Mixing properties of CSMA networks on partite graphs

We consider a stylized stochastic model for a wireless CSMA network. Experimental results in prior studies indicate that the model provides remarkably accurate throughput estimates for IEEE 802.11 systems. In particular, the model offers an explanation for the severe spatial unfairness in throughputs observed in such networks with asymmetric interference conditions. Even in symmetric scenarios, however, it may take a long time for the activity process to move between dominant states, giving rise to potential starvation issues. In order to gain insight in the transient throughput characteristics and associated starvation effects, we examine in the present paper the behavior of the transition time between dominant activity states. We focus on partite interference graphs, and establish how the magnitude of the transition time scales with the activation rate and the sizes of the various network components. We also prove that in several cases the scaled transition time has an asymptotically exponential distribution as the activation rate grows large, and point out interesting connections with related exponentiality results for rare events and meta-stability phenomena in statistical physics. In addition, we investigate the convergence rate to equilibrium of the activity process in terms of mixing times.

[1]  E. Olivieri,et al.  Markov chains with exponentially small transition probabilities: First exit problem from a general domain. I. The reversible case , 1995 .

[2]  Samuel Karlin,et al.  COINCIDENT PROPERTIES OF BIRTH AND DEATH PROCESSES , 1959 .

[3]  Vincenzo Mancuso,et al.  Measurement and Modeling of the Origins of Starvation in Congestion Controlled Mesh Networks , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[4]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[5]  Koushik Kar,et al.  Throughput modelling and fairness issues in CSMA/CA based ad-hoc networks , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[6]  Basil S. Maglaris,et al.  Throughput Analysis in Multihop CSMA Packet Radio Networks , 1987, IEEE Trans. Commun..

[7]  J. Keilson.  Markov Chain Models--Rarity And Exponentiality , 1979 .

[8]  Olle Häggström.  Ergodicity of the hard-core model onZ2 with parity-dependent activities , 1997 .

[9]  Johan van Leeuwaarden,et al.  Spatial fairness in linear random-access networks , 2012, Perform. Evaluation.

[10]  Rudolf Grübel,et al.  Rarity and exponentiality: an extension of Keilson's theorem, with applications , 2005, Journal of Applied Probability.

[11]  D. Aldous.  Markov chains with almost exponential hitting times , 1982 .

[12]  A. Girotra,et al.  Performance Analysis of the IEEE 802 . 11 Distributed Coordination Function , 2005 .

[13]  Yuri M. Suhov,et al.  A Hard-Core Model on a Cayley Tree: An Example of a Loss Network , 2004, Queueing Syst. Theory Appl..

[14]  den WThF Frank Hollander.  Metastability under stochastic dynamics , 2004 .

[15]  J. Keilson.  A review of transient behavior in regular diffusion and birth-death processes , 1964 .

[16]  Frank Kelly,et al.  Stochastic Models of Computer Communication Systems , 1985 .

[17]  G. Reuter,et al.  Spectral theory for the differential equations of simple birth and death processes , 1954, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[18]  David Aldous,et al.  Inequalities for rare events in time-reversible Markov chains II , 1993 .

[19]  Soung Chang Liew,et al.  Back-of-the-Envelope Computation of Throughput Distributions in CSMA Wireless Networks , 2007, 2009 IEEE International Conference on Communications.

[20]  Julian Keilson,et al.  A Limit Theorem for Passage Times in Ergodic Regenerative Processes , 1966 .

[21]  Patrick Thiran,et al.  A Packing Approach to Compare Slotted and Non-Slotted Medium Access Control , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

[22]  Julian Keilson.  Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes , 1971 .

[23]  Stan Zachary,et al.  Loss networks and Markov random fields , 1999 .

[24]  AARON KERSHENBAUM,et al.  An Algorithm for Evaluation of Throughput in Multihop Packet Radio Networks with Complex Topologies , 1987, IEEE J. Sel. Areas Commun..

[25]  Raúl Gouet,et al.  Blocking and Dimer Processes on the Cayley Tree , 2008 .