### 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 λ 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 λ<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.

Original language | English |
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Pages (from-to) | 3600-3628 |

Number of pages | 29 |

Journal | Annals of Applied Probability |

Volume | 28 |

Issue number | 6 |

DOIs | |

Publication status | Published - 8 Oct 2018 |

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## Profiles

## Vsevolod Shneer

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics - Associate Professor
- Research Centres and Themes, Energy Academy - Associate Professor

Person: Academic (Research & Teaching)

## Cite this

Shneer, V., & Stolyar, A. (2018). Stability conditions for a discrete-time decentralised medium access algorithm.

*Annals of Applied Probability*,*28*(6), 3600-3628. https://doi.org/10.1214/18-AAP1398