We consider an extension of the classical machine-repair model, where weassume that the machines, apart from receiving service from the repairman, alsoserve queues of products. The extended model can be viewed as a layeredqueueing network, where the first layer consists of the queues of products andthe second layer is the ordinary machine-repair model. Since the repair time ofone machine may affect the time the other machine is not able to processproducts, the downtimes of the machines are correlated. This correlation leadsto dependence between the queues of products in the first layer. Analysis ofthese queue length distributions is hard, since the exact dependence structurefor the downtimes, or the queue lengths, is not known. Therefore, we obtain anapproximation for the complete marginal queue length distribution of any queuein the first layer, by viewing such a queue as a single server queue withcorrelated server downtimes. Under an explicit assumption on the form of thedowntime dependence, we obtain exact results for the queue length distributionfor that single server queue. We use these exact results to approximate themachine-repair model. We do so by computing the downtime correlation for thelatter model and by subsequently using this information to fine-tune theparameters we introduced to the single server queue. As a result, weimmediately obtain an approximation for the queue length distributions ofproducts in the machine-repair model, which we show to be highly accurate byextensive numerical experiments.
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