On allocation of redundant components for systems with dependent components

In this paper we consider the problem of optimal allocation of a redundant component for series, parallel and k-out-of-n systems of more than two components, when all the components are dependent. We show that for this problem is naturally to consider multivariate extensions of the joint bivariates stochastic orders. However, these extensions have not been defined or explicitly studied in the literature, except the joint likelihood ratio order, which was introduced by Shanthikumar and Yao (1991). Therefore we provide first multivariate extensions of the joint stochastic, hazard rate, reversed hazard rate order and next we provide sufficient conditions based on these multivariate extensions to select which component performs the redundancy.     

Redundancy allocation at component level versus system level

In this paper, we treat the problem of stochastic comparison of standby [active] redundancy at component level versus system level. In the case of standby redundancy, we present some interesting comparison results of both series systems and parallel systems in the sense of various stochastic orderings for both the matching spares case and non-matching spares case, respectively. In the case of active redundancy, a likelihood ratio ordering result of series systems is presented for the matching spares case; and for the non-matching spares case, a counterexample is provided to show that there does not exist similar result even for the hazard rate ordering. The results established here strengthen and generalize some of those known in the literature. Some numerical examples are also provided to illustrate the theoretical results.     

On the hazard rate and reversed hazard rate orderings in two-component series systems with active redundancies

The lifetimes of two-component series systems with two active redundancies are compared using the hazard rate and the reversed hazard rate orders. We study the problem of where to allocate the spares in a system to obtain the best configuration. We compare redundancy at component level vs. system level using the likelihood ratio order. For this problem we find conditions under which there is no hazard rate ordering between the lifetimes of the systems.     

The failure rate and likelihood ratio orderings of standby redundant systems

There are various notions of partial ordering between lifetimes of systems; stochastic ordering, failure rate ordering, and likelihood ratio ordering. In this paper we show that for series systems with noni.i.d. exponential lifetimes of components, standby redundancy at component level is better than that at system level in failure rate ordering and likelihood ratio ordering. We also demonstrate that for 2-component parallel systems withi.i.d. exponential lifetimes of components, standby system redundancy is better than standby component redundancy in failure rate ordering and likelihood ratio ordering.     

Comparisons in the mean residual life order of coherent systems with identically distributed components

The comparisons of the performance of coherent systems (under different stochastic criteria) is an important task in the reliability theory. Several results have been obtained in the literature for the stochastic, hazard rate and likelihood ratio orders. In this paper, we obtain comparison results for the mean residual life order of coherent systems with identically distributed (ID) component lifetimes. These results can be applied not only to the usual case of systems with independent and identically distributed components but also to the case of systems with exchangeable components and to the more general case of just ID components. The results obtained are based on the representation of the system distribution as a distorted distribution of the common components' distribution. Some specific comparison results are given to illustrate the theoretical results. The comparison results for distorted distributions given here can also be applied to other statistical concepts such as order statistics, generalized order statistics or record values. Copyright © 2015 John Wiley & Sons, Ltd.     

Some positive dependence stochastic orders

In this paper we study some stochastic orders of positive dependence that arise when the underlying random vectors are ordered with respect to some multivariate hazard rate stochastic orders, and have the same univariate marginal distributions. We show how the orders can be studied by restricting them to copulae, we give a number of examples, and we study some positive dependence concepts that arise from the new positive dependence orders. We also discuss the relationship of the new orders to other positive dependence orders that have appeared in the literature.     

Stochastic orders of the Marshall-Olkin extended distribution

A general method of introducing a parameter, called tilt parameter, has been discussed by Marshall and Olkin (1997) to give more flexibility in modelling. In this paper, we take the tilt parameter of the Marshall-Olkin extended family as a random variable. The closure of this model under different stochastic orders viz. ageing intensity order, likelihood ratio order, shifted likelihood ratio orders and shifted hazard rate orders is discussed.     

Orderings of coherent systems with randomized dependent components

Consider a general coherent system with independent or dependent components, and assume that the components are randomly chosen from two different stocks, with the components of the first stock having better reliability than the others. Then here we provide sufficient conditions on the component’s lifetimes and on the random numbers of components chosen from the two stocks in order to improve the reliability of the whole system according to different stochastic orders. We also discuss several examples in which such conditions are satisfied and an application to the study of the optimal random allocation of components in series and parallel systems. As a novelty, our study includes the case of coherent systems with dependent components by using basic mathematical tools (and copula theory).     

