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 supplies 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, supplies, 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. Some arguments in the proofs rely on the so-called matching rates of specific stochastic matching models. As a by-product of these arguments, we show that, in several cases, the matching rates are independent of the matching policy, that is, the rule for choosing a match between various compatible elements.

本文讨论了图形及其应用中无处不在的显着措施。在很多排队系统中，需要考虑用户之间、供需之间的兼容性约束等。这样的系统的稳定区域可以被看作是图上的一组度量，其中所考虑的度量代表到达不同类别的用户、供应、需求等的流量，而图形代表这些类别之间的兼容性. 在本文中，我们证明了这些“稳定”措施总是可以很容易地构造为图形边缘上一系列权重的简单函数。其次，我们证明后一种测量总是与所考虑的图形上随机游走的不变测量一致。证明中的一些论点依赖于特定随机匹配模型的所谓匹配率。作为这些论点的副产品，我们表明，在一些情况下，匹配率独立于匹配策略，即在各种兼容元素之间选择匹配的规则。