Properties of an approximate hazard transform

The calculation of the exact reliability of complex systems is a difficult and tedious task. Consequently simple approximating techniques have great practical value. The hazard transform of a system is an invertible transformation of its reliability function which is convenient and useful in both applied and theoretical reliability work. A simple calculus for finding an approximate hazard transform for systems formed by series and parallel combinations of components is extended so that it can be used for any coherent system. The extended calculus is shown to lead to conservative approximations. A first order version of the extended calculus is also discussed. This method of approximation is even more simple to use, but is not always conservative. Examples of its application indicate that it is capable of giving quite accurate results.     

Reliability models and inference for systems operating in different environments

We consider a class of models for the reliability function of a series system. The models incorporate dependence of the function on environmental covariates. On the basis of censored data obtained by monitoring several series systems operating under various sets of values of the covariates, estimators are derived of the reliability function of a series system operating under a different set of values of the covariates. Estimators of various functionals of the reliability function, such as the trimmed mean and quantiles, are also presented. Asymptotic properties of the estimators are obtained with the use of the framework of counting processes and martingales. The loss in efficiency in estimating the regression parameters and the reliability function is also examined when one assumes a more general semiparametric model, when in fact the true model belongs to a more restricted model. We illustrate these procedures with the use of accelerated failure-time data. © 1996 John Wiley & Sons, Inc.     

Allocation of two redundancies in two‐component series systems

The allocation of redundancies in a system to optimize the reliability of system performance is an interesting problem in reliability engineering and system security. In this article, we focus on the optimal allocation of two exponentially distributed active (standby) redundancies in a two‐component series system using the tool of stochastic ordering. For the case of active redundancy, stochastic comparisons are carried out in terms of the likelihood ratio and reversed hazard rate orders. For the case of standby redundancy, a likelihood ratio ordering result is developed. The results established here generalize and strengthen corresponding results in the recent literature. In addition, several numerical examples are used to explicate the results. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

A general model for accelerated life testing with time‐dependent covariates

This paper introduces a general or “distribution‐free” model to analyze the lifetime of components under accelerated life testing. Unlike the accelerated failure time (AFT) models, the proposed model shares the advantage of being “distribution‐free” with the proportional hazard (PH) model and overcomes the deficiency of the PH model not allowing survival curves corresponding to different values of a covariate to cross. In this research, we extend and modify the extended hazard regression (EHR) model using the partial likelihood function to analyze failure data with time‐dependent covariates. The new model can be easily adopted to create an accelerated life testing model with different types of stress loading. For example, stress loading in accelerated life testing can be a step function, cyclic, or linear function with time. These types of stress loadings reduce the testing time and increase the number of failures of components under test. The proposed EHR model with time‐dependent covariates which incorporates multiple stress loadings requires further verification. Therefore, we conduct an accelerated life test in the laboratory by subjecting components to time‐dependent stresses, and we compare the reliability estimation based on the developed model with that obtained from experimental results. The combination of the theoretical development of the accelerated life testing model verified by laboratory experiments offers a unique perspective to reliability model building and verification. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 303–321, 1999     

On failure modeling

A promising approach to failure modeling, in particular to developing failure-time distributions, is discussed. Under this approach, system state or wear and tear is modeled by an appropriately chosen random process—for example, a diffusion process—and the occurrences of fatal shocks are modeled by a Poisson process whose rate function is state dependent. The system is said to fail when either wear and tear accumulates beyond an acceptable or safe level or a fatal shock occurs. This approach has significant merit. First, it provides revealing new insights into most of the famous and frequently used lifetime distributions in reliability theory. Moreover, it suggests intuitively appealing ways for enhancing those standard models. Indeed, this approach provides a means of representing the underlying dynamics inherent in failure processes. Reasonable postulates for the dynamics of failure should lend credence to the prediction and estimation of reliability, maintainability, and availability. In other words, accuracy of representation could lead to better, more reliable prediction of failure.     

