Analysis of D-policy discrete-time Geo/G/1 queue with second J-optional service and unreliable server^∗

School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, Sichuan, P.R. China.
sjlan89@163.com; tangyh@uestc.edu.cn

Received: 27 April 2015
Accepted: 21 January 2016

Abstract

This paper is concerned with a discrete-time Geo/G/ 1 queueing system with D-policy and J-optional services in which the service station may be subject to failures at random during serving the customers. All the arriving customers require the first essential service, whereas some of them may opt for a second service from the J additional services with some probability. As soon as the system becomes empty, the server will not restart the service until the sum of the service times of the waiting customers in the system reaches or exceeds some given positive integer D. Applying the total probability decomposition law, renewal theory, and probability generating function technique, the queueing indices and reliability measures are investigated simultaneously in our work. Both the probability generating function of the transient queue length distribution and the explicit formulas of the steady-state queue length distribution at time epoch n⁺ are derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and steady-state unavailability, the expected number of breakdowns during (0⁺,n⁺ ], and the equilibrium failure frequency, are discussed. Finally, the optimum value of D for minimizing the system cost is numerically discussed under a given cost structure.