Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

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.     

Reliability inference with extended sequential order statistics

In this article, we will address the complexity of non-identical components in multi-component systems. Most technical systems can be described as such since either component types or component functions within the system vary amongst components. While most reliability related work resorts to the assumption of homogeneous components, we aim to address the often more realistic assumption of heterogeneous components extending the model of Extended Sequential Order Statistics by two inferential methods. Firstly, the derivation of Maximum Likelihood Estimates including a simulation study demonstrating their good performance for large enough sample size. Secondly, we introduce a likelihood ratio test to test whether components can be assumed identical accompanied by a power study. Both methods are powerful tools in reliability contexts. The former increases our understanding of component behaviour, especially upon failure of other components. This knowledge empowers system operators to make better decisions regarding maintenance schedules and failure time prediction. The latter supports operators in their quest of identifying component equivalence. Therefore, both methods can be used to achieve meaningful results in real life applications.     

Applications of average and projected systems to the study of coherent systems

In this paper, we introduce the concepts of average and projected systems associated to a coherent (parent) system. We analyze several aspects of these notions and show that they can be useful tools in studying the performance of coherent systems with non-exchangeable components. We show that the average and projected systems are especially useful in studying the tail behavior of reliability, hazard rate and mean residual life functions of the parent system and also in obtaining the tail best systems (under different criteria) by permuting the components at the system structure. Moreover, they can be useful in assessing how the asymmetry of the joint distribution of the component lifetimes (with respect to permutations of the components in the system structure) affects the system performance.     

On redundant weighted voting systems with components having stochastic arrangement increasing lifetimes

As one generalization of the k-out-of-n structure, the weighted voting system has been paid much attention during the past two decades. This paper has a further study on active redundancies allocation to weighted voting reliability systems of components having LWSAI lifetimes. For redundancies with SAI lifetimes, allocating a more reliable redundancy to a weaker and more heavily weighted component is found to produce a more reliable system in the sense of having higher reliability. Also, in the context of redundancies with identically distributed lifetimes, we show that allocating more redundancies to a weaker and more heavily weighted component produces a more reliable system. Some numerical examples are presented to illustrate the main results as well.     

Stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components

This work considers stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components having lifetimes following the proportional odds (PO) model. The joint distribution of component lifetimes is modeled by Archimedean survival copula. We discuss some potential applications of our findings in system reliability and actuarial science.     

修理设备可更换且修理延迟的两相同部件冷储备可修系统(Ⅱ)

在文[1]的基础上,本文研究了修理有延迟和修理设备可更换的两单元冷储备可修系统.在假定单元的寿命服从指数分布、修理时间和延迟时间服从一般分布、修理设备的寿命和故障后的更换时间服从指数分布下,通过定义修理设备的"广义忙期",使用更新过程理论和全概率分解技术,提出一种新的分析技巧,讨论了修理设备的一些可靠性指标,获得了如修理设备的可用度和故障次数等可靠性结果.     

A two-unit general parallel system with (n − 2) cold standbys—Analytic and simulation approach

A parallel (2, n − 2)-system is investigated here where two units start their operation simultaneously and any one of them is replaced instantaneously upon its failure by one of the (n − 2) cold standbys. We assume availability of n non-identical, non-repairable units for replacement or support. The system reliability is evaluated by recursive relations with unit-lifetimes T_i (i = 1, … , n) that have a general joint distribution function F(t). On the basis of the derived expression, simulation techniques have been developed for the evaluation of the system reliability and the mean time to failure, useful when dealing with large systems or correlated unit-lifetimes and less mathematically manageable distributions. Simulation results are presented for various lifetime distributions and comparisons are made with derived analytic results for some special distributions and moderate values of n.     

On signature-based expressions of system reliability

The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular, it provides an interesting signature-based representation of the system reliability in terms of reliabilities of k-out-of-n systems. In the non-i.i.d. case, we show that, at any time, this representation still holds true for every coherent system if and only if the component states are exchangeable. We also discuss conditions for obtaining an alternative representation of the system reliability in which the signature is replaced by its non-i.i.d. extension. Finally, we discuss conditions for the system reliability to have both representations.     

Joint Reliability Function of Coherent Systems with Shared Heterogeneous Components

In this paper, we consider two coherent systems having shared components. We assume that in the two systems there are three different types of components; components of type one that just belong to the first system, components of type two that lie only in the second system and components of type three that are shared by the two systems. We use the concept of joint survival signature to assess the joint reliability function of the two systems. Using this concept, some representations for the joint reliability function of the system lifetimes are obtained under two different scenarios of component failures. In the first scenario, we assume that the components of the systems fail according to different counting processes such as non-homogeneous Poisson processes. In the second scenario, it is assumed that the component lifetimes of each type are exchangeable while the three types of component lifetimes can be independent or dependent. To illustrate the theoretical results, two systems with shared components are studied numerically and graphically.

