15 20 0 2 4 6 8 10 12 |N e c | |N coll e c | (c) Figure 5.6: Characteristics of the worst case topology with disk model. (a) Interference Ratio. (b) Scheduling Set Ratio. (c) Number of edges inN ec coll . 5.3.4 Characterizing the Worst Case Topology Recall that the worst case neighborhood topology derived in Section 5.2 without impos- ing any geometrical constraints from the physical layer had the following three charac- teristics: (i) Lowest interference factor possible. (ii) Maximum scheduling ratio possi- ble. (iii) All neighboring edges belonged to N ec coll . Now we characterize the worst case topology derived with the physical layer model assumed in this section. The intuition derived in Section 5.2 for the first two characteristics still holds. We numerically verify that the worst case topology has these two characteristics for all values of|N ec |≤ 20 by 78 plotting the interference factor and scheduling set ratio of the worst case topology in Fig- ures 5.6(a) and 5.6(b) respectively. Note that the maximum value of scheduling set ratio depends on the interference factor value (denoted by k Nec ) of the topology and|N ec |. For example, a topology with 9 nodes and k Nec = 4 will have a maximum scheduling set ratio of2/3, while withk Nec = 3, the maximum scheduling set ratio will be equal to 1. The third characteristic gets modified slightly. We explain the reason using an exam- ple. Lets consider the worst case neighborhood topology for|N ec | = 5. Recall that the maximum number of non-interfering edges possible inN ec coll is4, but the maximum pos- sible interference factor for|N ec | = 5 is equal to 1 (see Section 5.3.1). Thus, 5 edges cannot be placed in N ec coll while maintaining the interference factor to be 1. Thus the third characteristic now becomes: For a givenN ec and interference factor,N ec coll contains as many edges as possible, and the remaining edges are contained in N ec busy . Thus, for |N ec | = 5, N ec coll contains four edges and N ec busy contains the fifth one. We numerically verify this characteristic for all values of|N ec |≤ 20 by plotting the number of edges in N ec coll in the worst case topology in Figure 5.6(c). Finally, we now comment on how do the edges inN ec interfere with e c (how Theo- rem 2 gets modified). Edges inN ec coll still interfere as asymmetric edges. However, since non-interfering nodes placed in the range ofR ec belong to edges inN ec coll , edges inN ec busy cannot interfere as near hidden edges. Hence, edges in N ec busy interfere as coordinated stations (which has the next highest loss in throughput among the two-edge topologies belonging toN ec busy , see Section 4.3). As an example of how the worst case topology looks like, we plot this topology for |N ec | = 8 in Figure 5.4(b). There are 4 asymmetric edges, 4 coordinated station edges, and none of the edges inN ec interfere with each other. 79 5.3.5 Comparison with Related Work We now compare the bounds obtained in this section to bounds derived by prior works on maximal scheduling and asynchronous random access schemes. Chaporkar et al. [16] derived the bound for maximal scheduling to be 1/Δ where Δ denotes the interference degree of a multi-hop network. 6 Note that the schedule generated by IEEE 802.11 is also a maximal schedule, although it generates the maximal schedules randomly. Since IEEE 802.11 does not generate the worst maximal schedule every time, its performance should be better than the worst case bound on maximal scheduling. Chafekar et al. [13] derived the upper bound for asynchronous random access schemes to be also1/Δ. IEEE 802.11 also implements asynchronous random access, but also has additional features like exchange of RTS/CTS control messages, exponential backoffs and virtual carrier sensing. Hence, IEEE 802.11’s performance is expected to be better. For the assumptions made on physical layer in this section, Δ = 8 [16]. Hence, the worst case bound on maximal scheduling and asynchronous random access is12.5% of the optimal, while IEEE 802.11 is never worse than 30% of the optimal. 5.4 Typical Topologies The worst case topology constructed in the previous section requires a careful placement of nodes. For example, slight movement of any of the 8 nodes within the interference range ofT ec andR ec in Figure 5.4(b) will change the topology by bringing two of these nodes within each other’s range, and thus improving the ratio. Hence, the probability of the worst case neighborhood occurring in practical topologies is small. 6 Interference degree of an edge is the maximum number of edges which interfere with this edge but not with each other. Interference degree of a multi-hop network is the maximum interference degree over all edges in the multi-hop network. 80 To get an idea of how IEEE 802.11 does for typical topologies instead of worst-case ones, we next study the average throughput ratio of IEEE 802.11 and optimal scheduling as well as the throughput ratio seen in the congested neighborhood of a real topology. 