Failure Time Distribution Research Articles

Dynamic and dependent tree theory (D2T2): A framework for the analysis of fault trees with dependent basic events

• A fault tree framework is provided which overcomes limitations in conventional analysis methods. • Component failure events can have any distribution of failure or repair times. • Component dependencies can be accommodated. • Complex maintenance processes for components and sub-systems can be incorporated. • The algorithm identifies independent sub-problems which are solved accurately and efficiently. Fault tree analysis remains the most commonly employed method, particularly in the safety critical industries, to predict the probability or frequency of system failures. Whilst it has its origins back in the 1960s, the assumptions employed in the majority of commercial fault tree analysis codes have not changed significantly since this time and restrict the ability of the method to represent features of the design, operation and maintenance of modern industrial systems. The inability to include general dependencies between the basic events, the requirement for invariant failure and repair rates, and the inability to account for complex maintenance strategies are major limitations. This paper proposes a new fault tree analysis framework which can overcome these restrictions. Whilst retaining the fault tree structure to express the causality of the system failure, the internal calculation method is updated by exploiting features of the Binary Decision Diagram, Stochastic Petri Net and Markov methods. The key elements of the D 2 T 2 algorithm are described in detail and the framework demonstrated through application to a case study example of a pressure vessel cooling system.

A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components

<p style='text-indent:20px;'>Redundancy allocation problem (RAP) is a common technique for increasing the reliability of systems. In this paper, a new model for the RAP is introduced that takes into account the warm standby and mixed strategy, the model dynamics, and the type of the strategy in redundancy allocation problems. A recursive formula is first obtained for the reliability function in the dynamic warm standby and mixed redundancy strategies that leverages the success mode analysis and works for any arbitrary failure-time distribution. Failure rates for warm standby units change before and after their replacement with a damaged unit, and, therefore, the reliability function in warm standby varies with time (i.e., the model is dynamic). Although dynamic models are commonplace in practice, they are more challenging to assess than static models, which have been mainly considered in the literature. An optimization problem is then formulated to select the best redundancy strategy and redundancy levels. Genetic algorithm and particle swarm optimization are leveraged to solve the problem. Finally, the efficiency of the presented method is verified through a numerical example. The experimental results verify that the proposed model for RAP significantly improves the system reliability, which can be of vital importance for system designers.</p>

Stochastic modelling of a system having two non-identical class of units

This paper deals with the study of stochastic modelling of a system having two non-identical class of units. The whole system consists of two class L1 and L2 respectively. First class consists only one unit where as second class having two identical units. The whole system is used to process the jobs. Jobs are first processes on L1 unit (class one) and then goes to L2 unit (second class). All the failure time distribution are taken as arbitrarily distributed. Using regenerative point technique, various reliability characteristics of interest to the system managers have been evaluated. Finally, some graphs are also plotted in order to highlight the important results.

Optimal design of stress levels in accelerated degradation testing for multivariate linear degradation models

In recent years, more attention has been paid prominently to accelerated degradation testing (ADT) in order to characterize accurate estimation of reliability properties for systems that are designed to work properly for years or even decades. In this paper, we propose optimal experimental designs for repeated measures ADTs with competing failure modes that correspond to multiple response components. The marginal degradation paths are expressed using linear mixed effects models. The optimal design is obtained by minimizing the asymptotic variance of the estimator of some quantile of the failure time distribution at the normal use conditions. Numerical examples are introduced to ensure the robustness of the proposed optimal designs and compare their efficiency with standard experimental designs.

A novel failure time estimation method for degradation analysis based on general nonlinear Wiener processes

Nonlinear Wiener processes have been widely applied for degradation analysis in practical engineering. To improve the accuracy of failure time distribution approximation itself, a novel analytical approximation of the distribution under the concept of first hitting time (FHT) is constructed through a unique tangent treatment. A series of comprehensive simulation studies are implemented to demonstrate the effectiveness and accuracy improvement of the proposed approximation. Results show that the proposed method can guarantee improved accuracy under various nonlinear regulations and uncertainties. The feasibility and validity of the general deterioration model and reliability analysis methodology are illustrated by a typical real application of metal fatigue crack growth.

Mixture model for dependent competing risks data and application for diabetic retinopathy treatment

This paper proposes a flexible mixture modeling approach to analyze competing risks data with multiple causes of failure under a progressive first-failure censoring scheme where the causes may not be independent. The failure time distributions corresponding to different causes of failure are assumed to belong to the flexible proportional hazard family of distributions, some important members of which are Weibull, Burr XII and Gompertz distributions. We address the problem here from a classical viewpoint. The point estimates of the parameters can be easily obtained by employing the maximum likelihood principle. For interval estimation purposes, we resort to using the asymptotic confidence intervals. An extensive simulation study is carried out to validate our proposed model. Finally, a diabetic retinopathy treatment dataset is used as an application to study the effectiveness of our proposed model assuming some important members of the proportional hazard family.

