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Stability of the Bipartite Matching Model

Part of: Markov processes Computer system organization Special processes Graph theory

Published online by Cambridge University Press:  22 February 2016

Ana Bušić ,
Varun Gupta  and
Jean Mairesse
Ana Bušić*
Affiliation:
INRIA and École Normale Supérieure
Varun Gupta*
Affiliation:
Carnegie Mellon University
Jean Mairesse*
Affiliation:
CNRS and University Paris Diderot
*
∗ Postal address: INRIA, 23 Avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France. Email address: [email protected]
∗∗ Current address: Booth School of Business, University of Chicago, Chicago IL 60637, USA. Email address: [email protected]
∗∗∗ Postal address: University Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089 CNRS, Paris, France. Email address: [email protected]
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Abstract

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We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure μ on C× S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E⊂ C × S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), RANDOM (match uniformly), and PRIORITY. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and PRIORITY policies, we exhibit a bipartite graph with a non-maximal stability region.

Keywords

Markovian queueing theorystabilitybipartite matching

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling 05C21: Flows in graphs
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 2 , June 2013 , pp. 351 - 378
Copyright
© Applied Probability Trust 

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