Failure Time Distribution Research Articles

A gamma process based model for systems subject to multiple dependent competing failure processes under Markovian environments

The performance rules of many products that suffer from both natural degradation and random shocks may have discrepancies in different environments during their service life. This article proposes a reliability model for systems subject to multiple dependent competing failure processes affected by Markovian environments. The effect of dynamic environment is embodied in that the natural wear behavior of the system in different environments is controlled by distinct gamma processes. A maintenance model is developed in the case where switches of environment states take place only when systems are operating. Explicit formulas for system reliability indexes including the reliability function, first failure time distribution, mean time to first failure, and system availability are analytically derived in the form of Laplace transforms. In order to verify the correctness of the analytical formula, a simulation algorithm for evaluating the reliability function is proposed. Finally, they are applied to a study case of the micro-engine system, which illustrates the applicability of proposed models and the validity of obtained results.

Проверка значимости случайного эффекта для винеровской деградационной модели

This paper considers the Wiener degradation model with random effects. Random-effect models take into account the unit-to-unit variability of the degradation index. It is assumed that a random parameter has a truncated normal distribution. During the research, the expression for the maximum likelihood estimates and the reliability function has been obtained. Two statistical tests have been proposed to reveal the existence of random effects in degradation data corresponding to the Wiener degradation model. The first test is a well-known likelihood ratio test, and the second one is based on the variance estimate of the random parameter. These tests have been compared in terms of power with the Monte-Carlo simulation method. The result of the research has shown that the criterion based on the variance estimate of the random parameter is more powerful than the likelihood ratio test in the case of the considered pairs of competing hypotheses. An example of the analysis using the proposed tests for the turbofan engine degradation data has been considered. The data set includes the measurements recorded from 18 sensors for 100 engines. Before constructing the degradation model, the single degradation index has been obtained using the principal component method. The hypothesis of the random effect insignificance in the model has been rejected for both tests. It has been shown that the random-effect Wiener degradation model describes the failure time distribution more accurately than the fixed-effect Wiener degradation model.

Application of EM Algorithm to Mixture Cure Model for Grouped Relative Survival Data

The interest in estimating the probability of cure has been increas ing in cancer survival analysis as the cure of some cancer sites is becoming a reality. Mixture cure models have been used to model the failure time data with the existence of long-term survivors. The mixture cure model assumes that a fraction of the survivors are cured from the disease of interest. The failure time distribution for the uncured individuals (latency) can be mod eled by either parametric models or a semi-parametric proportional hazards model. In the model, the probability of cure and the latency distribution are both related to the prognostic factors and patients’ characteristics. The maximum likelihood estimates (MLEs) of these parameters can be obtained using the Newton-Raphson algorithm. The EM algorithm has been proposed as a simple alternative by Larson and Dinse (1985) and Taylor (1995). in various setting for the cause-specific survival analysis. This approach is ex tended here to the grouped relative survival data. The methods are applied to analyze the colorectal cancer relative survival data from the Surveillance, Epidemiology, and End Results (SEER) program.

A semiparametric mixture model approach for regression analysis of partly interval-censored data with a cured subgroup.

Failure time data with a cured subgroup are frequently confronted in various scientific fields and many methods have been proposed for their analysis under right or interval censoring. However, a cure model approach does not seem to exist in the analysis of partly interval-censored data, which consist of both exactly observed and interval-censored observations on the failure time of interest. In this article, we propose a two-component mixture cure model approach for analyzing such type of data. We employ a logistic model to describe the cured probability and a proportional hazards model to model the latent failure time distribution for uncured subjects. We consider maximum likelihood estimation and develop a new expectation-maximization algorithm for its implementation. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed method is examined through simulation studies. An application to a set of real data on childhood mortality in Nigeria is provided.

Multivariate failure time distributions derived from shared frailty and copulas

Multivariate failure time distributions derived from shared frailty and copulas

Discrete time series–parallel system and its optimal configuration

This paper is concerned with properties of series–parallel systems when the component lifetimes have discrete failure time distribution. For a series–parallel system consisting of a specified number of subsystems, we particularly focus on the number of failed components in each subsystem at the time when the system fails. Each subsystem is assumed to have identical components while different subsystems have different types of components. Assuming all components within the system are independent, we obtain exact distributions of the number of failed components at the time when the system fails. For the special case when the components have phase-type failure time distributions, matrix-based expressions are derived for the quantities under concern. The results are used to obtain optimal configuration of the series–parallel system which is replaced at failure.

