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A Random Multiple-Access Protocol with Spatial Interactions

Part of: Operations research and management science Special processes Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  14 July 2016

Charles Bordenave ,
Serguei Foss  and
Vsevolod Shneer
Charles Bordenave*
Affiliation:
University of California
Serguei Foss*
Affiliation:
Heriot-Watt University and Sobolev Institute of Mathematics
Vsevolod Shneer*
Affiliation:
Eindhoven University of Technology and EURANDOM
*
∗Current address: CNRS UMR 5219, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse, France.
∗∗Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK.
∗∗∗Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: shneer@eurandom.tue.nl
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Abstract

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We analyse an ALOHA-type random multiple-access protocol where users have local interactions. We show that the fluid model of the system workload satisfies a certain differential equation. We obtain a sufficient condition for the stability of this differential equation and deduce from that a sufficient condition for the stability of the protocol. We discuss the necessary condition. Furthermore, for the underlying Markov chain, we estimate the rate of convergence to the stationary distribution. Then we establish an interesting and unexpected result showing that the main diagonal is locally unstable if the input rate is sufficiently small. Finally, we consider two generalisations of the model.

Keywords

ALOHA protocolspatial interactionstability of processesfluid limits

MSC classification

Secondary: 60K25: Queueing theory 90B15: Network models, stochastic 68R10: Graph theory (including graph drawing)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 844 - 865
Copyright
Copyright © Applied Probability Trust 2009 

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