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A warranty product often has an interval structure, resulting in different warranty periods for each component due to their distinct degradation paths. From the manufacturer's perspective, it is reasonable to establish appropriate warranty periods for each part to achieve cost savings. This paper introduces a dependent series configuration consisting of a main component and an auxiliary component. The failure of the main component leads to system failure, while the failure of the auxiliary component only causes a random damage to the main component. These two components follow different Wiener degradation paths, and then their first passage times (FPTs) follow disparate inverse Gaussian distributions. Assuming a pre-deterministic warranty period, W , for the main component, and η W , 0 < η < 1 , for the auxiliary component, the warranty period can be divided into two stages: [ 0 , η W ] and [ η W , W ]. In this context, the paper proposes warranty claim policies for replacement, objective-oriented imperfect repair (OOIR), and refund. The corresponding warranty costs sustained by manufacturer and consumer within these two stages are considered. Finally, a numerical example is presented to illustrate the warranty costs sustained by manufacturer and consumer under different scenarios, as well as the total warranty costs. • The degradation paths, warranty periods and warranty costs of the series system is analyzed. • The warranty costs sustained by manufacturer and consumer under different scenarios is analyzed. • The interdependence between various components is considered. [ABSTRACT FROM AUTHOR]

• Propose a fully numerical iterative process to optimize the maintenance threshold for systems with non-monotonic degradation behavior subject to competing failures. • Consider lead time before maintenance implementation in the numerical framework. • Assess the impact of shock detection threshold on the expected cost. Shocks have attracted considerable interest in reliability and maintenance engineering because of their impact on vulnerable systems. Most industrial systems suffer from both internal degradation caused by fatigue and wear-out, and external shocks that often occur randomly due to harsh weather conditions, overloading, etc. Developing maintenance optimization models without taking these stochastic shocks into account is often ineffective. This paper develops a model to optimize the maintenance alarm threshold for a single-component continuously monitored system which is exposed to both fatal and non-fatal shocks in the presence of lead time for hard time maintenance. The shocks occur randomly according to a homogeneous Poisson process during the whole degradation process and have a stochastic impact on the degradation level, while the system resistance to shocks decreases as the system approaches failure. We propose a new numerical maintenance optimization model to find the solution without Monte-Carlo simulation and the model is compared to the Wiener process. A numerical example and a real-time experimental case study on roller bearings are used to demonstrate the effectiveness of the model. The results show that the model is capable of improving maintenance decision-making in terms of failure probability and risk perspective. [ABSTRACT FROM AUTHOR]

• A novel hybrid prognostic method for remaining useful life prediction is presented. • A new nonlinear health indicator is generated for the training bearings. • The condition monitoring interval of health state is determined by the 3σ criterion. • The degradation and RUL forecasting is achieved based on the nonlinear Wiener process with random effects method. This paper proposes a novel hybrid method aiming at the fault prognosis of bearings. A nonlinear health indicator (HI) is first constructed using Complete Ensemble Empirical Mode Decomposition with Adaptive Noise and Kernel Principal Component Analysis to reflect the health state of a bearing accurately and convincingly. Subsequently, multi-domain features are extracted from vibration signals and the Dual-Channel Transformer Network with the Convolutional Block Attention Module is applied for constructing HIs of the rest bearings. Moreover, the 3σ criterion is employed to establish the condition monitoring interval of health state and detect the First Prediction Time, with which degradation modeling and probabilistic Remaining Useful Life (RUL) prediction are conducted with the assistance of nonlinear Wiener process with random effects. The superior performance of the proposed hybrid prognostic method confirms that the method contributes to the accurate RUL prediction and uncertainty quantification. [ABSTRACT FROM AUTHOR]

This paper is focused on lifetime analysis based on accelerated degradation testing (ADT), considering the influence of stress and measurement errors. The traditional approach of ADT does not consider the random effects due to the influence of stress and measurement errors, and this can lead to inaccurate lifetime predictions. For this reason, we propose an ADT model that accounts for random effects. The model is based on Wiener process, which quantitatively expresses the relationship between the parameters of the ADT model and stress, as well as takes the measurement errors into consideration. Then, the proposed model is extrapolated to normal stress for lifetime analysis and a closed-form expression is deduced. Finally, ADT of the electrical cables is considered as case study to validate the effectiveness of the proposed algorithm. • A novel ADT model based on the Wiener process with random effects is proposed. • Measurement uncertainties are incorporated into the ADT model to account for realistic testing conditions. • The influence of stress is quantified and integrated into the model to enhance its predictive capability. • Lifetime analysis is conducted under normal stress levels, leveraging the proposed model to assess the longevity of the system accurately. [ABSTRACT FROM AUTHOR]

• A novel W-DATA method for mechanical degradation performance description and reliability estimation. • ApEn and P-EMD are introduced for degradation feature extraction and quantification. • An ApEn -based Wiener degradation model with a pseudo failure threshold is employed to estimate the reliability. • W-DATA can be effectively used for lubricating oil pump reducer reliability estimation. The Wiener-process-based Degradation Approach with Two-phase Algorithm (W-DATA) is introduced as a multi-dimensional framework explicitly designed for modeling mechanical system reliability. This approach leverages an enhanced Empirical Mode Decomposition, known as P-EMD, for degradation feature extraction, and utilizes Approximate Entropy (ApEn) for quantifying feature complexities. W-DATA seamlessly integrates these components into a time-resolved degradation model. To enhance its robustness, the proposed reliability model synergizes with a Box-Cox transformed Wiener model, allowing for improved adaptability to non-Gaussian data landscapes. The paper outlines a strategy for establishing a reliable threshold for this data-driven degradation model based on non-Gaussian data, a critical aspect for reliability modeling using the Wiener process. This strategy addresses the challenge of Wiener process verification through normality and independent testing of data increments. The integrative methodology is recommended for application in machinery reliability analytics and has undergone rigorous validation through a 770 h real-world test on a marine lubricating oil pump reducer. The results affirm its efficacy and feasibility in accurately estimating mechanical system reliability. [ABSTRACT FROM AUTHOR]

Due to the changeable internal mechanisms or external working conditions, the degradation trend of products usually presents multiple phase characteristics. However, most existing multi-phase degradation methods treat each phase as linear and rely on artificial designation to directly specify the forms of drift models, which may result in an inaccurate description of degradation characteristics for complex equipment. To this end, we formulate a general nonlinear multi-phase degradation model with three-source variability based on the Wiener process. Meanwhile, a stage division method is developed to automatically determine the degradation phase number, change-point locations, and the forms of drift models. Then, we obtain the expressions of remaining useful life (RUL) by considering the uncertainty of change-point degradation observations. Especially, we derive the approximate analytical solution of RUL based on the linear model. Furthermore, to fully consider the unit-to-unit heterogeneity and utilize the degradation observations of the in-service unit and prior information simultaneously, we propose a parameter estimation method based on variational Bayesian approach, which adaptively updates all parameters as random variables. Finally, two numerical examples and three practical examples are provided to verify the effectiveness of the proposed method. • We first consider a general nonlinear multi-phase model with three-source variability. • Our stage division method has higher model fitting accuracy than existing methods. • Our parameter estimation method fully considers unit variability and it is adaptive. • Identifying linear or nonlinear models ensures both prediction efficiency and accuracy. [ABSTRACT FROM AUTHOR]

As one of the most important aspects in prognostics and system health management (PHM), spare parts play significant roles in enhancing reliability, shortening downtime, and reducing operating cost. However, the performance degradation of spare parts, though frequently encountered in industrial practice, has been rarely concerned during lifetime estimation and the according inventory optimization. Based on an amenable Wiener-process-based degradation model, this paper integrates the performance degradation of spare parts into lifetime estimation for stochastic multi-component degrading systems with congeneric key parts. Under two prevalent replacement policies, i.e., unit-by-unit replacement policy and batch-by-batch replacement policy, the lifetime distribution of the system considering the performance degradation of spare parts in storage, has been derived and the corresponding cost function for inventory management optimization has been constructed. Simulations and case study of an inertial navigation platform with three gyros are conducted to illustrate and validate the proposed method. The results show that the performance degradation of spare parts should be considered during inventory management, and the presented method can achieve an accurate lifetime estimation of the concerned system. • Lifetime PDF of multi-component systems with degrading spare parts is derived. • Influence of different replacement policies on the lifetime PDF is investigated. • The expected cost function for inventory management optimization is constructed. [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]

Highlights • Consider a reliability demonstration method using accelerated degradation tests. • Use nonlinear random-coefficients modeling framework. • Derive cost-effective reliability demonstration plan meeting certain time constraints. • Apply two practical examples to validate the proposed method. Abstract Reliability demonstration tests (RDTs) have been widely adopted to verify reliability requirements of manufacturing products. In practice, due to the limited resource and tight development schedule for new products, it is preferable to determine the decision variables including the termination time and the sample size for the RDT in advance. Existing degradation models often fail to capture the nonlinear degradation characteristics of testing items with complicated degradation mechanisms. This paper proposes a reliability demonstration method using an accelerated degradation test (ADT) in the context of a nonlinear random-coefficients model. First, we present the capabilities of the proposed ADT model to degradation data. Then, the cost-effective RDT plan is derived based on two types of decision risks and reliability requirements from both producers and customers, while meeting certain testing time constraints. The proposed method is illustrated using two practical examples. Finally, sensitivity analysis is provided to evaluate the robustness of the proposed RDT plan using ADT data. [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 modeling is critical for health condition monitoring and remaining useful life prediction (RUL). The prognostic accuracy highly depends on the capability of modeling the evolution of degradation signals. In many practical applications, however, the degradation signals show multiple phases, where the conventional degradation models are often inadequate. To better characterize the degradation signals of multiple-phase characteristics, we propose a multiple change-point Wiener process as a degradation model. To take into account the between-unit heterogeneity, a fully Bayesian approach is developed where all model parameters are assumed random. At the offline stage, an empirical two-stage process is proposed for model estimation, and a cross-validation approach is adopted for model selection. At the online stage, an exact recursive model updating algorithm is developed for online individual model estimation, and an effective Monte Carlo simulation approach is proposed for RUL prediction. The effectiveness of the proposed method is demonstrated through thorough simulation studies and real case study. [ABSTRACT FROM AUTHOR]

