Transient solution of an M[X]/G/1 queueing model with feedback, random breakdowns, Bernoulli schedule server vacation and random setup time

We consider an M[X]/G/1 queue with Poisson arrivals, random server breakdowns and Bernoulli schedule server vacation. Both the service time and vacation time follow general distribution. After completion of a service, the server may go for a vacation with probability θ or continue staying in the system to serve a next customer, if any, with probability 1 − θ. With probability p, the customer feedback to the tail of original queue for repeating the service until the service becomes successful. With probability 1 − p = q, the customer departs the system if service be successful. The system may breakdown at random following Poisson process and the repair time follows exponential distribution. Also, we assume that at the end of a busy period, the server needs a random setup time before giving proper service. We obtain the probability generating function in terms of Laplace transforms and the corresponding steady state results explicitly.