On the series expansion for the stationary probabilities of an M/D/1 queue

In this paper, we give the series expansion for the stationary probabilities πn of the queue length of an M/D/1 queue based on analytic properties of the probability generating function π(z). We determine the poles and their associated residues of π(z), then give the partial fraction of π(z). The series expansion of πn is given by the poles and residues. We also give an upper bound and a lower bound for πn.