Jobs from a single Poisson input stream receive K independent stages of service, one at each stage in the pipeline. At stage i jobs are routed through one of the Ni available nodes, modelled as M/M/1 queues. These nodes are subject to random failure and repairs which leave their corresponding queues intact, but may affect the routeing of jobs arriving at that stage during the subsequent repair period. Two possible approximate solutions for the marginal queue size distributions are obtained using Marie's method and spectral expansion. Approximations are compared with solutions obtained by simulation techniques. Two routeing strategies are considered, fixed and selective, and the relative accuracy of the approximate solutions and predicted optimal routeing vectors are discussed. This method is obviously applicable to other, more general, network models and it is therefore interesting to observe the accuracy of the approximations and predictions of an optimal routeing vector. Models such as this have traditionally been studied through simulation. However, an exceedingly long runtime is needed to obtain steady state results, especially when failures are rare and repairs are slow. The method presented here gives a very rapid response and as such is clearly of great practical benefit, especially when optimising the routing of jobs.
Keywords: queueing networks, breakdowns, spectral expansionECS-LFCS-98-388.
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