A discrete-time single-server queueing system with an N-policy, an early setup and a generalization of the Bernoulli feedback

This paper studies a Geo/G/1/∞Geo/G/1/∞ queueing system under an (m,N)(m,N)-policy, i.e., the service station operates under an NN-policy with an early setup where the startup period begins when m(≤N)m(≤N) customers accumulate in the system. Moreover, it is assumed that the ii-th service of each customer is either unsuccessful (and then the customer joins the server for another service) with probability αiαi or successful (and then the customer leaves the system forever) with complementary probability 1−αi1−αi. We give the joint generating function of the server state and the system length as well as the main performance measures. The distributions of the lengths of the idle, setup, standby and busy periods, as well as the distribution of the number of customers served during a busy period, are also derived. We define a total expected cost function and present a tabu search algorithm as a procedure to find out the optimal values in cases where the convexity is difficult to prove. Finally, the significance of the cost model is discussed through several numerical results.