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

We consider an extension of the classical machine-repair model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, also supply service themselves to queues of products. The extended model can be viewed as a layered queueing network (LQN), where the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The second layer consists of a waiting buffer and a repairman, able to restore the machines into an operational state. When a machine breaks down, it waits in the repair buffer for the repairman to become available. Since the repair time of one machine may affect the period of time the other machine is not able to process products, the downtimes of the machines are correlated. We explicitly model the correlation between the downtimes, which leads to correlation between the queues of products in the first layer. Taking these correlations into account, we obtain approximations for the marginal distributions of the queue lengths in the first layer, by the study of a single server vacation queue. Extensive numerical results show that these approximations are highly accurate. Funded in the framework of the STAR-project “Multilayered queueing systems” by the Netherlands Organization for Scientific Research (NWO). The research of M. Vlasiou is also partly supported by an NWO individual grant through project 632.003.002. ∗EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands †Probability and Stochastic Networks, Centum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands