The physical basis for the roller-coaster hazard rate curve for electronics

This is the final installment of a series of three articles under the general topic ‘Off the bathtub onto the roller-coaster curve’. A short version was presented at the 1988 Annual Reliability and Maintainability Symposium in the U.S.A. The first article was entitled ‘The bathtub does not hold water any more’, and the second was entitled ‘The roller-coaster curve is in’. The three articles provide detailed discussions of the findings leading to the conclusions on the roller-coaster characteristics for the hazard rate curve for electronics. The first article discussed the problems with the bathtub and the second explored the shape of the hazard rate curve in view of some recent observations. This article suggests some plausible physical reasons for the formation of the roller-coaster shape. The generation of the shape starts with the basic failure mechanisms, which leads to the generally decreasing hazard rate. The humps could be caused by changing hazard conditions, wear-out failure distribution of flawed items, distribution of flaw sizes or residue small size flaws left in the equipment because of test and inspection limitations.     

The bathtub does not hold water any more

This is the first part of a three part article under the general heading of ‘Off the bathtub onto the roller-coaster curve’, a short version¹ of which was presented at the 1988 Annual Reliability and Maintainability Symposium in U.S.A. The other two parts, to be published in later issues of this journal, are entitled ‘The roller-coaster curve is in’ and ‘Physical basis for the roller-coaster hazard rate curve’. This three-part article provides detailed discussions of the findings leading to the conclusions on the roller-coaster characteristics for the hazard rate curve for electronics. Part 1 being presented here covers the discussions on problems with the bathtub curve. The problems include making erroneous decisions resulting in reliability demonstration risks against the manufacturers, windfall profits for satellite manufacturers, false claims in reliability growth and reliability prediction accuracy and stress screening shortfalls. More important than the erroneous short-term decisions is the long-term reliability engineering methodology development. It is critical that we must reshape the course of reliability methodology development away from the bathtub misconception.     

A virtual age model based on a bathtub shaped initial intensity

This paper presents a new reliability model for complex repairable systems, which combines a bathtub shaped ageing and imperfect maintenance. A bathtub shaped initial intensity function allows to take into account the burn-in period, the useful life and wear out of the systems. Repair effect is expressed by a reduction of the system virtual age, which depends on the ageing of the system. The main characteristics of the model are derived. The most important one is that the maintenance efficiency allows an extension of the system useful life duration. A statistical analysis of the model and an application to real failure data are presented.     

A modified Weibull extension with bathtub-shaped failure rate function

Models with bathtub-shaped failure rate function are useful in reliability analysis, and particularly in reliability related decision making and cost analysis. The traditional Weibull distribution is, however, unable to model the complete lifetime of systems with a bathtub-shaped failure rate function. In this paper, a new model, which is useful for modeling this type of failure rate function, is presented. The model can also be seen as a generalization of the Weibull distribution. Parameter estimation methods are studied for this new distribution. Examples and results of comparison are shown to illustrate the applicability of this new model.     

On the hazard rate process for imperfectly monitored multi-unit systems

The aim of this paper is to present a stochastic model to characterize the failure distribution of multi-unit systems when the current units state is imperfectly monitored. The definition of the hazard rate process existing with perfect monitoring is extended to the realistic case where the units failure time are not always detected (non-detection events). The so defined observed hazard rate process gives a better representation of the system behavior than the classical failure rate calculated without any information on the units state and than the hazard rate process based on perfect monitoring information. The quality of this representation is, however, conditioned by the monotony property of the process. This problem is mainly discussed and illustrated on a practical example (two parallel units). The results obtained motivate the use of the observed hazard rate process to characterize the stochastic behavior of the multi-unit systems and to optimize for example preventive maintenance policies.     

