We establish here an integral inequality for real log-concave functions, which can be viewed as an average monotone likelihood property. This inequality is then applied to examine the monotonicity of failure rates.

We study three classes of shock models governed by an inverse gamma mixed Poisson process (IGMP), namely a mixed Poisson process with an inverse gamma mixing distribution. In particular, we analyze (1) the extreme shock model, (2) the δ-shock model, and the (3) cumulative shock model. For the latter, we assume a constant and an exponentially distributed random threshold and consider different choices for the distribution of the amount of damage caused by a single shock. For all the treated cases, we obtain the survival function, together with the expected value and the variance of the failure time. Some properties of the inverse gamma mixed Poisson process are also disclosed.

More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k-out-of-n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\mathrm{e}}}^x)$. Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.

For torpedo electronic components tested by functional verification, there are characteristics of a few samples and few failures. During service life, it is difficult to analyse and predict the changes in reliability. At present, management's observation of quality is mainly base on failure data, and it is difficult to make predictions about the moments without failure in service life. In this paper, according to the failure data, we consider such factors as performance degradation and detection and use the model of instantaneous failure rate to evaluate the reliability of the detection moments periodically, and predict the reliability of stages through the results of detection moments. The method proposed in this paper, on the one hand, considers the service experience, and on the other combines the detection data, to make the final evaluation result more credible. In addition, this paper predicts the changing trend of reliability between adjacent detection moments, which can provide a useful reference for quality management work.

In the literature of stochastic orders, one rarely finds results characterizing non-comparability of random variables. We prove simple tools implying the non-comparability with respect to the convex transform order. The criteria are used, among other applications, to provide a negative answer for a conjecture about comparability in a much broader scope than its initial statement.

This paper presents a flexible family which we call the $\alpha$-mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The $\alpha$-mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for $\alpha$. We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.

We consider compound geometric approximation for a nonnegative, integer-valued random variable W. The bound we give is straightforward but relies on having a lower bound on the failure rate of W. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth–death processes and Poisson processes.

It is often claimed that 50–90% of strategic initiatives fail. Although these claims have had a significant impact on management theory and practice, they are controversial. We aim to clarify why this is the case. Towards this end, an extensive review of the literature is presented, assessed, compared and discussed. We conclude that while it is widely acknowledged that the implementation of a new strategy can be a difficult task, the true rate of implementation failure remains to be determined. Most of the estimates presented in the literature are based on evidence that is outdated, fragmentary, fragile or just absent. Careful consideration is advised before using current estimates to justify changes in the theory and practice. A set of guiding principles is presented for assisting researchers to produce better estimates of the rates of failure.

The Bayesian approach is a stochastic method, allowing to establish trend studies on thebehavior of materials between two periods or after a break in the life of these materials.It naturally integrates the inclusion of the information partially uncertain to support inmodeling problem. The method is therefore particularly suitable for the analysis of thereliability tests, especially for equipment and organs whose different tests are costly.Bayesian techniques are used to reduce the size of estimation tests, improving theevaluation of the parameters of product reliability by the integration of the past (dataavailable on the product concerned) and process, the case “zero” failure observed,difficult to treat with conventional statistical approach. This study will concern thereduction in the number of tests on electronic or mechanical components installed in amechanical lift knowing their a priori behavior in order to determine their a posterioribehavior.

Almost all populations existing in the real world are finite populations. Specifically, in the areas relevant to lifetime modeling and analysis, finite populations are frequently encountered. However, descriptions of failure/survival patterns of elements in the finite population have not yet been properly established. In particular, it is questionable whether the ordinary failure rate can be defined for finite populations in the same way and whether the corresponding interpretations are still valid. In this paper we consider two kinds of finite mixed population and provide new definitions for their failure rates. Then we clarify the notion of failure rate in finite populations.

La fiabilité des matériels mécaniques est largement conditionnée par les caractéristiques propres à chaque système, à ses conditions d'utilisation, son environnement... De plus, les mécanismes de dégradations comme la fatigue et le stress créent des phénomènes de vieillissement. Les bases de données de fiabilité, en fournissant des taux de défaillance génériques et constants, ne permettent pas de prendre en considération ces particularités. L'utilisation de telles bases entraîne par conséquent de grandes incertitudes quant aux résultats des évaluations fiabilistes. Nous proposons dans cet article une modélisation des taux de défaillance, fonction du temps, qui prend en compte les facteurs d'influence. Une application permet de valider le modèle.

We consider ageing properties of a general repair process. Under certain assumptions we prove that the expectation of an age at the beginning of the next cycle in this process is smaller than the initial age of the previous cycle. Using this reasoning, we show that the sequence of random ages at the start (end) of consecutive cycles is stochastically increasing and is converging to a limiting distribution. We discuss possible applications and interpretations of our results.

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.

Lifetime distributions for multicomponent systems are developed through the interplay of ageing and stress shocks to the system. The ageing process is explicitly modeled by an exponential function with rate affected by the magnitude of stresses from a compound Poisson process shock model. Applications of these life distributions and associated failure rates towards the study of multicomponent system survival are discussed. In particular, we illustrate the behavior of these survival functions in relevant subsets of the parameter space.

For systems subject to inspections at Poisson random times, we present an analytic method which gives upper and lower bounds for the reliability. We also study its asymptotic behaviour and derive the asymptotic failure rate.

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

Upper and lower bounds are derived for the tail probabilities of compound distributions using simple properties of the claim size distribution. General bounds are then obtained for various classes of claim size distributions. Some examples are given.

n candidates, represented by n i.i.d. continuous random variables X 1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X 1, …, Xn ) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn , and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak , let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.