Transient workload distribution in the M/G/1 finite-buffer queue with single and multiple vacations

A finite-buffer M/G/1-type queueing system with exhaustive service and server vacations is considered. The single and multiple vacation policies are investigated separately. In the former case a single vacation is being initialized every time when the system becomes empty. After the vacation the server waits on standby for packets to start the service process. In the multiple vacation policy the server begins successive single vacation periods until at least one packet is present in the system. For both vacation policies systems of integral equations for the transient virtual waiting time v(t) distributions, conditioned by the numbers of packets present in the system at the start epoch, are found. The representations for the twofold Laplace transforms of the conditional distributions of v(t) are obtained and written in compact forms, convenient in numerical treatment. From these formulae stationary distributions of v(t) and their means can be computed via usual operations on transforms. Some numerical illustrations of theoretical results are attached as well.