An Asymptotic Analysis of a Queueing System with Markov-Modulated Arrivals

We study the mean delay and the mean number in queue in a single-server system whose arrivals are given by a nonhomogeneous Poisson process with rate equal to a function of the state of an independent Markov process. Models of this sort arise naturally in the study of packet arrivals to a local switch. There are, in general, no closed form expressions for these quantities. We consider the normalized delay and the normalized number in queue with respect to the equivalent quantity for an M/M/1 queue with the same arrival and service rates. We show that in light and heavy traffic these quotients converge to finite, nonzero quantities that can be calculated in terms of the original model parameters. When the arrival stream is the superposition of arrival processes, the overall limit is related to the individual limits for each of the separate arrival processes. These results lead to approximations for the mean delay and number in queue for intermediate traffic. We present several examples showing the accuracy of this approximation.