Markov chain representations of discrete distributions applied to queueing models

We present applications of Markov chain based representations of discrete renewal distributions to queueing models, and extend the notion of that representation to some non-renewal discrete distributions. Two representations are considered; one based on remaining time, the other on elapsed time. These representations make it easier to use matrix-analytic methods for several stochastic models, especially queueing models, thereby allowing us to develop better algorithmically tractable procedures for their analysis. Specifically, they allow us to capitalize on the resulting special structures. We first discuss some key measures of these distributions using phase type distribution results, including some time reversibility relations between the elapsed and remaining time representations. We then show applications to the MAP/G/1, the GI/MSP/1 and the GI/G/1 systems, and briefly explain how the representations of the non-renewal types of discrete distributions can be used for the MRP/SMP/1 system. The emphasis of this paper is about efficient procedures for the R and G matrices associated with these queueing models.