5.4.1 Average Ratio We intend to average over all neighborhood topologies weighted by the probability of their occurrence in practice. In the absence of the probability distribution of the occur- rence of a neighborhood topology, we construct 100 random neighborhood topologies where an edgee c has|N ec | neighbors, compute the ratio of throughput achieved ate c at the max-min allocation by IEEE 802.11 and optimal scheduling and take the average. We first describe how a random neighborhood topology is constructed. We first fix the transmission range (and interference range) to be equal to L. (Recall that making the physical layer model less idealized improves the performance as we show in Sec- tion 5.6.1.) We then place the edgee c with the distance between the transmitter and the receiver chosen uniformly at random from (0,L]. We then place the|N ec | neighboring edges. Each edge is placed as follows. (a) Place a node labeledU uniformly at random within one hop from either the transmitter or the receiver. (b) Then place a nodeV uni- formly at random within one hop of nodeU. (c) With probability0.5, choose nodeU to be the transmitter and nodeV to be the receiver ; and with probability 0.5, chooseU to the receiver andV to be the transmitter. The dotted line in Figure 5.2 plots the average throughput ratio. The average perfor- mance is significantly better than the worst case performance, and is never worse than 55% even for topologies with20 neighbors. 81 5.4.2 A Real Topology: Houston Neighborhood We study the throughput ratio of the congested neighborhood in the real topology of the outdoor residential deployment of Figure 4.9. We assume that all links to be lossless in this section. When all nodes are sending packets to one of gateways (nodes 0 or 1), at the max-min rate allocation, the neighborhood around edge 6→ 1 is the congested neighborhood. Note that|N 6→1 | = 15. At the max-min allocation, the throughput ratio in the congested neighborhood is equal to 66.8%, which, not surprisingly, is closer to the average ratio rather than the worst case ratio. 5.5 End-to-End Rate Allocations in Multi-hop Topolo- gies We had chosen to study neighborhood topologies because the throughput performance of any multi-hop topology depends directly on the throughput achieved in the congested neighborhoods of the topology. However, one might wonder whether studying only the neighborhood topologies is sufficient to be able to comment on the performance of multi-hop networks. Global characteristics like how many hops does a flow go over, or how many congested neighborhoods does a flow go through may effect the end-to- end throughput performance and neighborhood topologies do not capture these global characteristics. In this section, we choose examples to illustrate that these global characteristics will have a negligible impact on the throughput ratio, which will be chiefly governed by the topology characteristics of the most congested neighborhood a flow passes through. 82 8 9 10 1 2 3 4 5 6 7 (a) 5 6 7 3 1 2 4 (b) 7 2 3 4 5 1 9 8 6 (c) Figure 5.7: (a)-(c) Multi-hop topologies which we use to demonstrate that end-to-end rate allocations in a multi-hop topology depend only on the topology characteristics of the congested neighborhoods. Congested queues are indicated with a symbol depicting a queue. 5.5.1 Throughput Ratio and Number of Hops In this section, we show that the number of hops a flow goes through has a negligible impact on the throughput ratio of IEEE 802.11 and optimal scheduling. Consider the multi-hop topology of Figure 5.7(a). Flow 1→ 8 passes over 7 hops and interferes with2 flows, 3→ 9 and3→ 10. At the max-min allocation, optimal and IEEE 802.11 scheduling allocate0.178 and0.171 Mbps respectively to all the three flows (throughput ratio =96%). The congested neighborhood in this topology is the neighborhood around the edge 2→ 3. Now, we modify the topology so that the congested neighborhood topology does not change, however the long flow goes over only 4 hops. The resultant topology is shown in Figure 5.7(b). At the max-min allocation, now optimal and IEEE 83 802.11 allocate 0.178 and 0.172 Mbps respectively to all the three flows. Thus the number of hops a flow passes through has a negligible impact on the throughput ratio. Note that this is not in disagreement with prior work [25, 56, 77] which had shown that IEEE 802.11’s performance deteriorates as the number of hops increase as prior works had assumed TCP or saturation conditions at the transport layer, while we find the rate achieved with the best possible transport. We next explain why we see slightly lower rates with IEEE 802.11 in the topology with more hops. Transmissions on edge 5→ 6 cause more collisions at edge 3→ 4 in the topology of Figure 5.7(a). This in turn increases the amount of data transmitted on edge 3→ 4, which in turn increases interference in the congested neighborhood. Hence, we observe slightly lower rates in the topology of Figure 5.7(a). 