Stochastic Non-Bipartite Matching Models and Order-Independent Loss Queues
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The need for matching items with one another while meeting assignment constraints or preferences has given rise to several well-known problems like the stable marriage and roommate problems. While most of the literature on matching problems focuses on a static setting with a fixed number of items, several recent works incorporated time by considering stochastic models, 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 continuous-time Markov chain 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.
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