A study on the behaviour of the server breakdown without interruption in a Mx/G(a, b)/1 queueing system with multiple vacations and closedown time

Abstract The objective of this paper is to study the behavior of the server breakdown without interruption in a M x /G(a, b)/1 queueing system with multiple vacations and closedown time. 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 is ξ , where ξ a , then the server performs closedown work at its closedown time C . After that, the server leaves for multiple vacation of random length. After a vacation, when the server returns, if the queue length is less than ‘ a ’, he leaves for another vacation and so on, until he finds ‘ a ’ customers in the queue. After a vacation, if the server finds at least ‘ a ’ customers waiting for service, say ξ , then he prefers to serve a batch of size min( ξ ,  b ) customers, where b  ⩾  a . The probability generating function of queue size at an arbitrary time and some important characteristics of the queueing system and a cost model are derived. An extensive numerical result for a particular case of the model is illustrated.