Mixing Properties of CSMA Networks on Partite Graphs

TL;DR

This paper analyzes the transition times and mixing behavior of CSMA networks modeled on partite graphs, revealing how network parameters influence throughput fairness and starvation, with implications for understanding metastability and rare events.

Contribution

It provides a detailed analysis of transition times and mixing times in CSMA networks on partite graphs, linking network structure to stochastic dynamics and asymptotic behaviors.

Findings

Transition times scale with activation rate and component sizes.

Scaled transition times are asymptotically exponential.

Convergence to equilibrium depends on network parameters.

Abstract

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.