Aging and other distributional properties of discrete compound geometric distributions

Distributional properties of some discrete reliability classes, including t he class of discrete compound geometric (D-CG) distributions, are discussed . The D-CG distribution is shown to be a subclass of the discrete strongly new worse than used class, and relations with discrete decreasing failure r ate classes are considered. Upper bounds for the tail probabilities of D-CG distributions are derived. These upper bounds are of discrete Lundberg-typ e, and are optimal for some choices of the compounded variable. Lower bound s are also obtained. Numerical examples are given to illustrate the calcula tions of the bounds. The results are then applied to obtain bounds and mono tonicity properties of the ruin probability in a discrete ruin model. Final ly, by exploiting connections with both compound geometric and mixed Poisso n distributions, reliability classifications and bounds are obtained for th e equilibrium M/C/1 queue length distribution. (C) 2001 Elsevier Science B. V. All rights reserved.