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Electromagnets are often used in indirect control for industrial applications. The ability of an electromagnet to control objects should decrease with performance degradation. And electromagnets product poses a danger to people and objects in the working environment. So, it is very difficult to analyze the reliability of electromagnetic performance degradation because of the complicated working condition. Failure Mode and Effect Analysis (FMEA) is the most commonly used tool for product reliability analysis. The new version of FMEA uses integer as evaluation value, which cannot represent the hesitation psychology of the evaluator. The Action Priority (AP) table of the FMEA describes the relationship between the evaluation of influencing factors and the risk level of the failure mode, which provides rules for determining the risk level of the failure mode. However, the AP table may result in multiple failure modes having the same ranking, which does not align with the intention of FMEA to prevent failures. Therefore, this paper proposes a reliability analysis method for electromagnetic performance degradation based on FMEA and FIS. Firstly, the Double Hierarchy Hesitant Fuzzy Linguistic Term Set (DHHFLTS) is used as the evaluation language to describe the hesitation psychology of evaluators. Secondly, the AP table of FMEA is used as FIS fuzzy inference rule. In this way, the idea of FMEA to determine the risk level of failure mode is retained and the problem of FIS fuzzy rule making is overcome. Then, FIS defuzzification AP table inference results to determine the risk ranking of failure modes. This avoids situations where the order of failure modes is equal. Finally, a performance degradation model of the electromagnet is constructed based on the Wiener process, and the calculation results of the new method are verified. [ABSTRACT FROM AUTHOR]

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 use of Constant‐stress accelerated degradation tests (CSADT) modeling is an effective way to assess the reliability of products. However, when conducting reliability experiments, the data collected may come from various sources such as different manufacturing batches, equipment, and operators. This can result in reduced accuracy of the test evaluations. In this research paper, we propose optimal designs for the Wiener constant‐stress accelerated degradation model that take into account the heterogeneity of manufacturing batches and individual differences. Our goal is to minimize the variability of the mean time to failure (MTTF) while working within a specified budget, test time, and available test units. To achieve this, we utilize particle swarm optimization (PSO) to find the best solution based on a complex objective function. We also compare our model with others using degradation data from LEDs, showing that our model has a better goodness‐of‐fit. Finally, we present examples of optimal CSADT plans under different constraints and conduct sensitivity analysis. [ABSTRACT FROM AUTHOR]

The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ (t) = A ⁢ u ⁢ (t) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ (t) , t > 0 , u ⁢ (0) = ξ . The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ (t) : t ≥ 0 } . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes. [ABSTRACT FROM AUTHOR]

A differential pursuit game in a stochastic descriptor linear system is analyzed. The dynamic of the system is described by Ito's stochastic differential algebraic equation. Solutions of the equation are presented by the stochastic formula of the variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct pursuer's control bringing the dynamic vector of the system to the terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transients in a radio engineering filter with random disturbances in the form of white noise. [ABSTRACT FROM AUTHOR]

In engineering practice, many products exhibit multiple and dependent degrading performance characteristics (PCs). It is common to observe that these PCs' initial measurements are nonconstant and sometimes correlated with the subsequent degradation rate, which typically varies from one unit to another. To accommodate the unit-wise heterogeneity, PC-wise dependency, and "initiation-growth" correlation, this article proposes a broad class of multi-dimensional degradation models under a framework of hierarchical multivariate Wiener processes. These models incorporate dual multi-normally distributed random effects concerning the initial values and degradation rates. To infer model parameters, expectation-maximization (EM) algorithms and several tools for model validation and selection are developed. Various simulation studies are carried out to assess the performance of the inference method and to compare different models. Two case studies are conducted to demonstrate the applicability of the proposed methodology. The of this article contain derivations of EM estimators, additional numerical results, and R codes. [ABSTRACT FROM AUTHOR]