In order to improve the performance and prolong the life of the equipment, it is usually necessary to maintain the equipment in practice. Remaining useful life prediction with imperfect maintenance activities is of great importance to prognostics and health management. In this paper, a multi-phase Wiener process-based degradation model is constructed to characterize the degradation process subjected to imperfect maintenance activities. The beta distribution is introduced to describe the residual degradation coefficient caused by the imperfect maintenance activity. The hyper-parameters of residual degradation coefficient are estimated through the maximum likelihood estimation and Newton iteration method. To reflect the unit heterogeneity, the drift coefficient and diffusion coefficient of each phase are synchronously defined as random variables. Furthermore, the analytical forms of remaining useful life are obtained by the convolution operator. The approximate expression of probability density function is derived by the Monte Carlo simulation approach. In the end, a numerical study and a practical study of gyroscopes are provided to demonstrate the practicality and effectiveness of the proposed method. • A multi-phase Wiener propose-based degradation model is improved. • Drift and diffusion coefficients are synchronously reviewed as random variables. • The approximate expressions of PDF for RUL are derived. • The method is accurate and superior based on a numerical study and gyro datasets. • The model realizes accurate RUL prediction under imperfect maintenance activities. [ABSTRACT FROM AUTHOR]

Degradation information reflecting the product or system health state plays an important role in assessing reliability and making maintenance schedule. Since degradation inspections are usually compounded and contaminated by measurement errors in real applications, the conventional Wiener process with identically distributed independent Gaussian error is usually adopted. However, in many situations, autocorrelation may probably exist among the measurement errors at sequential test points because of cyclic changes or modeling errors, especially when the time intervals are relatively short. Motivated by this practical issue, a Wiener process degradation model with one-order autoregressive (AR(1)) measurement errors is proposed for degradation analysis. Explicit forms of the probability distribution function (PDF), the cumulative distribution function (CDF) and the corresponding mean time to failure (MTTF) are derived based on the concept of first hitting time (FHT). Furthermore, maximum likelihood estimations (MLE) of unknown parameters are derived. The effects of model mis-specification regarding the estimation of MTTF are also discussed. Finally, a comprehensive simulation study and two practical applications are given to demonstrate the necessity and efficiency of the proposed model. [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]

In degradation test of some products such as plasma display panels (PDPs) and organic light emitting diodes (OLEDs), observed degradation paths tend to exhibit two-phase patterns over testing period. In this paper, we propose a change-point Wiener process (CPWP) model to fit the degradation paths with two-phase pattern mainly in a Bayesian framework. Considering the distinct degradation behaviors between testing units, we assume that degradation rates and change-points vary from unit to unit. Then hierarchical Bayesian approach is employed to estimate the parameters in the CPWP model. For comparison purpose, we also develop the maximum likelihood (ML) method. The results from simulation study show that the hierarchical Bayesian approach provides more robust inference on the model parameters than ML method. The analysis of OLED degradation data presents that the CPWP model outperforms three other existing models in terms of reliability prediction. [ABSTRACT FROM AUTHOR]

• A Wiener process considering measurable and unobservable external impacts is proposed. • The time-varying measurable covariate is modeled by an OU process. • The impact from unobservable factors is modeled as an time-varying degradation rate. • We develop the model estimation procedure and RUL prediction approach for the model. Degradation of products in field depends heavily on the external operating environment. The external factors, such as temperature and working load, are often time-varying or stochastic, which bring more uncertainty in the degradation process in addition to the inherent randomness. According to the availability of these factors in degradation modeling, they can be classified into the measurable covariates and the unobservable ones. In this study, we incorporate the influence of both kinds of factors into the degradation modeling using the Wiener process. We model the measurable time-varying covariate by an Ornstein-Uhlenbeck process and link it to the degradation rate through an exponential covariate-effect function, and model the impact of the unobservable factors through a time-varying degradation rate using a Brownian motion. Thus, the impacts of both the measurable covariate and the unobservable factors are accounted for in the degradation modeling. We develop the maximum likelihood estimation for the model, and propose a simulation-based algorithm for remaining useful life prediction. A simulation study is implemented to validate the performance of the proposed model, and applications to the battery degradation dataset and the outdoor coating weathering dataset are used to justify the advantages of the proposed model over the existing methods. [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]

• The analytical expressions of the RUL based on two-phase model have been derived. • Unit-to-unit variability is further investigated. • An iterative method is further obtained for multi-phase nonlinear Wiener model. • Lithium battery and capacitor data are adopted to verify our method. Owing to state transformation, operation switching, and degradation mechanisms, the processes of some deteriorating products often exhibit multiphase features. Unlike the traditional single-phase degradation model, the multiphase degradation model must consider the change-point variability and the associated degradation state. Therefore, degradation modeling and remaining useful life (RUL) estimation of the multiphase nonlinear deteriorating system is more challenging. With this consideration, we formulated two multiphase nonlinear degradation models based on the widely used nonlinear Wiener process-based model. Then, by modeling the uncertainty of the degradation state at the change-point, we obtained the exact analytical solutions for the lifetime and RUL estimation based on the two-phase model under the concept of the first passage time. Furthermore, we extended the results of the two-phase model to a multiphase model, and then provided an iterative method for lifetime and RUL estimations. Finally, we presented numerical and practical examples to verify the feasibility of the proposed method. [Display omitted] [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]

