Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.

Block, Savits and Singh previously presented a preservation result for bathtub-shaped functions. In this paper, we strengthen and extend their results to a class of functions which allow for more than one change of monotonicity. Various applications are also considered.

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.

We will state a general version of Simpson's paradox, which corresponds to the loss of some dependence properties under marginalization. We will then provide conditions under which the paradox is avoided. Finally we will relate these Simpson-type paradoxes to some well-known paradoxes concerning the loss of ageing properties when the level of information changes.

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

In this paper a unified derivation of the upper and lower bounds (in terms of the mean) of an IFR, DFR, IFRA, DFRA, NBU or NWU reliability function is presented. The method of proof provides a simpler alternative to the various proofs known in the literature. Moreover, this method can be used to generalize the existing results in two ways, as demonstrated here. First, the bounds for the reliability function are obtained in terms of any finite moment in all these cases. Subsequently we provide a moment bound on a reliability function which ages faster or slower than a known one in some sense. The existing literature does not offer any of these generalizations, except for a few results which are available in an unnecessarily complicated form.

Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

Analogues of some theorems of Brown (1980) concerning renewal measures with DFR or IMRL underlying distributions are proved for delayed renewal measures. Some related results, such as partial converses of the main theorems, are also presented.

Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.

Bounds are obtained for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the service distribution is either IFR, DFR, NBU or NWU.