This research paper tackles with the optical soliton solutions of stochastic the third-order nonlinear Schrödinger equation having multiplicative white noise via Ito calculus. Two efficient analytical procedures, which are the unified Riccati equation expansion method and the new Kudryashov method, are performed to structure the stochastic soliton solutions of introduced model. By making this choice, it has been intended to take advantage of the properties of these methods at the same time without using accessional operation and time in problem solving. Under favour of these techniques, bright and dark soliton forms are derived and their behavioral alterations on soliton dynamics under the effect of stochastic and the third-order dispersion terms are figured by 2D, contour and 3D diagrams. Considering the fact that the stochastic form of the equation has not been studied before in the absence of the chromatic dispersion term and the importance of noise effect for nonlinear models it is expected that the results which are obtained in this study will make a contribution to the studies in this field. [ABSTRACT FROM AUTHOR]

Axle box vibration serves as the main source of excitations for rail vehicles. Due to the wear of wheel/rail contact and the re-profiling procedure, the axle box vibration usually degrades periodically with the increased mileage in the service. This could significantly impact the estimation of vibration fatigue when the component is subjected to the axle box vibration. This paper develop a method to describe the periodic evolution of axle box vibration spectrum to better characterise the vibration spectrum of axle box. In this study, a Wiener process incorporating with four random parameters was employed to model the non-linearity of the degradation process. The maximum likelihood estimation (MLE) algorithm is used to estimate the initial values of the random parameters, and a Bayesian approach is employed to update the parameters based on newly obtained data. Finally, the proposed methodology is tested using long-term field test data from a high-speed train, and the results demonstrate that it accurately estimates the evolution of the axle box acceleration spectral density (ASD) spectrum. This could aid in predicting the residual service life of structures subjected to axle box vibration and further contribute to the development of maintenance strategies and top-down design of the structure. [ABSTRACT FROM AUTHOR]

Degradation modeling using condition monitoring (CM) data is fundamental for prognostics and health management (PHM). However, due to the variations in manufacturing materials and operating environments, degradation heterogeneity makes life prediction difficult. Motivated by this problem, a flexible nonlinear Wiener process with random drift diffusion is proposed for degradation modeling. Different from traditional methods, this approach regards both drift and diffusion coefficients as random parameters to describe the heterogeneity of the degradation rate and volatility simultaneously. In addition, the model supports the selection of appropriate distribution types for the random parameters according to the statistical characteristics of the actual data to improve fitting performance. To effectively overcome the parameter estimation difficulties caused by model assumptions, we propose a two-stage maximum likelihood estimation (MLE) algorithm embedded with a distribution selection strategy to estimate the model parameters. Specifically, the method is first used to estimate the drift and diffusion coefficients of each unit. Then, the estimated coefficients are used to select the distribution types and perform MLE for parameters in the selected distributions. The effectiveness of the proposed parameter estimation algorithm is demonstrated with both simulated datasets and real turbofan engine datasets. A comparison of results show that compared to simplified versions and traditional methods, the proposed degradation model improves the fitting performance and life prediction accuracy. [ABSTRACT FROM AUTHOR]

Remaining useful life prediction (RUL) is a critical procedure in the application of prognostics and health management for devices or systems. It is difficult to predict the RUL in a time‐varying external environment. Specifically, many mechanical systems typically experience various operating conditions, which have impacts on the degradation process and degradation rate. In particular, the linear degradation modeling of the Wiener process‐based RUL prediction method has attracted considerable attention recently. However, the dependency of degradation rate and operating conditions is generally ignored in the current degradation modeling, which leads to inaccurate issues in the RUL prediction. Therefore, to solve the above issues, a novel RUL prediction method based on the Wiener process considering parameter dependence is proposed in this paper. At first, a linear Wiener process degradation model considering parameter dependence is constructed to describe the dependency of the drift coefficient and operating conditions. Secondly, the probability density function of RUL is derived under the concept of first hit time. After that, the collaboration between the Bayesian update and expectation maximization algorithm is introduced to update and estimate the model parameters. Finally, the validity and applicability of the proposed method are verified by a numerical simulation and three case studies of bearings. [ABSTRACT FROM AUTHOR]

Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to obtain the analytic solutions for stochastic fourth-order (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with higher order odd and even terms. Our Study mainly depends on applying the improved modified extended tanh function method (IMETFM) to get the soliton wave solutions and other exact wave solutions of our model of interest. These solutions such as (bright, dark, singular) solitons, hyperbolic solutions, periodic solutions, singular periodic solutions, rational solutions, Jacobi elliptic functions solutions and Weierstrass elliptic functions solutions. Graphical representations of some of the extracted solutions using different noise intensities are shown to demonstrate the influence of the noise. [ABSTRACT FROM AUTHOR]

A low-voltage circuit breaker thermal trip plays a role in power systems, including opening and closing, control, protection, and more. Their reliability directly affects the security and stability of the power distribution system. The Wiener process model is established according to the accelerated degradation data of thermal trips, and remaining life prediction is carried out. In this paper, firstly, the constant stress accelerated degradation test is carried out on a thermal trip with temperature as the accelerated stress and specific thermal deflection as the degradation eigenquantity to characterise the degradation trajectory according to the degradation data and analyse the degradation law; then, it is verified by the test of normal distribution that the degradation data of the thermal trip conform to the Wiener process. The Wiener process, using the great likelihood estimation method to estimate the parameters of the remaining life, calculates the remaining life probability density function and reliability function under different temperature stresses and obtains the remaining life of the thermal trip under each accelerated stress conditions; finally, the remaining life of the thermal trip at the initial moment is taken as the pseudo-failure life, and the Arrhenius accelerated model is utilised to launch the external thermal trip under normal stress. Finally, the Arrhenius acceleration model is used to extrapolate the lifespan of the thermal trip under normal stress. [ABSTRACT FROM AUTHOR]

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-type drift coefficients. Moreover, we also assume that both drift f = f (t , x , z) and diffusion g = g (t , x , z) coefficient are Lipschitz continuous with respect to the space variable x , but only Hölder continuous with respect to the delay variable z. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Degradation tests are used when there is a quality characteristic related to the life of a product. In this paper, we investigate the model mis-specification effect on the estimation precision of product's mean time to failure (MTTF) and consider a degradation test design problem. The Wiener and inverse Gaussian (IG) processes are two possible models considered. We derive expressions for the mean and variance of the estimated product's MTTF when the true model is an IG process, but is wrongly fitted by a Wiener process. We further discuss the experimental design problem and derive the explicit functional form of the estimation variances. Using the derived functions, optimal degradation test plans assuming a Wiener process model is correctly or wrongly specified are both proposed. The derived plans are applied to a laser data example. We evaluate the test efficiency of the plans derived from a Wiener process assumption when the model is mis-specified. For many optimization criteria, we observe that the obtained plans are robust even when the fitted model is mis-specified. For some criteria that may result in very different test plans under different models, we use a weighted ratio criterion to find practically useful degradation plans under model uncertainty. [ABSTRACT FROM AUTHOR]

This work recovers optical soliton solutions to the concatenation model with differential group delay in the presence of white noise along both the components. Two integration algorithms have successfully recovered a full spectrum of solitons solutions to the model. It has been proved that the effect of white noise is visible only in the phase portion of the solitons along both components. [ABSTRACT FROM AUTHOR]

