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In this paper we study a Geo/T-IPH/1 queue model, where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countably phases. Using the operator-geometric solution method, we first give the expression of the operator and the joint stationary distribution. Then we obtain the probability generating function (PGF) for stationary queue length distribution and sojourn time distribution, respectively.

Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue