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-anddeath
(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.