Binomial service and multiple adaptive vacation schedules for $M^X/G/1$ queue with control policy on demand for re-service
The re-service policy in queueing theory has been always a new and hot research field. A $M^X/G/1$ type queueing system with Poisson arrival rate $\lambda$, and the general service random variable is considered here. In this paper, the server implements the notion of a binomial vacation schedule. Under this policy, there is no customer at a service completion epoch the server takes multiple vacations. The number of vacations follows a binomial distribution with parameter $p$, otherwise the server gives service with probability $1-p$ if any new customers arrive in the queue. At the completion of an essential service, the batch of customers may request for a re-service with probability $\pi$. However the re-service is rendered only when the number of customer waiting in the queue is less than one. If no request for re-service after an essential service and no customers in the queue, then the server avail a vacation of a random length with probability $(1-\pi)$. When the server returns from vacation and if there is no customer in the queue then he avails another vacation and so on until the server finds single customers in the queue. After completing of an essential service and the number of customers in the queue becomes one then the server will continue the service with general service rule.