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This study addresses the problem of performance reliability degradation and lifetime prediction in complex systems with high reliability but insufficient information. It comprehensively considers the effects of various uncertainties, such as measurement, cognitive, and stochastic uncertainties, and proposes a generalized normal grey cloud Bayesian Wiener process (GNGCBWP) stochastic degradation model. First, based on the historical degradation information of the system, the interval grey number sequence of the degradation increment is constructed, and the performance degradation path of the system is analyzed to determine the degradation model type. Then, combining the normal grey cloud and Bayesian estimation theory, the relevant definitions and theorems of generalized normal grey cloud Bayesian estimation (GNGCBE) are proposed. Finally, the parameter estimation results obtained using the GNGCBE method are substituted into the grey Wiener process degradation model to dynamically analyze performance reliability degradation and predict the remaining useful life (RUL) in real‐time. And the validity and scientificity of the proposed model is verified through a case study of laser equipment. Moreover, the comparison results of the models indicate that the proposed model has advantages in dynamic analysis of performance reliability degradation and real‐time RUL prediction for highly reliable complex systems with limited or poor degradation information. It can significantly reduce the impact of multiple uncertainties on the analysis results and improve the accuracy and reliability of the final prediction results. [ABSTRACT FROM AUTHOR]

The evidential variable shows the powerful ability on the parameter description, so we have cooperated it with the linear Wiener process to model linear degradation processes. However, the degradation paths of some products exhibit two‐phase patterns over the running time, which cannot be modeled by the linear Wiener process. Hence, it is developed into an associated evidential variable‐based two‐phase Wiener process. In order to utilize the proposed model, the related evidence obtaining, evidence fusion and performance inference approaches are constructed. A simulation study and an actual case are performed to verify the superiorities of the proposed method. The proposed method provides higher accuracies of the useful life, degradation, and reliability evaluations, due to utilizing the evidential variable and considering the two‐phase degradation conditions. [ABSTRACT FROM AUTHOR]

This paper presents two Bayesian hierarchical models—one utilizing the life-time data and other using the structural health monitoring (SHM) data, for degradation modeling and reliability assessment of rolling element bearings. The main advantage of the proposed life-time data based model is that, it accounts for the variability in failure times caused due to the difference in material properties, initial degradation, operating and environmental conditions by introducing Bayesian hierarchy in the model parameters. On the other hand, SHM data (such as vibration and strain) based model focuses on stochastic nature of bearing degradation, and models it using a two-phase Wiener process. In this model, the point of phase-transition is the time when the damage initiates. The detection of such a point is undertaken using Bayesian change point algorithms. For both the models, the model parameters and reliability are updated as more data becomes available. In this manner, the prior domain knowledge and life-time data or SHM data collected from the field can effectively be integrated to get updated reliability. Two case studies for rolling element bearings are presented to demonstrate the applicability to life-time as well as SHM data. [ABSTRACT FROM AUTHOR]

• A novel deterioration-integrated failure model with varying environments. • A matrix-based approximation method for health measures calculation. • An adaptive reliability-based maintenance policy with a switchable control chart. • An efficient computational algorithm to find the optimal decision variables. • Considerable better maintenance performance than published results. The performance of mechanical systems is greatly influenced by varying operational environments such as load, temperature, and humidity, which can result in accelerated degradation and failure. To address this issue, this article considers an adaptive maintenance policy based on updated conditional reliability for mechanical systems under variable environments. A model of deterioration-integrated failure process is presented, which integrates degradation process, failure events, and environmental variables of the system. The degradation process is modeled by a random effect Wiener process, with its drift parameter following a general distribution to represent the environmental random effects. The failure events are modeled by a proportional hazards (PH) model with environment-varying covariate. The health of the system is evaluated using a matrix-based approximation method that provides estimates of the system's conditional reliability and mean residual life. Using the prognostic information, an adaptive maintenance model with a switchable control chart is then developed to monitor and maintain system based on the status of system reliability. The decision variables of the adaptive reliability-based maintenance are determined by a computational algorithm formulated in a semi-Markov decision process (SMDP) framework, with the objective of minimizing the long-run expected average cost per unit time. Vibrational data of bearings under varying operational environments are presented to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

