On steady-state queue size distributions of the discrete-time GI/G/1 queue

Abstract

In this paper, we present results for the steady-state system length distributions of the discrete-timeGI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is ofGI/M/1 type if the embedding points are arrival epochs and is ofM/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For theGI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for theM/G/1 type chain, we develop a simple linear transformation that relates it to theGI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for theGI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.