Computational uses of the method of phases in the theory of queues

A probability density {p_k} on the positive integers is of phase type, if it is the probability density of the time till absorption in an (m + l)-state Markov chain with one absorbing state m + 1, and m transient states all of which lead to the absorbing state. If the service times in a discrete time queue have densities of phase type, several transient and stationary probability distributions of interest may be computed by highly efficient algorithms, which are presented in the paper.