基于BER和CTMC的航天器发射组织过程可靠度模型

在航天器发射工程中,存在组织过程可靠性评估难以量化的问题。基于航天器发射组织过程的时间特性呈现多个子过程并发执行且子过程具有Markov性和齐次性的特点,使用连续时间Markov链(CTMC)建立多吸收态的组织过程可靠度模型,利用互模拟等价关系(BER)简化组织过程的状态空间,并进一步给出各种状态转移率和组织过程可靠度计算方法。最后,通过数据分析说明模型在航天器发射工程进度计划评估中的作用和意义。     

Optimal allocation of minimal repairs in parallel and series systems

In reliability engineering, the concept of minimal repair describes that the repair brings the failed unit (eg, system or component) to the situation which is same as it was just before the failure. With the help of the well‐known Gamma‐Poisson relationship, this paper investigates optimal allocation strategies of minimal repairs for parallel and series systems through implementing stochastic comparisons of various allocation policies in terms of the hazard rate, the reversed hazard rate, and the likelihood ratio orderings. Numerical examples are presented to illustrate these findings as well. These results not only strengthen and generalize some known ones in the seminal work of Shaked and Shanthikumar, but also solve the open problems proposed by Chahkandi et al.'s study and Arriaza et al.'s study.     

Probabilistic models of decreasing failure rate processes

The “infant mortality” effect observed in the statistical treatment of reliability consists of a decreasing with age of the conditional probability of equipment failure (failure rate). One widely applicable explanatory hypothesis is that of population heterogeneity. This is developed here as a basis for several specific models of decreasing failure rate processes. Since, in the case of repairable devices, decreasing failure rate is often observed after the occurrence of failure and repair, consideration is extended to include repair in an explicit way. This union of failure and repair models is a fruitful one in view of the interaction between the two processes and gives a complete picture of the life of the device in terms of a stochastic process, usually with non-independent interfailure times. Four models, of particular significance due to their plausibility, mathematical tractability, and frugality of parameterization, are presented.     

屏蔽数据在新型截尾样本下系统的可靠性分析

当系统含有屏蔽数据时,在具有随机移走逐步增加型截尾模型下,讨论了部件寿命服从双参数指数分布的串联系统可靠性估计问题。设随机移走系统数服从二项分布,利用极大似然方法,Bayes理论及方法,推导出双参数指数部件参数、系统可靠性函数、失效率函数及移走概率的极大似然估计和Bayes估计。并利用Monte Carlo方法对两种估计结果进行了比较,表明Bayes估计较极大似然估计效果更优。     

Life distribution analysis based on Lévy subordinators for degradation with random jumps

For a component or a system subject to stochastic degradation with sporadic jumps that occur at random times and have random sizes, we propose to model the cumulative degradation with random jumps using a single stochastic process based on the characteristics of Lévy subordinators, the class of nondecreasing Lévy processes. Based on the inverse Fourier transform, we derive a new closed‐form reliability function and probability density function for lifetime, represented by Lévy measures. The reliability function derived using the traditional convolution approach for common stochastic models such as gamma degradation process with random jumps, is revealed to be a special case of our general model. Numerical experiments are used to demonstrate that our model performs well for different applications, when compared with the traditional convolution method. More importantly, it is a general and useful tool for life distribution analysis of stochastic degradation with random jumps in multidimensional cases. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 483–492, 2015     

系统可靠寿命的不确定性分析及其高效方法

为了分析元器件失效率的不确定性对系统可靠性的影响,借鉴Borgonovo的矩独立灵敏度分析思想,在充分考虑了系统可靠寿命完整不确定性信息的情况下,提出了基于系统可靠寿命的矩独立重要性测度,用来分析不确定性条件下系统元器件失效率对其可靠寿命的平均影响。但由于系统可靠寿命函数是系统可靠度函数的反函数,一般无法解析表达而以隐函数的形式存在,致使该矩独立重要性测度难以高效准确求解。为了解决这一问题,文章提出了一种新的Kriging自适应代理模型的高效算法,该算法以Kriging代理模型预测值的变异系数作为自适应学习函数,通过自主增加新的试验样本,增强代理模型的预测准确性。阀门控制系统和民用飞机电液舵机系统两个算例分析表明,在保证计算精度的情况下,通过变异系数自适应学习函数,仅需添加少量系统可靠寿命试验样本,就能够构建用来充分近似系统可靠寿命函数的Kriging代理模型,解决了重要性测度的高效求解问题,从而验证了所提方法的合理性和算法的高效性。     

Approximations to the reliability of phased missions

A system whose configuration (block diagram or fault tree) changes during consecutive time periods (phases) performs a “phased mission.” Recently, Esary and Ziehms have shown that any multiphase mission can be transformed into an equivalent, synthetic, single-phase system, and thus that the phased mission problem can in principle be solved by standard reliability methods. We employ these ideas here to study approximations to mission reliability and to develop an algorithm which may be of practical interest. In addition, we extend the reliability calculus of Rubinstein, and Esary and Hayne, based on an approximate hazard transform, to phased missions, and we show how this extended calculus can be used in situations where phases are not of known fixed duration.     

