Section:
New Results
Reversibility and further properties of FCFS infinite bipartite matching
[3]
The model of FCFS infinite bipartite matching was introduced in Caldentey, Kaplan,
& Weiss Adv. Appl. Probab., 2009. In this model, there is a sequence of items that are
chosen i.i.d. from a finite set and an independent sequence of items that are chosen i.i.d.
from a finite set , and a bipartite compatibility graph between and . Items of the
two sequences are matched according to the compatibility graph, and the matching is FCFS,
meaning that each item in the one sequence is matched to the earliest compatible unmatched
item in the other sequence. In Adan & Weiss, Operations Research, 2012, a Markov chain
associated with the matching was analyzed, a condition for stability was derived, and a
product form stationary distribution was obtained. In the current paper, we present several
new results that unveil the fundamental structure of the model. First, we provide a pathwise
Loynes' type construction which enables to prove the existence of a unique matching for
the model defined over all the integers. Second, we prove that the model is dynamically
reversible: we define an exchange transformation in which we interchange the positions of
each matched pair, and show that the items in the resulting permuted sequences are again
independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we
obtain product form stationary distributions of several new Markov chains associated with
the model. As a by product, we compute useful performance measures, for instance the link
lengths between matched items.