Exponential ergodicity in Markovian queueing and dam models

Abstract

We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.