On an M[X]/G/1 queuing system with random breakdowns, server vacations, delay times and a standby server

In this work, we study a batch arrival queue with a Bernoulli schedule vacation where after every service completion the server may take a vacation of random length with probability p or may stay in the system providing service with probability 1 − p. Moreover, we assume that the server may breakdown randomly, and the repair process does not start immediately after a breakdown; consequently, there is a delay time waiting for repairs to start. The main new contribution in this work is that we assume that the system is equipped with a standby server to serve the costumers during the vacation period of the main server. The service times, vacation times, repair times and delay times are assumed to follow different general arbitrary distributions while the service times of the standby server follows an exponential distribution.