5.5.2 Throughput Ratio and Topology Characteristics of the Con- gested Neighborhood In this section, we show that the throughput ratio in multi-hop topologies is dictated by the topology characteristics of the congested neighborhood. Consider the topology shown in Figure 5.7(c). It is similar to the topology of Figure 5.7(b), however the con- gested neighborhood (around edge 2→ 3) has been changed. Now, at the max-min allocation, optimal and IEEE 802.11 scheduling allocate0.2967 and0.192 Mbps respec- tively to all the flows (throughput ratio = 64.7%). We see a decrease in the throughput ratio because the interference factor of the congested neighborhood topology (around edge 2→ 3) becomes 2 as opposed to its value of 4 for the topology of Figure 5.7(b). Note that the throughput of both scheduling schemes has increased as the interference factor of the congested neighborhood has decreased, however, the improvement is larger with optimal scheduling. This is inline with our observations made in Section 5.2. 84 5.5.3 Throughput Ratio and Number of Congested Neighborhoods In this section, we show that the number of congested neighborhoods a flow passes through has no impact on the throughput ratio (as has been shown for wired networks, albeit with congested links). Consider the topology shown in Figure 5.1. It is simi- lar to the topology of Figure 5.7(a), except for an additional flow 8→ 11. There are two congested neighborhoods in the topology of Figure 5.1: around edges 2→ 3 and 6→ 7. Flow 1→ 8 passes through both the congested neighborhoods. However, the neighborhood around edge 2→ 3 hits congestion at a lower throughput as it has more flows passing through it. Hence, the throughput of flow 1→ 8 will be dictated by the throughput achieved in the neighborhood of 2→ 3. Hence the congested neighborhood topology dictating the performance of flow1→ 8 does not change. At the max-min allo- cation, optimal and IEEE 802.11 scheduling allocate0.178 and0.171 Mbps respectively to flows 1→ 8, 3→ 9 and 3→ 10, and allocate 0.534 and 0.422 Mbps respectively to flow8→ 11. Note that the throughput of flow1→ 8 remains the same as before. 5.6 Discussion: Effect of Our Assumptions In this section, we show that our results do not depend on our assumptions by showing that removing any of our assumptions will either improve the throughput ratio, or the effect will be negligible, or the corresponding negative effect on the throughput ratio can be easily removed by minor changes in the protocol. 5.6.1 Assumptions on the Physical Layer We have assumed a highly idealized physical layer model, one which is well known to be inaccurate in reality. Specifically, the physical layer models assumed in this chapter 85 do not model the following real effects: (i) Interference range is greater than the trans- mission range, (ii) Interference is neither binary nor pairwise, and (iii) Physical layer losses can occur in absence of collisions due to fading effects. Interference Range> Transmission Range In this section, we argue that as the interference range increases, the throughput ratio will improve. Hence, assuming the minimum value of the interference range (= the transmission range) will yield conservative results. We first study how the two constraints we derived in Section 5.3 change as the inter- ference range increases. First Constraint: As the interference range increases, the number of non-interfering nodes which can be fitted within the interference range of a node reduces. For example, if the interference range is twice the transmission range, then the two rules derived in Section 5.3.1 change as follows. (i) The maximum number of nodes which interfere with R ec but notT ec , as well as do not interfere with each other is equal to 3. (ii) The max- imum number of nodes which interfere with either T ec orR ec but do not interfere with each other is equal to 6. Fewer non-interfering nodes interfering with T ec orR ec imply a larger interference factor in the worst case neighborhood topology, which implies a better throughput ratio. Second Constraint: The second constraint does not change. Figure 5.8(a) compares the worst case and average throughput ratio for a disk model when the interference range is twice and equal to the transmission range. Its easy to see that both the worst case and average throughput ratios improve when the interference range is larger. 86 2 4 6 8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 |N e c | Throughput Ratio (IEEE 802.11/Optimal) Worst Case Worst Case (Interference Range = Trasmission Range) Worst Case (Interference Range = 2*Trasmission Range) Average Ratio (Interference Range = Trasmission Range) Average Ratio (Interference Range = 2*Trasmission Range) (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Rate at edge e 1 (in Mbps) Rate at edge e 2 (in Mbps) Binary Interference Model Non−Binary Interference Model (b) Figure 5.8: (a) Worst case and average throughput ratio when the interference range is twice the transmission range. (b) The achievable rate region of the topology of Fig- ure 4.2(c) with a binary and a non-binary interference model. Non-Binary and Non-Pairwise Interference We first discuss