Three Phase M/G/1 queue with Bernoulli Feedback and Multiple server vacation

摘要

In this paper, a three phase M/G/1 queueing system with Bernoulli feedback where the server takes multiple vacation is considered. All the poisson arrivals with mean arrival rate demand first essential service. Only some of them demand second optional service or third optional service. The service times of the first essential service are assumed to follow a general distribution B_1(v), the second optional service and the third optional service with general distribution with distribution function B_2(v) and B_3(v) respectively. After the completion of the first service or second service or third service, if the customer is dissatisfied he can join the tail of the queue for receiving another regular service with probability p. Otherwise the customer may depart from the system with the probability q = 1-p. If there is no customer in the queue, then the server can go for vacation and vacation periods are exponentially distributed with mean vacation time 1/γ. On returning from vacation, if the server again founds no customer waiting in the queue, then it again goes for vacation. The server continues to go for vacation until he finds at least one customer in the system. We find the time dependent probability generating function in terms of Laplace transforms and derive explicitly the corresponding steady state results. Moreover we find explicit expressions for the mean queue length and mean waiting time.