Stability conditions for a discrete-time decentralised medium access algorithm

TL;DR

This paper analyzes the stability of a discrete-time decentralized medium access algorithm in wireless networks with nodes arranged in line or circle topologies, proving stability under certain packet arrival rates.

Contribution

It provides the first stability conditions for a standard decentralized medium access algorithm in unsaturated wireless network models with specific topologies.

Findings

System is stable if packet arrival rate λ < 2/5 for both topologies.

Stability proof applies to unsaturated systems with non-zero queues.

Results align with intuitive throughput in circle topology, less so in line topology.

Abstract

We consider a stochastic queueing system modelling the behaviour of a wireless network with nodes employing a discrete-time version of the standard decentralised medium access algorithm. The system is {\em unsaturated} -- each node receives an exogenous flow of packets at the rate $\lambda$ packets per time slot. Each packet takes one slot to transmit, but neighboring nodes cannot transmit simultaneously. The algorithm we study is {\em standard} in that: a node with empty queue does {\em not} compete for medium access; the access procedure by a node does {\em not} depend on its queue length, as long as it is non-zero. Two system topologies are considered, with nodes arranged in a circle and in a line. We prove that, for either topology, the system is stochastically stable under condition $\lambda < 2/5$. This result is intuitive for the circle topology as the throughput each node receives in a saturated system (with infinite queues) is equal to the so called {\em parking constant}, which is larger than $2/5$. (The latter fact, however, does not help to prove our result.) The result is not intuitive at all for the line topology as in a saturated system some nodes receive a throughput lower than $2/5$.