Inference on a future reliability parameter with the weibull process model

An inferential procedure is presented which provides confidence intervals for a future reliability parameter when reliability growth testing is only partially completed. Hypothesis tests based on this method are uniformly most powerful unbiased. These results are applicable if (1) the system failure rate can be modeled as the intensity function of a Weibull process and (2) efforts to improve reliability are assumed to continue at a steady rate throughout the intervening period of testing. The usefulness of this methodology is illustrated by evaluating the risk of not reaching some future reliability milestone. If such risk is unacceptably high, program management may have time to identify problem areas and take corrective action before testing has ended. As a consequence, a more reliable system may be developed without incurring overruns in the scheduling or cost of the development program.     

Inference from accelerated life tests using eyring type re-parameterizations

Least squares estimators of the parameters of the generalized Eyring Model are obtained by using data from censored life tests conducted at several accelerated environments. These estimators are obtained after establishing that the Gauss-Markov conditions for least squares estimation are satisfied. Confidence intervals for the hazard rate at use conditions are obtained after empirically showing that the logarithm of the estimate of the hazard rate at use conditions is approximately normally distributed. The coverage probabilities of the confidence intervals are also verified by a Monte Carlo experiment. The techniques are illustrated by an application to some real data.     

逐步增加Ⅱ型截尾样本下k/N(G)系统可靠性指标的置信区间

在逐步增加Ⅰ型截尾样本下,研究k/N(G)系统可靠性指标的区间估计问题。假设部件寿命服从指数分布,利用极大似然法和Bayes方法,首先给出了部件失效率的Bayes近似置信区间,其次推导出了系统可靠度和平均寿命的Bayes近似置信区间的计算公式。最后给出随机模拟例子,并对置信区间的精度进行了讨论。     

A general approach to the shape of failure rate and MRL functions

Failure rate and mean residual life are two important characteristics for studying reliability of products. In literature, some work studied the shape of failure rate function based on the knowledge of the associated probability density function; some other work investigated the shape of mean residual life function based on the shape of the associated failure rate function separately for continuous case and discrete case. In this article, a general approach is developed which can be applied to the aforementioned studies. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Two novel critical shock models based on Markov renewal processes

In this paper, we investigate systems subject to random shocks that are classified into critical and noncritical categories, and develop two novel critical shock models. Classical extreme shock models and run shock models are special cases of our developed models. The system fails when the total number of critical shocks reaches a predetermined threshold, or when the system stays in an environment that induces critical shocks for a preset threshold time, corresponding to failure mechanisms of the developed two critical shock models respectively. Markov renewal processes are employed to capture the magnitude and interarrival time dependency of environment-induced shocks. Explicit formulas for systems under the two critical shock models are derived, including the reliability function, the mean time to failure and so on. Furthermore, the two critical shock models are extended to the random threshold case and the integrated case where formulas of the reliability indexes of the systems are provided. Finally, a case study of a lithium-ion battery system is conducted to illustrate the proposed models and the obtained results.     

Warranty costs: An age‐dependent failure/repair model

An age‐dependent repair model is proposed. The notion of the “calendar age” of the product and the degree of repair are used to define the virtual age of the product. The virtual failure rate function and the virtual hazard function related to the lifetime of the product are discussed. Under a nonhomogeneous Poisson process scenario the expected warranty costs for repairable products associated with linear pro‐rata, nonrenewing free replacement and renewing free replacement warranties are evaluated. Illustration of the results is given by numerical and graphical examples. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.     

Shock model in Markovian environment

A Markov modulated shock models is studied in this paper. In this model, both the interarrival time and the magnitude of the shock are determined by a Markov process. The system fails whenever a shock magnitude exceeds a pre‐specified level η. Nonexponential bounds of the reliability are given when the interarrival time has heavy‐tailed distribution. The exponential decay of the reliability function and the asymptotic failure rate are also considered for the light‐tailed case. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005