In this study, for the first time, a study on optical soliton solutions of the stochastic perturbed Schrödinger-Hirota equation which models the propagation of solitons through optical fibers having the Kerr law is carried out. For this purpose, the nonlinear partial differential equation of the examined model is first reduced to the nonlinear ordinary differential equation form using the complex wave transformation. Then, using the new Kudryashov and the Kudryashov methods, bright, dark and singular soliton solutions are derived as a basic step. The next part is devoted to the examination of the noise effect, and first of all, soliton solutions are obtained in the absence of noise strength. Afterwards, the same soliton solutions are obtained, taking into account the noise effect, and comparison and comments are made. In addition to the fact that the form of the examined equation has not been presented and studied before. Considering that, in today's world, taking into account the effect of noise is an inevitably important factor for engineering models to be designed correctly and to operate efficiently, it is obvious that the results obtained will contribute to the literature. [ABSTRACT FROM AUTHOR]

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk's fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). [ABSTRACT FROM AUTHOR]

AbstractIn engineering practice, some products start deteriorating only after a period of usage. Moreover, the degradation-free periods of different products show a phenomenon of clustering, that is, there are several heterogeneous subpopulations. Both the degradation-free period and its subpopulation label are often unobservable, which poses great challenges to the reliability modeling and evaluation. In this paper, a randomly delayed degradation model considering the heterogeneous initiation time is developed, where the mixture lognormal distribution and the Wiener process are integrated to characterize the overall degradation process. The model parameters are estimated by the stochastic expectation maximization algorithm which also produces the estimator of the degradation-free period for each product simultaneously. In addition, inferences on the choice of parameters’ initial values, interval estimation, determination of the number of subpopulations and classification of products are also presented. Finally, simulation studies are carried out to validate the proposed formulas and methods, and a practical example is provided for illustration. [ABSTRACT FROM AUTHOR]

The evolution of performance parameter states of civil aviation engines presents multi-stage characteristics, and the current evaluation methods for performance parameters cannot effectively use the big data of state monitoring, or accurately identify their influence on the performance of civil aviation engines. Therefore, a new reliability evaluation model considering the phase correlation of dynamic evolution of civil aviation engine performance parameter states is proposed. The model uses dynamic Wiener process to characterize the states of the performance parameters of civil aviation engines according to the phase characteristics of the change point, and the joint probability distribution function model of the phased dynamic Wiener process is established by using Copula function. Monte Carlo simulation sampling method is used to evaluate the reliability based on the mathematical property of the first reach threshold of the evolution of performance parameter states. The actual condition monitoring data of airlines are used to verify the superiority of the proposed model, and the results show that compared with the model without considering the correlation between evolution stages, the average error of the proposed model for evaluating the state reliability of the performance parameters of civil aviation engines is reduced by about 12.9%. [ABSTRACT FROM AUTHOR]

The improved modified extended tanh‐function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger‐Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stochastic singular periodic solutions, stochastic exponential solutions, stochastic rational solutions, and stochastic Jacobi elliptic doubly periodic solutions. Constraints on the parameters were taken into account to ensure the existence of the obtained stochastic soliton solutions. Additionally, selected solutions were presented graphically to illustrate the physical characteristics of the stochastic solutions. In this paper, we used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs. [ABSTRACT FROM AUTHOR]

In the traditional switching policy of a standby system, the standby unit is switched to the normal operation state only after the active unit fails. In contrast, in satellite engineering, a new periodic switching strategy is used by technicians in the satellite gyroscope standby system. Due to the performance degradation of the gyroscope in the long‐term operation and standby process, satellites may fail on orbit. For this problem, an operation optimization model is established for a two‐unit warm standby degradation system considering the periodic switching strategy. First, the degradation process of the units is described by a multiphase Wiener process. Second, based on the first‐hitting time theory, the analytic expressions of reliability and mean time to failure (MTTF) of the degradation system are derived. Third, the optimal periodic switching interval is determined to maximize the system MTTF by using the particle swarm optimization algorithm (PSO). Finally, a numerical example of a gyroscope warm standby subsystem is provided to validate the proposed model. [ABSTRACT FROM AUTHOR]

