Heavy traffic analysis of a Markov-modulated queue with finite capacity and general service times

We consider a set of N independent sources, each of which alternates between "on" and "off" states. When a source is on, it generates a Poisson arrival stream to a finite-capacity queue with a general server. We derive the balance equations satisfied by the joint steady-state distribution of the queue length and the number of "on" sources. Then we analyze the problem in the heavy traffic limit where N --> infinity and the average arrival rate is nearly equal to the mean service rate. The capacity is scaled to be O(root N). The first two terms in the asymptotic series are characterized as solutions to elliptic partial differential equations (PDEs) with appropriate boundary conditions. We then develop numerical and asymptotic methods for solving these PDEs. The analysis makes use of singular perturbation techniques and special functions.