Transient properties of many-server queues and related QBD's

Søren Asmussen and Mats Pihlsgård

Abstract:: The time t(n) of first passage from queue length x to queue length n > x in an MAP/M/c queue is considered. The mean and the Laplace transform is computed as solutions of systems of linear equations coming out byoptional stopping of a martingale obtained as an stochastic integral of the exponential Wald martingale for Markov additive processes. Compared to existing techniques for QBD's, the approach has the advantage ofbeing far more efficient for large n.

Key words:: birth-death process, buffer overflow, exponential martingale, first passage problem, heterogeneous servers, Kella-Whitt martingale, Laplace transform, Lévy process, MAP/M/c queue, Markov additive process, optional stopping