Reliability analysis using exponentiated Weibull distribution and inverse power law

Today in reliability analysis, the most used distribution to describe the behavior of devices is the Weibull distribution. Nonetheless, the Weibull distribution does not provide an excellent fit to lifetime datasets that exhibit bathtub shaped or upside‐down bathtub shaped (unimodal) failure rates, which are often encountered in the performance of products such as electronic devices (ED). In this paper, a reliability model based on the exponentiated Weibull distribution and the inverse power law model is proposed, this new model provides a better approach to model the performance and fit of the lifetimes of electronic devices. A case study based on the lifetime of a surface‐mounted electrolytic capacitor is presented in this paper. Besides, it was found that the estimation of the proposed model differs from the Weibull classical model and that affects the mean time to failure (MTTF) of the capacitor under analysis.     

The omega probability distribution and its applications in reliability theory

A new three‐parameter probability distribution called the omega probability distribution is introduced, and its connection with the Weibull distribution is discussed. We show that the asymptotic omega distribution is just the Weibull distribution and point out that the mathematical properties of the novel distribution allow us to model bathtub‐shaped hazard functions in two ways. On the one hand, we demonstrate that the curve of the omega hazard function with special parameter settings is bathtub shaped and so it can be utilized to describe a complete bathtub‐shaped hazard curve. On the other hand, the omega probability distribution can be applied in the same way as the Weibull probability distribution to model each phase of a bathtub‐shaped hazard function. Here, we also propose two approaches for practical statistical estimation of distribution parameters. From a practical perspective, there are two notable properties of the novel distribution, namely, its simplicity and flexibility. Also, both the cumulative distribution function and the hazard function are composed of power functions, which on the basis of the results from analyses of real failure data, can be applied quite effectively in modeling bathtub‐shaped hazard curves.     

Reliability analysis incorporating exponentiated inverse Weibull distribution and inverse power law

In reliability research, electronic devices are an important part of our lives and modelling their lives is the most difficult and fascinating area. To investigate the failure functioning of electronic equipments, reliability monitoring of systems is widely used. However, it is stated in the literature that one in five electronic system collapses are a consequence of degradation and saving energy and forecasting future losses, it is necessary to summarize the data through certain versatile models of probability . In current article, a model of reliability formed on inverse power law and generalized inverse Weibull model is suggested. This current distribution presents a clearer framework to modelling the efficiency and functionality lifespan of electronic equipments. In this article, an empirical analysis is discussed related to life cycle of a surface-mounted electrolytic capacitor (SMEC). In addition, it has noticed that evaluation of suggested distribution varies from classical model of inverse Weibull and that influences average time to failure (ATTF) of the studied capacitor.     

An application of a bathtub failure model to imperfectly repaired systems data

In this paper we use an exponential power distribution to develop a bathtub failure model for repairable systems. These results are compared with those using the more familiar Weibull distribution. The analysis is extended to three types of repair scenarios: good-as-new repairs (in which the repair following a failure completely refreshes the failure intensity), bad-as-old repairs (in which the repair has no effect on the failure intensity) and imperfect repairs (in which the failure intensity is only partially reset). We apply this analysis to hydro-electric turbine data and find that an exponential power distribution produces a well-defined bathtub failure model when used with an imperfect repair assumption. We furthermore find that this bathtub model produces an improved fit over the equivalent Weibull model.     

A reliability analysis for electronic devices under an extension of exponentiated perks distribution

This paper presents a reliability analysis for electronic devices (ED) with bathtub curve-shaped failure times. An extension of the exponentiated perks distribution (EPD) is proposed for the analysis. The extension of this new distribution is based on the Alpha Power Transformation, so the Alpha Exponentiated Perks Distribution (AEXP) is introduced. The AEXP has three shape parameters and one scale parameter, allowing greater flexibility to represent failure rates in an increasing, decreasing, or bathtub curve form. Some useful properties in the reliability engineering context are presented. AEXP parameters were estimated via the Maximum Likelihood Method. Finally, two case studies focused on ED are used to compare the proposed distribution and other distributions with similar failure rate representation properties. The obtained results show that the AEXP better describes the behavior of ED than the distributions considered in the analysis.     