Stochastic Modeling of Three Non Identical Complex System With Single Service Facility Available In The System

The paper deals with the System comprising of three components in which are in parallel configuration and in series with unit .The system fails if either or both units fails. A single server takes some time to arrive the system to carry out repair activities.The repair of the system is based on first come first serve (fcfs). The failure time distribution and time to repair of all the units is taken exponential of the form. The arrival time of the server is taken as general.

System reliability analysis using component-level and system-level accelerated life testing

Accelerated life testing (ALT) has been widely used to expedite the analysis of a product's reliability. Systems consisting of multiple components could be tested at component level and/or system level. Each testing level requires different resources to be performed and a specific approach to analyze the information carried with it in order to draw reliability conclusions. In addition, each of these two levels: component-level tests and system-level tests have their own advantages and disadvantages. Systems of multiple components undergoing a system-level test could be expensive, but it considers the dependence of components failure times of the system. The component-level test, consists of testing each component separately, being cheap and allowing testing customization. However, it does not include any of failure time correlations of components when assembled together in one system. This paper introduces a novel framework to analyze the reliability of systems with multiple components using ALT. The framework includes shared frailty models to model the dependence between failure time distributions of the components of a system. A Bayesian method is proposed to fuse both component-level testing information and system-level testing information to calculate system reliability by propagating and minimizing the uncertainty incurred from each testing level. Results of numerical examples show the efficacy of the proposed ALT-based system reliability analysis methods.

Reliability Analysis of Accelerated Destructive Degradation Testing Data for Bi-Functional DC Motor Systems

An accelerated degradation test (ADT) has become a popular method to accelerate degradation mechanisms by stressing products beyond their normal use conditions. The components of an automobile are degraded over time or cycle due to their constant exposure to friction or wear. Sometimes, the performance degradation can be measured only by destructive inspection such as operating torques of return-springs in a bi-functional DC motor system. Plastic deformation of the return-spring causes the degradation of actuating forces for shield movement, resulting in deterioration of the shield moving speed in a headlight system. We suggest a step-by-step procedure for a reliability analysis for a bi-functional DC motor in a headlight system, based mainly on accelerated destructive degradation test (ADDT) data. We also propose nonlinear degradation models to describe the ADDT data of the return-springs. Exposure effects of high temperatures on the return-springs are quantitatively modeled through the ADDT models. We compare the estimation results from both the closed-form expression and Monte Carlo simulation to predict the failure–time distribution at normal use conditions, showing that the lifetime estimation results from the closed-form formulation are more conservative.

Evaluation of mean-time-to-failure based on nonlinear degradation data with applications

In reliability engineering, obtaining lifetime information for highly reliable products is a challenging problem. When a product quality characteristic whose degradation over time can be related to lifetime, then the degradation data can be used to estimate the first-passage (failure) time distribution and the Mean-Time-To-Failure (MTTF) for a given threshold level. To model the degradation data, the commonly used Lévy process modeling approach assumes that the degradation measurements are linearly related to time throughout the lifetime of the product. However, the degradation data may not be linearly related to time in practice. For this reason, trend-renewal-process-type models can be considered for degradation modeling in which a proper trend function is used to transform the degradation data so that the Lévy process approach can be applied. In this article, we study several parametric and semiparametric models and approaches to estimate the first-passage time distribution and MTTF for degradation data that may be not linearly related to time. A Monte Carlo simulation study is used to demonstrate the performance of the proposed methods. In addition, a model selection procedure is proposed to select among different models. Two numerical examples of lithium-ion battery degradation data are applied to illustrate the proposed methodologies.

Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States

AbstractWe study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.

Discrete Weibull-geometric distribution

The failure-time distribution is used to describe the life of a device, material, or structure. One of the most commonly used distributions to analyze failure-time data is Weibull distribution. The Weibull distribution is very flexible for modeling varying types of failure rates data with constant, increasing and decreasing shapes, but cannot be used for modeling data with unimodal failure rates. Because of this, we developed the Weibull distribution using compounding method to produce a Weibull-Geometric distribution. The Weibull-Geometric distribution is useful for modeling failure rates data with a unimodal shape. In studies of failure, time to failure is frequently measured in the number of cycles to failure or shocks to failure and become a discrete random variable. Hence, this paper discusses the formation of distribution that more appropriate to modeling discrete failure data. The discrete distribution is obtained by discretizing the continuous Weibull-Geometric distribution. The discretization is carried out by maintaining one of characteristics of the continuous distribution, that is, its survival function. The result distribution, called Discrete Weibull-Geometric distribution (DWG) has unimodal failure rates, right skew, and more appropriate at modeling discrete failure data. At the end of this paper, the Discrete Weibull-Geometric distribution is used to illustrate the real dataset and show that DWG distribution is the appropriate model.