The local time (LT) of a given stochastic process { X t : t ≥ 0 } is defined informally as L X ⁢ (t , x) = ∫ 0 t δ x ⁢ (X s) ⁢ d s , where δ x denotes the Dirac function; actually, it counts the duration of the process's stay at 푥 up to time 푡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ⁢ (u) ⁢ d B u for a given deterministic function f ∈ L 2 ([ 0 , + ∞ [) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments. [ABSTRACT FROM AUTHOR]

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called "noise" that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method. [ABSTRACT FROM AUTHOR]

Simultaneous degradation of multiple dependent performance characteristics (PCs) is a common phenomenon for industrial products. The associated degradation modeling is of practical importance yet challenging. The dependence of the PCs can usually be attributed to two sources, one being the overall system health status and the other the common operating environments. Based on the observation, this study proposes a parsimonious multivariate Wiener process model whose number of parameters increases linearly with the dimension. We introduce a common stochastic time scale shared by all the PCs to model the dependence from the dynamic operating environment. Conditional on the time scale, the degradation of each PC is modeled as the sum of two independent Wiener processes, where one represents the common effects shared by all the PCs, and the other represents degradation caused by randomness unique to this PC. An EM algorithm is developed for model parameter estimation, and extensive simulations are implemented to validate the proposed model and the algorithms. For efficient reliability evaluation under a multivariate degradation model, including the proposed one, a bridge sampling-based algorithm is further developed. The applicability and the advantages of the proposed methods are demonstrated using a multivariate degradation dataset of a coating material. for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper we proceed with the multiscale analysis of semilinear damped stochastic wave motions. The analysis is made by combining the well-known sigma convergence method with its stochastic counterpart, associated to some compactness results such as the Prokhorov and Skorokhod theorems. We derive the equivalent model, which is of the same type as the micro-model. One of the novelties of the work is that the corrector problem is solved in the classical sense of distributions, thereby allowing numerical computations of the homogenized coefficients. [ABSTRACT FROM AUTHOR]

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics' fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative–soliton-like waves, showing how the solutions changed as the values of the studied system's physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences. [ABSTRACT FROM AUTHOR]

For high‐reliability and long‐life electronic devices, reliability analyses based on degradation data with small sample sizes are an outstanding question. Recently, increasing attention has been paid to the abundant historical degradation data. Exploiting the information in these historical data efficiently and integrating them to benefit the current reliability analysis is an important issue. This study proposes a new integrating method for the historical and current degradation data based on the Wiener process by considering the consistency of failure mechanisms between the different batches of degradation data. Simulation studies show that the new method has superior reliability estimation, and is robust to the assumption of consistent failure mechanisms. Finally, the proposed method is used to analyze a real data set consisting of the metal‐oxide‐semiconductor field‐effect transistor (MOSFET) degradation data. [ABSTRACT FROM AUTHOR]

This paper evaluates the remaining useful life (RUL) for the accelerated Wiener process with mixed random effects and measurement errors. In the discussed model, the random effects are included in both the drift and diffusion parameters of the Wiener process to model the heterogeneity among different degradation paths of products. The measurement errors are introduced to describe the inaccuracy during the measurement process. The effect of the accelerated stress level is included in the drift parameter. The expectation‐maximization (EM) algorithm is applied to obtain the estimation of the model parameters. The expressions of the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the lifetime and RUL distributions of products under the normal usage stress level are derived theoretically. Finally, an example of stress relaxation data is used to illustrate the effectiveness of the developed model and method. [ABSTRACT FROM AUTHOR]