• An approximation of FHT's distribution for nonlinear Wiener processes is proposed. • Comprehensive simulations demonstrate the effectiveness of the approximation. • Real application shows the usefulness of the proposed reliability analysis method. 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. [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]

• An optimal sampling plan is designed by maximizing the parameter estimation accuracy. • The average degradation rate θ is taken to simplify the acceptance test problem. • The existence of the acceptance sampling plan is deduced from the monotonicity of θ. • The decision-making criterion is obtained by solving the risk constraint equations. • Simulation study and real case study show the effectiveness of the proposed method. The acceptance sampling plan (ASP), designed with the degradation data following the Wiener process, is widely used to verify the reliability requirements of products. Previous studies mainly designed ASPs under the given sampling plan, which lacked the justification of the sampling plan; furthermore, they ignored the unit heterogeneity of products in the degradation modeling, which affects the description accuracy of risks. To solve the above problems, this paper proposes an optimal ASP design method considering the unit heterogeneity. Firstly, the determinant of the Fisher information matrix is used to characterize the parameters estimation accuracy, under the cost requirement, the optimal test time and sample size are determined by maximizing the determinant, and an optimal sampling plan is obtained. Then, with the likelihood ratio order and the monotonicity of the average failure time on the average degradation rate, the average degradation rate is taken as the acceptance index to effectively simplify the original acceptance test problem. On this basis, the decision-making criterion considering the existence of ASP is obtained by solving the risk constraint equations of both parties. Finally, the simulation and a real example are presented to demonstrate the implementation and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, lumen degradation is described by using a modified Brownian motion process for mid-power white-light LED packages, which were aged under step stress accelerated degradation test (SSADT). First, a SSADT model has been established based on the theory of equivalent accumulative damage. Then, a method was proposed to improve the accuracy of the parameter estimation by carefully modifying the estimator, which was proposed in the previous research. Experimental data show that parameters estimated by using SSADT model are very close to those estimated by using constant stress accelerated degradation test (CSADT) model, indicating the feasibility of the SSADT model. The experiment also indicates that SSADT can be used as an alternative to CSADT, as it enables comparable estimation accuracy, while using less testing time, a smaller sample size and less test capacity. [ABSTRACT FROM AUTHOR]

The degradation of mechanical components is non-stationary and their degradation level can only be detected through offline discrete inspection in most cases. Aiming at the effect of the non-stationary feature and the delay of state detection caused by discrete inspection on long-run operation cost, this paper focuses on proposing an optimal inspection-based maintenance policy for three-state mechanical components subject to competing failure modes. A double-Wiener-process degradation model is established to describe the two operation states, which includes two Wiener process models under the same law but with different parameters. In addition, a preventive maintenance policy including a degradation threshold, an age threshold and a degradation control limit are adopted. A five-scenario probability-based model is proposed to describe the state evolution during one inspection interval and then the analytical model of the impact of delay on detecting state-transition and degradation-level on preventive maintenance policy is proposed. An optimization model of an inspection- based maintenance is established. Finally, sensitivity analysis in terms of the decision variables for state-transition time distribution and parameters in wear-out state is carried out based on numerical examples and the efficiency and superiority of the proposed policy is demonstrated by comparing with the current two-state maintenance policy. [ABSTRACT FROM AUTHOR]

• A condition-based maintenance policy is presented. • Maintenance time and maintenance threshold are the decision variables. • Characteristic function is used to find the distribution of the maintenance cost. • Expected utility theory is used to optimize the policy using a utility function. • The financial risk tolerance of the decision maker affects the results. Reasonably accurate remaining-useful-life (RUL) predictions allow for the introduction of maintenance policies where resources, such as spare parts and personnel, are only acquired based on the predicted need. For some assets, such a policy will help reduce the cost of renewals but will also increase the probability of renewal cycles with long downtime and associated large losses. From a decision theoretical point of view decision makers are often risk-averse and therefore their financial risk tolerance should be considered. This paper presents a procedure based on expected utility theory for the optimization challenge. To calculate the expected utility the characteristic function is used to find the full probability mass function of the maintenance cost in a finite time interval. A numerical example and a case study, based on data from an offshore oil and gas platform, are presented to illustrate the proposed model. These examples show that using the long-run cost rate to optimize the presented maintenance policy may lead to decisions that are not in line with the preferences of a risk-averse decision maker. [ABSTRACT FROM AUTHOR]

• The improved FIDES reliability model is established for accelerated degradation test. • The model considers the degradation volume, stress, installation, and application. • The model can predict the service life under a single acceleration stress level. A bearing life prediction method based on the improved FIDES reliability model was proposed in this paper to solve the problem of difficulty in predicting the service life of products under the action of single acceleration stress level. Firstly, the degradation model of the bearing is established on the basis of the nonlinear Wiener process to fit and predict the degradation data of the bearing. Secondly, transient failure rate function is introduced in the FIDES reliability model by combining the Weibull distribution function. The prediction of bearing life is achieved by considering stress, application and installation factors. Finally, the proposed method is validated with actual cases of bearings. Results showed that the bearing life prediction method based on the improved FIDES reliability model can accurately predict the service life of bearings, and the superiority of the model is demonstrated by comparative analysis. [ABSTRACT FROM AUTHOR]

