Stability regions of systems with compatibilities, and ubiquitous measures on graphs

TL;DR

This paper explores the stability regions of systems with compatibility constraints modeled as measures on graphs, revealing that stabilizing measures can be constructed from edge weights and correspond to invariant measures of random walks.

Contribution

It demonstrates that stabilizing measures in such systems are easily constructed from edge weights and are equivalent to invariant measures of associated random walks.

Findings

Stabilizing measures can be derived from simple functions of edge weights.

These measures coincide with invariant measures of random walks on the graphs.

The approach simplifies analysis of stability regions in systems with compatibility constraints.

Abstract

This paper addresses the ubiquity of remarkable measures on graphs, and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supply and demands, and so on. The stability region of such systems can then be seen as a set of measures on graphs, where the measures under consideration represent the arrival flows to the various classes of users, supply, demands, etc., and the graph represents the compatibilities between those classes. In this paper, we show that these `stabilizing' measures can always be easily constructed as a simple function of a family of weights on the edges of the graph. Second, we show that the latter measures always coincide with invariant measures of random walks on the graph under consideration.