Optimisation of Preventive Maintenance Regime Based on Failure Mode System Modelling Considering Reliability

Today, the ability to maintain a continuous complex system operation is viewed as a key attribute for ensuring uninterrupted revenue contribution and the survival of a business. Many industrial organisations have come to understand that by having an effective plan of maintenance, the efficiency and reliability of a system can be improved, while costs can be minimised and revenue-generating production can continue. The novelty of this paper is based on using failure modes and effects analysis (FMEA) to develop a holistic preventive maintenance schedule for a complete system. A system can be modelled as a series and parallel arrangement of subsystems and components, and failure of different components of the system can be determined from their life expectancy. The objective is to ensure continuity of production output while maintaining a high level of system reliability and minimising the total maintenance costs. The reliability of a complete system is analysed using the Weibull failure-time distribution. By using the exhaustive search optimisation method, the maintenance cost is minimised by determining the optimal replacement interval for each FMEA block, subject to system reliability. Based on the results obtained from a case study, it is shown that the approach in this paper can ensure the continuity of production output during maintenance activities, reduce system maintenance costs, and achieve maximum system reliability. This holistic approach can be applied to any form of complex system, and at every step of the process.

A Three-Component Additive Weibull Distribution and Its Reliability Implications

We consider a new failure time distribution by combining three Weibull distributions. The proposed distribution exhibits decreasing, increasing, and bathtub-shaped failure rate functions. Some statistical properties of the proposed distribution are presented. The maximum likelihood method is used to estimate the parameters, reliability, and failure rate functions of the proposed distribution, and the uncertainties of the estimates are obtained by using asymptotic confidence intervals. The cross-entropy method, a stochastic optimization procedure, is used to find the maximum of the log-likelihood function. The superiority of the proposed distribution is demonstrated on two benchmark datasets.

Development of New Control Charts to Monitor Product Failure Times with Normal and Log Normal Distributions

Reliability is referred to as quality over time and one of the dimensions of quality. Improving product reliability is one of the major concerns in manufacturing and service processes that can be achieved by applying statistical process control. Statistical control chart is a process monitoring tool that is widely used in the manufacturing industry and can be used to monitor the failure process. The development of control charts for this purpose is one of the topics of interest to researchers in the field of reliability. However, research in this area has been limited to the development of control charts for separate monitoring of statistical indicators. In the present study, a new control chart will be proposed to monitor failure times and product reliability. The purpose of this chart is to simultaneously monitor various parameters of failure time distribution. For this purpose, in this research, normal distribution and normal log are investigated.

A Comprehensive Comparison of Additional Benefit Assessment Methods Applied by Institute for Quality and Efficiency in Health Care and European Society for Medical Oncology for Time-to-Event Endpoints After Significant Phase III Trials—A Simulation Study

ObjectivesAfter a successful Marketing Authorization Application for clinical trials with time-to-event endpoints, the degree of the added benefit from new treatments remains unknown and needs to be assessed. Unfortunately, until now no clear definition for added benefit determination of a treatment exists. Nevertheless, European authorities / societies have developed 2 “additional benefit assessment” methods, which have up to now not been compared: the European Society for Medical Oncology (ESMO) developed a dual rule considering relative and absolute benefit. The German Institute for Quality and Efficiency in Health Care (IQWiG) developed a method using upper 95% hazard ratio confidence interval. MethodsWe evaluate and compare both methods in an extensive simulation study including different censoring rates, failure time distributions, and treatment effects for sample size calculation. The methods’ performance is assessed via Receiver Operating Characteristic curves, Spearman correlation, and percentage of achieved maximal scores. ResultsThe results show that IQWiG’s method has in many situations a lower maximal scoring proportion than ESMO’s rule, that is, up to 28.5% versus 94.7%. Various failure time distributions lead to strongly changed maximal scoring percentages for ESMO. High positive correlation between the methods is present for moderate treatment effects. ConclusionsIQWiG’s method is usually more conservative than ESMO’s. ESMO’s rule tends to be more susceptible for various failure time distributions. Using the lower confidence interval limit seems to be a better solution resulting in a higher true-positive rate without increasing the false-positive rate. Thus, IQWiG’s method might need to be adapted accordingly to achieve a better overall classification.

Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies

Survival and reliability analyses deal with incomplete failure time data, such as censored and truncated data. Recently, the classical left-truncation scheme was generalized to analyze “field data”, defined as samples collected within a fixed period. However, existing competing risks models dealing with left-truncated field data are not flexible enough. We propose copula-based competing risks models for latent failure times, permitting a flexible parametric form. We formulate maximum likelihood estimation methods under the Weibull, lognormal, and gamma distributions for the latent failure times. We conduct simulations to check the performance of the proposed methods. We finally give a real data example. We provide the R code to reproduce the simulations and data analysis results.

