Summary: This paper considers an \(\mathrm{M}/\mathrm{M}/1\) vacation queue with T-SPH vacation time, where T-SPH denotes the discrete-time phase type distribution defined on a birth and death process with countable many states. The queue model can be described by a quasi-birth-and-death (QBD) process with infinite phases. Firstly, concrete forms of rate operator and operator-geometric solution of the QBD process are given. Secondly, the stochastic decomposition structure of the stationary queue length is presented. Finally, it is proved that the additional queue length follows discrete-time infinite phase type distribution.