• A two-stage degradation system with two performance characterizes is considered. • A correlated bivariate Wiener process is adopted in modeling. • The explicit expressions of system reliabilities and availabilities are derived. • Two algorithms for the simulations of system reliabilities are given. It is still a challenge to study the degradation mechanisms of complex systems with multiple performance characteristics. This paper develops a two-stage stochastic degradation model. The degeneration processes of two correlated performance characteristics are described by a correlated bivariate Wiener process in the first stage, in which neither of the degradation levels of performance characteristics reaches their thresholds. When one of the degradation levels of performance characteristics reaches its threshold, the system operates defectively in the second stage, and the system degradation process is described by the other performance characteristic which is modeled by a univariate Wiener process. The system fails completely when both the degradation levels of performance characteristics reach their thresholds. Explicit expressions of the system reliability and availabilities are given based on different inspection and maintenance policies. Simulation is also presented, and it shows that the analytical results and the simulation results are in good agreement. Finally, a detailed case study of the rail track geometry degradation is given to demonstrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper proposes a condition-based maintenance (CBM) policy for a deteriorating system whose state is monitored by a degraded sensor. In the literature of CBM, it is commonly assumed that inspection of system state is perfect or subject to measurement error. The health condition of the sensor, which is dedicated to inspect the system state, is completely ignored during system operation. However, due to the varying operation environment and aging effect, the sensor itself will suffer a degradation process and its performance deteriorates with time. In the presence of sensor degradation, the Kalman filter is employed in this paper to progressively estimate the system and the sensor state. Since the estimation of system state is subject to uncertainty, maintenance solely based on the estimated state will lead to a suboptimal solution. Instead, predictive reliability is used as a criterion for maintenance decision-making, which is able to incorporate the effect of estimation uncertainty. Preventive replacement is implemented when the estimated system reliability at inspection hits a specific threshold, which is obtained by minimizing the long-run maintenance cost rate. An example of wastewater treatment plant is used to illustrate the effectiveness of the proposed maintenance policy. It can be concluded through our research that: 1) disregarding the sensor degradation while it exists will significantly increase the maintenance cost and 2) the negative impact of sensor degradation can be diminished via proper inspection and filtering methods. Note to Practitioners—This paper was motivated by the observation of sensor degradation in wastewater treatment plants but the developed approach also applies to other systems such as manufacturing systems, chemical plants, and pharmaceutical factories, where sensors are dedicated to a long-time operation in a harsh environment. This paper investigates the impact of sensor degradation on CBM and suggests that the effect of sensor degradation should be carefully addressed while making maintenance decisions. Otherwise, it will lead to a suboptimal maintenance decision and increase the operating cost. An optimal maintenance decision, which contains the optimal inspection interval and reliability threshold, is achieved via minimizing the long-run cost rate. In the presence of measurement noise and intrinsic uncertainty from degradation, a stochastic filtering approach is employed to estimate the system and sensor state. Based on the estimated states and the calculated reliability, a dynamic maintenance decision is obtained at each inspection. This paper can be further extended considering non-Gaussian noise and alternative degradation processes. [ABSTRACT FROM AUTHOR]

For reliability assessment based on accelerated degradation tests (ADTs), an appropriate parameter estimation method is very important because it affects the extrapolation and prediction accuracy. The well‐adopted maximum likelihood estimation (MLE) method focuses on interpolation fitting and obtains results via maximizing the likelihood of the observations. However, a best interpolation fitting does not necessarily yield a best extrapolation. In this paper, therefore, a pseudo‐MLE (P‐MLE) method is proposed to improve the prediction accuracy of constant‐stress ADTs by considering the degradation mechanism equivalence under Wiener process. In particular, the degradation mechanism equivalence is characterized by a mechanism equivalence factor which presents the proportional relationship between degradation rate and variation. Then, the mechanism equivalence factor is determined via a two‐step method. The other model parameters can be estimated by the general MLE method. The asymptotic variances of acceleration factors and the p‐quantile of product failure time under normal condition are adopted to compare the statistical properties of the proposed method and the general MLE approach. Numerical examples show that the novel P‐MLE method may not achieve a maximum likelihood but can provide more benefits regarding prediction accuracy enhancement especially when the sample size is limited. [ABSTRACT FROM AUTHOR]

Highlights • Three types of thresholds including alarm line and two different failure thresholds are used to model the system reliability. • Two degradation process models, Wiener-Gamma and Gamma-Gamma, are developed. • Explicit formulas for the system reliability in terms of duration thresholds for the degradation models are presented. Abstract Stochastic degradation process models are developed in terms of cumulative degradation signals of systems under three types of thresholds including alarm line and two different failure thresholds. One of the failure thresholds corresponds to the degradation amount, and the other corresponds to the duration. We assume that the degradation process can be separated into two distinctive stages by a change-point which is the first passage time of the degradation process with respect to the alarm line. For simplicity, the first stage is regarded as an age-dependent (or non-stationary) Wiener process or an age-dependent (or non-stationary) Gamma process, and the second stage is regarded as an age-dependent (or non-stationary) Gamma process; thus, two special degradation models, Wiener-Gamma and Gamma-Gamma, are constructed. Specially, two-stage stationary Wiener-Gamma and Gamma-Gamma models can be viewed as special cases. The system reliability is defined as the probability that the degradation signals do not exceed a failure threshold and the duration of exceeding the alarm line is less than the duration threshold. Some reliability results including two cases in terms of duration thresholds (constants and random variables) are derived. The moments of lifetime are also given based on the results. In addition, simulation is carried out to verify the given results. And some numerical examples are presented to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