Degradation tests have been used for the purpose of assessing reliability information. In this paper, we consider a degradation test design problem under a statistical model mis-specification scenario. The Wiener and gamma processes are considered in this research. A gamma process is suitable for describing degradation paths that exhibit monotone behavior. However, if one fits a Wiener process model to the data, the resulting statistical inferences may be affected. Tsai, Tseng, and Balakrishnan studied the effect on the estimated mean time to failure. However, the experimental design problem was not discussed. A lack of the explicit functional form of the estimation variances makes it difficult to find degradation plans that can improve test efficiency. In this paper, functional forms of optimal degradation test plans are proposed under model mis-specification. Furthermore, a weighted ratio objective function considering the prior probability of the true model is used to find robust test plans for practical use. The results from a numerical example show that, by using an appropriate degradation test plan, the estimation variance can be reduced, and the test efficiency can be improved. A simulation study is conducted to investigate the performance of the parameter estimates when the sample size is small. [ABSTRACT FROM AUTHOR]

In this paper we propose an objective Bayesian method to analyze the accelerated degradation model based on the new Wiener process stated in Ye, Chen, and Shen (Reliab. Eng. Syst. Safe., 2015, 139: 58–67). The Jeffreys prior and reference priors are derived after reparameterization, and the posterior properties under each prior are validated. A Metropolis random walk algorithm is then employed to generate samples from the posterior distributions, and a simulation study is carried out to reveal the frequentist properties of these priors. Finally, we apply the approach to a real data and obtain the mean-time-to-failure (MTTF) under the usage condition. [ABSTRACT FROM AUTHOR]

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations. [ABSTRACT FROM AUTHOR]

Dissolution studies are a fundamental component of pharmaceutical drug development, yet many studies rely upon the f1 and f2 model‐independent approach that is not capable of accounting for uncertainty in parameter estimation when comparing dissolution profiles. In this paper, we deal with the issue of uncertainty quantification by proposing several model‐dependent approaches for assessing the similarity of two dissolution profiles. We take a statistical modeling approach and allow the dissolution data to be modeled using either a Dirichlet distribution, gamma process model, or Wiener process model. These parametric forms are shown to be reasonable assumptions that are capable of modeling dissolution data well. Furthermore, based on a given statistical model, we are able to use the f1 difference factor and f2 similarity factor to test the equivalency of two dissolution profiles via bootstrap confidence intervals. Illustrations highlighting the success of our methods are provided for both Monte Carlo simulation studies, and real dissolution data sets. [ABSTRACT FROM AUTHOR]

In this paper, we propose degradation test sampling plans (DTSPs) used to determine the acceptability of a product in a Wiener process model. A test statistic and the acceptance criterion based on Wiener process parameter estimates are proposed. The design of a degradation test is investigated using a model incorporating test cost constraint to minimize the asymptotic variance of the proposed test statistic. Some important variables, including the sample size, measurement frequency, and the total test time, are chosen as decision variables in a degradation test plan. The asymptotic variance of the test statistic and the approximate functional forms of the optimal solutions are derived. A search algorithm is also represented in a flow chart for the purpose of finding the optimal DTSPs. In addition, we assess the minimum cost requirement for the result of the test procedure to satisfy the minimum requirements for the producer's risk and the consumer's risk. When the given test budget is not large enough, we suggest some methods to find appropriate solutions. Finally, a numerical example is used to illustrate the proposed methodology. Optimum DTSPs are obtained and tabulated for some combinations of commonly used producer and consumer risk requirements. A sensitivity analysis is also conducted to investigate the sensitivity of the obtained DTSPs to the cost parameters used. [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]

In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where "generalized" indicates that a boundary function has a finite number of first kind discontinuity curves. The algorithm consists of the following main stages: (1) the transition from an infinite domain to a finite domain by an inversion; (2) the consideration of a new Dirichlet generalized harmonic problem on the basis of Kelvin's theorem for the obtained finite domain; (3) the numerical solution of the new problem for the finite domain by the MPS, which in turn is based on a computer simulation of the Weiner process; (4) finding the probabilistic solution of the posed generalized problem at any fixed points of the infinite domain by the solution of the new problem. For illustration, numerical examples are considered and results are presented. [ABSTRACT FROM AUTHOR]

