On the N-policy of M/G/1 feedback queue with varying arrival rates

An M/G/1 queue with Bernoulli feedback and general setup time under a control policy is analyzed. The customers' arrival rate at service station varies according to the system status: buildup, setup and busy states. The probability generating function for system size and mean number of customers in the system are obtained under steady-state conditions. During the idle period of the server, the probability generating function for the queue length is expressed as a convex combination of the probability generating function of queue size during buildup period and probability generating function of the queue size during setup period. The Laplace–Stieltjes transform of waiting time and its mean are studied. The optimal N-value from which the server starts his setup is determined by minimizing the total operation cost of the system.