EURANDOM PREPRINT SERIES 2010-025 Exact FCFS matching rates for two infinite multi-type sequences

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. 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 assume that the types of items in the two sequences are i.i.d. with given probability vectors α, β. We describe this system by a Markov chain, obtain conditions for ergodicity, and derive its stationary distribution which is 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.