Abstract
Jobs generated by a single Poisson source can be routed through N alternative gateways, modelled as parallel M / M /1 queues. The servers at those queues are subject to random breakdowns and repairs. When a breakdown occurs, all jobs present in the corresponding queue are lost; moreover, no incoming jobs are directed to that queue during the subsequent repair period. The marginal queue size distributions are determined by finding the roots of a polynomial inside the unit disc, and solving a set of simultaneous linear equations. The optimal splitting of the input stream between the servers, so as to minimize the job loss rate, is examined. In the case N = 2, it is also possible to find the joint equilibrium distribution of the numbers of jobs in the two queues, by a reduction to a Dirichlet boundary value problem on a circle.
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