Bathtub failure rate and upside-down bathtub mean residual life

This paper shows that: (1) the mean residual life (MRL) of a component has an upside-down bathtub-shape if the component has a bathtub-shape failure-rate function, but the converse does not hold; and (2) there is an optimal burn-in policy to maximize the MRL when the underlying lifetime distribution has a bathtub-shape failure rate     

Useful periods for lifetime distributions with bathtub shaped hazard rate functions

We propose computationally tractable formal mathematical definitions for the 'useful period' of lifetime distributions with bathtub shaped hazard rate functions. Detailed analysis of the reduced additive Weibull hazard rate function illustrates its utility for identifying such useful periods. Examples of several other bathtub shaped hazard rate functions are also presented with applications to lifetime data. The suggestion is made of defining and considering analogous 'stable periods' in the case of the corresponding upside-down bathtub shaped mean residual life functions.     

Mean residual life of lifetime distributions

This paper characterizes the general behaviors of the MRL (mean residual lives) for both continuous and discrete lifetime distributions, with respect to their failure rates. For the continuous lifetime distribution with failure rates with only one or two change-points, the characteristic of the MRL depends only on its mean and failure rate at time zero. For failure rates with “roller coaster” behavior, the subsequent behavior of the MRL depends on its MRL and failure-rates at the change points. Using the characterization, their behaviors for the: Weibull; lognormal; Birnbaum-Saunders; inverse Gaussian; and bathtub failure rate distributions are tabulated in terms of their shape parameters. For discrete lifetime distributions, for upside-down bathtub failure rate with only one change point, the characteristic of the MRL depends only on its mean and the probability mass function at time zero     

Mean residual life and its association with failure rate

The life time distributions having decreasing, increasing, or upside-down bathtub shaped MRL (mean residual life) are used as models in many applications. Mi (1995) has shown that if a component has a bathtub shape failure rate function, then the MRL is unimodal but the converse does not hold. This paper develops sufficient conditions for the unimodal MRL to imply that the failure rate function has a bathtub shape     

Analysis of burn-in time using the general law of reliability

Empirical lifetime distributions sometimes have a bathtub-shaped failure rate. This paper deals with some models having a bathtub-shaped failure rate. The root-mean-square criterion is proposed for selection of the best model. Besides two criteria of optimum burn-in time are proposed. The comparison of some models with the general law of reliability is given to determine a burn-in time in a number of examples.     

A modified Weibull distribution

A new lifetime distribution capable of modeling a bathtub-shaped hazard-rate function is proposed. The proposed model is derived as a limiting case of the Beta Integrated Model and has both the Weibull distribution and Type 1 extreme value distribution as special cases. The model can be considered as another useful 3-parameter generalization of the Weibull distribution. An advantage of the model is that the model parameters can be estimated easily based on a Weibull probability paper (WPP) plot that serves as a tool for model identification. Model characterization based on the WPP plot is studied. A numerical example is provided and comparison with another Weibull extension, the exponentiated Weibull, is also discussed. The proposed model compares well with other competing models to fit data that exhibits a bathtub-shaped hazard-rate function.     

Minimizing cost-functions related to both burn-in and field-operation under a generalized model

Summary and Conclusions-Burn-in is a method used to improve the quality of products. In field operation, only those units which survived the burn-in procedure will be used. This paper considers various additive cost structures related to both burn-in procedure and field operation under a general failure model. The general failure model includes two types of failures. Type I (minor) failure is removed by a minimal repair, whereas type II failure (catastrophic failure) is removed only by a complete repair (replacement). We introduce the following cost structures: (i) the expenses incurred until the first unit surviving burn-in is obtained; (ii) the minimal repair costs incurred over the life of the unit during field use; and (iii) either the gain proportional to the mean life of the unit in field operation or the expenditure due to replacement at a catastrophic failure during field operation. We also assume that, before undergoing the burn-in procedure, the unit has a bathtub-shaped failure rate function with change points t/sub 1/ & t/sub 2/. The optimal burn-in time b/sup */ for minimizing the cost function is demonstrated to be always less than t/sub 1/. Furthermore, a large initial failure rate is shown to justify burn-in, i.e. b/sup */>0. A numerical example is presented.     

Lifetime Distribution Identities

Five ways of representing the distribution of a continuous nonnegative random variable T are used extensively in the reliability literature: the probability density function, the reliability (survivor function), the hazard rate, the cumulative hazard function, and the mean residual life function. Properties, identities, and intuitive interpretations of the five representations are discussed. Several examples are given. Although there are other functions, such as normalized mean residual life for studying replacement policies, these five distribution representations have surfaced as vehicles for representing a lifetime distribution. The choice of which distribution representation to use depends on whether ? 1. The representation has a tractable form 2. Intuition is gained concerning the distribution by seeing a plot of the representation.     

