Basic Theorems on the Backoff Process in 802.11

1. Basic Theorems on the Backoff Process in 802.11 JEONG-WOO CHO Q2S, Norwegian University of Science and Technology (NTNU), Norway Joint work with YUMING JIANG Q2S, Norwegian University of Science and Technology (NTNU), Norway A part of this work was done when J. Cho was at EPFL, Switzerland.

2. Understanding 802.11 • Single-cell802.11 network • Every node interferes with the rest of the nodes. • CSMA synchronizes all nodes. • User activity is determined by whether there is a carrier in the medium or not. • Sufficiency of the backoff analysis • The kernel lies in backoff analysis • Backoff process is simple • Every node in backoff stage k attempts transmission with probability pk for every time-slot. • If it succeeds, k changes to 0; otherwise, k changes to (k+1)mod (K+1) where K is the index of the highest backoff stage.

3. Why Mean Field Theory? • Markov chain models of the backoff process • Due to their irreversibility, mathematically intractable. • Decoupling approximation • Backoff process at a node is asymptotically independent from those at other nodes. • Q: Decoupling approximation is valid? • Exactly under which conditions? • Recent advances in Mean Field Theory [BEN08] [BOR07] • If the following nonlinear ODEs are globally stable, it is valid; otherwise, oscillations may occur. [BEN08] M. Benaim and J.-Y. Le Boudec, “A class of mean field limit interaction models for computer and communication systems”, Perf. Eval., Nov. 2008. [BOR07] C. Bordenave, D. McDonald, and A. Proutiere, “A particle system in interaction with a rapidly varying environment: Mean Field limits and applications”, to appear in NHM.

4. Decoupling Approximation Validated • Bianchi’s Formula • Representative formula exploiting decoupling approximation. • A set of fixed-point equations to compute collision probability.

5. Beyond Throughput Analysis • New Interest in Backoff Distribution • How much backoff time should a packet wait for transmission? • Possible misunderstanding for N=2 • Based on extensive simulations, for the case N=2, [BRE09] and [BER04] concluded that Ω is exponentially and uniformly distributed, resp. • Possible misunderstanding about the distribution of Ω. [BRE09] M. Bredel and M. Fidler, “Understanding fairness and its impact on quality of service in IEEE 802.11”, IEEE Infocom, Apr. 2009. [BER04] G. Berger-Sabbatel et al., “Fairness and its impact on delay in 802.11 networks”, IEEE Globecom, Nov. 2004.

6. Outline Mean Field Technique Revisited Supports us to apply decoupling approximation in the following principles Per-Packet Backoff Principle One of the two works is incorrect? Power-Tail Principle What is the distribution type of the delay-related variables? Is there long-range dependence inherent in 802.11? Inter-Transmission Principles Can we develop an analytical model for short-term fairness? When does the short-term fairness undergo a dramatic change? Conclusion

7. Per-Packet Backoff Principle • Misunderstandings cleared up: both works [BRE09] [BER04] are correct. • The contradicting conclusions are due to the different contention window size in 802.11b and 802.11a/g. • For N=2, • In the sense that • 802.11b leads to approx. uniform backoff distribution, while 802.11a/g leads to approx. exponential backoff distribution

8. Long-range Dependence (LRD) Self-similar processes Processes w/ finite 2nd moment LRD Processes • There are LRD processes that are • either not self-similar • or with infinite variances. • Correctly speaking, harmful is LRD. • Why LRD, termed “Joseph Effect” [MAN68], is harmful? • [Bible, Genesis 41] “Seven years of great abundance are coming throughout the land of Egypt, but seven years of famine will follow them.” • long periods of overflow followed by long periods of underflow • hard to derive efficient bandwidth (envelope) of the traffic and to decide buffer size [MAN68] B. Mandelbrot and J Wallis, “Noah, Joseph and operational hydrology”, Water Resources Research, 1968.

9. Bridging between Maths on LRD and 802.11 Black Box Approach • Empirical studies based on high volume data sets of traffic measurements Getting to Know Your Network Approach • Qualitative studies based on rigorous mathematical theories • “Focuses on understanding of LRD and providing physical explanations.” [WIL03] • Developed by Kaj & Taqqu et al. (around 2005) Theoretical Gap The state of the art in 802.11 • A bridge between this approach and 802.11 is required. [WIL03] W. Willinger, V. Paxson, R. Riedi, and M. Taqqu, “Long-Range Dependence and Data Network Traffic”, Theory and Applications of Long-Range Dependence, Birkhäuser Boston, 2003.

10. Power-Tail Principle • Per-packet backoff has a truncated form of Pareto-type distribution. • Sketch of proof: • Discovery of recursive relation in LST of • The quantifier set in regular variation theory is dense. • Application of advanced Karamata Tauberian Theorem • A bridge between recent mathematical theories on LRD and 802.11

11. LRD in 802.11 Identified Ω • Backoff process of each node can be viewed as a renewal counting process. ∑ Superpose Intermediate Telecom Process  LRD process that is not self-similar • Long-range dependence in 802.11 is identified. [KAJ05] I. Kaj, “Limiting fractal random processes in heavy-tailed systems”, Fractals in Engineering, 2005.

12. Short-Term Fairness in 802.11 • Long-term Fairness in 802.11 (without enhanced functionalities) • the total throughput shared equally. • Short-term Fairness in 802.11: not quantified yet. • Inter-transmission probability • Node N is the tagged node.

13. Inter-Transmission principles General formula for (i) small K (ii) large K and α>2 Doubly stochastic Poisson process : a Poisson process on the line with random intensity The resultant dist. is approx. Gaussian. General formula for (iii) large K and α<2 The resultant dist. is approx. Lévian entailing skewness. Leaning: dist. is leaning to the left Directional: dist. has heavy-tail on its right part and decays faster than exponentially on its left part.

14. Collision Dominates Aggregation Given by Per-Packet Backoff Principle Aggregation Effect : Poisson Limit for Superposition Process : Decreases with N Collision Effect : Gaussian Intensity : Increases with N • Gaussian (collision effect) dominates Poisson (aggregation effect).

15. Conclusion Decoupling Approximation Revisited Per-Packet Backoff Principle Possible misunderstanding removed. Power-Tail Principle Backoff distribution formula: truncated Pareto-type. Inter-Transmission Principles Short-term fairness formulas: approximately Gaussian or Lévian

16. Self-Similarity and Long-Range Dependence • Roughly, a self-similar process with finite 2nd moment is long-range dependent if H>1/2, in the sense r(k) possesses non-summability. • Self-similarity doesn’t have negative implications. It is long-range dependence which has a serious impact on the network performance.

17. NS-2 Simulation Results 802.11b K=6 802.11b K=6 • Estimated slopes on log-log scale show a good match with analytical formulae.

18. NS-2 Simulation Results 802.11b K=6 • Leaning tendency and directional unfairness can be observed as predicted by analysis.