A general stochastic matching model on multigraphs

TL;DR

This paper extends a stochastic matching model to multigraphs with self-loops, providing stability conditions and explicit stationary distributions for the First Come, First Matched policy.

Contribution

It introduces a generalized stochastic matching model on multigraphs, analyzing its stability and stationary distribution, which was not previously addressed.

Findings

Derived necessary and sufficient stability conditions.

Explicit stationary distribution for FCFS matching policy.

Extended the model to include self-loops in multigraphs.

Abstract

We extend the general stochastic matching model on graphs introduced in (Mairesse and Moyal, 2016), to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is `First Come, First Matched'.