In this paper, we analyze a finite T-SPH$/M/1/N$ queue model with state-dependent service rate, where T-SPH denotes the continuous time phase type distribution defined on a birth and death process with countable number of states. The queue system investigated can be described by a quasi-birth-and-death (QBD) process with infinite levels and finite number of phases. By analyzing the QBD process with the method of generalized eigenvalues, we derive the analytic expression of the stationary queue length distribution of the queue model. Meanwhile, to explain the validity of our method, we also present several numerical examples to illustrate the effect of the varying parameters on the system performance.