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.     

More on the mis‐specification of the shape parameter with Weibull‐to‐exponential transformation

When lifetimes follow Weibull distribution with known shape parameter, a simple power transformation could be used to transform the data to the case of exponential distribution, which is much easier to analyze. Usually, the shape parameter cannot be known exactly and it is important to investigate the effect of mis‐specification of this parameter. In a recent article, it was suggested that the Weibull‐to‐exponential transformation approach should not be used as the confidence interval for the scale parameter has very poor statistical property. However, it would be of interest to study the use of Weibull‐to‐exponential transformation when the mean time to failure or reliability is to be estimated, which is a more common question. In this paper, the effect of mis‐specification of Weibull shape parameters on these quantities is investigated. For reliability‐related quantities such as mean time to failure, percentile lifetime and mission reliability, the Weibull‐to‐exponential transformation approach is generally acceptable. For the cases when the data are highly censored or when small tail probability is concerned, further studies are needed, but these are known to be difficult statistical problems for which there are no standard solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Reliability Data Analysis for Life Test Designed Experiments with Sub‐Sampling

The ability to model lifetime data from life test experiments is of paramount importance to all manufacturers, engineers and consumers. The Weibull distribution is commonly used to model the data from life tests. Standard Weibull analysis assume completely randomized designs. However, not all life test experiments come from completely randomized designs. Experiments involving sub‐sampling require a method for properly modeling the data. We provide a Weibull nonlinear mixed models (NLLMs) methodology for incorporating random effects in the analysis. We apply this methodology to a reliability life test on glass capacitors. We compare the NLLMs methodology to other available methods for incorporating random effects in reliability analysis. A simulation study reveals the method proposed in this paper is robust to both model misspecification and increasing levels of variance on the random effect. Copyright © 2012 John Wiley & Sons, Ltd.     

Reliability Model for Electronic Devices under Time Varying Voltage

Present reliability models, which estimate the lifetime of electronic devices, work under the assumption that the voltage level must be constant when an Accelerated Life Testing is performed. Nevertheless, in a real operational environment, electronic devices are subjected to electrical variations present in the power lines; that means the voltage has a time‐varying behavior, which breaks the assumption of reliability models. Thus, in this paper, a reliability model is presented, which describes the lifetime of electronic devices under time‐varying voltage via a parametric function. The model is based on the Cumulative Damage Model with random failure rate and the modified Inverse Power Law. In order to estimate the parameters of the proposed model, the maximum likelihood method was employed. A case study based on the time‐varying voltage induced by electrical harmonics when Alternate Current/Direct Current (AC/DC) transformer is connected to the power line is presented in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Model for Repairable Systems with Nonmonotone Intensity Function

In this study, a new model based on the mixture of bounded Burr XII failure intensity and bounded intensity process is proposed to describe the failure intensity of minimally repaired systems with approximate bathtub behavior. The estimates of the model parameters are easily obtained using the maximum likelihood estimation method. The confidence intervals for the model parameters are also provided. Other existing models, such as superposed power law process, log‐linear process–power law process, and bounded bathtub intensity process are used to compare with our proposed model. Through numerical examples, the results show that our proposed model performs well regarding the Akaike information criterion value and the mean of square errors. Copyright © 2014 John Wiley & Sons, Ltd.     

The mean time between failures for an LCD panel

The calculation of mean time between failures is very important in reliability life data analysis. For different distributions, the values of mean time between failures are always different. The two‐parameter Weibull distribution is widely used in reliability engineering. However, some distributions may offer a better fit of data. This paper aims to develop an algorithm for determining the best‐fitted distribution of a liquid crystal display panel based on the field return data. The two‐parameter and three‐parameter Weibull distributions and other distributions such as the Burr XII distribution, the Pareto distribution and the Log‐logistic distribution are compared to provide a better characterization of the life data which is based on the maximum value of all log‐likelihood functions. We also provide a goodness‐of‐fit test for the best‐fitted distribution. It is recommended that the Burr XII distribution could be used to characterize the reliability life of a liquid crystal display panel. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.     

