The threshold policy in an M/G/1 queue with an exceptional first vacation

This paper treats the threshold policy for an M/G/1 queue with an exceptional first vacation. Under this policy, the server takes a vacation when he finishes serving all customers in the system. Upon returning from the vacation, the server takes another vacation if the number in the system is below a certain level, otherwise, he resumes service. The distribution function of the first vacation is different from that of the subsequent vacations. There is a startup cost for starting a series of vacations, a linear customer waiting cost, and type-dependent vacation reward rates. Renewal theory is used to develop expressions for the average system cost, and a provably finite search procedure is developed for determining the optimal threshold policies.