Consider an M/M/1 queueing system with server vacations where the server is turned off as soon as the queue gets empty. The authors assume that the vacation durations form a sequence of i.i.d. random variables with exponential distribution. At the end of a vacation period, the server may either be turned on if the queue is non empty or take another vacation. The following costs are incurred: a holding cost of h per unit of time and per customer in the system and a fixed cost of γ each time the server is turned on. The authors show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach they use was first proposed in Blanc et al. and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters.
