An M/G/1 Queue with Second Optional Service and General Randomized Vacation Policy

This paper studies a continuous time queue system with second optional service where all the arriving customers demand the first "essential" service while only some of them demand the second "optional" service with probability a. The service time of the first essential service and the second optional service both are independent and arbitrarily random variables. Whenever a busy period is completed, the server takes a vacation. If there is at least one customer waiting at a vacation, the server immediately serves the customer. Otherwise, the server takes another vacation with probability p, or remains idle with probability 1 - p. We give some performances analysis of this model. Finally, it gives some numerical examples to illustrate the effect of the probabilities. and p on the mean system size, waiting time, the probabilities when the server is idle and is on a vacation.