In this paper, the authors present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. They 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 of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, the authors identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, they derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, the authors develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. They 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 the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.