Updated hazard rate equations for dual safeguard systems

A previous paper by this author [M.J. Rothschild, Updated hazard rate equation for single safeguards, J. Hazard. Mater. 130 (1-2) (2006) 15-20] showed that commonly used analytical methods for quantifying failure rates overestimates the risk in some circumstances. This can lead the analyst to mistakenly believe that a given operation presents an unacceptable risk. For a single safeguard system, a formula was presented in that paper that accurately evaluates the risk over a wide range of conditions. This paper expands on that analysis by evaluating the failure rate for dual safeguard systems. The safeguards can be activated at the same time or at staggered times, and the safeguard may provide an indication whether it was successful upon a challenge, or its status may go undetected. These combinations were evaluated using a Monte Carlo simulation. Empirical formulas for evaluating the hazard rate were developed from this analysis. It is shown that having the safeguards activate at the same time while providing positive feedback of their individual actions is the most effective arrangement in reducing the hazard rate. The hazard rate can also be reduced by staggering the testing schedules of the safeguards.     

Short-term reliability of a brief hazard perception test

Hazard perception tests (HPTs) have been successfully implemented in some countries as a part of the driver licensing process and, while their validity has been evaluated, their short-term stability is unknown. This study examined the short-term reliability of a brief, dynamic version of the HPT. Fifty-five young adults (M_age = 21 yrs) with at least two years of post-licensing driving experience completed parallel, 21-scene HPTs with a one-month interval separating each test. Minimal practice effects (∼0.1 s) were manifested. Internal consistency (Cronbach's alpha) averaged 0.73 for the two forms. The correlation between the two tests was 0.55 (p < 0.001) and correcting for lack of reliability increased the correlation to 0.72. Thus, a brief form of the HPT demonstrates acceptable short-term reliability in drivers whose hazard perception should be stable, an important feature for implementation and consumer acceptance. One implication of these results is that valid HPT scores should predict future crash risk, a desirable property for user acceptance of such tests. However, short-term stability should be assessed over longer periods and in other driver groups, particularly novices and older adults, in whom inter-individual differences in the development of hazard perception skill may render HPT tests unstable, even over short intervals.     

The roller-coaster curve is in

This is the second part of a three-part article under the general heading of ‘Off the bathtub onto the roller-coaster curve’, a short version¹ of which was presented at the 1988 Annual Reliability and Maintainability Symposium in the U.S.A. The first part was entitled ‘The bathtub does not hold water any more’,² and the last part, to be published in a later issue of this journal, is entitled ‘Physical basis for the roller-coaster hazard rate curve’. This three-part article provides detailed discussions of the findings leading to the conclusions on the roller-coaster characteristics for the hazard rate curve for electronics. Part 1 discussed problems with the bathtub curve. This part explores the shape of the failure rate curve in view of some recent observations. For more than twenty years investigators studying semiconducting device burn-in have subscribed to the concept of freak failures, which manifest themselves as a hump on the hazard rate curve. In addition to semiconducting device data several sets of screening data, as well as operational data dating all the way back to 1961, also showed that indeed one can expect electronic systems to have generally decreasing failure rate curves with failure humps on them. The author of this paper calls these curves the roller-coaster curves.     

Applications of hazard rate profile to software reliability prediction. Growth modelling and allocation

The concept of a computer program's ‘hazard rate profile’ is introduced. A software fault's hazard rate is the amount the fault contributes to the overall program failure rate. The hazard rate profile describes the relationship between the program's failure rate and fault content by summarizing the relative proportion of software fault hazard rates that fall into different hazard rate classes. The usefulness of the hazard rate profile concept is shown by applying it to the tasks of software reliability prediction, growth modelling and allocation.     

On changing points of mean residual life and failure rate function for some generalized Weibull distributions

The failure rate function and mean residual life function are two important characteristics in reliability analysis. Although many papers have studied distributions with bathtub-shaped failure rate and their properties, few have focused on the underlying associations between the mean residual life and failure rate function of these distributions, especially with respect to their changing points. It is known that the change point for mean residual life can be much earlier than that of failure rate function. In fact, the failure rate function should be flat for a long period of time for a distribution to be useful in practice. When the difference between the change points is large, the flat portion tends to be longer. This paper investigates the change points and focuses on the difference of the changing points. The exponentiated Weibull, a modified Weibull, and an extended Weibull distribution, all with bathtub-shaped failure rate function will be used. Some other issues related to the flatness of the bathtub curve are discussed.     

