Quasi-Stationary Distributions of Single-Server Phase-Type Queues

Let Q̃ be the infinitesimal generator governing a PH/PH/1 queueing process and let Q be the lossy generator obtained by deleting the states corresponding to no customers in the queue. In this paper, after uniformization P = I + Q/ν for sufficiently large ν where I denotes the identity matrix, we study the quasi-stationary distribution of the lossy Markov chain in discrete time governed by P. The convergence norm r of P, i.e., r = R⁻¹ and R = sup{z^· ∑_k=0^∞P^kz^k < ∞}, which is of independent interest, is determined and P is shown to be R-transient. It is proved that if the traffic intensity is strictly less than 1 then the quasi-stationary distribution x, from which the quasi-stationary queue length and virtual waiting time distributions of the phase-type queue can be determined, exists and is a unique solution to the system rx^T = x^TP (T denotes the transpose). Finally, we demonstrate that results for M/PH/1 and PH/M/1 queues obtained in Kyprianou (1971, 1972) for more slightly general settings are easily derived by our method.