Sequential Order Statistics: Ageing and Stochastic Orderings

Ordered random variables play an important role in statistics, reliability theory, and many applied areas. Sequential order statistics provide a unified approach to a variety of models of ordered random variables. We investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics given some well known stochastic orderings, such as the usual stochastic, the hazard rate and the likelihood ratio orders, among others. Also, we derive sufficient conditions under which the sequential order statistics are increasing hazard rate, increasing hazard rate average or decreasing hazard rate average. Applications of the main results involving nonhomogeneous pure birth processes are also given.     

Stochastic comparisons of multivariate mixture models

In this paper we consider sufficient conditions in order to stochastically compare random vectors of multivariate mixture models. In particular we consider stochastic and convex orders, the likelihood ratio order, and the hazard rate and mean residual life dynamic orders. Applications to proportional hazard models and mixture models in risk theory are also given.     

Comparisons Between Largest Order Statistics from Multiple-outlier Models with Dependence

We study stochastic comparisons between the largest order statistics from samples which may contain outliers. The data in each sample can also be dependent. Under these assumptions we study three cases. In the first one we consider the general case without additional assumptions. In the second we assume that the data come from two different distributions. In the third one we assume that the data come from a proportional hazard rates model. The results obtained here can be applied to compare parallel systems. Some illustrative examples are provided.     

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??The mean residual life (MRL) function plays a very important role in the area of reliability engineering, survival analysis, and many other fields. In this paper, we introduce and study a new stochastic order which gives stochastic comparison for mean residual life of strictly increasing concave function of two random variables. We show that this new stochastic order lies between the hazard rate and mean residual life orders. The preservation properties under mixtures are presented here. Finally, we give some applications of this new order in reliability theory.     

Comparisons of series and parallel systems with components sharing the same copula

The paper is devoted to study stochastic comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula. We show that, under some conditions on the common copula, the series system with heterogeneous components is worse than the series system with homogeneous components having a common reliability function, which is equal to the average of the reliability functions of the heterogeneous components. However, we show that this property is not necessarily true for arbitrary copulas. We obtain similar properties for parallel systems and for general coherent systems. For these purposes, we introduce in our analysis the notion of the mean function of a copula. Copyright © 2009 John Wiley & Sons, Ltd.     

Component and system active redundancies for coherent systems with dependent components

The prevailing engineering principle that redundancy at the component level is superior to redundancy at the system level is generalized to coherent systems with dependent components. Sufficient (and necessary) conditions are presented to compare component and system redundancies by means of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orderings. Explicit numerical examples are provided to illustrate the theoretical findings. Some related results in the literature are generalized and extended. Copyright © 2017 John Wiley & Sons, Ltd.     

两比例失效率元件组成串联系统元件冗余与系统冗余的随机比较

针对两个比例失效率元件组成的串联系统,在热冗余的情形下,讨论了串联系统的元件冗余与系统冗余两种方案,并基于随机序的方法,对普通随机序、失效率序、反失效率序建立了元件冗余优于系统冗余的随机比较理论.     

Optimal allocation of redundancies in series systems

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

On fail‐safe systems under random shocks

In practical situations, systems often suffer shocks from external stressing environments, stressing the system at random. These random shocks may have non‐ignorable effects on the system's reliability. In this paper, we provide sufficient (and necessary) conditions on components' lifetimes and their surviving probabilities from random shocks for comparing the lifetimes of two fail‐safe systems by means of the usual stochastic, hazard rate, and likelihood ratio orderings. Numerical examples are presented to highlight these theoretical results as well.     

Limit of hazard rate function of coherent system with discrete life

In this paper we study the limiting behavior of the hazard rate function of a coherent system 𝒮 whose components are independent and have discrete lifetimes. The lifetime of the system 𝒮 can be viewed as the lifetime of a parallel system consisting of the series systems obtained from the minimal paths of 𝒮. If we appropriately interpret the ‘strongest’ minimal path(s) of the system 𝒮, it will be shown that the limit of the hazard rate functions of 𝒮 is the same as that of the ‘strongest’ path(s) of 𝒮. The application of the main results are also provided. Copyright © 2010 John Wiley & Sons, Ltd.     

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.