Degradation studies are often used to assess reliability of products subject to degradation-induced soft failures. Because of limited test resources, several test subjects may have to share a test rig and have their degradation measured by the same operator. The common environments experienced by subjects in the same group introduce significant interindividual correlations in their degradation, which is known as the block effect. In the present article, the Wiener process is used to model product degradation, and the group-specific random environments are captured using a stochastic time scale. Both semiparametric and parametric estimation procedures are developed for the model. Maximum likelihood estimations of the model parameters for both the semiparametric and parametric models are obtained with the help of the EM algorithm. Performance of the maximum likelihood estimators is validated through large sample asymptotics and small sample simulations. The proposed models are illustrated by an application to lumen maintenance data of blue light-emitting diodes. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This paper presents a reliability modeling and analysis framework for load-sharing systems with identical components subject to continuous degradation. It is assumed that the components in the system suffer from degradation through an additive impact under increased workload caused by consecutive failures. A log-linear link function is used to describe the relationship between the degradation rate and load stress levels. By assuming that the component degradation is well modeled by a step-wise drifted Wiener process, we construct maximum likelihood estimates (MLEs) for unknown parameters and related reliability characteristics by combining analytical and numerical methods. Approximate initial guesses are proposed to lessen the computational burden in numerical estimation. The estimated distribution of MLE is given in the form of multivariate normal distribution with the aid of Fisher information. Alternative confidence intervals are provided by bootstrapping methods. A simulation study with various sample sizes and inspection intervals is presented to analyze the estimation accuracy. Finally, the proposed approach is illustrated by track degradation data from an application example. [ABSTRACT FROM AUTHOR]

Lifetime prognostic based on the degradation data has been widely investigated and adopted for reliability assessment and maintenance policy. However, the measurement error (ME) is usually inevitable, which leads to the bias of lifetime estimation and erroneous evaluation of the safety risk. In this paper, we mainly focus on an inverse issue: how to specify the sensor's performance (i.e., the ME range) for satisfying a given requirement of the lifetime estimation. Under this consideration, we first analyze the probability distribution functions of the lifetime estimation with/without the ME based on Wiener process degradation model. Then a distance measure based on the relative entropy is formulated to evaluate the difference between these two lifetime estimations. Furthermore, the permissible ranges of the time-dependent and time-independent ME are attained under a given allowable bias of lifetime estimation according to the proposed distance measure. In addition, the influence of the ME on maintenance policy is discussed. Finally, numerical examples and a case study are provided to illustrate. [ABSTRACT FROM AUTHOR]

Degradation test is an effective method in assessing product reliability when measurements of degradation leading to failure can be observed. The accuracy of reliability inference in degradation analysis highly depends on the fitted model to the observed degradation data. Sometimes, observed degradation paths of the products exhibit multiphase pattern over the testing period. In this paper, we propose a change-point Wiener process with measurement errors (CPWPME) to fit two-phase degradation paths of organic light-emitting diodes (OLEDs). We assume the unit-specific parameters of the CPWPME model by using hierarchical Bayesian method. Based upon the proposed approach, the failure-time distribution and the remaining useful life distribution along with mean time to failure and mean residual life function are derived in closed form. A simulation study shows the utility of the proposed CPWPME model and the validity of the hierarchical Bayesian approach for the degradation data possessing two-phase degradation characteristics. In the analysis of OLED degradation data, the hierarchical Bayesian CPWPME model provides higher modeling flexibility and prediction power for future testing units than existing three degradation models. [ABSTRACT FROM AUTHOR]

Wiener degradation processes with fixed failure threshold have recently received extensive applications due to their desirable mathematical properties. However, a system may experience various environments during the overall degradation life cycle. This may lead to abrupt changes of both the failure thresholds and drift parameters due to the fluctuation of endurance of the deterioration system. Motivated by the practical needs of such multi-phase degradation systems, in this paper, a multi-phase degradation model based on the Wiener process is developed to describe the phenomenon of abrupt changes of failure thresholds and drift parameters. Both the two-phase and multi-phase ( n  ≥ 3) cases are considered under changing failure thresholds and drift parameters in this paper. Analytical expression of system reliability is given for the Wiener process model with two successive different failure thresholds. Meanwhile, a recursive formula is developed to get the analytical expression of system reliability for the Wiener process model with multiple successive different failure thresholds ( n  ≥ 3). In addition, simulation procedures are carried out to verify the given results. And some numerical examples are presented to illustrate the proposed model. [ABSTRACT FROM AUTHOR]

Because of the existence of uncertainty measurement in the process of bearings degradation, it is difficult to carry out the reliability analysis. The random performance degradation model is used to analyze the reliability life of high-speed train bearing with the characteristics of slow degradation process and relatively stable degradation path. The unknown coefficients are viewed as random variables in the model. According to the analysis to a bearing testing data, the reliability analysis of the bearing life is finally completed. The results show that the method can assess reliability of the bearing life with zero failure by taking full advantage of the performance degradation data in the small sample size. [ABSTRACT FROM AUTHOR]

Observations on degradation performance are often used to analyze the underlying degradation process of highly reliable products. From the two-phase degradation path of the bearing performance observations, we observed that there exists an abrupt increase in degradation measurement at a change point. Then, the following degradation process started with the abrupt degradation measurement will degrade in a higher degradation rate. Here, a stochastic process-based degradation model is constructed to interpret the jump at the change point in the degradation process which is governed by the linear Wiener process. Meanwhile, the distribution of the first passage time over a prespecified threshold for the process is discussed. In addition, to get the estimates of the model parameter, the expectation-maximization algorithm is utilized since the change points are unobservable. Furthermore, to demonstrate the model's advantages over estimate, a comparison is made between the proposed and the existing known models from the literature. The results reveal that considering the jump in the degradation process can improve the accuracy of estimations in real applications. [ABSTRACT FROM PUBLISHER]