• A Wiener process-based degradation model is proposed to obtain remaining useful life prediction. • Model universality among different types of degradation data is considered. • An uncertain description of remaining useful life prediction is obtained. • Model parameters are updated online by transfer learning. • Accurate remaining useful life predictions are acquired. An unsuitable type of degradation trend function in the Wiener process-based degradation model will negatively influence its performance when calculating remaining useful life (RUL) predictions. To solve this problem, we propose a Wiener process-based degradation model that can adaptively learn the degradation trend in different degradation data, which avoids the selection of degradation trend function. First, based on the degradation trends extracted by empirical mode decomposition, a long short-term memory (LSTM) neural network is trained and used as the degradation trend function of a Wiener process-based degradation model. Then, transfer learning is used to update the parameters of the LSTM neural network online. Concurrently, the diffusion coefficient of the Wiener process-based degradation model is obtained via maximum likelihood estimation. Finally, using the concept of first hitting time, the analytical formulation to the probability density function of RUL can be derived in a closed form. Two numerical examples are presented to demonstrate the implementation and the achieved parameter estimation accuracy of the proposed model. In addition, a real battery dataset is used to demonstrate the superior performance of the proposed model against previous Wiener process-based degradation models in RUL prediction. [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]

• Multi-stage division method is developed using real-time clustering with sliding window. • The weights of three models are dynamically updated by means of AIC-based model selection method. • R2 and Pearson coefficient are introduced to explain the weight setting and model selection results. • The convergence and generalization of presented model are proved. • Proposed combined model has better prediction performance and interpretability. Due to mass uncertain issues affecting health status of aero-engines, their degradation process commonly exhibits multi-stage features. Also, key characteristics underlying degradation process cannot be precisely described by traditional remaining useful life (RUL) prediction methods. Thus, for multi-stage RUL prediction, we develop a novel real-time combined approach that effectively explains the weights of each base model. The degradation process is divided into multiple stages through real-time clustering. Then, a combined prediction model including Wiener process, LSTM network, and XGBoost is introduced, which can optimally select multiple models for prediction according to AIC. The convergence and generalization of proposed model are proved. Besides, we adopt R2 and Pearson correlation to analyze model selection, further explaining the weights of every base model. The effectiveness of proposed model is validated by comparing with state-of-art methods available. RMSE results of three-stage and four-stage decrease by 2.72% and 2.69% respectively, compared with results of other models. Especially, RMSE of the stage with lowest value reduces by about 4.27%. From 90% whole life analysis, RE results are reduced by 2.77% (three stages) and 2.75% (four stages). Experimental results show that proposed method enhances prediction accuracy and improves model interpretability. [ABSTRACT FROM AUTHOR]

Burn-in is an effective tool to improve product reliability and reduce field failure costs before a product is sold to customers. As many products are becoming highly reliable, traditional burn-in that tests a batch of a product until most weak units fail requires an unaffordable testing duration. If the product failure can be associated with an underlying degradation process and a weak unit degrades faster than a normal one, then degradation-based burn-in can be implemented. Due to such various factors as human errors and limited precision of the measurement device, measurement errors are often inevitable. Ignoring measurement errors in the degradation observations would lead to inferior burn-in decisions. This study uses the Wiener process to model the underlying degradation and considers Gaussian measurement errors in the observations. Two burn-in models with different cost structures are studied and the optimal cutoff level for each model is obtained analytically. The relation between the two models is discussed, leading to a new cost model. [ABSTRACT FROM AUTHOR]

Abstract: We consider a repairable system with continuous output that alternates between states of operation and repair. The output of the system in the operating state is represented by a continuous, decreasing function of time. We assume that during the repair state, the system can produce output that is modelled by an increasing stochastic process. The repair action gradually restores the output of the system to its initial level and it returns to the operating state. We obtain and analyse expressions for the generalized availability and related characteristics of systems with gradual repair and consider several meaningful examples. [Copyright &y& Elsevier]

Due to the powerful ability of artificial neural network in data fitting, it has been applied to describe the mean function in Wiener process for degradation modeling and estimating reliability. However, the previously published method neglects the measurement error and individual difference, which need to be considered in reliability estimation. In order to handle the above issues, the artificial neural network supported Wiener process is improved. Both the individual difference and measurement error are considered. The measurement error is described by a normal distribution with zero mean. The individual difference is described by introducing hyper parameters. In order to demonstrate the proposed method, a simulation study and a case study have been performed. It can be seen that, considering the measurement error and individual difference can improve the reliability and lifetime estimation accuracies, resulting in the proposed method is more powerful and more practical in engineering practice. [ABSTRACT FROM AUTHOR]

