A Single-Server Queue with Markov Modulated Service Times

We study a queueing system with a Poisson arrival process and Markov modulated, exponential service requirements. For a modulating Markov Chain with two states, we show that the distribution of the number-in-system is a superposition of two matrix-geometric series and provide a simple algorithm for computing the rate and coefficient matrices. These results hold for both finite and infinite waiting space systems, as well as for cases in which eigenvalues of the rate matrices’ characteristic polynomials have multiplicity grater than one. We make the conjecture that the Markov-modulated system performs better than its M/G/1 analogue if and only if the switching probabilities between the two states satisfy a simple condition. We give an intuitive argument to support this conjecture.