Stability conditions for a discrete-time decentralised medium access algorithm

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 unsaturated—each node receives an exogenous flow of packets at the rate of $\lambda$ packets per time slot. Each packet takes one slot to transmit, but neighbouring nodes cannot transmit simultaneously. The algorithm we study is standard in the following sense: a node with an empty queue does not compete for medium access; the access procedure by a node does not depend on its queue length as long as it is nonzero. 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 the condition $\lambda<2/5$. This result is intuitive for the circle topology as the throughput each node receives in the saturated system (with infinite queues) is equal to the so-called parking constant, which is larger than $2/5$. (This fact, however, does not help us to prove the result.) The result is not intuitive for the line topology as in the saturated system some nodes receive a throughput lower than $2/5$.