Optional services in a non-Markovian queue

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.