Stochastic non-bipartite matching models and order-independent loss queues

Abstract

The problem of appropriately matching items subject to compatibility constraints arises in a number of important applications. While most of the literature on matching theory focuses on a static setting with a fixed number of items, several recent works incorporated time by considering a stochastic model in which items of different classes arrive according to independent Poisson processes and assignment constraints are described by an undirected non-bipartite graph on the classes. In this paper, we prove that the Markov process associated with this model has the same transition diagram as in a product-form queueing model called an order-independent loss queue. This allows us to adapt existing results on order-independent (loss) queues to stochastic matching models and, in particular, to derive closed-form expressions for several performance metrics, like the waiting probability or the mean matching time, that can be implemented using dynamic programming. Both these formulas and the numerical results that they allow us to derive are used to gain insight into the impact of parameters on performance. In particular, we characterize performance in a so-called heavy-traffic regime in which the number of items of a subset of the classes goes to infinity while items of other classes become scarce.