Markov Processes Whose Steady State Distribution is Matrix- Exponential with an Application to the GI/PH/1 Queue.

This paper is concerned with a bivariate Markov Process {X sub t, N sub {t}; t >= 0} with a special structure. The process X sub t may either increase linearly or have jump (downward) discontinuities. The process X sub t takes values in [0, inf) and N sub t takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {X sub t, N sub {t}; t >= 0} has a matrix exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a nonlinear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand. Such a model may be used to determine the manufacturing capacity for a factory which produces goods whose demand fluctuates seasonally.