Analysis of MX / G /1 feedback queuewith two-phase service, compulsory server vacation and random breakdowns

This paper investigates a batch arrival queueing system with two-phase heterogeneous service under compulsory vacation schedule. Upon arrival any batch that finds the server busy, on vacation or under repair joins the queue. Otherwise one customer from the arriving batch enters the server for service immediately while the rest join the queue. After completion of the second phase of service for each customer, the server goes for a vacation. Busy servers are interrupted by breakdowns and repair has to be done to resume service. If a customer is unsatisfied with his service he is allowed to join the tail of the queue as a feedback customer. We construct the mathematical model and derive the probability generating functions of transient state probabilities in terms of their Laplace transforms. The steady state distributions of the server state are deduced and the average number of customers in the system, the mean waiting time of a customer are obtained. Particular cases of interest of the model are discussed in order to verify the results. Numerical computations are provided to visualize the effect of parameters on system performance measures and also to validate our analytical results.