A Batch Arrival Feedback Queue with M-Optional Service and Multiple Vacations Subject To Random Breakdown

In this paper, we present a batch arrival non- Markovian queuing model with m-optional service, subject to random break downs and multiple vacations. Batches arrive in Poisson stream with mean arrival rate, such that all customers demand the first essential service, whereas only some of them demand the second optional service from m kinds of different service. The service times of the both first essential service and the second optional m kind of service are assumed to follow general (arbitrary) distribution with distribution function B_0 (v) and B_k (v) (k=1, 2, m) respectively. After the completion of second service, customer may feedback to the tail of original queen to repeat the service until it is successful or may depart forever from the system. The server may undergo breakdowns which occur according to Poisson process with breakdown rate. Once the system encounter break downs it enters the repair process and the repair time is followed by exponential distribution with repair rate. . The server takes vacation each time the system becomes empty and the vacation period is assumed to be exponential distribution. On returning from vacation if the server finds no customer waiting in the system, then the server again goes for vacation until he finds at least one customer in the system. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly