The optimal service policies in an M/G/1 queueing system with multiple vacation types

In this paper, a single server queueing system with Poisson arrivals and multiple vacation types, in which the server can choose one of several types of vacations to take when he finishes serving all customers in the system, is considered. Upon completion of a vacation, the server checks the number of customers waiting in the system. If the number of customers is greater than a critical threshold, the server will resume serving the queue exhaustively; otherwise, he will take another vacation. The cost structure consists of a constant waiting cost rate, fixed costs for starting up service, and reward rates for taking vacations. It is shown that this infinite buffer queueing system can be formulated as a finite state Semi-Markov decision process. With this finite state model, one can determine the optimal service policy to minimize the long-term average cost of this vacation system.