Stationary waiting time distribution in queues with phase type service and in quasi-birth-and-death processes

We consider the stationary waiting time distribution under the FIFO discipline for multi-server queues with (possibly heterogeneous) phase type servers and renewal arrivals, and for queues which can be represented as a quasi-birth-and-death process. It is shown that the complementary distribution function of the stationary waiting time in these queues is of the form where θ is positive and the dj are probabilities. A complete characterization of θ and dj , which lends itself to interesting probabilistic interpretations, is given for all these cases. Except in the case of the multiserver GI/PH/c queue which is to be discussed in detail elsewhere, we provide an efficient algorithm to compute the waiting time distribution.