Parallel constant-stress accelerated degradation testing (PCSADT) is widely used to assess the reliability of highly reliable products in a timely manner when the products’ degradation can be measured. Under a time-censored PCSADT, several groups of units are tested simultaneously, but under different stress levels, until a prespecified censoring time is reached. At this time, degradation values from the censored units, and failure times of the failed units are obtained. When the degradation follows a Wiener process where the parameters depend on the stress level through a life-stress model containing an unknown nuisance parameter, estimating this parameter often biases the maximum likelihood and least-squares estimators of the lifetime parameters. In this paper, we propose a two-stage procedure to address this problem. In the first stage, we transform the data under the different stress levels of a PCSADT so that the resulting data can be considered to have been obtained under normal stress. In the second stage, we introduce a latent variable for the unobserved degradation after the failure time for each failed unit to obtain a pseudodegradation value at the censoring time. We then use all degradation values (pseudo or observed) at the censoring time to develop latent variable estimators for all model parameters. Unlike other existing estimators, the proposed estimators are shown to be s-consistent, have closed-form expressions, and are easy to interpret. We use a real example of light-emitting diodes to illustrate the proposed method. In addition to proving s-consistencies, we conduct a simulation study to demonstrate that the proposed estimators also perform well in finite samples. [ABSTRACT FROM PUBLISHER]

The use of degradation model to perform the reliability analysis has drawn much attention due to the fact that the performance of numerous highly reliable systems degrades over time. To describe the unit-to-unit variability for a population of systems, the random effect has been incorporated into the degradation model that plays an important part in assessing the reliability of deteriorating systems. In the existing literature, the normal distribution is commonly adopted to represent the random effect, but the assumption can be unsuitable for some practical applications, such as the degradation process of train wheels. In this paper, we present a degradation modeling and reliability estimation approach by using truncated normal distribution to characterize the unit-to-unit variability. A Wiener process with truncated normal distribution is firstly applied to model the degradation process of the deteriorating system, and the analytical expressions of probability density function and reliability function are derived. Expectation maximization algorithm is then used to estimate the model parameters. The effectiveness and feasibility of the presented approach are illustrated through a numerical example and practical case studies for laser devices and train wheels. The results indicate that the presented approach can obtain better reliability estimation results by considering the truncated normal distribution when the unit-to-unit variability has significant difference. [ABSTRACT FROM AUTHOR]

Complex characteristics such as non-linearity and multi-stage are usually presented during the degradation process of aluminum electrolytic capacitors (AECs). Therefore, it is difficult to accurately predict the capacitors' remaining useful life (RUL), which is significant to the system's reliability. Based on the model of Wiener process with linear drift, a simple two-stage RUL predicting method for AECs is proposed in this paper. In the offline stage, the Box–Cox transformation (BCT) is first utilized to linearize the degraded data. Then the model parameters of BCT are identified, and the Shapiro–Wilk test is employed to verify whether the transformed data meet the linearity. In the online stage, based on similarity measurement, the drift coefficient of the Wiener process is updated in real-time. By combining BCT and similarity measurement, the nonlinear and multi-stage problems are effectively solved. Finally, the proposed method is verified on NASA's accelerated degradation data set. Experimental results show that the proposed method can automatically learn the capacitor's complex degradation characteristics and accurately predict its RUL. • The BCT (Box–Cox Transformation) is applied to the degradation of AECs (aluminum electrolytic capacitor), which makes it possible to model the degradation process by Wiener process with linear drift. • Based on similarity measurement, an online updating method of DC is proposed, which can automatically learn the multi-stage degradation characteristics of failure samples. • Based on BCT and similarity measurement, a two-stage RUL prediction method for AECs is proposed. [ABSTRACT FROM AUTHOR]

We consider degrading engineering systems monitored by degrading sensors. Since accurate information is crucial for predicting system health condition and the subsequent decision-making, considering the effect of sensor degradation is highly important to determine the justified reliability characteristics of systems such as the remaining useful life (RUL). Although the concept of sensor degradation has been introduced previously in the literature, the remaining useful life estimation in this case or parameter estimation in the presence of sensor degradation has not been studied in detail. To fill the gap, this study aims to estimate the RUL of a system that is continuously monitored by a degrading sensor. In this work, to distinguish sensor degradation from that of the main system, an additional calibration sensor is used to accurately inspect the system health condition at certain points of time. Subsequently, maximum-a-posteriori estimation technique is employed to estimate the parameters for the system degradation process and maximum likelihood estimation is used to estimate the parameters of sensor degradation. A Kalman filter is then used to estimate the system and sensor states, followed by system RUL evaluation. A numerical example with simulated data is employed to illustrate the effectiveness of the proposed method. It is shown through the numerical study that neglecting sensor degradation can result in significant errors in RUL estimation, which can further impact the subsequent maintenance decisions. • We evaluate the remaining life of degrading systems monitored by degrading sensors. • We estimate the parameters of the degradation process for the sensor and system. • We estimate the system states with Kalman filter. • We highlight the importance of sensor degradation in system reliability evaluation. [ABSTRACT FROM AUTHOR]