how a non-binary interference model will effect the throughput ratio. Any such model will incorporate capture and deferral probabilities to model that a col- lision may not result in a packet loss and the channel might not be sensed busy at a node inspite of the ongoing interfering transmission [70]. Incorporating both will increase the number of successful simultaneous transmissions, which will improve the through- put achieved by IEEE 802.11 scheduling. We compare the entire achievable rate region with a binary and a non-binary interference model for the worst case neighborhood topology for|N ec | = 1 in Figure 5.8(b). The worst case neighborhood topology with only one neighboring edge is the asymmetric two-edge topology of Figure 4.2(c). The derivation of the rate region with the non-binary interference model assumes that the distance betweenR e 2 andT e 2 is 1.75 times the distance betweenT e 1 andR e 1 , which is the same as the distance betweenT e 2 andR e 2 , and assumes a distance-based attenuation model for the channel with the attenuation parameter equal to 4. Its easy to see that the introduction of capture and deferral probabilities has increased the rate region with 87 IEEE 802.11 scheduling. Note that the worst case topology with a non-binary interfer- ence model may change. However, the performance of the new worst case topology is better than the worst case topology with the binary model, and is going to only become even better with the more realistic model. Note that exploiting capture/deferral in opti- mal scheduling requires exact knowledge of the channel state on not only the edges on which transmissions are taking place, but also on all the interference edges in the neighborhood. So, exploiting capture/deferral with optimal scheduling requires an extra overhead to achieve the same gains as IEEE 802.11 (which requires no overhead). Thus, we conclude that the results derived in this work are conservative. For a non-pairwise interference model, many simultaneous transmissions can cause deferral/collision at a node even though each of them individually may not have the same effect. But this happens only rarely in practice [23, 58], and not incorporating this effect in the analysis does not result in a significant deviation from reality [49]. Physical Layer Losses due to Fading In the current IEEE 802.11 protocol, the transmitter does not distinguish between losses due to fading and collisions. In either case, the transmitter performs an exponential backoff, and tries to retransmit again. Optimal scheduling will also retransmit lost pack- ets, but will not perform an exponential backoff. Hence, exponential backoffs cause extra throughput loss in IEEE 802.11. However, exponential backoffs are meant to resolve collisions, and should not occur for losses due to fading. This throughput loss can be easily fixed by accurately differentiating between losses due to fading and col- lisions. Performing this differentiation has been an active field of research [3, 45, 81], albeit in a different context, however these techniques can be applied directly in this context also. 88 Note that there will be some additional throughput loss in IEEE 802.11 because the transmitter waits for a certain backoff duration before each transmission. Avoiding exponential backoffs for fading losses keeps the backoff duration small for each retrans- mission, hence, this throughput loss will not have any significant impact on our results. For example, assuming a perfect way to differentiate between losses due to fading and collision, a20% data loss rate due to fading (excluding collisions) drops the throughput ratio to 78.5% from 84.7% for the worst case neighborhood topology for|N ec | = 1. (Note that in real networks, links with more than20% data loss due to fading will not be included for routing packets.) 5.6.2 Multiple Outgoing Edges from a Node If two edges share a transmitter, then they share the same queue which can result in a head of line blocking effect. We use the topology of Figure 5.1 to illustrate. Node 3 has three outgoing edges: 3→ 4, 3→ 9 and 3→ 10. If the first packet in the queue at node 3 is for the edge 3→ 4, and there is a transmission ongoing at edge 5→ 6, then the transmission on edge 3→ 4 will not go through till the transmission on edge 5→ 6 ceases. Now, if there is a packet later in the queue at node 3 for edges 3→ 9 or 3→ 10, then its successful transmission is possible. However, since it is blocked behind the packet for edge 3→ 4, no attempt for its transmission will take place either. This head of line blocking effect will result in a throughput loss for IEEE 802.11. However, this effect can be easily rectified by giving each edge a separate queue, maintaining a separate backoff counter at each queue and allowing each queue to access the PHY layer. Thus, there may be several queues and backoff counters running at the same node. Note that when the node is busy sending a packet on one of the edges, the backoff counters of all the other edges emanating from this node will remain frozen by virtual carrier sensing. Note that this is essentially equivalent to each of these edges having a separate 89 transmitter, where all these transmitters interfere with each other and have exactly the same interference relationship with other nodes in the network. 