Hazard Versus Renewal Rate of Electronic Items

Two different indexes, the hazard rate and the renewal rate, which are implied by conventional uses of the bathtub-shaped curve, are often noted in reliability. The hazard rate is applicable for a single failure time of each item, such as that of a nonrepairable part; the renewal rate is applicable for multiple failure times of each item, such as those of repairable equipment. Occasionally, remarks are made in the literature concerning the mathematical models for the bathtub-shaped hazard rate but not for the renewal rate. Furthermore, bathtub-shaped hazard and renewal curves as conventionally used are each based on certain assumptions concerning failure time distributions. Little data have been recorded for electronic parts and equipment which would substantiate the widespread use of the conventional implications of the bathtub-shaped hazard and renewal rates. The validity of the assumptions concerning the underlying distributions of failure times affects the accuracy of the results of reliability analyses, such as prediction, data analysis, formal assurance tests, operational planning, and maintenance planning. A study of the applications-oriented literature suggests that the distinction between the hazard rate and the renewal rate, as well as some associated implications, are not generally appreciated. Thus the existing situation is apt to lead engineers astray as well as others with application interests. Basic concepts and definitions are emphasized and extensions and implications are sketched. References are selected and noted for those interested in further pursuit.     

A New Nonparametric Growth Model

This paper proposes a new nonparametric reliability growth model for the analysis of the failure rate of a system that is undergoing development test. The only restrictions on the actual, unknown failure distribution for each stage of testing is that it be continuous, have only one unknown parameter ?, and have an associated unimodal likelihood function. No assumptions regarding the parametric form of the failure rate of the development process are made, only that there is no decay in the reliability of the system during the design changes. The parameters are assumed to be ordered from one test stage to the next such that ?1 ? ?2 ? ... ? ?m. The new model performs very well based on relative error and mean square error. The model is generally superior to the popular AMSAA model, regardless of the actual underlying failure process. In addition, the results indicate a notable bias in the AMSAA model, early in the development process, regardless of the actual underlying failure process.     

A Birnbaum-Saunders accelerated life model

The 2-parameter family of probability distributions introduced by Birnbaum and Saunders characterizes the fatigue failure of materials subjected to cyclic stresses and strains. It is shown that the methods of accelerated life testing are applicable to the Birnbaum-Saunders distribution for analyzing accelerated lifetime data, and the (inverse) power law model is used due to its justification for describing accelerated fatigue failure in metals. This paper develops the (inverse) power law accelerated form of the Birnbaum-Saunders distribution, and explores the corresponding inference procedures-including parameter estimation techniques and the derivation of the s-expected Fisher information matrix. The model approach in this paper is different from an earlier work, which considered a log-linear form of a model with applications to accelerated life testing. Here, using an example data set, the fitted model is effectively used to estimate lower distribution percentiles and mean failure times for particular values of the acceleration variable. The benefits of having an operable closed form of the Fisher information matrix, which is unique to this article for this model, include interval estimation of model parameters and LCB on percentiles using relatively simple computational procedures     

Preservation of partial orderings under the formation ofk-out-of-n:G systems of i.i.d. components

The authors discuss the preservation of certain partial orderings by a k-out-of-n:G system of i.i.d. components. If the lifetime of a component A is larger than that of a component B in the likelihood ratio, failure rate, or stochastic ordering, then a k-out-of-n:G system formed by n i.i.d. components of type A has a larger lifetime, in that ordering, than that of a similar system consisting of n i.i.d. components of type B. However, if the lifetime of a component A is larger than that of a component B in mean residual life, harmonic-average mean residual life, or variable orderings, it is not necessary that a k-out-of-n:G system formed by n i.i.d. components of type A has a larger lifetime, in that ordering, than that of a similar system consisting of n i.i.d. components of type B     

Residual lifetime distribution and its applications

Let T be a continuous positive random variable representing the lifetime of an entity. This entity could be a human being, an animal or a plant, or a component of a mechanical or electrical system. For nonliving objects the lifetime is defined as the total amount of time for which the entity carries out its function satisfactorily. The concept of aging involves the adverse effects of age such as increased probability of failure due to wear. In this paper, we consider certain characteristics of the residual lifetime distribution at age t, such as the mean, median, and variance, as describing aging. Gamma and Weibull families of distributions are studied from this point of view. Explicit asymptotic expressions for the mean, variance and the percentiles of corresponding residual lifetime distributions are found. Finally these families of distributions are fitted to four sets of actual data, two of which are entirely new. The results can be used in discriminating different shape parameters.     

Critical time of the lognormal distribution

The critical time is the time point as the failure rate starts to decrease and also as the mean residual lifetime starts to increase. The estimated critical time is useful for determining the duration of a burn-in process. The method for estimating the critical time of the failure rate for lognormal lifetime distribution is discussed. A single time censored data is used as a example for illustration.     

