Quasi-stationary distributions in a PH/PH/c queue

Explicit representations of the quasi-stationary (QS) distributions in a multi-server phase-type queue are obtained by extending the results for a single server queue in Kijima. Two types of QS distributions are treated, one for fully busy periods and one for partially busy periods. It is shown that these distributions are given as positive solutions of vector equations γmx=xQm (m=1,c), where Qm is the lossy generator governing the queueing process with at least m customers and γm is its decay parameter. The paper first develops the method to determine γm, and then obtains explicit expressions for the QS distributions. It turns out that the QS distribution for partially busy periods has a matrix-geometric structure in some cases. By investigating the asymptotic behaviors of these distributions, it is also shown that the QS distributions have longer tails than the stationary distribution.