90 Chapter 6: WCP-CAP: Distributed Congestion Control for IEEE 802.11-Scheduled Multi-Hop Networks The results in the previous chapter show that it is possible to achieve the same degree of performance with IEEE 802.11 as with distributed approximations to optimal schedul- ing in multi-hop networks. Recall that backpressure-based techniques yield a distributed congestion control algorithm which achieves optimal throughput with optimal schedul- ing. However, its not clear if its possible to achieve the same good performance with IEEE 802.11 using a distributed congestion control protocol. Maybe the complexity gets shifted to the transport layer, and it may not be possible to build a distributed congestion control protocol which can achieve this good performance. To dispel this concern, this chapter proposes a new distributed congestion control protocol which we label WCP-CAP (Wireless Congestion Protocol - Capacity based). WCP-CAP estimates the available capacity within each neighborhood, and apportions this capacity to contending flows. Thus it provides the sources with explicit and precise feedback, in much the same way that XCP [43] and RCP [24] do for wired networks. WCP-CAP uses the analysis presented in Chapter 4 to directly estimate the exact capac- ity of a edge as a function of the edge rates at the neighboring edges. We next describe 91 how each intermediate node apportions the estimated capacity to contending flows, and notifies the flow-sources of the rate. 6.1 The Algorithm Conceptually, each router maintains, for each outgoing edgee, a rateX e which denotes the maximum rate allowable for a flow passing through the edge. However, a flow traversing e is actually only allowed to transmit at the minimum (denoted X min e ) of all rates X e j such that e j belongs toN e (intuitively, at the most constraining rate over all edges that share channel capacity with e). The rate feedback is carried in the packet header. When a packet traverses e, the router sets the feedback field to X min e if X min e is lower than the current value of the field. This feedback rate is eventually delivered to the source in an end-to-end acknowledgement packet, and the source uses this value to set its rate. Thus the source sets its rate to the smallest allowable rate in the wireless neighborhoods that it traverses. X e for each edge is updated everyk·rtt Smax avg e , wherertt Smax avg e is the shared RTT andk is a parameter which trades-off the response time to dynamics for lower overhead. The shared RTT (rtt Smax avg e ) denotes the maximum average RTT among all edges in the neighborhood of e and is calculated using the following equation: rtt Smax avg e = max ∀e j ∈Ne P ∀m∈Fe rtt avg m |Fe| where F e is the set of flows traversing edge e and rtt avg m is the average RTT of flow m. The duration between two successive updates of X e is referred to as an epoch. During each epoch, transmitter T e measuresx e , the actual data rate over edge e and n e , the number of flows traversing edge e. Using x e j and n e j , ∀e j ∈ N e , transmitter T e computes the new value of X e (denoted by X new e ) to be used in the next time epoch, and broadcastsx e ,n e , andX new e to all edges inN e . 92 We now describe how X new e is determined (Figure 6.1). Note that the transmitter T e hasx e j andn e j ,∀e j ∈N e . It uses this information, and the methodology described in Chapter 4, to determine the maximum value of δ such that the rate vector ~ x shown in Figure 6.1 is achievable. (δ can have a negative value if the current rates in the neighborhood are not achievable.) Then, node T e setsX new e toX e +ρδ ifδ is positive, elseX new e is set toX e +δ. We use a scaling factorρ while increasing the rate to avoid big jumps, analogous to similar scaling factors in XCP and RCP. On the other hand, we remain conservative while decreasing the rate. Each transmitter independently computes X e for its edges. These computations do not need to be synchronized, and nodes use the most recent information from their neighbors for the computation. We now comment on the computational complexity of the WCP-CAP algorithm. To determineX new e , we perform a binary search to find the maximum value of δ such that the rate vector ~ x is achievable. Thus, we perform a logarithmic number of iterations whose complexity is polynomial in|N e |. In practical topologies the cardinality ofN e is small and the overall complexity is quite low. For example, in our experiments (run in 3.06GHz Linux boxes) determiningX new e takes as much time as it takes to send a data packet. Since each epoch consists of about 30 data packet transmissions and a single X new e computation, the computational overhead is very low. Finally, we note that, if naively designed, WCP-CAP can impose significant