Exact time-dependent solutions for the M/D/1M/D/1 queue

Time-dependent solutions to queuing models are very useful for evaluating the performance of real-world systems. However, because of their mathematical complexity, few available results exist. In this paper, we derive the time-dependent performance measures for an M/D/1M/D/1 queue starting with a positive number of initial customers. Using the limiting property of an Erlang distribution, we obtain closed-form time-dependent formulas for the queue length and the waiting time. Furthermore, the time-dependent queue length probability in a busy period is derived.