Reservicing some customers in M/G/1 queues under three disciplines

Consider an M/G/1 production line in which a production item is failed with some probability and is then repaired. We consider three repair disciplines depending on whether the failed item is repaired immediately or first stockpiled and repaired after all customers in the main queue are served or the stockpile reaches a specified threshold. For each discipline, we find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period.