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

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.

Stochastic dynamic matching: A mixed graph-theory and linear-algebra approach

“…The recent paper [5] is perhaps the closest to ours, and we provide a detailed discussion to highlight the relation with our paper. The equivalence of statements (ii) and (iv) in [5, Theorem 1] is synonym to the equivalence of statements 2 and 3 in Proposition 3.2.…”

“…Our work is part of a broader research effort on the stochastic matching model that will be described in details in Section 2.1 [5,6,11,14,20,23,24]. Among these works, the following are particularly relevant because directly related to our results on stability.…”

“…Our proof is significantly shorter because it relies more heavily on existing results. [5,Theorem 4] is a special case of our observation at the beginning of Section 4 that the conservation equation has a unique solution if and only if the graph is bijective (and not surjective-only). Several other formulas derived in [5,Section 7] are special cases of the formulas derived in Proposition 4.1.…”

“…[5,Theorem 4] is a special case of our observation at the beginning of Section 4 that the conservation equation has a unique solution if and only if the graph is bijective (and not surjective-only). Several other formulas derived in [5,Section 7] are special cases of the formulas derived in Proposition 4.1. The model in [5] is slightly more general because it consider graphs with loops, that is, an edge can have identical endpoints, but this paper does not adopt the mixed graph-theory and linear-algebra approach that supports most of our results.…”

“…Several other formulas derived in [5,Section 7] are special cases of the formulas derived in Proposition 4.1. The model in [5] is slightly more general because it consider graphs with loops, that is, an edge can have identical endpoints, but this paper does not adopt the mixed graph-theory and linear-algebra approach that supports most of our results.…”