Fluid limits for Queue-based CSMA with polynomial rates, homogenization and reflection

TL;DR

This paper analyzes a variation of CSMA with polynomial back-off rates, proving convergence to a fluid limit under certain conditions and addressing challenges at queue depletion, using a novel stochastic averaging approach.

Contribution

It introduces a new method for coupled stochastic averaging in queue-based CSMA with polynomial rates, extending understanding of its fluid limits and behavior at queue depletion.

Findings

Proves convergence of scaled process to a deterministic fluid limit.

Addresses the challenge of queue depletion in the fluid limit.

Develops a new stochastic averaging method for coupled processes.

Abstract

We study in this paper a variation of the acclaimed CSMA random access protocol. We will focus on the case where back-off rates at each node is polynomial in the size of the queue. Under a condition relating the exponent in the polynomial rates and the geometry of the interference graph, we prove convergence of the scaled process to a deterministic fluid limit up to the time a queue reaches $0$ on the fluid scale . We outline the difficulties arising at that time and solve them in the case of a complete interference graph. This paper relies on a new method to obtain a fully coupled stochastic averaging principle and can hopefully lead to more result in heavy load situations.