commu- nication overhead. For each edgee, the following information needs to be transmitted to all nodes inN e once every epoch: the maximum RTT across the flows passing through the edge, the actual data rate at the edge, the number of flows passing through the edge andX e . There are ways to optimize this, by quantizing the information or reducing the frequency of updates. But our objective is to merely show that distributed rate control schemes can achieve good performance, so we leave reducing the overhead as future 93 Every k·rtt Smax avg e sec Find max δ such that ~ x← x ej +n ej δ for e j ∈N e is achievable X new e ← X e +ρδ δ > 0 X e +δ δ≤ 0 Broadcast X new e , x e and n e to all edges in N e Figure 6.1: Pseudo-code for rate controller at edge e work. Also, in our simulations, we assume that all the relevant information is available at each node without cost. 6.2 Properties To understand the design rationale of the WCP-CAP algorithm, we characterize the fair- ness properties of an idealized WCP-CAP algorithm. The idealized WCP-CAP algo- rithm assumes that all control message broadcasts are exchanged instantaneously and without loss, and each node has complete information about the entire network instead of just its neighborhood. Specifically, each node is aware of the data rate at each edge in the network and the global network topology. The last assumption is needed because residual capacity at an edge depends on the global topology and not merely the local neighborhood topology. Hence WCP-CAP obtains an approximate value of the residual capacity while idealized WCP-CAP will obtain an exact value of the residual capacity. We prove fairness properties for idealized WCP-CAP here, and evaluate how do the non-idealities impact performance of WCP-CAP through simulations in Section 6.3. Theorem 6. Idealized WCP-CAP allocates max-min fair rates to the flows. Proof. At the max-min allocation, let the edges whose queues are nearly fully utilized be referred to as congested edges, and neighborhood of congested edges be referred to 94 as congested regions. Each flow may pass through several congested regions, however, the most congested region a flow passes through (the region which gets congested at the lowest throughput) dictates the throughput of that flow. The next lemma states a property of the max-min rate allocation relating the throughputs of flows which share the most congested region they traverse. Lemma 24. If the most congested region two flows pass through is the same, the max- min rate allocation will allocate equal rates to these two flows. Proof. Let there be n flows passing through the congested region C e . Additionally, let k of thesen flows haveC e as the most congested region they pass through. Consider the following rate allocation. Fix the rate of the other n−k flows as dictated by the most congested region they pass through, and then assign the maximum possible equal rate to thek flows. Label the equal rate assigned to thesek flows asR eq . Then, by definition of the most congested region a flow passes through, the other n−k flows have a rate smaller thanR eq . Lete denote the congested edge in the congested regionC e . (That is, edgee operates at full utilization.) Increasing the rate of any of thek flows will either increase the busy probability or the collision probability at edge e, making its queue unstable (follows from Lemma 21). To keep the rate allocation feasible, the rates of one of the other flows (which have either smaller or equal rates) will have to be reduced. Hence, by definition, allocating the maximum possible equal rate to the k flows sharing the same most congested region is the max-min rate allocation. Now, idealized WCP-CAP will also allocate equal rates to flows which share the most congested region they traverse because the rate of these flows will be set to the maximum allowable rate in that region. 95 6.3 Evaluation of WCP-CAP In this section, we evaluate the performance of WCP-CAP using simulations. We first describe the simulation methodology, and then present the simulation results on different topologies. 6.3.1 Methodology We have implemented WCP-CAP in the Qualnet simulator. Our implementation closely follows the description of the protocol in Section 6.1. However, it does not simulate the exchange of control messages at the end of each epoch; rather, this control information is made available to the relevant simulated nodes through a central repository. This ignores the control message overhead in WCP-CAP, so our simulation results overestimate its performance. This is consistent with our goal, which is to show that it is possible to implement a distributed rate control protocol which achieves fair and efficient rates. All our simulations are conducted using an unmodified 802.11b MAC (DCF). We use default parameters for 802.11b in Qualnet stated in Table 3.2. We fix the data rate to 11 Mbps, and assume all packets are of size 512 bytes. Buffer size is set to 64 packets per queue, and the MAC retry limit is set to its default value. All our