Jackson Networks with Innite Supply and Unreliable Nodes

We study generalized Jackson networks with single server stations, where some nodes have an in nite supply of work. We allow in our model simultaneous breakdown of groups of servers, and group repair strategies, thereby capturing the feature that repairing several servers simultaneously may lead to more ecient repair actions and thus may reduce the repair time. In addition, we allow for servers to have innite supply. We will establish the existence of a steady-state distribution of the queue-length vector at stable nodes for dierent type of failure-regimes. As it turns out, the distribution of the failure/repair regime and the steady-state distribution of the queue-length vector at stable nodes decouple in a product-form way, and we will provide closed-form solutions for the long run throughput of the network. It is a well known problem in reliability, that the distribution of the time between breakdowns is hard to estimate. We illustrate the impact of our results with an analysis of parameter insecurity of the throughput of the systems for dierent breakdown regimes