Marginal queue length approximations for a two-layered network with correlated queues

We consider an extension of the classical machine-repair model, where we assume that the machines, apart from receiving service from the repairman, also serve queues of products. The extended model can be viewed as a layered queueing network, where the first layer consists of the queues of products and the second layer is the ordinary machine-repair model. As the repair time of one machine may affect the time the other machine is not able to process products, the downtimes of the machines are correlated. This correlation leads to dependence between the queues of products in the first layer. Analysis of these queue length distributions is hard, as the exact dependence structure for the downtimes, or the queue lengths, is not known. Therefore, we obtain an approximation for the complete marginal queue length distribution of any queue in the first layer, by viewing such a queue as a single server queue with correlated server downtimes. Under an explicit assumption on the form of the downtime dependence, we obtain exact results for the queue length distribution for that single server queue. We use these exact results to approximate the machine-repair model. We do so by computing the downtime correlation for the latter model and by subsequently using this information to fine-tune the parameters we introduced to the single server queue. As a result, we immediately obtain an approximation for the queue length distributions of products in the machine-repair model, which we show to be highly accurate by extensive numerical experiments.