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Stability of the stochastic matching model

Part of: Markov processes Computer system organization Graph theory

Published online by Cambridge University Press:  09 December 2016

Jean Mairesse  and
Pascal Moyal
Jean Mairesse*
Affiliation:
CNRS and UPMC
Pascal Moyal*
Affiliation:
Université de Technologie de Compiègne and Northwestern University
*
* Postal address: Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6, 4 place Jussieu, 75252 Paris Cedex 05, France.
** Postal address: LMAC, Université de Technologie de Compiègne, 60203 Compiègne Cedex, France. Email address: pascal.moyal@utc.fr
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Abstract

We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.

Keywords

Markovian queueing theorystabilitymatchinggraph

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68M20: Performance evaluation; queueing; scheduling 05C75: Structural characterization of families of graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1064 - 1077
Copyright
Copyright © Applied Probability Trust 2016 

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