Reversibility and Further Properties of FCFS Infinite Bipartite Matching

Abstract

The model of FCFS infinite bipartite matching was introduced in Caldentey, Kaplan and Weiss, 2009. In this model, there is a sequence of items that are chosen i.i.d. from a finite set 𝒞 and an independent sequence of items that are chosen i.i.d. from a finite set 𝒮, and a bipartite compatibility graph G between 𝒞 and 𝒮. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, meaning that each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In Adan and Weiss, 2012, a Markov chain associated with the matching was analyzed, a condition for stability was derived, and a product form stationary distribution was obtained. In the current paper, we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes’ type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by-product, we compute useful performance measures, for instance the link lengths between matched items.