Stochastic dynamic matching problems have recently gained attention in the stochastic-modeling community due to their diverse applications, such as supply-chain management and kidney exchange programs. In this paper, we study a matching problem where items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility between items is represented by a simple graph, where items can be matched if their classes are connected.We analyze matching policies in terms of stability, delay, and long-term matching rate optimization. Our approach relies on the conservation equation, which ensures a balance between arrivals and departures in any stable system. Our main contributions are as follows.We establish a link between the existence of stable policies, the dimensionality of the solution set of the conservation equation, and the compatibility graph's structure.We describe the convex polytope formed by non-negative solutions to the conservation equation, and we design policies that can achieve or closely approximate the vertices of this polytope.Lastly, we discuss potential extensions of our results beyond the main assumptions of this paper.

翻译：随机动态匹配问题因其在供应链管理和肾脏交换计划等领域的广泛应用，近年来在随机建模学界受到关注。本文研究一类匹配问题，其中不同类别的项目依据独立泊松过程到达。未匹配的项目存储于队列中，项目间的兼容性由简单图表示，当项目类别在图中有边相连时即可匹配。我们从稳定性、延迟及长期匹配率优化的角度分析匹配策略。我们的方法基于守恒方程，该方程确保任何稳定系统中到达与离开的平衡。主要贡献如下：我们建立了稳定策略的存在性、守恒方程解集的维度与兼容图结构之间的联系；描述了由守恒方程非负解构成的凸多面体，并设计了能够实现或逼近该多面体顶点的策略；最后探讨了本文主要假设之外的结果扩展可能性。