This paper considers a parallel queueing system with multiple stations, each of which contains many statistically identical servers and has a dedicated queue. Upon each customer arrival, the system manager must decide to which station the customer should be routed, with the objective of minimizing the system’s long-run average delay cost. One feature of this paper is that a customer’s delay cost depends not only on his/her delay, but also on the routed station. Considering this heterogeneity across stations, we propose a routing policy, which can be regarded as an extension of the MED–FSF policy. Under this policy, any arriving customer will be routed to: (i) the station with the minimum value, which depends on the station’s expected delay and the station index when servers in all stations are fully occupied; or otherwise (ii) the station with a fastest idle server. Using asymptotic analysis, we derive diffusion limits of queue-length processes and their stationary distributions under the proposed policy in the Halfin–Whitt regime. Combined with an asymptotic lower bound result for the long-run average delay cost, we show that the proposed routing policy is asymptotically optimal under the considered objective. Finally, we provide numerical experiments to validate the accuracy of our diffusion approximation, and we compare the performance metrics under the proposed policy with those under other commonly used routing policies.

中文翻译：

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具有长期平均标准的多服务器并行排队系统的渐近最优路由


本文考虑了一个具有多个站点的并行排队系统，每个站点都包含许多统计上相同的服务器，并有一个专用队列。在每次客户到达时，系统经理必须决定客户应路由到哪个站点，目的是最大限度地减少系统的长期平均延迟成本。本文的一个特点是，客户的延误成本不仅取决于他/她的延误，还取决于路由的车站。考虑到这种跨站的异构性，我们提出了一种路由策略，它可以被视为 MED-FSF 策略的扩展。根据此政策，任何到达的客户都将被路由到：（i） 具有最小值的站点，这取决于站点的预期延迟和所有站点服务器都已满时的站点索引;或者 （ii） 具有最快空闲服务器的工作站。使用渐近分析，我们推导出了 Halfin-Whitt 制度下拟议政策下队列长度过程及其平稳分布的扩散极限。结合长期平均延迟成本的渐近下界结果，我们表明，在所考虑的目标下，所提出的路由策略是渐近最优的。最后，我们提供了数值实验来验证扩散近似的准确性，并将拟议策略下的性能指标与其他常用路由策略下的性能指标进行了比较。