Abstract: The M/T-SPH/1 queueing model is studied, and the probability generating function for the stationary queue length of the model is given based on the quasi-birth-and-death process and operator-geometric method. Furthermore,  distribution of queue length is not a PH distribution, while the tail of the distribution has the characteristic of geometric decay under certain conditions.

Key words:  M/T-SPH/1 queue, operator-geometric solution, stationary queue length, discrete PH distribution, geometric tail