Algorithmic Solution of Some Queues with Overflows

The overflow stream from an M/PH/1 queue of finite capacity is used as the input to an unbounded queue with one or more exponential servers. It is shown that the combined system, consisting of the two queues, may be studied as a highly structured Markov process. In the stable case, this Markov process has a matrix-geometric invariant vector. Particular features of the infinitesimal generator of the process may be used to simplify the numerical computation of various steady-state features of the model. Several variants and numerical examples are discussed.