Modeling reliability with a two-sided power distribution

Modeling the reliability of a system typically involves the piecewise combination of mathematically disparate functions covering distinct periods (e.g., decreasing, constant, and increasing time-dependent failure rates defining the traditional bathtub curve) in the life of an engineered system. This work explores the two-sided power distribution, a nonlinear extension of the triangular distribution, for describing failure time and rendering a statistically valid bathtub curve with a single probability density function. The parameters of the two-sided power distribution also provide some flexibility to describe decreasing and increasing failure rates with a single distribution that are not available with similar distributions.     

航天器氦质谱真空容器总漏率检测的灵敏度及可信度探讨

依据航天器氦质谱真空容器总漏率检测原理,讨论了检漏仪和真空容器直接相连、同真空容器主真空泵并联、在前级泵和主真空泵间等连接方式,逐项研究了总漏率测试的最小可检漏率、反应时间,分析各种连接方式的应用特点,认为对大型航天器氦质谱真空容器检漏系统,检漏仪连接在前级泵和主泵之间可得到较好的灵敏度和反应时间.通过测量不确定度的分析与评定,确定一般航天器氦质谱真空容器总漏率检测的相对标准不确定度小于10％.     

Modeling the reliability of threshold weighted voting systems

In many applications, ranging from target detection to safety monitoring systems, we are interested in determining whether or not to accept a hypothesis based on the information available. In this paper we model the reliability of threshold weighted voting systems (WVS) with multi-failure-modes, where a general recursive reliability function of the WVS is presented. We also develop approximation formulas for calculating the reliability of WVS based on a large number of units. We also develop reliability functions of time-dependent threshold weighted voting systems, where each unit is a function of time. Finally, the optimal stopping time that minimizes the total cost of the systems subject to a reliability constraint is discussed.     

Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.     

Continuum shape sensitivity and reliability analyses of nonlinear cracked structures

A new method is proposed for shape sensitivity analysis of a crack in a homogeneous, isotropic, and nonlinearly elastic body subject to mode I loading conditions. The method involves the material derivative concept of continuum mechanics, domain integral representation of the J-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is required in the proposed method. Since the governing variational equation is differentiated before the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques. Based on the continuum sensitivities, the first-order reliability method was employed to perform probabilistic analysis. Numerical examples are presented to illustrate both the sensitivity and reliability analyses. The maximum difference between the sensitivity of stress-intensity factors calculated using the proposed method and the finite-difference method is less than four percent. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently.     

A two-parameter parsimonious bathtub model for the analysis of system failure times with illustrations to reliability data

In this study, a two-parameter, upper-bounded probability distribution called the tau distribution is introduced and its applications in reliability engineering are presented. Each of the parameters of the tau distribution has a clear semantic meaning. Namely, one of them determines the upper bound of the distribution, while the value of the other parameter influences the shape of the cumulative distribution function. A remarkable property of this new probability distribution is that its probability density function, survival function, hazard rate function (HRF), and quantile function can all be expressed in terms of its cumulative distribution function. The HRF of the proposed probability distribution can exhibit an increasing trend and various bathtub shapes with or without a low and long-flat phase (useful time phase), which makes this new distribution suitable for modeling a wide range of real-world problems. The constraint maximum likelihood estimation, percentile estimation, approximate Bayesian computation, and approximate quantile estimation computation are proposed to calculate the unknown parameters of the model. The suitability of the estimation methods is verified with the aid of simulation and real-world data results. The modeling capability of the tau distribution was compared with that of some well-known two- and three-parameter probability distributions using two data sets known from the literature of reliability engineering: time between failures data of a machining center, and time to failure of data acquisition system cards. Based on empirical results, the new distribution may be viewed as a viable competitor to the Weibull, Gamma, Chen, and modified Weibull distributions.     

