An M/G/1 queue with multiple types of feedback, gated vacations and FCFS policy

We consider an M/G/1 queueing system with multiple types of feedback, gated vacations and first-come-first-served policy where the first service of a new customer is either successful (and then the customer leaves the system) or unsuccessful (and then the customer joins to the end of the queue for another service as old customer with different Bernoulli feedback parameter and different service distribution), and customers are served in the order of joining the tail of the queue. By applying a new method developed by authors we obtain joint probability generating function of system sizes of new and old customers at steady state and Laplace–Stieltjes transform of total response time. We also give algorithms for calculation of moments of system size and total response time.