A Batch Arrival Bulk Service Queue with Multiple Vacations, Optional Re-Service, Closedown time and Setup time under Restricted Admissibility of Arriving Batches

Abstract

We consider an /G(a,b)/1  with multiple vacations, optional re-service, closedown time, setup time under a restricted admissibility policy of arriving batches is considered. A batch of customers arrive according to Poisson with rate λ, whereas the general bulk service is rendered by a single server with a minimum batch size of ‘a’ and maximum of ‘b’. The service times follow a general distribution. At the completion of an essential service, the leaving batch of customers may request for a re-service with probability π. After completing re-service or if no request for re-service with probability is made the number of customers in the queue is less than ‘a’ then the server performs closedown work at its closedown time C. After that, the server leaves for multiple vacation of a random length irrespective of queue length. When the server returns from vacation and if the queue length is still less than ‘a’ he avails another vacation so on until the server finds at least ‘a’ customers in the queue. At a vacation completion epoch, if the server finds at least ‘a’ customers waiting for service, he requires a setup time to start the service. On service completion or on re-service completion, if the server finds at least ‘a’ customers waiting for service say ξ ), he serves a batch of min customers, where. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all time. The probability generating function of queue size at a random epoch is obtained. Some important performance measures such as expected queue size, expected busy period and idle period are derived. Along with the cost model, particular cases are discussed. Numerical illustration for particular values of parameter is presented.