Nonlinear degradation model and reliability analysis by integrating image covariate

Degradation-based reliability analysis is an essential method for reliability prediction and prognosis of critical systems. Recently, with the advances of information technology, high-dimensional data such as images are available to improve system modeling and analysis. In this research, we propose a new nonlinear degradation model that integrates material microstructure image covariates. Based on the proposed model, product reliability can be precisely predicted and the failure time distribution is calculated. A maximum likelihood estimation method and an expectation-maximization method are developed to estimate the model parameters. Simulation studies are conducted to exam the effectiveness of the developed model and the model parameter estimation method. In the case study, the developed methodology is applied to a real-world problem of dual-phase advanced high strength steel (AHSS), which is now widely used in the automotive and aerospace industries. The results show that the proposed model can effectively model the nonlinear degradation trend with material image covariates, and the model greatly outperforms multiple existing methods.

Self-reporting and screening: Data with right-censored, left-censored, and complete observations.

We consider survival data that combine three types of observations: uncensored, right‐censored, and left‐censored. Such data arises from screening a medical condition, in situations where self‐detection arises naturally. Our goal is to estimate the failure‐time distribution, based on these three observation types. We propose a novel methodology for distribution estimation using both semiparametric and nonparametric techniques. We then evaluate the performance of these estimators via simulated data. Finally, as a case study, we estimate the patience of patients who arrive at an emergency department and wait for treatment. Three categories of patients are observed: those who leave the system and announce it, and thus their patience time is observed; those who get service and thus their patience time is right‐censored by the waiting time; and those who leave the system without announcing it. For this third category, the patients' absence is revealed only when they are called to service, which is after they have already left; formally, their patience time is left‐censored. Other applications of our proposed methodology are discussed.

A two-stage stochastic programming model for selective maintenance optimization

The problem of selective maintenance exists in many multicomponent systems carrying out an alternating sequence of missions, with scheduled breaks where only a limited number of components can be maintained due to time limit. In this paper, a stochastic programming approach is proposed for determining an optimum maintenance plan to minimize maintenance costs and expected failure costs, while maximizing the probability of successful accomplishment of the next mission under uncertainties in future operating conditions. Traditionally, future operating conditions that affect failure time distribution when calculating reliability in selective maintenance models were assumed as deterministic. In this study, future operating conditions are assumed to be uncertain. The system is subject to several uncertain condition scenarios of exposure, conditional, usage, stress, etc. Each scenario is modeled with its associated occurrence probability. The presented model is a two-stage stochastic mixed-integer nonlinear programming model with fixed recourse, where the first stage is associated with maintenance decisions made before uncertainties are revealed, and the second stage is modeled as a recourse function which is related to the occurrence probability of system failure. A numerical example of a series-parallel system is used to demonstrate the effectiveness of the suggested model.

Cross-comparative Analysis of Frequentist and Bayesian Perspectives on Failure Time Distributions

In reliability analysis, both the Weibull and the lognormal distributions could be analyzed using the observed data logarithms. While the Weibull data logarithm is skewed, the lognormal data logarithm is symmetrical. This review work initiates discussions on and syntheses of the various views held on the use of the frequentist and Bayesian approaches to drawing statistical inferences on failure time distributions of survival models. Of greater concern was the discussion on the use of the exponential, Weibull and log-normal distributions in reliability analysis. Various methods have been used to discriminate between the two most important distributions. They include: Coefficients of variation (CV); the standard deviation of the data logarithms (SD); the percentile position of the mean of the data logarithm (PP); the cumulated logarithm dispersion before and after the mean (CLD); ratio of the maximum likelihood (RML); Kullback-Leibler Divergence (KLD); and Minimized Kullback-Leibler Divergence (RMKLD). In the (CV, SD, PP and CLD) study, a stress-strength data set was used for the analysis. The stress data followed a lognormal distribution, while the strength data followed a Weibull distribution, therefore for the stress-strength analysis the lognormal-Weibull combination was used. In the ratio of the maximum likelihood (RML) study, it was averred that though each of the distributions had great applications, none of them produced a good fit. In the study using Kullback-Leibler Divergence (KLD); and Minimized Kullback-Leibler Divergence (RMKLD), test results revealed that RML=0.345>0. Hence the lognormal distribution was selected. Similarly, the RMKLD=0.6028>0, therefore the Weibull distribution was selected. In the final analysis, none of the study results could reject the Weibull in favour of the lognormal distribution model. In respect of the frequentist and the Bayesian approach to conducting statistical inferences, it came out strongly that it was high time psychologist who had adopted the use of the frequentist framework moved away from it and started using the Bayesian approach. A shift towards the use of the Bayesian, lognormal-Weibull approach is thus recommendable.