• A combined mechanism-data-driven degradation model is proposed to augment small samples for oil condition prediction. • A two-stage data augmentation method is proposed, including the initialization and update of parameters. • Two types of small samples are predicted with the time-varying probability density function. Due to insufficient monitoring data, the reliability and accuracy of oil condition predictions are not guaranteed. Data-driven models provide data augmentation with small samples to solve this problem. However, the absence of degradation mechanisms would introduce unpredictable uncertainties in a long-term prediction. To address this, a data augmentation method is proposed for improved prediction by integrating degradation mechanisms and monitoring data. Primarily, a degradation model is established considering the degradation mechanisms. The model parameters are estimated with the time-vary probability distribution of the monitoring data. Therefore, the evidential variables are used to describe parameters with small samples. Then, the detailed parameters are estimated by integrating small-sample and prior parameters. With this well-trained model, the augmented data can be obtained with a particle filtering method for prediction. For validation, both the sparse and truncated samples from real-world monitoring are used to demonstrate the superiority of the proposed method. The high predicted accuracy demonstrates that the reliability of oil condition prediction can be guaranteed even with small samples. [ABSTRACT FROM AUTHOR]

• A SSADT optimal design method is developed. • A generalized Wiener process-based model considering the distribution feature is proposed. • An optimization criterion for products' degradation distribution is proposed. • Statistical inference of the proposed generalized Wiener process-based model is studied. • A set of metal-film resistor data is applied to illustrate the proposed method The Step-stress accelerated degradation test (SSADT) is an efficient tool to assess the reliability of products. To ensure an accurate evaluation of reliability-related indexes with limited cost, the optimal design for SSADTs is widely applied. Conventional optimal design methods ideally assume the degradation process to be linear and distribution-free, which may cause the designed test plans to be drastically different from the optimal one. To overcome the drawbacks of the unreasonable assumption, this paper develops a new optimal design method for SSADTs, including a generalized Wiener process-based degradation model (GWPDM) considering the distribution feature and a corresponding optimization criterion aiming at accurate prediction of degradation distributions. The GWPDM is first discussed, where the influence of the nonlinearity, unit-to-unit variability, measurement error and stress-related temporal variability on the degradation process is clarified. The statistical inference of the developed GWPDM is conducted to deduce the proposed optimization criterion. Thus, the optimization model is established to determine the optimal SSADT plan. The proposed method is used to design a SSADT plan for the metal-film resistor. Finally, the necessity and the robustness of the proposed method are verified. [ABSTRACT FROM AUTHOR]

Abstract: The issue of residual life (RL) estimation, which has been widely studied recently, plays an important role in scheduling maintenance. In this work, we present an adaptive method of RL estimation based on a generalized Wiener degradation process which subsumes several existing models as limiting cases. The nonlinearity, the temporal uncertainty, and the product-to-product variability of the degradation are jointly taken into account in the proposed degradation model. Under a mild assumption, an analytical approximation to the probability density function of the RL is derived in a closed-form, which becomes quite useful in maintenance decision making. The unknown parameters of the model that characterize the population-based degradation characteristics are obtained by using the maximum likelihood approach, while the parameters that describe the online product-specific characteristic are estimated by using the Markov chain Monte Carlo (MCMC) method. Once new degradation data information of the target product becomes available, the degradation model is first updated based on the degradation history up to the current time through a strong tracking filter, and then the RL is estimated sequentially. In this way, the RL of a product can be estimated in an adaptive manner. Finally, the validity of the proposed method is demonstrated with an illustrative example concerning fatigue cracks. [Copyright &y& Elsevier]

• 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]

In degradation analysis, there exist two natural features for the degradation data. The first is the dependence of degradation rate and degradation volatility, and the other is the time-varying mean-to-variance ratio. Ignoring them may lead to a significant bias in assessing lifetime information of materials. This paper proposes a generalized Wiener degradation model, which puts these two characteristics and unit-to-unit variation into consideration simultaneously. The proposed model includes many existing Wiener degradation models as special cases. Then, a generalized closed-form approximated residual lifetime distribution is given for the proposed model. Statistical inference for the model parameters is conducted based on the expectation maximizing (EM) method, and two simple auxiliary frameworks are developed for the determination of initial values and the forms of the time-scaled functions. The developed methodologies are then illustrated and verified in a simulation study and two real data analysis. • The proposed generalized Wiener model encompasses many existing Wiener models. • Owning dependent degradation-rate-volatility and no-constant mean-to-variance ratio. • A general approximate residual lifetime distribution for our model is developed. • Two auxiliary methods ship with EM for conducting statistical inference. [ABSTRACT FROM AUTHOR]

• Arrhenius based stochastic degradation rate model • A capacity conversion path from random temperature to reference temperature • A new battery aging model under time-varying temperature condition • A novel online RUL prediction method under Bayesian framework Time-varying temperature condition has a significant impact on discharge capacity and aging law of lithium-ion battery. Consequently, a novel remaining useful life (RUL) prediction method for lithium-ion battery under time-varying temperature condition is developed in this paper. Firstly, a stochastic degradation rate model based on Arrhenius temperature model is proposed, and an interesting battery capacity conversion path from random temperature condition to reference temperature condition is established. Secondly, the aging model of lithium-ion battery under time-varying temperature condition is developed based on Wiener process, and a two-step unbiased estimation method based on maximum likelihood estimation (MLE) combined with genetic algorithm (GA) is proposed. Next, the random parameter is online updated under Bayesian framework. Then the probability density function (PDF) of the RUL for lithium-ion battery under time-varying temperature condition is derived. Finally, a case study is implemented to verify the effectiveness, and the results show that the proposed prediction method has higher accuracy and smaller uncertainty. [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]