Efficient reliability analysis of dynamic k‐out‐of‐n phase‐AND mission systems

AbstractMotivated by real‐world applications of satellites and wireless sensor networks, this paper models and evaluates a dynamic k‐out‐of‐n phase‐AND mission system (k/n‐PAMS). The mission task conducted by a k/n‐PAMS involves multiple consecutive phases; the mission is successful as long as the task is successful in any of the phases. Due to factors, such as scheduled maintenance, location changes in task execution during different phases, and resource sharing with other tasks, the total number of available components n for the considered mission task and the required number of working components k may change from phase to phase. In addition, due to varying load and working environments, component failure time distributions are also phase dependent. This paper proposes an analytical modeling approach based on multivalued decision diagrams (MDDs) for assessing reliability of the considered k/n‐PAMS. The approach encompasses a new and fast MDD model generation algorithm that considers behaviors of all the mission phases simultaneously based on node labeling. As demonstrated through empirical studies on k/n‐PAMSs with different sizes (different numbers of phases and different numbers of system components), the proposed algorithm is more efficient than the traditional phase‐by‐phase model generation method.

A software reliability growth model with Gompertz-logarithmic failure time distribution

PurposeThe Gompertz curve has been used in industry to estimate the number of remaining software faults. This paper aims to introduce a family of distributions for fitting software failure times which subsumes the Gompertz distribution.Design/methodology/approachThe mean value function of the corresponding non-homogenous Poisson process software reliability growth model is presented. Model parameters are estimated by the method of maximum likelihood. A comparison of the new model with eight models that use well-known failure time distributions of exponential, gamma, Rayleigh, Weibull, Gompertz, half normal, log-logistic and lognormal is performed according to the several statistical and informational criteria. Moreover, a Shannon entropy approach is used for ranking and model selection.FindingsNumerical experiments are implemented on five real software failure datasets varying from small to large datasets. The results exhibit that the proposed model is promising and particularly outperforms the Gompertz model in all considered datasets.Originality/valueThe proposed model provides optimized reliability estimation.

Kernel machines for current status data

In survival analysis, estimating the failure time distribution is an important and difficult task, since usually the data is subject to censoring. Specifically, in this paper we consider current status data, a type of data where the failure time cannot be directly observed. The format of the data is such that the failure time is restricted to knowledge of whether or not the failure time exceeds a random monitoring time. We propose a flexible kernel machine approach for estimation of the failure time expectation as a function of the covariates, with current status data. In order to obtain the kernel machine decision function, we minimize a regularized version of the empirical risk with respect to a new loss function. Using finite sample bounds and novel oracle inequalities, we prove that the obtained estimator converges to the true conditional expectation for a large family of probability measures. Finally, we present a simulation study and an analysis of real-world data that compares the performance of the proposed approach to existing methods. We show empirically that our approach is comparable to current state of the art, and in some cases is even better.

Lithium Ion Battery Life Analysis in Railway Application

It is well-known that the life of battery is influenced by the temperature, current rate/ pattern, depth of discharge, and chemistry of cell. Capacity and power fade over cycling and storage of lithium ion batteries are still major barriers as their calendar life is 3-5 years whereas target automotive and stationary applications require 10-20years of life time. Battery lifetime is highly dependent on application conditions. In railway application the power pattern is usually similar and repeated, and needs at least 5 years-life warrant. Understanding the relationship between the operating conditions and aging behavior of the batteries is an important task to increase the lifespan. However, it is not easy to quantify the effect of each condition on aging behavior as the aging of batteries involves multiple closely interrelated factors. Accelerated degradation testing with associated modeling and data analysis can effectively be used to predict failure-time distributions. The behaviors of lithium ion cells and aging mechanisms are strongly dependent to the battery cell chemistry and technology so should be chosen in the right application. The lithium ion battery in railway is operated during the whole day time in the mainly outside, and needs high power, high energy characteristics like PHEV and should store the regenerative breaking energy from the traction driving motor.In this study by using an accelerated method the life of railway application battery is evaluated. This study attempts to quantify the strength of the stress factors and the relationship between aging behavior and operating conditions. The accelerating factor is a temperature and depth of discharge, current pattern to understand the effect on life.

