We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.

The comparison of lifetimes has been treated extensively during the last decade. A wide variety of mathematical objects have been defined, which, in reliability theory, are used to quantify ageing properties. In this work, using the equilibrium variable, we give a new viewpoint on ageing properties. Moreover, we give new bounds on the moments of series systems.

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

Sharp upper and lower reliability bounds for the ℒ- and -classes of life distributions with a given mean are provided. Bounds are also given for a distribution which is related to a known distribution (not necessarily exponential) through the Laplace order.

We consider two weaker partial orderings among non-negative random variables. We derive some characterizations of the equivalence of two life distributions under the weaker orderings. These results lead to the characterizations of the equality of two non-negative random vectors in distributions by moments of extremes and sample mean. As the application of these results, we get various characterizations of the exponential distribution among HNBUE and HNWUE life distribution classes. The main results of Bhattacharjee and Sethuraman (1990), and Basu and Kirmani (1986) are special cases of these more general results.

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0∞Q̄ℙ[N(t)=k], where Q̄k is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk, ∊ ℕ+, are independent and NBU, and Q̄k+j ≦ Q̄k· Q̄j holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

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.

The concept of positive ageing describes the adverse effects of age on the lifetime of units. Various aspects of this concept are described in terms of conditional probability distributions of residual lifetimes, failure rates, equilibrium distributions, etc. In this paper we further analyse this concept and relate it to stochastic dominance of first and higher orders. In the process we gain many insights and are able to define several new kinds of ageing criteria which supplement those existing in the literature.

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.