simulations are conducted with zero channel losses, although packet losses due to collisions do occur. We run bulk transfer flows for 200 seconds for WCP-CAP and TCP. We use TCP with SACK and ECN, but with Nagle’s algorithm and the delayed ACK mechanism turned off. (We have also evaluated TCP-Reno on our topologies. The results are qual- itatively similar.) To keep the comparison fair with TCP, we also implement reliability in WCP-CAP, and while finding the optimal, we incorporate the back-flow of transport layer ACK packets of size 40 bytes. We choose to keep the per-queue utilization to be less than 100% as operating at a very high utilization causes small non-idealities to 96 drive the network beyond the capacity region. The per-queue utilization is kept below a fraction U by modifying Equation (4.3) to P e∈Ov λ e E[S e ] < U,∀v∈ V . We choose U to be 0.7 in our simulations, the choice of the value of U is discussed later in this section. We measure the goodput achieved by each flow in a given topology by TCP and WCP-CAP, and compare these goodputs with the optimal max-min rate allocations. For each topology we simulate, we show results averaged over 10 runs. 6.3.2 Performance of WCP-CAP We evaluate the performance of WCP-CAP on four different topologies. Topology 1: Flow in the Middle We first evaluate the performance of WCP-CAP for the topology shown in Figure 4.6(a). Table 6.1 summarizes the simulation results. The optimal max-min rate allocation for this topology is to allocate 300 Kbps to all the flows. TCP starves the middle flow, while WCP-CAP assigns close to 263 Kbps to all the three flows. Thus WCP-CAP is within 15% of the optimal achievable rate for this topology. WCP-CAP yields lower throughput than the optimal because we have set the per-queue utilization to0.7. Flow TCP WCP-CAP Optimal 1→ 3 677.8 Kbps 263 Kbps 300 Kbps 4→ 6 8.2 Kbps 262.6 Kbps 300 Kbps 7→ 9 665.7 Kbps 263 Kbps 300 Kbps Table 6.1: Simulation Reults for the Flow in the Middle Topology Topology 2: Diamond We next evaluate the performance of WCP-CAP for the topology shown in Figure 6.2(a). Table 6.2 summarizes the simulation results. The optimal max-min rate allocation is to 97 9 1 2 3 6 5 4 7 8 (a) 9 1 2 3 6 5 4 7 8 (b) Figure 6.2: (a) The Diamond topology. (b) The Half-Diamond topology. There are three flows in both these topologies: 1→ 3,4→ 6 and7→ 9. allocate325 Kbps to all the flows. TCP starves the outer flows, while WCP-CAP assigns close to 300 Kbps to all the flows. Thus, WCP-CAP is within 10% of the optimal for this topology. Flow TCP WCP-CAP Optimal 1→ 3 4.8 Kbps 300.8 Kbps 325 Kbps 4→ 6 678.1 Kbps 300.2 Kbps 325 Kbps 7→ 9 4.8 Kbps 300.9 Kbps 325 Kbps Table 6.2: Simulation Reults for the Diamond Topology Topology 3: Half-Diamond We next evaluate the performance of WCP-CAP for the topology shown in Figure 6.2(b). Table 6.3 summarizes the simulation results. The optimal max-min rate allocation is to allocate 335 Kbps to flow 1→ 3, and allocate 315 Kbps to the rest. Relative to other topologies, TCP performs fairly well for this topology. WCP-CAP assigns max-min fair rates to all the flows, and is within14% of the optimal. 98 Flow TCP WCP-CAP Optimal 1→ 3 424.9 Kbps 276.3 Kbps 335 Kbps 4→ 6 267.9 Kbps 276.5 Kbps 315 Kbps 7→ 9 286.8 Kbps 277.6 Kbps 315 Kbps Table 6.3: Simulation Reults for the Half-Diamond Topology 11 3 7 5 6 9 8 2 1 4 10 Figure 6.3: The Chain-cross topology. There are five flows in this topology: 1→ 2, 1→ 7,6→ 7,8→ 9 and10→ 11. Topology 4: Chain-cross Finally, we evaluate the performance of WCP-CAP for the topology shown in Figure 6.3. Table 6.4 summarizes the simulation results. The optimal max-min rate allocation is to allocate 420 Kbps to flow6→ 7, and allocate 255 Kbps to the rest. TCP starves flows 1→ 2 and 1→ 7, while WCP-CAP achieves close to max-min throughputs which are within 15% of the optimal. Flow TCP WCP-CAP Optimal 1→ 2 18.5 Kbps 222.5 Kbps 255 Kbps 1→ 7 1.4 Kbps 221.5 Kbps 255 Kbps 6→ 7 1581 Kbps 477.2 Kbps 420 Kbps 8→ 9 1564.5 Kbps 222.5 Kbps 255 Kbps 10→ 11 1565.6 Kbps 222.5 Kbps 255 Kbps Table 6.4: Simulation Reults for the Chain-cross Topology 99 Topology Flow Delay 1→ 3 6.2 ms Flow in the Middle 4→ 6 15.6 ms 7→ 9 6.2 ms 1→ 3 5.1 ms Diamond 4→ 6 4.2 ms 7→ 9 5.3 ms 1→ 3 6.2 ms Half-Diamond 4→ 6 10.1 ms 7→ 9 13.7 ms 1→ 2 6.1 ms 1→ 7 39.2 ms Chain-cross 6→ 7 2.9 ms 8→ 9 1.5 ms 10→ 11 1.5 ms Table 6.5: Delay results for WCP-CAP. Delay Since WCP-CAP keeps the network within the achievable rate region, it is able to main- tain smaller queues. Hence, WCP-CAP achieves good througput with small average end-to-end delays. Table 6.5 summarizes the average end-to-end delay for all the four topologies studied in this chapter. To estimate the queueing delay per queue, note that the transmission delay for the network parameters assumed in this chapter is equal to 1.3 ms. Choice ofU In simulations, we set the value of U which governs the per-queue utilization value to 0.7. We now justify this choice. The analysis described in Chapter 4 derives the achievable rate region without losses and