### 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 (US) |
---|---|

Pages (from-to) | 3600-3628 |

Number of pages | 29 |

Journal | Annals of Applied Probability |

Volume | 28 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2018 |

### Fingerprint

### Keywords

- Carrier-sense multiple access
- Discrete parking process
- Medium access protocols
- Nonmonotone process
- Queueing networks
- Stochastic stability
- Wireless systems

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Applied Probability*,

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

**Stability conditions for a discrete-time decentralised medium access algorithm.** / Shneer, Seva; Stolyar, Alexander.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 28, no. 6, pp. 3600-3628. https://doi.org/10.1214/18-AAP1398

}

TY - JOUR

T1 - Stability conditions for a discrete-time decentralised medium access algorithm

AU - Shneer, Seva

AU - Stolyar, Alexander

PY - 2018/12

Y1 - 2018/12

N2 - 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.

AB - 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.

KW - Carrier-sense multiple access

KW - Discrete parking process

KW - Medium access protocols

KW - Nonmonotone process

KW - Queueing networks

KW - Stochastic stability

KW - Wireless systems

UR - http://www.scopus.com/inward/record.url?scp=85055008485&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055008485&partnerID=8YFLogxK

U2 - 10.1214/18-AAP1398

DO - 10.1214/18-AAP1398

M3 - Article

AN - SCOPUS:85055008485

VL - 28

SP - 3600

EP - 3628

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 6

ER -