Performance and reliability analysis of a repairable discrete‐time Geo/G/1 queue with Bernoulli feedback and randomized policy

This paper is concerned with a discrete‐time Geo/G/1 repairable queueing system with Bernoulli feedback and randomized p,N‐policy. The service station may be subject to failures randomly during serving customers and therefore is sent for repair immediately. The p,N‐policy means that when the number of customers in the system reaches a given threshold value N, the deactivated server is turned on with probability p or is still left off with probability 1−p. Applying the law of total probability decomposition, the renewal theory and the probability generating function technique, we investigate the queueing performance measures and reliability indices simultaneously in our work. Both the transient queue length distribution and the recursive expressions of the steady‐state queue length distribution at various epochs are explicitly derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and the steady‐state unavailability of the service station, the expected number of the service station breakdowns during the time interval 0+,n+ and the equilibrium failure frequency of the service station are also discussed. Finally, an operating cost function is formulated, and the direct search method is employed to numerically find the optimum value of N for minimizing the system cost. Copyright © 2017 John Wiley & Sons, Ltd.