In this paper, we discuss an interesting but challenging bilateral stochastically matching problem: A more general matched queue with matching batch pair (m, n) and two types (i.e., types A and B) of impatient customers, where the arrivals of A- and B-customers are both Poisson processes, m A-customers and n B-customers are matched as a group which leaves the system immediately, and the customers' impatient behavior is to guarantee the stability of the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process. Based on this, we provide a detailed analysis for this matched queue, including the system stability, the average stationary queue lengthes, and the average sojourn time of any A-customer or B-customer. We believe that the methodology and results developed in this paper can be applicable to dealing with more general matched queueing systems, which are widely encountered in various practical areas, for example, sharing economy, ridesharing platform, bilateral market, organ transplantation, taxi services, assembly systems, and so on.

中文翻译：

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具有匹配批次对 (m, n) 的匹配队列

在本文中，我们讨论了一个有趣但具有挑战性的双边随机匹配问题：具有匹配批次对 (m, n) 和两种类型（即类型 A 和 B）的不耐烦客户的更一般的匹配队列，其中 A-和B-客户都是泊松过程，m个A-客户和n个B-客户匹配成一组立即离开系统，客户的不耐烦行为是为了保证系统的稳定性。我们表明，这个匹配的队列可以表示为一个新的双向水平依赖准生死 (QBD) 过程。在此基础上，我们对这个匹配的队列进行了详细的分析，包括系统稳定性、平均静止队列长度以及任何A客户或B客户的平均逗留时间。