Optimality of a Threshold Policy in theM/M/ queue with repeated vacations

Consider an M/M/1 queueing system with server vacations where the server is turned oo as soon as the queue gets empty. We 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 xed cost of each time the server is turned on. We show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach we use was rst proposed in Blanc et al. (1990) and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters. Queueing systems with vacations of the server have already received much attention in the literature and a comprehensive discussion can be found in the survey papers by Doshi (1986) and Teghem (1986). These models are commonly used for modeling and tuning various systems ranging from manufacturing systems to communication and computer systems (cf. Doshi (1986)). Vacation models can be classiied into two categories: models with repeated (multiple) vacations of the server and models with a removable server. In the former case the length of a vacation period is driven by an external process (e.g., the vacation lengths are i. the latter case the length of a vacation period is driven by the arrival process (see Heyman and