Construction of event-tree/fault-tree models from a Markov approach to dynamic system reliability

While the event-tree (ET)/fault-tree (FT) methodology is the most popular approach to probability risk assessment (PRA), concerns have been raised in the literature regarding its potential limitations in the reliability modeling of dynamic systems. Markov reliability models have the ability to capture the statistical dependencies between failure events that can arise in complex dynamic systems. A methodology is presented that combines Markov modeling with the cell-to-cell mapping technique (CCMT) to construct dynamic ETs/FTs and addresses the concerns with the traditional ET/FT methodology. The approach is demonstrated using a simple water level control system. It is also shown how the generated ETs/FTs can be incorporated into an existing PRA so that only the (sub)systems requiring dynamic methods need to be analyzed using this approach while still leveraging the static model of the rest of the system.     

Dynamic reliability analysis for non-stationary non-Gaussian response based on the bivariate vector translation process

Calculation of probability of exceedance for nonstationary non-Gaussian responses remains a great challenge to researchers in the field of structural reliability. In this paper, an analytical solution is proposed for calculating the mean upcrossing rate (MCR) of the non-stationary non-Gaussian responses by approximating the displacement and velocity responses with the bivariate vector translation process, in which the unified Hermite polynomial model (UHPM) is selected as the mapping function. The first four moments (i.e., mean value, standard deviation, skewness, and kurtosis) and cross-correlation function of the displacement and velocity responses needed in UHPM are estimated from some representative samples generated by random function-spectral representation method (RFSRM) and time-domain analysis. Under the Poisson assumption of the upcrossing events, the calculation of extreme value distribution or probability of exceedance for structural response can be determined with the proposed method. The proposed method is applicable to a wide range of structural responses, including asymmetric and hardening or softening responses. Three numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method. It can be concluded that the proposed method provides an accurate and useful tool for dynamic reliability assessment in engineering applications.     

Dynamic response and reliability analysis of non-linear stochastic structures

A new probability density evolution method is proposed for dynamic response analysis and reliability assessment of non-linear stochastic structures. In the method, a completely uncoupled one-dimensional governing partial differential equation is derived first with regard to evolutionary probability density function (PDF) of the stochastic structural responses. This equation holds for any response or index of the structure. The solution will put out the instantaneous PDF. From the standpoint of the probability transition process, the reliability of the structure is evaluated in a straightforward way by imposing an absorbing boundary condition on the governing PDF equation. However, this does not induce additional computational efforts compared with the dynamic response analysis. The computational algorithm to solve the PDF equation is studied. A deterministic dynamic response analysis procedure is embedded to compute coefficient of the evolutionary PDF equation, which is then numerically solved by the finite difference method with total variation diminishing scheme. It is found that the proposed hybrid algorithm may deal with non-linear stochastic response analysis problem with high accuracy. Numerical examples are investigated. Parts of the results are illustrated. Some features of the probabilistic information of the response and the reliability are observed and discussed. The comparisons with the Monte Carlo simulations demonstrate the accuracy and efficiency of the proposed method.     

含运动副间隙的平面函数机构运动点可靠性分析

运动副间隙是影响机构运动与动力性能的一类最重要的不确定性.为精确估计含运动副间隙的机构运动可靠度,建立机构运动间隙合理、有效的概率模型是关键．为此,以含运动副间隙的函数生成机构为例,在其运动误差函数中综合考虑机构的结构误差和随机误差,应用截尾混合降维法对运动副间隙变量进行处理并建立机构运动误差的概率等效模型,采用一次二阶矩法实现等效模型的求解．数值实例验证了截尾混合降维法在考虑结构误差和随机误差含运动副间隙的函数机构运动可靠性分析中的有效性．