Exact FCFS Matching Rates for Two Infinite Multitype Sequences

Motivated by queues with multi-type servers and multi-type customers, we consider an infinite sequence of items of types C = {c1, . . . , cI}, and another infinite sequence of items of types S = {s1, . . . , sJ}, and a bipartite graph G of allowable matches between the types. We assume that the types of items in the two sequences are i.i.d. with given probability vectors α, β. Matching the two sequences on a first come first served basis defines a unique infinite matching between the sequences. For (ci, sj) ∈ G we define the matching rate rci,sj as the long term fraction of (ci, sj) matches in the infinite matching, if it exists. We describe this system by a multi-dimensional countable Markov chain, obtain conditions for ergodicity, and derive its stationary distribution which is, most surprisingly, of product-form. We show that if the chain is ergodic, then the matching rates exist almost surely, and give a closed form formula to calculate them. We point out the connection of this model to some queueing models.