Detecting the abnormal degradation rate of Wiener processes is important for quality control and reliability management. In this paper, we propose upper-side CUSUM control schemes for monitoring the abnormal degradation rate of three classes of Wiener processes. Average run length and standard deviation run length are applied to evaluating the efficiency of CUSUM control schemes. The results show that the proposed CUSUM control schemes can efficiently give out-of-control signals when the degradation rate of Wiener process accelerates. It is also shown that initial values of degradation rate and sampling frequency have significant influence on the performance of CUSUM control schemes. [ABSTRACT FROM AUTHOR]

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters. [ABSTRACT FROM AUTHOR]

Summary: We propose a low‐complexity graph‐based iterative algorithm for the detection of coherent optical OFDM signals in the presence of residual carrier frequency offset and phase noise. We exploit the circular properties of the frequency offset and phase noise and derive graphical messages using the Gaussian probability density function. The proposed algorithm shows similar error rate performance as the existing algorithm at a much lower implementation complexity. [ABSTRACT FROM AUTHOR]

Abstract The appropriate control of maintenance effort in the edge computing based on open-source software (OSS) relates to stable and reliable operation, because the fault detection phenomenon depends on the effort expenditure. Actually, several software reliability growth models with testing-effort have been proposed in the past. In this paper, we propose cyclic Wiener process models to consider the network environment under the edge OSS operation. Then, we assume that the sawtooth wave and square one as the cyclic noises. Thereby, the proposed models will be able to comprehend two characteristics of OSS and network environment. Then, it will be useful to assess the operation effort expenditures with network environment of edge OSS service. Moreover, actual effort data sets are analyzed to show numerical examples of the proposed model considering the network environment under the edge OSS operation. [ABSTRACT FROM AUTHOR]

The degradation model and the remaining useful life (RUL) prediction are two key metrics for machine health prognostics. However, the precision of RUL prediction highly depends on the validity of the degradation model. Consequently, the uncertainty of RUL prediction based on the two‐phase degradation process may cause the estimated product reliability to differ from its actual value and lead to a faulty predictive maintenance strategy. However, recent studies have rarely compared the RUL predictions based on two‐phase and single‐phase degradation models. In this study, a real‐time RUL prediction approach was developed for a two‐phase linear Wiener degradation process. A two‐phase Wiener degradation model was proposed along with an estimation method for the change point and model parameters. Based on a state‐space model and Markov chain Monte Carlo, the drift coefficient of the degradation model was updated, and a real‐time RUL prediction model was developed. To examine whether the two‐phase Wiener degradation model was incorrectly assumed to be single‐phase, a case study was conducted to analyze RUL prediction based on the original and incorrect models. The results demonstrated that to obtain a more accurate RUL prediction, a sufficient number of samples should be used in addition to updating the drift coefficient of the degradation model when new measurement data are available. [ABSTRACT FROM AUTHOR]

Timely planning and scheduling of railway infrastructure maintenance interventions are crucial for increased safety, improved availability, and reduced cost. We propose a data‐driven decision‐support framework integrating track condition predictions with tactical maintenance planning and operational scheduling. The framework acknowledges prediction uncertainties by using a Wiener process‐based prediction model at the tactical level. We also develop planning and scheduling algorithms at the operational level. One algorithm focuses on cost‐optimisation, and one algorithm considers the multi‐component characteristics of the railway track by grouping track segments near each other for one maintenance activity. The proposed framework's performance is evaluated using track geometry measurement data from a 34 km railway section in northern Sweden, focusing on the tamping maintenance action. We analyse maintenance costs and demonstrate potential efficiency increases by applying the decision‐support framework. [ABSTRACT FROM AUTHOR]

