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 {Xt, Nt; t . 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,.) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t . 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear 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.