Queueing system with vacations after a random amount of work

This paper considers an M/G/1 queue with the following vacation discipline. The server takes a vacation as soon as it has served a certain amount of work since the end of the previous vacation. If the system becomes empty before the server has completed this amount of work, then it stays idle until the next customer arrival and then becomes active again. Such a vacation discipline arises, for example, in the maintenance of production systems, where machines or equipment mainly degrade while being operational. We derive an explicit expression for the distribution of the time it takes until the prespecified amount of work has been served. For the case the total amount of work till vacation is exponentially distributed, we derive the transforms of the steady-state workload at various epochs, busy period, waiting time, sojourn time, and queue length distributions.