An Optimum Control of a Batch Arrival Queue with Second Optional Service and Setup time under Bernoulli Vaction Schedule

An M x /G/1 queueing system with second optional service (SOS)is studied under N policy and Bernoulli Vacation. The system remains idle until the queue size reaches or exceeds N ( ≥ 1). When the queue size reaches at leastN, the server may begin his setup operation with probability s or may start the service with probability (1-s). The server provides two phases of heterogeneous services, of which, first phase of service is essential and second phase of service is optional. As soon as the first essential service (FES) of a unit is completed, the customer may leave the system with probability (1-r) or may immediately opt for SOS with probability r. Wheneverthe service of each unit is completed, the server will have the option of taking vacation(Bernoulli). Thus a customer completes his service, by undergoing FES alone, the server may take a vacation with probability p1orstays idle or continue the next service to the new customer if any,with probability (1-p1). If the customer who finishes FES proceeds to SOS then the server may take vacation at the end of SOS with probability p2 or stays idle or continues with the FES for the new customer with probability (1-p2) according as the system is empty or having customers in the system. The queue size distribution at a random epoch is obtained for this model using SVT and various particular cases are deduced. Further various performance measures and the optimum management policy are also derived.