On the exponential decay rate of the tail of a queue length distribution

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially the radius of convergence and the number of poles on the circle of convergence. The result is applied to an M/G/1 type Markov chain to provide a weak sufficient condition for the exponential decay of the tail of the stationary distribution. We give a counter example for the Proposition 1 of Glynn and Whitt (1994), which insists a “better result” than in this paper. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially