Product-form results for two-station networks with shared resources

Queueing networks are studied with two stations: either in tandem or in parallel, and with a common service resource shared among the two stations. First, a necessary and sufficient criterion, called adjoint reversibility, is provided to decide whether the system possesses a product form or not. This criterion unifies both the parallel (a reversible) and the tandem (a non-reversible) system in one product-form theorem. Next, the criterion is applied separately for the parallel and tandem system to obtain a number of new product-form examples which also includes non-balanced capacity sharing. Despite, but also due to, the different parallel and tandem mechanisms we observe that for certain examples the product form has the same structure, while for others there are essential differences. In addition, it is also proven that several models cannot have a product-form result. The results provide new insights and a step forward in understanding the behavior of multi-layered queueing networks in which resources are shared among stations.