Aging and other distributional properties of discrete compound geometric distributions

Distributional properties of some discrete reliability classes, including the 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 rate classes are considered. Upper bounds for the tail probabilities of D-CG distributions are derived. These upper bounds are of discrete Lundberg-type, and are optimal for some choices of the compounded variable. Lower bounds are also obtained. Numerical examples are given to illustrate the calculations of the bounds. The results are then applied to obtain bounds and monotonicity properties of the ruin probability in a discrete ruin model. Finally, by exploiting connections with both compound geometric and mixed Poisson distributions, reliability classifications and bounds are obtained for the equilibrium M/G/1 queue length distribution.