In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.

We discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

The solutions of a certain class of first order linear differential equations are shown to be either completely or absolutely monotone depending on the nature of its coefficients. This is a simple theorem which is used to deduce a number of new and interesting results dealing with the complete and absolute monotonicity of functions. In particular, a partial answer is supplied to a question posed by Askey and Pollard: “When is completely monotone?”