A study on the behaviour of the server breakdown without interruption in a M[X]/G(a, b)/1 queueing system with Multiple vacations, Accessible batches and Closedown time

In this paper we consider M[X]/G(a, b)/1 queueing system with accessible service, server breakdown without interruption, multiple vacation and closedown time. The service rule is as follows: the server starts service only when a minimum number of customers ′a′ is available in the queue and the maximum service capacity is ′b′ units such that late entries are allowed to join a service batch, without affecting the service time, if the size of the batch being served is less than ′d′ (maximum accessible limit). After completing a batch of service, if the server is breakdown with probability (π) then the renovation of service station will be considered. After completing the renovation of service station or if there is no breakdown of the server with probability (1− π), if the queue length δ < a, then the server performs a closedown work. Following closedown work, the server leaves for a vacation of random length irrespective of queue length. When the server returns from a vacation and if the queue length is still less than ′a′ he leaves for another vacation and so on until he finds minimum ′a′ customers waiting for service in the queue. The probability generating function of the queue size at an arbitrary time, some important characteristic of the queueing system and cost model are derived.