A Non-Markovian Bulk Queue with Multiple Vacations and Control Policy on Request for Re-Service

Abstract

The steady state behaviour of a single server non-Markovian queue with multiple vacations and controlled optional re-service is analysed. A batch of customers arrive according to Poisson with rate A, whereas the 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 n. However, the re-service is rendered only when the number of customers waiting in the queue is less than ‘a’. If no request for re-service is made after the completion of an essential service and numbers of customers in the queue is less than ‘a, then the server will avail a vacation of a random 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 ‘a’ customers in the queue. After the completion of an essential service and numbers of customers in the queue is greater than ‘a then the server will continue the batch service with general bulk service rule. 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. Cost model is discussed with numerical illustration.