Inspired by quantum switches, we consider a discrete-time multi-way matching system with two classes of arrivals: requests for entangled pair of qubits between two nodes, and qubits from each node that can be used to serve the requests. An important feature of this model is that qubits decohere and so abandon over time. In contrast to classical server-based queueing models, the combination of queueing, server-less multi-way matching, and abandonment make the analysis a challenging problem. The primary focus of this paper is to study a simple system consisting of two types of requests and three types of qubits operating under a Max-Weight policy. In this setting, we characterize the stability region under the Max-Weight policy by adopting a two-time scale fluid limit to get a handle on the abandonments. In particular, we show that Max-Weight is throughput optimal and that it can achieve throughputs larger than the ones that can be achieved by non-idling policies when the requests are infinitely backlogged. Moreover, despite the use of the Max-Weight policy, we show that there can be a counter-intuitive behavior in the system: the longest requests queue can have a positive drift for some time even if the overall system is stable.

翻译：受量子交换机的启发，我们考虑一个离散时间的多向匹配系统，该系统包含两类到达：节点间纠缠量子比特对的请求，以及来自每个节点可用于服务这些请求的量子比特。该模型的一个重要特征是量子比特会发生退相干并随时间放弃。与经典的基于服务器的排队模型不同，排队、无服务器多向匹配和放弃的组合使得分析成为一个具有挑战性的问题。本文主要研究一个由两类请求和三类量子比特组成的简单系统，该系统在最大权重策略下运行。在此设置中，我们通过采用双时间尺度的流体极限来处理放弃问题，从而刻画了最大权重策略下的稳定性区域。特别地，我们证明最大权重策略具有吞吐量最优性，并且在请求无限积压的情况下，它能实现比非空闲策略更高的吞吐量。此外，尽管采用了最大权重策略，我们仍证明系统中可能存在反直觉行为：即使整个系统是稳定的，最长请求队列也可能在一段时间内具有正漂移。