• Multivariate Wiener processes combining time-scale transformed was proposed. • The correlation of four copula functions and the reliability analysis was studied. • The RUL prediction model combining copula function was proposed. • The degradation failure of the aircraft lock mechanism was studied in detail. The failure mechanism and degradation analysis of an aircraft cabin door lock mechanism are studied. A data-driven degradation analysis is conducted based on the characteristics of two types of degradation data for the lock mechanism. The degradation process of the fluctuating elevation is described by Wiener process, and the lifetime is described using the Inverse Gauss (IG) process. The correlation between the two degradation processes is studied, and the correlation analysis is carried out with four copula functions. Moreover, the optimal model is obtained for reliability analysis using three model selection methods and Bayesian inference based on the Markov chain Monte Carlo (MCMC) method for parameter estimation. Finally, a residual life prediction model is proposed, and the accuracy of the model is tested with the error. The results show that the RUL prediction accuracy can be accepted, which helps to improve the reliability and service life of a type of aircraft lock mechanism. [ABSTRACT FROM AUTHOR]

• Importance measures for components with continuous time degradation are proposed. • The component reliability model is constructed by Wiener process. • The system reliability model is developed within Copula Hierarchical Bayesian Network considering the characteristics of dependency, uncertainty, nonlinearity and randomness. • Relationships among different importance measures are discussed. • A case study on the harmonic gear drive is conducted to demonstrate the use of proposed importance measures. In order to identify the vulnerable components and ensure the required reliability of mechatronics systems, importance measures of critical components are crucially used in the early design of systems. However, complex mechatronics systems have the properties of hierarchy, nonlinearity, dependency, uncertainty, and randomness, which make it difficult to analyze the coupling failure mechanisms, model the system, estimate its reliability, and complete importance measures of its components. This paper proposes importance measures for components with continuous time degradation. The Wiener process model is used to describe the continuous-time degradation process, and the Copula Hierarchical Bayesian Network (CHBN) is developed for system reliability estimation. Six importance measures are proposed for continuous-time degrading components. These importance measures provide a time-dependent analysis of the criticality of components, thus adding insights on the contributions of the components on the system reliability or performance over time. A case study on the harmonic gear drive is then conducted to demonstrate the use of the proposed importance measures. The results of the study show that the CHBN-based importance measures can be a valuable decision-support tool for designers in the early design of systems. [ABSTRACT FROM AUTHOR]

For newly developed, highly reliable, and long-lifespan products, it is quite difficult to implement effective remaining useful life (RUL) prediction in the early usage under limited time cost. However, accelerated degradation testing (ADT) is generally used for lifetime evaluation for such products with harsher test conditions and shorter test time in the late research and development phase. Thus, in this paper, we propose a life prediction framework to integrate the information from ADT to conduct field RUL prediction for highly reliable products. Because ADT belongs to reliability testing used for inferring the population information from the selected test samples, we at first present the modified Wiener process (MWP) model. Different from traditional methods that embody both the random variability and unit-to-unit variability into the diffusion coefficient, the proposed method describes them separately in ADT analysis. Then, the MWP model from ADT is used as a prior for field RUL prediction of the target product during which the strong tracking filtering algorithm is introduced for updating the hidden state and computing the RUL prediction results when the new monitoring data are available. Because of the complexity of the MWP model, the Markov chain Monte Carlo method is provided to estimate the unknown parameters. Finally, the simulation study and the light-emitting diode application verify the effectiveness of the proposed framework that can achieve reasonable life prediction results for highly reliable products for both linear and nonlinear scenarios. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Accelerated degradation analysis plays an important role in assessing reliability and making maintenance schedule for highly reliable products with long lifetime. In practical engineering, degradation data, especially measured under accelerated condition, are often compounded and contaminated by measurement errors, which makes the analysis more challenging. Therefore, a Wiener process model simultaneously incorporating temporal variability, individual variation and measurement errors is proposed to analyze the accelerated degradation test (ADT). The explicit forms of the probability distribution function (PDF) and the cumulative distribution function (CDF) are derived based on the concept of first hitting time (FHT). Then, combining with the acceleration models, the maximum likelihood estimations (MLE) of the model parameters are obtained. Finally, a comprehensive simulation study involving two examples and a practical application are given to demonstrate the necessity and efficiency of the proposed model. [ABSTRACT FROM AUTHOR]

Random effects are ubiquitous in the degradation processes of multiple performance characteristics (PCs) of a complex system, and these PCs are not commonly independent. Neglecting either of the above features will lead to bias in system reliability estimation. Therefore, this study develops a reliability estimation method for the complex systems with multiple PC degradations considering both of the features. First, a Wiener process-based marginal degradation model with random effects (including the random initial PC, unit-to-unit variability, temporal uncertainty, and measurement errors) is proposed to track the degradation path of each PC. A model based on D-vine copulas is put forward to describe the correlation among multiple PCs. Then, a system reliability function is derived based on the two models. Furthermore, a two-step statistical inference is developed to estimate the unknown parameters in the system reliability function. Finally, using a practical case, the effectiveness of the proposed method is illustrated. • A reliability estimation method is proposed for complex systems. • A marginal Wiener process-based degradation model with random effects is proposed. • The correlation among multiple performance characteristics is modeled by D-vine copulas. • A two-step statistical inference is developed to perform the parameter estimation. [ABSTRACT FROM AUTHOR]

