An M-x / G(a, b)/1 queue with breakdown and delay time to two phase repair under multiple vacation

In this paper, we consider an M-x /G (a, b)/1 queue with active breakdown and delay time to two phase repair under multiple vacation policy. A batch of customers arrive according to a compound Poisson process. The server serves the customers according to the "General Bulk Service Rule" (GB SR) and the service time follows a general (arbitrary) distribution. The server is unreliable and it may breakdown at any instance. As the result of breakdown, the service is suspended, the server waits for the repair to start and this waiting time is called as 'delay time' and is assumed to follow general distribution. Further, the repair process involves two phases of repair with different general (arbitrary) repair time distributions. Immediately after the repair, the server is ready to start its remaining service to the customers. After each service completion, if the queue length is less than a', the server will avail a multiple vacation of random length. In the proposed model, the probability generating function of the queue size at an arbitrary and departure epoch in steady state are obtained using the supplementary variable technique. Various performance indices, namely mean queue length, mean waiting time of the customers in the queue etc. are obtained. In order to validate the analytical approach, we compute numerical results.