A semiparametric regression model under biased sampling and random censoring: A local pseudo‐likelihood approach

AbstractMethodologies developed for left‐truncated right‐censored failure time data can mostly be categorized according to the assumption imposed on the truncation distribution, i.e., being completely unknown or completely known. While the former approach enjoys robustness, the latter is more efficient when the assumed form of the truncation distribution can be supported by the data. Motivated by data from an HIV/AIDS study, we consider the middle ground and develop methodologies for estimation of a regression function in a semiparametric setting where the truncation distribution is parametrically specified while the failure time, censoring and covariate distribution are left completely unknown. We devise an estimator for the regression function based on a local pseudo‐likelihood approach that properly accounts for the bias induced on the response variable and covariate(s) by the sampling design. One important spin‐off from these results is that they yield the adjustment for length‐biased sampling and right‐censoring; the so‐called stationary case. We study the small and large sample behaviour of our estimators. The proposed method is then applied to analyze a set of HIV/AIDS data.

Performance Assessment of Deteriorating Bridge Channels in the Presence of Hurricane Failures: The Competing Risks Scenario

Natural disasters such as hurricanes, earthquakes, and storm surges among other hazards affect communities and large geographic areas of the United States, resulting in negative repercussions on civil infrastructure. The impacts of these hazards on bridges affect the operability and functional integrity of transportation networks. This study focuses on two failure modes, namely the deterioration of bridge channels in the presence of competing hurricane hazards (hurricane-induced failure of channels) and the complex interactions between the two modes of failure, to evaluate the performance of bridge channels. To achieve the aforementioned objective, competing risks models are developed for the already-mentioned failure modes using the product limit survival analysis, cause-specific hazard rates, and the cumulative incidence functions (CIF). Failure time distributions are developed for standard commonly recognized bridge channel elements using legacy inspection data for bridges, annual probabilities of occurrence of hurricanes, and damage matrices from existing hurricane records. The method of maximum likelihood estimation (MLE), weighting techniques, and inference procedures are employed to describe competing risks event data for the analysis. From the results, it is observed that bridge channels deteriorate faster in the urban areas compared with rural areas. The deterioration and hurricane failures generally yield significantly different impacts based on the log-rank chi-square outputs. Also, it was observed that the impact of hurricane Categories 3 and 2 on bridge channels were more significant (based on the hazard ratios) at the coast than noncoastal areas and also generally high for urban bridge channels compared with rural channels.

Parametric models for combined failure time data from an incident cohort study and a prevalent cohort study with follow-up.

A classical problem in survival analysis is to estimate the failure time distribution from right-censored observations obtained from an incident cohort study. Frequently, however, failure time data comprise two independent samples, one from an incident cohort study and the other from a prevalent cohort study with follow-up, which is known to produce length-biased observed failure times. There are drawbacks to each of these two types of study when viewed separately. We address two main questions here: (i) Can our statistical inference be enhanced by combining data from an incident cohort study with data from a prevalent cohort study with follow-up? (ii) What statistical methods are appropriate for these combined data? The theory we develop to address these questions is based on a parametrically defined failure time distribution and is supported by simulations. We apply our methods to estimate the duration of hospital stays.

Statistical Inference for Burr Type X Distribution using Geometric Process in Accelerated Life Testing Design for Time censored data

In accelerated life testing researcher generally use a life stress relationship between life characteristic and stress to estimate the parameters of failure time distributions at use condition which is just a re-parameterization of original parameters but from statistical point of view it is easy and reasonable to deal with original parameters of the distribution directly instead of developing inference for the parameters of the life stress relationship. So, an attempt is made here to estimate the parameters of Burr Type X life distribution directly in accelerated life testing by assuming that the lifetimes at increasing stress levels forms a geometric process. A mathematical model for the analysis of constant stress accelerated life testing for type-I censored data is developed and the estimates of parameters are obtained by using the maximum likelihood method. Also a Fisher information matrix is constructed in order to get the asymptotic variance and interval estimates of the parameters. Lastly, a simulation study is performed to illustrate the statistical properties of the parameters and the confidence intervals.