In this paper a M X /G (a, b)/1 queueing system with multiple vacations, setup time with N-policy and closedown times is considered. On completion of a service, if the queue length is n, where n < a, then the server performs closedown work. Following closedown the server leaves for multiple vacations of random length irrespective of queue length. When the server returns from a vacation and if the queue length is still less than ÔNÕ, he leaves for another vacation and so on, until he finds ÔNÕ (N > b) customers in the queue. That is, if the server finds at least ÔNÕ customers waiting for service, then he requires a setup time ÔRÕ to start the service. After the setup he serves a batch of ÔbÕ customers, where b P a. Various characteristics of the queueing system and a cost model with the numerical solution for a particular case of the model are presented.
In this paper, a single server queue with variable batch size service, Poisson bulk arrival with multiple working vacations and server breakdown is considered. In working vacation, the server works with different rates rather than completely stoping the service during the vacation period. In this model, during the working vacation the server starts the service if it finds at least one customer in the queue with a maximum of 'b' customers, otherwise the server serves with variable batch size. Service time in working vacation and in regular period follows general distribution. The probability generating function of a queue size at an arbitrary time epoch as well as other completion epochs is derived. Expected queue length in a steady state is obtained. Also a numerical illustration is presented.
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 O, whereas the bulk service is rendered by a single server with a minimum batch size of and maximum ofThe 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 probabilityS. However, the re-service is rendered only when the number of customers waiting in the queue is less than 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 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 he avails another vacation so on until the server finds customers in the queue. After the completion of an essential service and numbers of customers in the queue is greater than 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. . a , a a a a
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