We consider a general class of closed product-form queueing networks, consisting of single-server queues and infinite-server queues. Even if a network is of product-form type, performance evaluation tends to be difficult due to the potentially large state space and the dependence between the individual queues. To remedy this, we analyze the model in a Halfin–Whitt inspired scaling regime, where we jointly blow up the traffic loads of all queues and the number of customers in the network. This leads to a closed-form limiting stationary distribution, which provides intuition on the impact of the dependence between the queues on the network’s behavior. We assess the practical applicability of our results through a series of numerical experiments, which illustrate the convergence and show how the scaling parameters can be chosen to obtain accurate approximations.

我们考虑一类封闭的产品形式排队网络，由单服务器队列和无限服务器队列组成。即使网络是产品形式类型的网络，由于潜在的大状态空间和各个队列之间的依赖性，性能评估往往也很困难。为了解决这个问题，我们在受 Halfin-Whitt 启发的扩展机制中分析模型，在这种情况下，我们共同增加了所有队列的流量负载和网络中的客户数量。这导致了一个封闭形式的限制平稳分布，它提供了关于队列之间的依赖关系对网络行为的影响的直觉。我们通过一系列数值实验评估我们结果的实际适用性，