Modelling and analysis of a M x /G(a,b)/1 queue with multiple vacations, setup time, closedown time and server breakdown without interruption

In this paper, a Mx/G(a,b)/1 queueing system with multiple vacations, setup time, closedown time and server breakdown without interruption are considered. After completing a batch of service, if the server breaks down 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, irrespective of queue 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 the server performs set up work at its set up time G, then he serves 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.