Markovian bulk service queue with delayed vacations

Scope and purpose This model deals with the study of a general bulk service queueing system with repeated server vacation and changeover time. Whenever the server finds a−1 customers waiting in the queue, instead of going for vacation immediately, the server will wait in the system for some time which is called the changeover time. If there is an arrival during the changeover time the server starts service, otherwise the server will go for a vacation. Such bulk service queueing models are useful in many real life transportation systems such as shuttle bus service, taxi stand, express elevators, tour guides and so on. An M/M(a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a−1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.