On the number of failed components in a coherent system consisting of multiple types of components

In this paper, we investigate the number of failed components in an operating coherent system. We assume that its components are of multiple types. That is, the system consists of components having nonidentical failure time distributions. We extend the results which are well-known in the literature for k-out-of-n systems. We formulate the optimization problem to determine the optimal values of the number of components of each type together with their arrangement in the system.

Recent Advances in Robust Design for Accelerated Failure Time Models with Type I Censoring

Many fields including clinical and manufacturing areas usually perform life-testing experiments and accelerated failure time models (AFT) play an essential role in these investigations. In these models the covariate causes an accelerant effect on the course of the event through the term named acceleration factor (AF). Despite the influence of this factor on the model, recent studies state that the form of AF is weakly or partially known in most real applications. In these cases, the classical optimal design theory may produce low efficient designs since they are highly model dependent. This work explores planning and techniques that can provide the best robust designs for AFT models with type I censoring when the form of the AF is misspecified, which is an issue little explored in the literature. Main idea is focused on considering the AF to vary over a neighbourhood of perturbation functions and assuming the mean square error matrix as the basis for measuring the design quality. A key result of this research was obtaining the asymptotic MSE matrix for type I censoring under the assumption of known variance regardless the selected failure time distribution. In order to illustrate the applicability of previous result to a study case, analytical characterizations and numerical approaches were developed to construct optimal robust designs under different contaminating scenarios for a failure time following a log-logistic distribution.

Particular aspects of determining reliability indicators of thermoelectric generator modules using experimental data

Resource tests allowed finding that the relative degradation of output power and efficiency of thermoelectric generator modules is not subject to linear law. This means that the distribution law for the failure time of such modules does not «copy» the distribution of their initial parameters, i.e. is neither normal nor logarithmically normal. Therefore, the aim of this paper is to find or select from among the existing such a failure time distribution law, which would clearly take into account the scattering of the rates of relative degradation of the parameters of thermoelectric generator modules. The paper substantiates the need to use diffusion-nonmonotonic failure time distribution for processing the results of resource tests of thermoelectric generator modules in order to determine their standardized reliability indicators and relative errors of the obtained values. It is proposed to determine the point estimates of the parameters of the law, namely the average failure time and the parameter of variation of the rate of degradation processes not by formulas obtained by the method of maximum likelihood, but by smoothing the probability of failure-free operation obtained by tests. The least squares method and Newton's method are used. Estimates obtained by the method of maximum likelihood serve as an initial approximation for Newton's method. This allows achieving significantly less error in determining standardized reliability indicators than when using the method of maximum likelihood.

Порівняння надійності термоелектричних генераторних модулів з різними схемами сполучення термоелементів

The study compares reliability indicators of thermoelectric generator modules using the nonmonotonic-diffusion law of failure time distribution. The authors use the law to propose for the first time an analytical expression of the failure time distribution of an individual thermoelement, which fundamentally differs from the traditional simple exponential law with a constant failure intensity. The law is used to calculate the mean time between failures (MTBF), the 95% resource and equivalent failure rate of thermoelectric generator modules with series-parallel and parallel-series connection of thermocouples. Considered are the variants of schemes with different number of elements in rows and different number of rows for serial-parallel circuit and with different number of sequential groups and elements in a group for parallel-serial circuit. Coefficients of increase of the average failure time, 95% resource and coefficients of decrease of the equivalent failure intensity for a purely series connection of thermocouples are determined for the considered thermocouple schemes. It is established that these indicators can be improved by more than three orders of magnitude.

Minimum cost replacement and maintenance scheduling in dual-dissimilar-unit standby systems

Recent studies have shown that reusing standby elements during the mission may improve the mission success probability significantly. However, such a benefit cannot be effectively achieved without a careful design of the replacement and maintenance schedule (RMS), which determines work period durations of operating elements and types of maintenance performed for idle elements. This paper makes contributions by modeling and optimizing the RMS for a heterogeneous dual-unit warm standby system with the aim to minimize the total expected mission cost, covering operation, standby and maintenance costs as well as mission failure penalty cost. The two system elements are dissimilar, characterized by different performance, failure time distribution and cost parameters. For a successful mission, a specified amount of work must be accomplished before both elements become failed or unavailable. We propose a new probabilistic model-based methodology for assessing the mission success probability and expected mission cost (EMC) of the considered system. An optimization problem is further formulated and solved to find the optimal RMS minimizing the EMC. A case study of a two-pump oil transfer system is conducted to demonstrate the proposed model and effects of different cost parameters on the optimal RMS solution and corresponding mission success probability and EMC.