A Single Server M/G/1 Feedback Queue with Two Types of Service Having General Distribution

Abstract We consider an M/G/1 feedback queue with optional server vacations based on Bernoulli schedule and a single vacation policy. We assume that the server provides two types of service, type 1 with probability p1 and type 2 with probability p2 with the service times following general distribution and each arriving customer may choose either type of service. However after the completion of each type 1 or type 2 service, the customer can feedback to the tail of the original queue with probability p to repeat the service until it is successful or may depart the system with probability q if service happens to be successful. The feedback customer also has the option to choose either type 1 or type 2 service with probability p1 and p2 respectively. Again at the completion of each type of service, the server can take a vacation with probability θ or may continue to stay in the system with probability 1 − θ. Further, we assume that whenever, the server takes a vacation, it is always a single vacation with exponentially distributed vacation period. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also the mean queue length and the mean waiting time are computed.