Stability of the bipartite matching model

TL;DR

This paper analyzes the stability of a bipartite matching model with customers and servers, examining how different matching policies affect the system's stability and identifying conditions for maximal stability regions.

Contribution

It establishes conditions under which the stability region is maximal for various matching policies in the bipartite matching model.

Findings

Maximal stability regions are identified for certain policies and graphs.

ML policy guarantees maximal stability region for any bipartite graph.

MS and priority policies may have non-maximal stability regions for some graphs.

Abstract

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan, and Weiss (Adv. Appl. Probab., 2009). Customers and servers play symmetrical roles. There is a finite set C resp. S, of customer, resp. server, classes. Time is discrete and at each time step, one customer and one server arrive in the system according to a joint probability measure on CxS, independently of the past. Also, at each time step, pairs of matched customer and server, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph. 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), priorities. 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.