On optimal burn-in procedures - a generalized model

Burn-in is a manufacturing technique that is intended to eliminate early failures. In this paper, burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure), which can be removed by a minimal repair or a complete repair; and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. During the burn-in process, two types of burn-in procedures are considered. In Burn-In Procedure I, the failed component is repaired completely regardless of the type of failure; whereas, in Burn-In Procedure II, only minimal repair is done for the Type I failure, and a complete repair is performed for the Type II failure. Under the model, various additive cost functions are considered. It is assumed that the component before undergoing the burn-in process has a bathtub-shaped failure rate function with the first change point t/sub 1/, and the second change point t/sub 2/. The two burn-in procedures are compared in cases when both the procedures are applicable. It is shown that the optimal burn-in time b/sup */ minimizing the cost function is always before t/sub 1/. It is also shown that a large initial failure rate justifies burn-in, i.e., b/sup */>0. The obtained results are applied to some examples.     

Estimation in a random censoring model with incomplete information:exponential lifetime distribution

Analysis of methods and simulation results for estimating the exponential mean lifetime in a random-censoring model with incomplete information are presented. The instant of an item's failure is observed if it occurs before a randomly chosen inspection time and the failure is signaled. Otherwise, the experiment is terminated at the instant of inspection during which the true state of the item is discovered. The maximum-likelihood method (MLM) is used to obtain point and interval estimates for item mean lifetime, for the exponential model. It is demonstrated, using Monte Carlo simulation, that the MLM provides positively biased estimates for the mean lifetime and that the large-sample approximation to the log-likelihood ratio produces accurate confidence intervals. The quality of the estimates is slightly influenced by the value of the probability of failure to signal. Properties of the Fisher information in the censored sample are investigated theoretically and numerically     

Reliability analysis and comparison of several structures

In this study, we consider the redundant structure with the function of swithover processing which is assumed to cause the increase of the failure rate of the system. A single component system and a simple redundant system are compared in terms of the four reliability measures such as the reliability function, the MTBF, the failure rate and the mean residual life (MRL). We find the relations of the MTBF, the failure rate, and the MRL between two systems. We also consider a quad configuration system and a parallel string configuration system and evaluate the four reliability measures for two systems. As a numerical example, the total system down of an ATM switching system is considered.     

Preservation of some partial orderings under non-homogeneous poisson shock model and laplace transform

In this paper, we study the preservation of some partial orderings under non-homogeneous Poisson shock model and Laplace transform. These partial orderings are: likelihood ratio ordering (LR); failure rate ordering (FR); stochastic ordering (ST); variable ordering (V); mean residual life ordering (MR); variance residual life ordering (VR); concave ordering (CV); harmonic average mean residual life ordering (HAMR).     

荧光寿命分析法鉴别机油品种特征信息提取

由于机油的品种鉴别在环境监测、事故认定等领域有着重要的作用,结合非线性最小二乘法的荧光寿命衰减曲线提取技术,能解决不同机油混合后的分类识别问题。研究通过激光诱导荧光雷达系统,利用时间分辨荧光实验,采用非线性最小二乘法将荧光寿命衰减曲线按照不同指数拟合,从相关指数及残差分布确定最佳拟合曲线。实验结果表明,三种不同机油产品的荧光寿命衰减曲线按照不同指数衰减其拟合效果不同,选取各自最佳拟合曲线,按照合成规律可以拟合出不同机油的荧光平均寿命参数,通过误差分析可信赖程度在95%以上。当以所有激发区域内像素点的平均荧光寿命为中心,利用2倍标准差作为置信限,统计其中两种不同机油激发区域中像素点荧光寿命值落入各自置信区间内的概率,二者的概率分别为87%和68%。通过三种不同油品的对比试验,验证了利用荧光寿命分析法实现不同机油产品油种识别的可能性,并可通过荧光寿命成像判断不同油种的空间分布信息。荧光寿命分析法为机油产品油种鉴别、成分分析及检测空间分布提供了一种新途径。     

On the Mean Residual Life Function of Coherent Systems

We consider a coherent structure consisting of $n$ components having the property that if it is known that at most $r$ components $(r≪n)$ have failed, the system is still operating with probability 1. Some examples of the systems having this property are $(n-k+1)$-out-of- $n$, some parallel-series, and some series-parallel structures. Depending on the structure, and the number of active components of the coherent systems at time $t$ , the mean residual life function of the system is studied, by several authors. This paper investigates more properties of the mean residual life function of the coherent systems sharing the described property. We will show that, when the components of the system have increasing failure rate, the mean residual life function of the system is decreasing in time. Several examples, and illustrative graphs are also provided.