Analysis of a two incongruent arrivals and services M/G/1 queue with random feedback

A queuing system with two incongruent arrivals and services is considered. Two kinds of customers enter the system by Poisson process and the service times are assumed to have general distribution. After first kind service completion, it may feedback to repeat the first service, leave the system or go to give second service. The same policy is applied for the other kind of customer. All stochastic processes involved in this system are independent. We derive the probability generating function for each kind and for the system that yield the performance measures. Some numerical approaches examined the validity of the results.