Abstract: The paper presents a Bayesian framework consisting of off-line population degradation modeling and on-line degradation assessment and residual life prediction for secondary batteries in the field. We use a Wiener process with random drift, diffusion coefficient and measurement error to characterize the off-line population degradation of secondary battery capacity, thereby capturing several sources of uncertainty including unit-to-unit variation, time uncertainty and stochastic correlation. Via maximum likelihood, and using observed capacity data with unknown measurement error, we estimate the parameters in this off-line population model. To achieve the requirements for on-line degradation assessment and residual life prediction, we exploit a particle filter-based state and static parameter joint estimation method, by which the posterior degradation model is updated iteratively and the degradation state of an individual battery is estimated at the sametime. A case study of some Li-ion type secondary batteries not only shows the effectiveness of our method, but also provides some useful insights regarding the necessity of on-line updating and the apparent differences between the population and individual unit degradation modeling and assessment problems. [Copyright &y& Elsevier]

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]

Abstract: We address the problem of determining inspection and maintenance strategy for a system whose state is described by a multivariate stochastic process. We relax and extend the usual approaches. The system state is a multivariate stochastic process, decisions are based on a performance measure defined by the values of a functional on the process, and the replacement decision is based on the crossings of a critical levels. The critical levels are defined for the performance measure itself and also as the probability of never returning to a satisfactory level of performance. The inspection times are determined by a deterministic function of the system state. A non-periodic policy is developed by evaluating the expected lifetime costs and the optimal policy by an optimal choice of inspection function. The model thus gives a guaranteed level of reliability throughout the life of the project. In the particular case studied here, the underlying process is a multivariate Wiener process, the performance measure is the norm, and the last exit time from a critical set rather than the first hitting time determines the policy. [Copyright &y& Elsevier]

• A nonlinear-drift-driven wiener process degradation model considering three sources of variability. • The analytical expression for the probability distribution function of RUL is derived. • The proposed model is general and covers the previous simplified models. • A down-sampling strategy is introduced to avoid the numerical underflow issue. The estimation of remaining useful life (RUL) for a degrading system has gained increasing attention both in academia and in industry for decades. In this study, a nonlinear-drift-driven Wiener process model considering three common sources of uncertainty is constructed by an age-dependent state-space model for the RUL estimation of degrading systems. Analytical expressions approximating the probability distribution function of RUL of the above-described model are derived for both online estimation and offline estimation scenarios. It is shown that the derived expressions are more general and cover the simplified cases discussed in previous woks. A model parameter estimation method is proposed based on unbalanced historical degradation measurements, and a down-sampling strategy is introduced to avert the underflow issue. The prognostic performance of the proposed method against previous similar works under the online and offline estimation scenarios is demonstrated on two publicly available datasets by comparisons in terms of three prognostic metrics and the probabilistic distribution of RUL at different condition monitoring points. The results show that it is necessary to include the nonlinear degradation characteristics and the three sources of uncertainty into the RUL estimation especially for the offline estimation scenario. [ABSTRACT FROM AUTHOR]

• Evidential variable and evidence theory are introduced in Wiener process. • The measurement error is considered. • Lifetime and degradation testing data is used to estimate model parameters. • Difference of reliability degrees among the samples is considered. • The proposed method shows high accuracy in reliability prediction. Evidential variable has been applied in Wiener process based reliability estimation due to its powerful ability on parameter describing. The previously published evidential variable and stochastic process based reliability estimation methods neglect measurement error and cannot utilize lifetime testing data. However, in practical applications, lifetime testing is an important approach and measurement error is an inevitable factor. Hence, in this paper, the evidential variable and Wiener process based reliability estimation method is improved to handle the above issues. A simulation study is used to verify the effectiveness of the proposed reliability estimation method. Furthermore, an actual engineering case on piston pump is also studied to demonstrate the proposed method in engineering practice. It is concluded that utilizing lifetime testing data and considering measurement error can improve the accuracies of model parameter evaluation, degradation prediction, reliability estimation and etc., in evidential and Wiener process based reliability estimation. [ABSTRACT FROM AUTHOR]

• We propose a novel degradation-rate-volatility related effect Wiener process model. • Model-RV has high accuracy in ageing process modeling and RUL prediction. • We extend Model-RV to AAT by incorporating stress-acceleration models. • A set of 3S-60 silicon rubber accelerated ageing data is used for illustration. For some materials, positive correlations between degradation rate and volatility are common observations affecting health evolution processes, referred as degradation-rate-volatility related effect. Ignoring such correlations may weaken the stability and precision of prognosis outcomes, ultimately misleading maintenance decision-makings. However, few attempts explore physical integrations of rate-volatility correlations, while quantify the correlation pattern polluted by random effect. Besides, achieving high accuracy of degradation modeling emphasizes the influence of systematic error. We address such technical challenges by constructing a Wiener-based degradation model integrating: (a) time-variant drift and diffusion coefficients linked by a linear function and (b) a systematic error term independent of process variances. Joint statistical inference employing expectation maximization (EM) is executed, based on which the analytical structure of RUL distribution is constructed. We further extend the prognosis approach to accelerated ageing test (AAT) by incorporating the stress-acceleration models. A case study on degradation data of 3S-60 silicon rubber states the superior performance of the proposed approach in reducing fitting error of degradation trajectory as well as improving the RUL prediction accuracy. [ABSTRACT FROM AUTHOR]

