Exact FCFS Matching Rates for Two Infinite Multitype Sequences

Motivated by queues with multitype servers and multitype customers, we consider an infinite sequence of items of types 𝒞 = {c₁,…,c_I}, and another infinite sequence of items of types 𝒮 = {s₁,…,s_J}, and a bipartite graph G of allowable matches between the types. We assume that the types of items in the two sequences are independent and identically distributed (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 (c_i,s_j) ∈ G we define the matching rate r_{ci, sj} as the long-term fraction of (c_i, s_j) matches in the infinite matching, if it exists. We describe this system by a multidimensional 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 we give a closed-form formula to calculate them. We point out the connection of this model to some queueing models.