Stability of the stochastic matching model

TL;DR

This paper introduces a stochastic matching model where items arrive and are matched in pairs according to a graph, analyzing the conditions for the system's stability based on the graph's properties.

Contribution

The paper establishes a new stochastic matching model and characterizes its stability condition as dependent on the non-bipartiteness of the matching graph.

Findings

Model stability iff the matching graph is non-bipartite.

Buffer content process forms a Markov chain under i.i.d. arrivals.

Provides a criterion for stability based on graph properties.

Abstract

We introduce and study a new model that we call the {\em matching model}. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be {\em matched}. There is a finite set of classes $\maV$ for the items, and the allowed matchings depend on the classes, according to a {\em matching graph} on $\maV$. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a {\em matching policy}. When the sequence of classes of the arriving items is i.i.d., the sequence of buffer-contents is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is non-bipartite.