• An evidential variable based reliability estimation method is proposed. • The evidential variable is used to describe the model parameters. • The proposed method can infer population performance without adding parameters. • Bayesian inference method is used to evaluate and update the evidential variables. • The proposed method is more accurate and stable than random variable based method. Based on Wiener process and evidential variable, a reliability estimation and degradation modeling method is proposed in this paper. Wiener process is combined with evidence theory by applying evidential variable to describe model parameters. The basic probability assignments of evidential model parameters are evaluated based on likelihoods. Bayesian inference method is used to fuse degradation data and update evidential model parameters. Based on a wildly used actual degradation dataset, the distinguish features of the proposed method are verified. Due to powerful uncertainty describing ability of evidential variable, the proposed method shows superiority on online prediction and population evaluation compared to random variable based method, such as improvements in stability and accuracy. Furthermore, the proposed method can be used to evaluate population reliability without increasing parameter size, resulting in stochastic process become more practical under small sample conditions. [ABSTRACT FROM AUTHOR]

• A general iterative method of SSL estimation for stochastic degradation processes is first proposed. • Analytical expressions of SSL's PDF based on Wiener-process-based model are obtained. • The unit-to-unit variability is described by a Gaussian Mixture Model. • Some analytical results of SSL's mean and variance are provided under a nonstorage-failure assumption. Standby redundancy is an effective fault-tolerant technique for enhancing reliability and prolonging the standby system's operating lifetime. How to estimate the total lifetime of a standby system with a predetermined number of standby components (i.e., spare parts) presents an interesting practical issue. Most existing studies, however, have mainly focused on the lifetime or remaining useful lifetime prediction of a single online product. Moreover, spare parts usually deteriorate in storage, which will worsen their performance and even lead to failure. This makes lifetime estimation for a standby system more challenging. This study, therefore, focused on how to estimate a standby system's lifetime (SSL) with deteriorating spare parts. Unlike prior work, we fully considered the uncertainty and randomness caused by the spare parts' storage degradation in the SSL estimation. By establishing the transition probability function of the storage degradation process, we first proposed a general iterative algorithm for SSL estimation under the concept of the first passage time (FPT) and provided the proof based on mathematical induction. Then, we extended this result to a Wiener-process-based model and obtained the iterative result in a single integral form. Moreover, we attained the analytical expressions of SSL mean and variance under a non-storage-failure hypothesis and further provided the requirement for the establishment of the hypothesis. Finally, a numerical case and a practical case of the gyroscope are introduced for illustration. [ABSTRACT FROM AUTHOR]

• A Weibull proportional hazards model is adopted to model the hazard rate of the hard failure. • The degradation level is treated as a multiplicative time-varying covariate. • The closed-form of the RUL distribution is derived based on the Brownian bridge theory. • The optimal maintenance schedule is updated when new degradation signals are available. The remaining useful lifetime (RUL) estimated from the in-situ degradation data has shown to be useful for online predictive maintenance. In the literature, the RUL is often estimated by assuming a soft-failure threshold for the degradation data. In practice, however, systems may not be subject to the degradation-induced soft failures. Instead, the systems are deemed to be fail when they cannot perform the intended function, and such failures are known as hard failures. Because there are no fixed thresholds for hard failures, the corresponding RUL estimation is not an easy task, which causes difficulties in finding the optimal maintenance schedule. In this study, a Weibull proportional hazards model is proposed to jointly model the degradation data and the failure time data. The degradation data are treated as the time-varying covariates so that the degradation does not directly lead to system failures, but increases the hazard rate of hard failures. A random-effects Wiener process is proposed to model the degradation data by considering the system heterogeneities. Based on the developed proportional hazards model, closed-form distribution of the RUL is derived upon each inspection and the optimal maintenance schedule is then obtained by minimizing the system maintenance cost. The proposed maintenance strategy is successfully applied to predictive maintenance of lead-acid batteries. [ABSTRACT FROM AUTHOR]

• A new scope of sequential RUL prediction compared with conventional point-wise prediction. • A surrogate modeling framework for uncertainty propagation using drifted Wiener processes. • Hybrid modeling between machine learning and statistics • Improve the near-failure prediction accuracy by sacrificing the far-to-failure prediction accuracy. In modern industrial systems, sensor data reflecting the system health state are commonly used for the remaining useful lifetime (RUL) prediction, which are increasingly processed by modern deep learning based approaches recently. But these deep learning models do not automatically provide uncertainty information for the RUL prediction, hence this paper is motivated to introduce a novel approach that allows to control trade-off between prediction performance and knowledge about the uncertainty of the RUL prediction. The key aspect of our approach is to use a long short-term memory (LSTM) network as an expressive black-box predictor and the Wiener process as a surrogate to model the propagation of prediction uncertainty. The uncertainty propagation model is used to interactively train the RUL predictor. Our empirical results in a turbofan engine degradation simulation use case show that the surrogate Wiener propagation model can improve the near-failure prediction accuracy by sacrificing the far-to-failure prediction accuracy. [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]