• A hybrid WKN-BiGRU and Wiener process model is proposed for reliability evaluation of rolling bearing. • A hybrid WKN-BiGRU model is utilized to deep feature extraction and health index construction. • The Wiener process has been introduced to model the health index for reliability evaluation. Reliability evaluation is highly significant for the safe and reliable service of rolling bearings. It is to accurately reflect degradation states of rolling bearings. However, traditional methods have difficulties in solving the problems resulted from the lack of measured data, while the deep learning techniques are insufficient in dealing with uncertainties. This paper proposes a new reliability evaluation schedule based on the WaveletKernelNet (WKN), bidirectional gated recurrent unit (BiGRU), and Wiener process model. The proposed method consists of two parts: a health index construction model by the WKN-BiGRU and a Wiener process-based reliability evaluation method. The WKN-BiGRU network is to extract deep features and construct the health index of the rolling bearings. The Wiener process is to achieve the reliability evaluation of rolling bearings and to quantify uncertainties. The effectiveness of the proposed methodology is confirmed by a real case study of rolling bearings. Overall, the proposed methodology contributes to effectively deep features extraction and reliability estimation of rolling bearings. [ABSTRACT FROM AUTHOR]

In this article, the population reliability modeling and individual residual life prediction are discussed. Firstly, three kinds of different Wiener process models are used to characterize the degradation data, and the unknown parameters are estimated by using the Markov Chain Monte Carlo (MCMC) method. Secondly, under those degradation models, the individual residual life prediction method is also obtained on the basis of current degradation quantity. Finally, a fatigue cracks data example is given to illustrate the usefulness and validity of the proposed model and method. Numerical results show that the random effect model is well fitted with the actual degradation data, and this model has smallest prediction error. Meanwhile the prediction accuracy is acceptable, and this prediction method provides a foundation for maintenance decision. [ABSTRACT FROM AUTHOR]

Repeated degradation testing data have been widely used to assess lifetime of highly-reliable products or components with scarce failures. In this paper, we consider the problem of planning repeated degradation testing for products exhibiting three-source variability in their degradation characteristics, including the temporal variability, unit-to-unit variability, and measurement variability. The primary objective of this paper is centered on deciding the amount of units and the measurement schedule to achieve required estimation precision for some important statistics of interest. In such a planning process, the testing budget is limited and used as a constraint of an optimization model. To do so, a kind of Wiener degradation process, which has a random drift coefficient and a constant volatility coefficient, is used to model the repeated degradation testing data, where the measurement errors are considered and described as additive zero-mean random variables. Under the presented modeling framework, the lifetime distribution is formulated under the concept of the first passage time of the stochastic degradation process. Then, the large-sample approximate standard errors of the maximum likelihood estimations for the mean failure time and the quantile of the degradation distribution are derived, respectively. Furthermore, the constrained optimization model, which incorporates the cost of each degradation measurement, is proposed to plan the degradation testing by minimizing the testing cost under the condition of a maximum acceptable approximate standard error. Finally, an example is provided to illustrate the procedure and advantages of the presented method. [ABSTRACT FROM AUTHOR]

Precisely predicting the remaining life for an individual plays an important role in condition-based maintenance, so Bayesian inference method, which can integrate useful data from several sources to improve the prediction accuracy, has became a research hot. Aiming at the situation that accelerated degradation tests have been widely applied to assess the reliability of products, a remaining life prediction method based on Bayesian inference by taking accelerated degradation data as prior information is proposed. A Wiener process with random drift, diffusion parameters is used to model degradation data, and conjugate prior distributions of random parameters are adopted. To solve the problem that it is hard to estimate the hyper parameters from accelerated degradation data using an Expectation Maximization algorithm, a data extrapolation method is developed. With acceleration factors, degradation data are extrapolated from accelerated stress levels to the normal use stress level. Acceleration factor constant hypothesis is used to deduce the expression of acceleration factor for a Wiener degradation model. Besides, simulation tests are designed to validate the proposed method. The method of constructing the confidence levels for the remaining life predictions is also provided. Finally, a case study is used to illustrate the application of our developed method. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Degradation is a common phenomenon for many products. Because of a variety of reasons, the degradation rates of units from the same population are often heterogeneous. In addition, when the degradation process is monitored using dedicated sensors, the measurements are often inaccurate because of various noisy factors. To account for the heterogeneous degradation rate and the non-negligible measurement errors, we model the degradation observations using a random-effects Wiener process with measurement errors. Under the model, direct estimation of current degradation and prediction of future degradation are difficult. We thus develop a filtering algorithm that recursively estimates the joint distribution of the degradation rate and the current degradation levels. Based on the estimates, the distribution of the remaining useful life can be timely predicted. Our method is both computational efficient and storage efficient. Its effectiveness is demonstrated through simulation and real data. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Head wear becomes inevitable as the physical clearance between the head and disk in current hard disk drives is close to 1 nm or lower. Accurate characterization of the wear process is essential in understanding the wear mechanism, evaluating the reliability of hard disk drives (HDDs), and predicting the future evolution of wear as well as guiding the design of HDDs of later vintages. Due to the variations in the material and geometry of magnetic head, the wear process is dynamic and stochastic. To cope with the dynamics, a Wiener process with a positive drift is used to model the wear process. Furthermore, to evaluate the reliability quickly, an accelerated degradation model using the wear measurements is developed, from which the parameters/failure times under light loads can be effectively predicted. The developed model is then applied to a real wear problem of magnetic head to demonstrate the effectiveness. [ABSTRACT FROM AUTHOR]

