On the Inter-Departure Times in $M/\widetilde{D}/1/{B}_{on}$ Queue With Queue-Length Dependent Service and Deterministic/Exponential Vacations

We derive the distribution of inter-departure times of a finite-buffer single-server queue with Poisson arrival process and queue-length-dependent service times, where the server goes to vacation if either the queue is emptied or a limited number
$(R_{1})$
of packets are served, whichever occurs first, in the current busy period. We consider two types of vacation distributions: 1) deterministic and 2) exponential. Queue-length distribution at embedded points is derived first, then, the distribution and variance of inter-departure times are derived, for both types of vacations. The simulation results are in good agreement with the derived analytical results. The above framework would be useful at the receiver in modeling and analyzing the jitter and the waiting time of time-division multiplexing (TDM) emulated packets in TDM over packet-switched network (TDM over PSN) technology as a function of a buffer size.