Representations for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues

Abstract

Neuts has shown that a Markov chain of GI/M/1 type has a matrix-geometric steady-state distribution, extending earlier work by Evans [13] and Wallace [31]. Various authors later observed that, under this steady-state distribution, the “level” component is phase type. We give a probabilistic interpretation of this result by constructing, from a sample path of the original process, a Markov chain with the correct distribution of time to absorption. We give a similar construction for the matrix-exponential steady-state distribution of Sengupta's continuous-level chain [27]. We extend Sengupta's analysis of the GI/PH/1 queue to the many-server case and thereby prove that the steady-state waiting time distribution in a GI/PH/c queue is phase type. This leads to a proof of a recent conjecture of Abate, Choudhury and Whitt [1] and gives an alternative approach to Neuts and Takahashi's [20] results on the exponential decay parameter of the waiting-time distribution