• A new PHM with environmental factors and degradation trend as covariates is built. • The complex integrand in reliability function is solved by the Taylor formula. • A real-time updated reliability prediction method is developed based on the PHM. • Numerical study and real case studies show the superiority of the proposed method. Degradation data and lifetime data have been broadly used for assessing product and system reliability. To utilize the two kinds of data simultaneously, the proportional hazard (PH) model with degradation data as a covariate is proposed for reliability analysis. However, most existing works focus on modeling the PH model with degradation state as a covariate, while the degradation trend is ignored, which makes the PH model unable to carry out reliability prediction directly. Confronted with that, a new PH model with the degradation trend and environmental factor as covariates is developed in this paper. The Wiener process is firstly applied to depict the degradation trend, then the degradation trend and temperature are used as covariates to establish the PH model, and a closed-form of the reliability is derived by the Taylor approximation. Based on the degradation data under actual ambient conditions, the real-time updated reliability prediction is provided to guide the health management of products and systems. The simulation study validates that the proposed model outperforms two existing models in terms of Mean Square Error (MSE). Finally, a real-world example of MOSFET is presented to demonstrate the implementation of the proposed method. [ABSTRACT FROM AUTHOR]

As an effective way to enhance the system reliability, the standby redundancy technique has been widely applied in many industrial systems. Finding methods to determine the maintenance policy and spare-part inventory management is an interesting practical issue. However, due to the imperfect storage and inner mechanism, the spare parts usually deteriorate over time, which not only degrades their performance but also may lead to storage failure. In addition, the immediate burst failure caused by the external shock should not be neglected in storage. These two aspects make the joint optimization of preventive maintenance and inventory policies for a standby system more challenging. Thus, in this study, we first propose a general iterative approach for the lifetime estimation of standby systems with a hybrid spare-part degradation process, which includes both stochastic degradation and immediate burst failure. We then take an example of the Wiener-process-based model to explain how to obtain the analytical results of the lifetime estimation. On this basis, we further establish a joint optimization model, in which the preventive block replacement interval and the inventory number are treated as two decision variables to minimize the expected cost per unit time. Finally, a numerical example and practical case are provided for illustration. • Both spare-part storage degradation and immediate burst storage failure are considered. • A general iterative approach for the lifetime estimation of the standby system is proposed. • A joint optimization model of the maintenance interval and the inventory number is established. [ABSTRACT FROM AUTHOR]

In this paper, a new Radiation Hardness Assurance methodology applicable to cumulative effects is proposed to assess the impact of radiation on device reliability. This approach, that relies on the stochastic interpolation of experimental data, is based on the notion of Wiener process. An example of application of this methodology is presented and a comparison with a more conventional Radiation Hardness Assurance is discussed. [ABSTRACT FROM PUBLISHER]

Degradation models are widely used to assess the lifetime information of highly reliable products possessing quality characteristics that both degrade over time and can be related to reliability. The performance of a degradation model largely depends on an appropriate model description of the product's degradation path. Conventionally, the random or mixed-effect model is one of the most well-known approaches presented in the literature in which the normal distribution is commonly adopted to represent unit-to-unit variability in the degradation model. However, this assumption may not appropriately signify accurate projections for practical applications. This paper is motivated by laser data wherein the normal distribution is relaxed with a skew-normal distribution that consequently provides greater flexibility as it can capture a broad range, non-normal, asymmetric behavior in unit-to-unit variability. Based on the proposed degradation model, we first derive analytical expressions for a product's lifetime distribution along with its corresponding mean-time-to-failure (MTTF). We then utilize the laser data to illustrate advantages gained by the proposed model. Finally, we address effects from the skewness parameter with regard to the accuracy of both a product's MTTF and its qth failure quantile; especially when the underlying skew-normal distribution is mis-specified as a normal distribution. The result demonstrates that effects from the skewness parameter on the tail probabilities of a product's lifetime distribution are not negligible when the random effect of the true degradation model follows a skew-normal distribution. [ABSTRACT FROM AUTHOR]