hence assumes infinite buffer sizes and infinite MAC retrry limit. Assuming no losses, operating very close to the capacity region will result in a large delays. However, in practice both the buffer sizes and MAC retransmit 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 U Average End−toEnd Delay (in sec) Figure 6.4: Average End-to-End Delay as a function ofU for the flow4→ 6 in the Flow in the Middle topology. limits are finite. Hence, these large delays can result in significant losses. For this reason, we operate the network well within the capacity region; the parameterU controls how far the network is from the boundary of the capacity region. To understand how to set its value, we plot the end-to-end delay of the middle flow,4→ 6, in the Flow in the Middle topology (which has the highest delay) in Figure 6.4. We varyU from0 to1 and assume very large buffer sizes and MAC retransmit limits. Ideally one would setU near the knee of the curve, which is around 0.8. We choose a conservative estimate and set it to0.7. Summary WCP-CAP achieves max-min fair rates in all the topologies we study, and is always within 15% of the optimal. In addition, WCP-CAP also has low delay as it keeps the queues small. And it achieves this good performance while being distributed and exchanging messages only within neighborhoods. Thus, we conclude that it is possible to achieve the same degree of performance with IEEE 802.11 as achieved with dis- tributed approximations to optimal scheduling in multi-hop networks with distributed rate control schemes. 101 Chapter 7: Conclusions and Future Work In this work, we first characterize the achievable rate region of an arbitrary multi-hop network with IEEE 802.11 scheduling by deriving a methodology to characterize the achievable edge-rate region. We then formally show that IEEE 802.11 achieves the same degree of performance as state-of-the-art distributed approximations to optimal scheduling. Two important observations made while deriving the worst case bounds for IEEE 802.11 are: (i) Congested neighborhoods dictate the throughput performance in wireless multi-hop networks. (ii) Larger the interference factor, higher the throughput ratio achieved by IEEE 802.11 compared to optimal. Finally, we propose a distributed explicit and precise feedback-based rate control algorithm which we label WCP-CAP. WCP-CAP achieves max-min rates within 15% of the optimal for all the topologies studied in this work. We now put our results in perspective by commenting on what our results do not imply. Our results only compare IEEE 802.11 and distributed approximations of opti- mal scheduling. They do not imply that IEEE 802.11 is sufficient to build multi-hop networks without new physical layer techniques. Even if one can implement optimal scheduling, new physical layer techniques like MIMO [8], use of multiple channels [7], network coding [17], interference cancellation [33] etc are needed to increase the overall 102 network capacity. Since use of IEEE 802.11 will not solve problems which are present when optimal scheduling is used, these techniques will still be required. Similarly, our results are not counter to order of capacity results which prove that the capacity of wire- less networks do not scale [36]. Much work still remains. This work motivates the use of existing CSMA-CA based random access schemes like IEEE 802.11 at the MAC layer. However, how to modify IEEE 802.11 to make it more more suited to multi-hop networks is still an open problem. Also much work is needed before WCP-CAP can be used at the transport layer. More efficient implementations with lower communication overhead are needed. 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Abstract (if available)

Abstract This work formally establishes that IEEE 802.11 yields exceptionally good performance in the context of wireless multi-hop networks. A common misconception is that existing CSMA-CA random access schemes like IEEE 802.11 yield unfair and inefficient rates in wireless multi-hop networks. This misconception is based on works which study IEEE 802.11-scheduled multi-hop networks with TCP or in saturation conditions both of which grossly underutilize the available capacity that IEEE 802.11 provides, or use topologies which cannot occur in practice due to physical layer limitations.

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Creator Jindal, Apoorva (author)

Core Title IEEE 802.11 is good enough to build wireless multi-hop networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 03/03/2009

Defense Date 12/02/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag capacity region,IEEE 802.11,multi-hop networks,OAI-PMH Harvest,random access scheduling,wireless

Language English

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Advisor Psounis, Konstantinos (committee chair), Govindan, Ramesh (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email apoorvaj@gmail.com,apoorvaj@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1995

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