Retrial Queueing system with Second Optional Service, Random break down, Set up time and Bernoulli vacation

In this paper, we investigate a single server batch arrival non-Markovian retrial queueing model with random break down and Bernoulli vacation. Customers arrive in batches accordingly Poisson process with arrival rate λ but are served one by one with first come first served basis. All customers demand the first essential service, whereas only some of them demand the second optional service. The server is assumed to be unreliable so that it may encounter break down at any time. As the server has to be repaired, the repair process start immediately. The customer, who finds the server busy upon arrival, can either join the orbit with probability p or he/she can leave the system with probability 1-p. Upon completion of a service the server may go for a vacation with probability θ or stay back in the system to serve a next customer with probability 1 − θ, if any. After finishing a vacation, the server could not provide service as the it should go for a setup time. Assuming that, the retrial time, the service time, the repair time, the vacation completion time and the set up time of the server are all arbitrarily distributed. We obtain the transient solution and 24 G.Ayyappan and S.Shyamala steady solution of the model by using supplementary variable technique. Also we derive the system performance measures, reliability indices for the prescribed model.