Optimal control of an M/G/1 queue with impatient priority customers

An optimal operating policy is characterized for the infinite–horizon average–cost case of a single server queueing control problem. The server may be turned on at arrival epochs or off at departure epochs. Two classes of customers, each of them arriving according to an independent Poisson process, are considered. An arriving 1–customer enters the system if the server is turned on upon his arrival, or if the server is on and idle. In the former case, the 1–customer is selected for service ahead of those customers waiting in the system; otherwise he leaves the system immediately. 2–Customers remain in the system until they complete their service requirements. Under a linear cost structure, this paper shows that a stationary optimal policy exists such that either (1) leaves the server on at all times, or (2) turns the server off when the system is empty. In the latter case, we show that the stationary optimal policy is a threshold strategy, this feature being commonplace in most of priority queueing systems and inventory models. However, the optimal policy in our model is determined by two thresholds instead of one.