Analysis of Batch Arrival Queue with Second Optional and Repair Under Optional Vacation, Balking

This paper analyze a batch arrival queue with second optional service and repair under optional vacation and balking is considered. All coming customers demand the first essential service where in only some of them demand the second optional service. After completion of service, the server will go for vacation with probability p or remain staying back in the system for providing the service to the next customer with probability 1-p if any. The service times of both essential and optional services and vacation time follow arbitrary (general) distribution. The system may breakdown at random time and the breakdowns occur according to poisson stream. Once the server breakdown, it must be spend to repair process immediately. The most realistic aspect in modeling of a unreliable server, multi optional repair is required. Both essential and optional repair times follow exponential distribution. In addition to this, due to the annoyance of seeing long queue in the system the customer may decide not to join the queue, called balking. Such a customer behavior is considered in both busy time and vacation time of the system. The time dependent probability generating functions has been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly