1/15 Basic Theorems on the Backoff Process in 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.

2/15 Basic Theorems on the Backoff Process in Understanding Single-cell network Every node interferes with the rest of the nodes. CSMA synchronizes all nodes. whether User activity is determined by whether there is a carrier in the medium or not. Sufficiency Sufficiency of the backoff analysis The kernel lies in backoff analysis Backoff process is simple (i)Every node in backoff stage k attempts transmission with probability p k for every time-slot. (ii)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.

3/15 Basic Theorems on the Backoff Process in MFT Why Mean Field Theory? Markov chain models of the backoff process irreversibility Due to their irreversibility, mathematically intractable. Decoupling approximation Backoff process at a node is asymptotically independent from those at other nodes. [BEN08] M. Benaim and J.-Y. Le Boudec, “A class of mean field limit interaction models for computer and communication systems”, Perf. Eval., Nov [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. Q: Decoupling approximation is valid? Exactly under which conditions? Recent advances in Mean Field Theory [BEN08] [BOR07] Recent advances in Mean Field Theory [BEN08] [BOR07] If the following nonlinear ODEs are globally stable, it is valid; otherwise, oscillations may occur.

4/15 Basic Theorems on the Backoff Process in Decoupling Approximation Validated Bianchi’s Formula Representative formula exploiting decoupling approximation. A set of fixed-point equations to compute collision probability.

5/15 Basic Theorems on the Backoff Process in Beyond Throughput Analysis New Interest in Backoff Distribution How much backoff time should a packet wait for transmission? [BRE09] M. Bredel and M. Fidler, “Understanding fairness and its impact on quality of service in IEEE ”, IEEE Infocom, Apr [BER04] G. Berger-Sabbatel et al., “Fairness and its impact on delay in networks”, IEEE Globecom, Nov misunderstanding 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.

6/15 Basic Theorems on the Backoff Process in Outline Mean Field Technique Revisited Supports us to apply decoupling approximation in the following principles 1.Per-Packet Backoff Principle One of the two works is incorrect? 2.Power-Tail Principle What is the distribution type of the delay-related variables? Is there long-range dependence inherent in ? 3.Inter-Transmission Principles Can we develop an analytical model for short-term fairness? When does the short-term fairness undergo a dramatic change? Conclusion

7/15 Basic Theorems on the Backoff Process in 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 b and a/g. For N=2, In the sense that b leads to approx. uniform backoff distribution, while a/g leads to approx. exponential backoff distribution

8/15 Basic Theorems on the Backoff Process in Long-range Dependence (LRD) Self-similar processes Processes w/ finite 2 nd moment LRD Processes There are LRD processes that are not self-similar – either not self-similar with infinite variances – or with infinite variances. Correctly speaking, harmful is LRD. “Joseph Effect” [MAN68], 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.

9/15 Basic Theorems on the Backoff Process in Bridging between Maths on LRD and Black Box Approach Empirical studies Empirical studies based on high volume data sets of traffic measurements Getting to Know Your Network Approach Qualitative studies Qualitative studies based on rigorous mathematical theories physical explanations “Focuses on understanding of LRD and providing physical explanations.” [WIL03] Developed by Kaj & Taqqu et al. (around 2005) A bridge between this approach and is required. A bridge between this approach and is required. Theoretical Gap The state of the art in [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.

10/15 Basic Theorems on the Backoff Process in Power-Tail Principle a truncated form of Pareto-type distribution Per-packet backoff has a truncated form of Pareto-type distribution. Sketch of proof: (1)Discovery of recursive relation in LST of (2)The quantifier set in regular variation theory is dense. (3)Application of advanced Karamata Tauberian Theorem bridge A bridge between recent mathematical theories on LRD and

11/15 Basic Theorems on the Backoff Process in LRD in Identified [KAJ05] I. Kaj, “Limiting fractal random processes in heavy-tailed systems”, Fractals in Engineering, Long-range dependence Long-range dependence in is identified. Backoff process of each node can be viewed as a renewal counting process. Ω ∑ Superpose Intermediate Telecom Process  LRD process  LRD process that is not self-similar

12/15 Basic Theorems on the Backoff Process in Short-Term Fairness in Long-term Long-term Fairness in (without enhanced functionalities) the total throughput shared equally. Short-term Short-term Fairness in : not quantified yet. Inter-transmission probability Inter-transmission probability Node N is the tagged node.

13/15 Basic Theorems on the Backoff Process in Inter-Transmission principles Doubly stochastic Poisson process : : a Poisson process on the line with random intensity The resultant dist. is approx. Gaussian. General formula for (i) small K (ii) large K and α>2 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.

14/15 Basic Theorems on the Backoff Process in Collision Dominates Aggregation Aggregation Effect : Poisson Limit for Superposition Process Decreases with N : Decreases with N Collision Effect : Gaussian Intensity Increases with N : Increases with N Gaussian (collision effect) dominates Poisson (aggregation effect). Given by Per-Packet Backoff Principle

15/15 Basic Theorems on the Backoff Process in 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 L é vian

16/15 Basic Theorems on the Backoff Process in Self-Similarity and Long-Range Dependence self-similarlong-range dependent Roughly, a self-similar process with finite 2 nd moment is long-range dependent if H>1/2, in the sense r(k) possesses non-summability. long-range dependence Self-similarity doesn’t have negative implications. It is long-range dependence which has a serious impact on the network performance.

17/15 Basic Theorems on the Backoff Process in NS-2 Simulation Results – Estimated slopes on log-log scale show a good match with analytical formulae b K= b K=6

18/15 Basic Theorems on the Backoff Process in NS-2 Simulation Results – Leaning tendency and directional unfairness can be observed as predicted by analysis b K=6