Abstract: Over the past few decades, condition-based maintenance (CBM) has attracted many researchers because of its effectiveness and practical significance. This paper deals with mission-oriented systems subject to gradual degradation modeled by a Wiener stochastic process within the context of CBM. For a mission-oriented system, the mission usually has constraints on availability/reliability, the opportunity for maintenance actions, and the monitoring type (continuous or discrete). Furthermore, in practice, a mission-oriented system may undertake some preventive maintenance (PM) and after such PM, the system may return to an intermediate state between an as-good-as new state and an as-bad-as old state, i.e., the PM is not perfect and only partially restores the system. However, very few CBM models integrated these mission constraints together with an imperfect nature of the PM into the course of optimizing the PM policy. This paper develops a model to optimize the PM policy in terms of the maintenance related cost jointly considering the mission constraints and the imperfect PM nature. A numerical example is presented to demonstrate the proposed model. The comparison with the simulated results and the sensitivity analysis show the usefulness of the optimization model for mission-oriented system maintenance presented in this paper. [Copyright &y& Elsevier]

Burn-in test is a manufacturing process applied to products to eliminate latent failures or weak components in the factory before the products reach customers. The traditional burn-in test over a short period of time to collect time-to-failure or go/no-go data is rather inefficient. This decision problem can be solved if there exists a suitable quality characteristic (QC) whose degradation over time can be related to the lifetime of the product. Recently, optimal burn-in policies have been discussed in the literature assuming that the underlying degradation path follows a Wiener process. However, the degradation model of many materials (especially in the case of fatigue data) may be more appropriately modeled by a gamma process that exhibits a monotone-increasing pattern. Here, motivated by laser data, we first -propose a mixed gamma process to describe the degradation path of the product. Next, we present a decision rule for classifying a unit as typical or weak. A cost model is used to determine the optimal termination time of a burn-in test, and a motivating example is then presented to illustrate the proposed procedure. Finally, a simulation study is carried out to examine the effect of wrongly treating a mixed gamma process as a mixed Wiener process, and the obtained results reveal that the effect on the probabilities of misclassification is not negligible. [ABSTRACT FROM AUTHOR]

Today, many products are designed to function for a long period of time before they fail. For such highly-reliable products, collecting accelerated degradation test (ADT) data can provide useful reliability information. However, it usually requires a moderate sample size to implement an ADT. Hence, ADT is not applicable for assessing the lifetime distribution of a newly developed or very expensive product which only has a few available test units on hand. Recently, a step-stress ADT (SSADT) has been suggested in the literature to overcome the above difficulty. However, in designing an efficient SSADT experiment, the issue about how to choose the optimal settings of variables was not discussed, such as sample size, measurement frequency, and termination time. In this study, we first use a stochastic diffusion process to model a typical SSADT problem. Next, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal settings of these variables are obtained by minimizing the asymptotic variance of the estimated 100pth percentile of the product's lifetime distribution. Finally, an example is used to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

The reliability data are very useful in maintenance operations since they are used to calculate the Mean Time To Failure. However, they are rarely used for the online calculation of the Remaining Useful Life (RUL). This is because the features measured in the reliability tests are not always measurable online. In this paper, the proposed solution is usage of physical knowledge of the components to build models linking these features to the variables measurable online. Then, the physical models are used to generate health indices whose evolution can be estimated and predicted online, and the reliability data used for initializing the trend models of the health indices. To guarantee the robustness of the remaining useful life estimation to changes in Condition Monitoring, this paper proposes to update online the drift parameter of Wiener process. The latter is used to model the trend of the health indices. The results obtained by the most used updating methods in the literature are analyzed and compared in order to highlight the influence of the model updating on prognosis performance. Experimental results, obtained by an application to estimate the RUL of MOS Field-Effect Transistor, show the effectiveness of the proposed method. • Degradation trend modeling based on physical knowledge and Reliability data • Remaining Useful Life prediction based on updating Wiener process coefficient using most used estimators • Fault prognosis of MOS Field-Effect Transistors • Prognosis performance analysis using universal metrics [ABSTRACT FROM AUTHOR]

• A fixed change point model with a jump based on a Wiener process is developed. • The fixed change point model is extended to a random change point model. • Two phase changing patterns for the random change point model are proposed. • Analytical results of the system reliability indexes are given for Model 1 and the HPP case of PCP 2 in Model 2. A two-phase degradation system with a change point is commonly seen in many real-world degradation systems due to influence of internal mechanisms and external environments, such as lithium-ion batteries, light-emitting diodes (LEDs), plasma display panels (PDPs), and vacuum fluorescent displays (VFDs). Random jumps exist at the change points of the degradation processes. Motivated by this phenomenon, this paper attempts to model such kinds of systems based on the Wiener process. Specifically, In Model 1, a constant change point model is first considered. The analytical results of the system reliability, lifetime distribution, and the remaining useful lifetime (RUL) are obtained. Then, based on Model 1, a random change point model, i.e., Model 2, is proposed to explain the change point phenomenon which is influenced by the accumulative effect of a shock process. Two phase-change patterns (PCPs) are considered in Model 2. For PCP I, the change point is triggered when the number of shocks reaches a predetermined threshold. For PCP II, the change point is triggered when the accumulative shock damage exceeds the corresponding threshold. The system reliability indexes are studied by using analytical and simulation methods. Finally, several numerical examples are given to illustrate the approaches and results. [ABSTRACT FROM AUTHOR]