Stability of the bipartite matching model

Abstract

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan, and Weiss (Adv. Appl. Probab., 2009). 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 ¼ on C×S , independently of the past. Allowed matchings are given by a fixed bipartite graph ( C, S,E ? C × S ), and customers/servers that cannot be matched are stored in a buffer. In this paper, we study the stability of the associated discrete time Markov chain under various admissible matching policies including: ML (Match-the-Longest), MS (Match-the-Shortest), FIFO (match the oldest), priorities. For the ML policy, we prove via a Lyapunov function argument that the stability region is maximal for any bipartite graph. On the other hand, we exhibit a bipartite graph for which the MS and priority policies have non-maximal stability regions. Finally, we prove that the stability of the bipartite model is implied by the stability of the associate fluid limits, and use this result to establish maximal stability region for the FIFO matching policy.