     

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0$t>0$ with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.     

具有修理延迟的两不同部件冷贮备系统的可靠性分析

研究了有修理延迟的两个不同部件和两个修理工组成的冷贮备系统.假定部件的工作寿命服从一般分布,故障后的延迟修理时间和修理时间均服从指数分布.利用马尔可夫更新过程、拉普拉斯变换和拉普拉斯-司梯阶变换工具,得到了系统的首次故障前时间、可用度和平均故障次数等可靠性指标.     

Estimating cumulative incidence functions when the life distributions are constrained

In competing risks studies, the Kaplan-Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure.     

On the optimality of a class of designs with three concurrences

In the present paper we consider a class of unequally replicated designs having concurrence range 2 and spectrum of the form μ₁(μ₂)^v−3μ₃. Now, Jacroux’s [Some sufficient conditions for the type I optimality of block designs, J. Statist. Plann. Inference 11 (1985) 385-396] Proposition 2.4 says that a design with spectrum of the above form, if satisfies some further conditions, is type 1 optimal. Unfortunately, this proposition does not apply to our designs since they have a poor status regarding E-optimality. Yet we are able to prove the A-optimality (in the general class) of these designs using majorisation technique. A method of construction of an infinite series of our A-optimal designs has also been given.The first and only known infinite series of examples of designs satisfying Jacroux’s conditions appears to be the first one in Section 4.1 of Morgan and Srivastav [On the Type-1 optimality of nearly balanced incomplete block designs with small concurrence range, Statist. Sinica 10 (2000) 1091-1116] - hitherto referred to as [MS]. In this paper, we use majorisation technique to prove stronger optimality properties of the above mentioned designs of [MS] as well as to present simpler proof of another optimality result in [MS].     

Semi-varying coefficient models with a diverging number of components

Semiparametric models with both nonparametric and parametric components have become increasingly useful in many scientific fields, due to their appropriate representation of the trade-off between flexibility and efficiency of statistical models. In this paper we focus on semi-varying coefficient models (a.k.a. varying coefficient partially linear models) in a “large n, diverging p” situation, when both the number of parametric and nonparametric components diverges at appropriate rates, and we only consider the case p=o(n). Consistency of the estimator based on B-splines and asymptotic normality of the linear components are established under suitable assumptions. Interestingly (although not surprisingly) our analysis shows that the number of parametric components can diverge at a faster rate than the number of nonparametric components and the divergence rates of the number of the nonparametric components constrain the allowable divergence rates of the parametric components, which is a new phenomenon not established in the existing literature as far as we know. Finally, the finite sample behavior of the estimator is evaluated by some Monte Carlo studies.     

开关寿命连续型n部件冷贮备系统的可靠性分析

为了讨论由n个同型部件和一个转换开关组成的冷贮备不可修系统的可靠度,假设部件的工作时间和转换开关的寿命均服从指数分布、所有随机变量均相互独立,首先根据系统的特征和可靠度的定义研究并求出三部件冷储备系统可靠度的解析式,然后在三部件冷储备系统可靠度研究的基础上建立了n部件冷贮备系统的可靠性关于部件个数n的递推关系式,最终得到系统的可靠度的解析表达式.计算过程对其他多部件系统可靠度的推导有一定的借鉴作用,计算结果具有一定的理论价值.     

n个同型部件k个修理设备的冷贮备可修系统的可靠度研究

在设定元件的工作寿命和维修时间的分布都是负指数分布,全部随机变量都互相独立,发生故障的元件能够修成新的前提下,对由n个同种型号的元件和k(k≤n)个设备修理构成的冷贮存可修体系的可靠性方面进行分析,构建关于此体系的数学模型,获得了n个同种型号元件的冷贮存体系在k个设备修理的情况下的可靠度,且探讨在修理设备k值不相同的情况的可靠度,并针对n=3的情况下,分析修理设备数不相同情况下的可靠度。     

Optimal redundancy allocation problems in engineering systems with dependent component lifetimes

This paper deals with the problem of allocating spare components for improving the system reliability when lifetimes of components are dependent. Two policies, called active and standby redundancies, are investigated in details. The optimal allocations are given without specific assumptions on the dependency structure among the component lifetimes. When there exist the (positive/negative) quadratic dependence and (upper/lower) orthant order among the component lifetimes, the derived results are simplified. Various illustrative examples are also given. Copyright © 2015 John Wiley & Sons, Ltd.     

On the reliability function of a double redundant system with general repair time distribution

The reliability function is considered for a hot double redundant repairable heterogeneous system with exponentially distributed lifetimes and generally distributed repair times of its components. The problem of its sensitivity to the repair time distribution is discussed.