Reversibility and further properties of FCFS infinite bipartite matching

Ivo Adan, Ana Busic, Jean Mairesse, Gideon Weiss

Published 2015-07-21Version 1

The model of FCFS infinite bipartite matching was introduced in caldentey-kaplan-weiss 2009. In this model there is a sequence of items that are chosen i.i.d. from $\mathcal{C}=\{c_1,\ldots,c_I\}$ and an independent sequence of items that are chosen i.i.d. from $\mathcal{S}=\{s_1,\ldots,s_J\}$, and a bipartite compatibility graph $G$ between $\mathcal{C}$ and $\mathcal{S}$. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In adan-weiss 2011 a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and and the rates $r_{c_i,s_j}$ of matches between compatible types $c_i$ and $s_j$ were calculated. In the current paper we present further results on this model. We use a Loynes type scheme to show that if the system is stable there is a unique matching of the sequence over all the integers. We show dynamic reversibility: if in every matched pair we exchange the positions of the items the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. We use these results to derive stationary distributions of various Markov chains associated with the model. We also calculate the distribution of the link lengths between matched items.