The stochastic dynamic matching problem has recently drawn attention in the stochastic-modeling community due to its numerous applications, ranging from supply-chain management to kidney exchange programs. In this paper, we consider a matching problem in which items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility constraints are described by a simple graph on the classes, so that two items can be matched if their classes are neighbors in the graph. We analyze the efficiency of matching policies, not only in terms of system stability, but also in terms of matching rates between different classes. Our results rely on the observation that, under any stable policy, the matching rates satisfy a conservation equation that equates the arrival and departure rates of each item class. Our main contributions are threefold. We first introduce a mapping between the dimension of the solution set of this conservation equation, the structure of the compatibility graph, and the existence of a stable policy. In particular, this allows us to derive a necessary and sufficient stability condition that is verifiable in polynomial time. Secondly, we describe the convex polytope of non-negative solutions of the conservation equation. When this polytope is reduced to a single point, we give a closed-form expression of the solution; in general, we characterize the vertices of this polytope using again the graph structure. Lastly, we show that greedy policies cannot, in general, achieve every point in the polytope. In contrast, non-greedy policies can reach any point of the interior of this polytope, and we give a condition for these policies to also reach the boundary of the polytope.

翻译：随机动态匹配问题近年来因其在供应链管理、肾脏交换计划等众多应用中的重要性，引起了随机建模领域的广泛关注。本文考虑一个匹配问题，其中不同类别的物品按照独立的泊松过程到达。未匹配物品存储在一个队列中，兼容性约束由类别间的一个简单图描述，因此两个物品仅当它们的类别在图上是邻居时才能匹配。我们不仅从系统稳定性的角度，还从不同类别之间匹配率的角度，分析了匹配策略的效率。我们的结果基于一个观察：在任意稳定策略下，匹配率满足一个守恒方程，该方程平衡了每个物品类别的到达率和离开率。我们的主要贡献有三点。首先，我们引入了一个映射，将这一守恒方程解集的维度、兼容图的结构以及稳定策略的存在性联系起来。特别地，这使我们能够推导出一个充要的稳定性条件，该条件可在多项式时间内验证。其次，我们描述了守恒方程非负解构成的凸多面体。当该多面体缩减为一个单点时，我们给出解的闭合形式；一般情况下，我们再次利用图结构来刻画该多面体的顶点。最后，我们证明贪心策略通常无法达到多面体中的每一个点。相反，非贪心策略可以到达该多面体内部的任何点，并且我们给出了这些策略也能到达多面体边界的条件。