This paper deals with the steady state analysis of a single server batch arrival queueing system with three stages of compulsory service. An additional supposition of a fourth stage optional service is well thought-out. The server may take a vacation after completion of service. In this model, the vacation is of predetermined duration. A busy server may break down at any moment. It is tracked by a repair process. Service time, vacation time and repair time follow general distribution. The steady state probability generating function for the system and also system recital measures is obtained by using a supplementary variable technique. Some special cases of the model are also discussed. The model is justified by means of numerical illustrations followed by graphical representation.

Abstract:We study a single server queue with Poisson arrivals, two optional services following a general service time distributions. The first service is essential. Other two services are optional. Only some of the arriving customers demand the first optional service or the second optional service. We derive the system size distribution at random points and at departure points and other performance indices such as average number of customers and the average waiting time in the queue and the system by employing generating function and supplementary variable techniques. By taking numerical illustration, the sensitivity analysis is also conducted to determine the effects of the system parameters on various performance measures of system state are derived.