Delay performance in random-access grid networks

A. Zocca*, S. C. Borst, J. S.H. Van Leeuwaarden, F. R. Nardi

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We examine the impact of torpid mixing and meta-stability issues on the delay performance in wireless random-access networks. Focusing on regular meshes as prototypical scenarios, we show that the mean delays in an L×L toric grid with normalized load ρ are of the order  1/(1-ρ)L. This superlinear delay scaling is to be contrasted with the usual linear growth of the order 1/(1-ρ) in conventional queueing networks. The intuitive explanation for the poor delay characteristics is that (i) high load requires a high activity factor, (ii) a high activity factor implies extremely slow transitions between dominant activity states, and (iii) slow transitions cause starvation and hence excessively long queues and delays. Our proof method combines both renewal and conductance arguments. A critical ingredient in quantifying the long transition times is the derivation of the communication height of the uniformized Markov chain associated with the activity process. We also discuss connections with Glauber dynamics, conductance, and mixing times. Our proof framework can be applied to other topologies as well, and is also relevant for the hard-core model in statistical physics and sampling from independent sets using single-site update Markov chains.

Original languageEnglish
Pages (from-to)900-915
Number of pages16
JournalPerformance Evaluation
Volume70
Issue number10
DOIs
Publication statusPublished - 19 Sept 2013
Externally publishedYes

Keywords

  • Conductance
  • Delay scaling
  • graph theory
  • Hitting times
  • Mixing times
  • random-access networks
  • Low temperature
  • interacting particle systems
  • queueing theory
  • hard-core model

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