Two‐Stage Reliability Sampling Plans with Bogey Tests

This article considers the design of two‐stage reliability test plans. In the first stage, a bogey test was performed, which will allow the user to demonstrate reliability at a high confidence level. If the lots pass the bogey test, the reliability sampling test is applied to the lots in the second stage. The purpose of the proposed sampling plan was to test the mean time to failure of the product as well as the minimum reliability at bogey. Under the assumption that the lifetime distribution follows Weibull distribution and the shape parameter is known, the two‐stage reliability sampling plans with bogey tests are developed and the tables for users are constructed. An illustrative example is given, and the effects of errors in estimates of a Weibull shape parameter are investigated. A comparison of the proposed two‐stage test with corresponding bogey and one‐stage tests was also performed. Copyright © 2012 John Wiley & Sons, Ltd.     

Weibull and inverse Weibull mixture models allowing negative weights

This article presents the finding that when components of a system follow the Weibull or inverse Weibull distribution with a common shape parameter, then the system can be represented by a Weibull or inverse Weibull mixture model allowing negative weights. We also use an example to illustrate that the proposed mixture model can be used to approximate the reliability behaviours of the consecutive-k-out-of-n systems. The example also shows data analysis procedures when the parameters of the component life distributions are either known or unknown.     

An improved failure‐free time estimation method

In this paper a simple order‐statistics‐based iterative procedure for estimating the failure‐free time is put forward. The method is applicable when the population lifetime follows the Weibull, C‐Weibull or Gompertz–Makeham distribution. Copyright © 1999 John Wiley & Sons, Ltd.     

A new variable control chart under failure‐censored reliability tests for Weibull distribution

In this paper, we propose a new variable control chart under type II or failure‐censored reliability tests by assuming that the lifetime of a part follows the Weibull distribution with fixed and stable shape parameter. The purpose is to monitor the mean and the variance of a Weibull process. In fact, the mean and the variance are related to the scale parameter. The necessary measures are given to calculate the average run length (ARL) for in‐control and shifted processes. The tables of ARLs are presented for various shift constants and specified parameters. A simulation study is given to show the performance of the proposed control chart. The efficiency of the proposed control chart is compared with a control chart based on the conditional expected value under type II censoring. An example is also given for the illustration purpose.     

The Weibull log-logistic mixture distributions: Model,theory and application to lifetime data

Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull-log-logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull-log-logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log-logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.     

An improved modified Weibull distribution applied to predict the reliability evolution of an aircraft lock mechanism

A new improved modified Weibull distribution (IMW), which consists of two tandem failure models is proposed in this paper. The order statistics, moment estimate and maximum likelihood estimate of the new distribution are investigated. The distribution is flexible for modeling the three phases of the modified bathtub-shaped hazard function. The modified distribution allows for enhanced statistical simulation. The new distribution is analyzed and applied to two sets of known lifetime data to demonstrate the advantages of the model. To verify the practical engineering applications of the model, reliability evolution test data for an aircraft door lock mechanism are analyzed and fitted. Based on the obtained trends of the lock hook angle and lock hook distance, we predict the evolutionary data for numerous mechanism samples, thus reducing the test cost. By considering the subdistribution and removing one parameter, the fitting effect is improved over that of the traditional distribution model.     

Distribution of time between failures of machining center based on type I censored data

This paper applies a type I censor likelihood function to make the fitting of Weibull distribution of time between failures of machining center (MC). The paper also gives Goodness-of-fit tests by Hollander's method and proves that the time between MC failures follows the Weibull distribution. The conclusion not only deeply analyzes the MC failure law, but also establishes the basis of calculation for the mean time between failures of MC with censored lifetime data.