An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate λ (> 0), all demand the first ‘essential’ service, whereas only some of them demand the second ‘optional’ service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B1(v) and that of the second optional service with general (arbitrary) distribution with distribution function B2(v). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.