To enhance the performance of a system, a common practice employed by reliability engineers is to use redundant components in the system. In this paper we compare lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies. These comparisons are made with respect to the increasing concave (convex) order, the hazard rate order, and the stochastic precedence order. The main results extend some related conclusions in the literature.

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In this paper we study these orders when h is nondecreasing. We note that all these orders are location independent, and we characterize the dispersive order, and the location-independent riskier order, by means of the generalized TTT orders with nondecreasing h. Further properties, which strengthen known properties of the dispersive order, are given. A useful nontrivial closure property of the generalized TTT orders with nondecreasing h is obtained. Applications in poverty comparisons, risk management, and reliability theory are described.

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.

Brown and Proschan (1983) introduced the imperfect repair model, in which an item, upon failure, is replaced with a new one with probability α, and is minimally repaired with probability 1 − α. In this paper we equip the imperfect repair model with preventive maintenance, and we obtain stochastic maintenance comparisons for the numbers of failures under different policies via a point-process approach. We also obtain some results involving stochastic monotonicity properties of these models with respect to the unplanned complete repair probability α.

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.

In this paper we describe a model for survival functions. Under this model a system is subject to shocks governed by a Poisson process. Each shock to the system causes a random damage that grows in time. Damages accumulate additively and the system fails if the total damage exceeds a certain capacity or threshold. Various properties of this model are obtained. Sufficient conditions are derived for the failure rate (FR) order and the stochastic order to hold between the random lifetimes of two systems whose failures can be described by our proposed model.

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.

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.

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

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.

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

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.

We consider models of systems of components with dependent life-lengths having certain multivariate distributions. Upon failure, components are repaired. Two types of repair are possible. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. Different sources of failure are distinguished and affect the probabilities of perfect and imperfect repair.

In this paper, we study a similar replacement model in which the successive survival times of the system form a process with non-increasing means, whereas the consecutive repair times after failure constitute a process with non-decreasing means. The system is replaced at the time of the Nth failure since the installation or last replacement. Based on the long-run average cost per unit time, we determine the optimal replacement policy N∗ and the maximum of the long-run average reward explicitly. Under additional conditions, the policy N∗ is even optimal among all replacement policies.

Consider a population of distinct species Sj, j∈J, members of which are selected at different time points T1, T2,· ··, one at each time. Assume linear costs per unit of time and that a reward is earned at each discovery epoch of a new species. We treat the problem of finding a selection rule which maximizes the expected payoff. As the times between successive selections are supposed to be continuous random variables, we are dealing with a continuous-time optimal stopping problem which is the natural generalization of the one Rasmussen and Starr (1979) have investigated; namely, the corresponding problem with fixed times between successive selections. However, in contrast to their discrete-time setting the derivation of an optimal strategy appears to be much harder in our model as generally we are no longer in the monotone case.

This note gives a general point process formulation for this problem, leading in particular to an equivalent stopping problem via stochastic intensities which is easier to handle. Then we present a formal derivation of the optimal stopping time under the stronger assumption of i.i.d. (X1, A1) (X2, A2), · ·· where Xn gives the label (j for Sj) of the species selected at Tn and An denotes the time between the nth and (n – 1)th selection, i.e. An = Tn – Tn–1. In the case where even Xn and An are independent and An has an IFR (increasing failure rate) distribution, an explicit solution for the optimal strategy is derived as a simple consequence.

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ(Sk). If Tj is the lifetime of component j, it is shown that (T1, · ··, Tn) belongs to various multivariate non-parametric life classes depending on the life class of .