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Stability of the stochastic matching model

Part of: Computer system organization Markov processes 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

Information

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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References

[1] Adan, I. and Weiss, G. (2012).Exact FCFS matching rates for two infinite multitype sequences.Operat. Res. 60,475–489.Google Scholar
[2] Adan, I.,Bušić, A.,Mairesse, J. and Weiss, G. (2016).Reversibility and further properties of FCFS infinite bipartite matching.Available at http://arxiv.org/abs/1507.05939v1.Google Scholar
[3] Brémaud, P. (1999).Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues(Texts Appl. Math. 31).Springer,New York.Google Scholar
[4] Brualdi, R. A.,Harary, F. and Miller, Z. (1980).Bigraphs versus digraphs via matrices.J. Graph Theory 4,51–73.Google Scholar
[5] Bušić, A.,Gupta, V. and Mairesse, J. (2013).Stability of the bipartite matching model.Adv. Appl. Prob. 45,351–378.Google Scholar
[6] Caldentey, R.,Kaplan, E. H. and Weiss, G. (2009).FCFS infinite bipartite matching of servers and customers.Adv. Appl. Prob. 41,695–730.CrossRefGoogle Scholar
[7] Fayolle, G.,Malyshev, V. A. and Menshikov, M. (1995).Topics in the Constructive Theory of Countable Markov Chains.Cambridge University Press.Google Scholar
[8] Gans, N.,Koole, G. and Mandelbaum, A. (2003).Telephone call centers: tutorial, review, and research prospects.Manufacturing Serv. Operat. Manag. 5,79–141.Google Scholar
[9] Gurvich, I. and Ward, A. (2014).On the dynamic control of matching queues.Stoch. Systems 4,479–523.Google Scholar
[10] Kumar, P. R. (1993).Re-entrant lines.Queueing Systems Theory Appl. 13,87–110.Google Scholar
[11] McKeown, N.,Mekkittikul, A.,Anantharam, V. and Walrand, J. (1999).Achieving 100% throughput in an input-queued switch.IEEE Trans. Commun. 47,1260–1267.CrossRefGoogle Scholar
[12] Tassiulas, L. and Ephremides, A. (1992).Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks.IEEE Trans. Automatic Control 37,1936–1948.Google Scholar