Stochastic ordering properties for systems with dependent identically distributed components

In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution‐free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non‐exchangeable joint distribution. Copyright © 2012 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.     

Statistical inference for coherent systems with Weibull distributed component lifetimes under complete and incomplete information

Point estimators for the parameters of the component lifetime distribution in coherent systems are evolved assuming to be independently and identically Weibull distributed component lifetimes. We study both complete and incomplete information under continuous monitoring of the essential component lifetimes. First, we prove that the maximum likelihood estimator (MLE) under complete information based on progressively Type‐II censored system lifetimes uniquely exists and we present two approaches to compute the estimates. Furthermore, we consider an ad hoc estimator, a max‐probability plan estimator and the MLE for the parameters under incomplete information. In order to compute the MLEs, we consider a direct maximization of the likelihood and an EM‐algorithm–type approach, respectively. In all cases, we illustrate the results by simulations of the five‐component bridge system and the 10‐component parallel system, respectively.     

Some results on series and parallel systems of randomized components

We study the system (series/parallel) where the components are randomly chosen from two different batches. We assume that one batch is more reliable than the other in some stochastic sense. In the case of series systems we show that, under certain conditions, lifetime of one system dominates that of the other in different stochastic orders viz. hazard rate, down shifted hazard rate and likelihood ratio orders. Further, we show that the same principle holds for the reversed hazard rate and the likelihood ratio orders in the case of parallel systems.     

Some properties and applications of cumulative Kullback–Leibler information

The cumulative Kullback–Leibler information has been proposed recently as a suitable extension of Kullback–Leibler information to the cumulative distribution function. In this paper, we obtain various results on such a measure, with reference to its relation with other information measures and notions of reliability theory. We also provide some lower and upper bounds. A dynamic version of the cumulative Kullback–Leibler information is then proposed for past lifetimes. Furthermore, we investigate its monotonicity property, which is related to some new concepts of relative aging. Moreover, we propose an application to the failure of nanocomponents. Finally, in order to provide an application in image analysis, we introduce the empirical cumulative Kullback–Leibler information and prove an asymptotic result. Copyright © 2015 John Wiley & Sons, Ltd.     

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X _Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX _Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX _Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX _Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system. This work is supported by the National Natural Science Foundation of China.     

Some stochastic comparisons of conditional coherent systems

This paper investigates coherent systems with independent and identical components. Stochastic comparison on the residual life and the inactivity time of two systems with stochastically ordered signatures is conducted. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Existence results of stochastic impulsive systems with expectations‐dependent nonlinear terms

In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov‐like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.     

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??Coherent systems are very important in reliability,survival analysis and other life sciences. In this paper, we consider the number of working components in an $(n-k+1)$-out-of-$n$ system, given that at least $(n-m+1)$ components are working at time $t$, and the system has failed at time $t$. In this condition, we compute the probability that there are exactly $i$ working components. First the reliability and several stochastic properties are obtained. Furthermore, we extend the results to general coherent systems with absolutely continuous and exchangeable components.     

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.     

Some new results involving general standby systems

This article presents a stochastic comparison on the total lifetime of the general standby system and a discussion of the optimal allocation of a general standby component in a series system with two independent components. Several examples are also presented to justify the main results, which provide nice generalizations of some existing conclusions in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

单调关联系统的条件失效元件数量

Coherent systems are very important in reliability,survival analysis and other life sciences.In this paper,we consider the number of failed components in an(n-k+1)-out-of-n system,given that at least m(m相似文献   

Some results for repairable systems with minimal repairs

In this paper, we consider a repairable system with minimal repairs whose number of repairs is a positive random variable with a given probability vector. Some preservation theorems and aging properties of repairable systems are established. Under the condition that at time t the system is working, a new random variable for the residual lifetime of the system is proposed. Some stochastic ordering results among the lifetimes and residual lifetimes of two systems are obtained. Similar results for coherent systems with independent components and exchangeable components were obtained in the previous literature. Copyright © 2013 John Wiley & Sons, Ltd.     

Bounds for the reliability functions of coherent systems with heterogeneous components

The computation of the reliability function of a (complex) coherent system is a difficult task. Hence, sometimes, we should simply work with some bounds (approximations). The computation of these bounds has been widely studied in the case of coherent systems with independent and identically distributed (IID) components. However, few results have been obtained in the case of heterogeneous (non ID) components. In this paper, we derive explicit bounds for systems with heterogeneous (independent or dependent) components. Also some stochastic comparisons are obtained. Some illustrative examples are included where we compare the different bounds proposed in the paper.     

Inactivity times of coherent systems with dependent components under periodical inspections

If we know that a coherent system has failed before a time t, the inactivity time is the period (from t) in which the system has been broken. If T is the system lifetime, the inactivity time at t is (t?T|T<t). Under periodical inspections, we may typically know that the system was working at a time t₁, but that is broken at another time t₂>t₁. Under this assumption, we obtain representations for the reliability function of the system inactivity time (t₂?T|t₁<T<t₂). We consider both the cases of independent and dependent components. Similar representations are obtained under other assumptions with partial information about component failures at times t₁ and t₂. These representations are used to compare stochastically the inactivity times under different assumptions. Some illustrative examples are provided.     

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比例机率模型在生存数据分析中发挥着重要作用. 在本文中, 我们建立了此模型的一些随机比较和年龄性质的结果. 所得结果很好的补充和扩展了Kirmani and Gupta,(2001)中的相关结果.     

Allocations of cold standbys to series and parallel systems with dependent components

In the context of industrial engineering, cold‐standby redundancies allocation strategy is usually adopted to improve the reliability of coherent systems. This paper investigates optimal allocation strategies of cold standbys for series and parallel systems comprised of dependent components with left/right tail weakly stochastic arrangement increasing lifetimes. For the case of heterogeneous and independent matched cold standbys, it is proved that better redundancies should be put in the nodes having weaker [better] components for series [parallel] systems. For the case of homogeneous and independent cold standbys, it is shown that more redundancies should be put in standby with weaker [better] components to enhance the reliability of series [parallel] systems. The results developed here generalize and extend those corresponding ones in the literature to the case of series and parallel systems with dependent components. Numerical examples are also presented to provide guidance for the practical use of our theoretical findings.     

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).     

Some results on the proportional reversed hazards model

The proportional reversed hazards model consists in describing random failure times by a family {[F(x)]^θ, θ>0} of distribution functions, where F(x) is a baseline distribution function. We show various results on this model related to some topics in reliability theory, including ageing notions of random lifetimes, comparisons based on stochastic orders, and relative ageing of distributions.