Analysis of an Admission Control $$M^{[X]}/G(a,b)/1$$ Queue with Multiple Vacation, Restricted Re-service, Closedown and Setup Times

Abstract

The objective of this paper is to analyze an $$M^{[X]}/G(a,b)/1$$ queueing model with admission control, multiple vacation, restricted re-service, setup and closedown times. At the moment of service completion, the leaving batch of customers may request for re-service with probability $$\eta $$ and it is not compulsory to accept the request. The server accepts the request with probability $$\zeta $$ and restricts it with probability $$1-\zeta $$ . All the batches of customers are not allowed all the time into the system. On completion of a service or re-service, if the queue length is $$\omega $$ , where $$\omega <a$$ , then the server resumes closedown work. After that, the server leaves for a vacation of random length. When the server returns from a vacation if the queue length is still less than a, he leaves for another vacation and so on until he finds at least a customers in the queue. If the queue length is at least a, then the server commences setup work before starting the service. Using supplementary variable technique, the steady-state probability generating function of the system size at an arbitrary time is obtained. The performance measures and cost model are also derived. Numerical illustrations are presented to visualize the effect of system parameters.