Remaining useful life prediction (RUL) is a key component in the application of prognostics and health management associated with devices or systems. But such RUL predictions are cumbersome owing to complexities from external effects and internal degradation mechanisms within systems. Specifically, it is common for degradation processes to comprise distinct multiple stages rather than just one uniform stage in many mechanical systems. In particular, the two‐stage degradation modeling for RUL prediction based on the Wiener process with linear drift has received significant attention in recent years. However, negative effects of measurement errors and stochasticity of the degradation states are generally excluded from current degradation modeling, which causes inaccuracy problems that can impact system maintenance schedules and operational efficiency. Therefore, to solve such problems, measurement errors are considered in this paper and a two‐stage degradation model is proposed, in which an adaptive term is also characterized by the Wiener process. The transition probability density function (TPDF) of the degradation state at the two‐stage changing point is derived and an analytical solution for the RUL is obtained under the concept of the first hit time (FHT). A Kalman filter and smoothing algorithm are introduced to estimate variables, and the expectation maximization (EM) algorithm is applied to update and estimate model parameters. Finally, the effectiveness and applicability of the proposed model in RUL predictions are verified through numerical simulation and a case study of bearings. [ABSTRACT FROM AUTHOR]

• An analytical parameters estimation results by the MLE method for the Wiener process with random effects is obtained. • The comparative study between different MLE methods is implemented. • An unbiased parameters estimation method is proposed and applied to other types of Wiener process-based degradation models. • An empirical unbiased parameters estimation results when the degradation data is measured at different times is developed. • Mis-specification issue about whether considering the random effects is analyzed in theory. Accurate parameters estimation is one of the most crucial components in remaining useful life (RUL) prediction. This paper proposes an unbiased parameters estimation method and analyzes the impact of model mis-specification of Wiener process-based degradation model with random effects. First, we obtain an analytical expression of parameters estimation for the linear Wiener process. Then, by analyzing the natures of parameters estimation, an unbiased parameters estimation method is proposed and applied to other types of Wiener process-based degradation models. This unbiased estimation method is very similar to the sample variance of normal distribution. Since the sample size of degradation data is typically small, hence the proposed unbiased estimation method is suggested. After that, we present an empirical unbiased parameters estimation method with a closed-form solution when the degradation data is measured at different times. The proposed unbiased estimation method could increase computational speed and avoid falling into local minimum. In addition, based on parameters estimation results, the mis-specification issue about whether the random effects should be considered in the degradation modeling is analyzed in theory. Finally, several numerical examples and case studies are used for experimental verification. The experimental results demonstrate that the proposed unbiased parameters estimation method is superior to other parameters estimation methods and the random effects are suggested to be considered. [ABSTRACT FROM AUTHOR]

Real-time monitoring has received considerable attention recently due to its potential in automatic driving, tele-surgery, and factory automation in the 6G era. In remote estimation or reconstruction of stochastic processes, the statistical properties of stochastic processes to be monitored play a central role. Among the stochastic processes interested by real-time applications, the Brownian motion, also known as the Wiener processes is a typical one. In this paper, we are interested in how to monitor Brownian motions efficiently, timely, and reliably. To achieve this goal, we reveal that the real-time estimation error is jointly determined by the quantization error and freshness of data samples. Based on this observation, we present an optimal joint sampling and quantization scheme that efficiently balances the quantization distortion and the age-of-information (AoI). Furthermore, we find that the error accumulation will lead to infinite distortion as monitoring time increases. To overcome this, a multi-layer error correction method is presented for infinite-time monitoring, in which bounded distortion can be achieved with limited data rate. Finally, to conquer the accumulation of transmission errors in unreliable channels, we present an error correction mechanism based on periodic feedback. Diffusion approximation is then adopted to determine the optimal feedback rate and interval. [ABSTRACT FROM AUTHOR]

In this paper, we provide a construction of two algorithms for approximation of the system of the jump-diffusion SDEs. Both methods are based on the classical Milstein steps and use equidistant discretization points. We construct implementable algorithm and their derivative-free version. We show that constructed methods are asymptotically optimal in the class of methods that use equidistant discretization of the interval [ 0 , T ]. Moreover, we report numerical experiments performed for some test equations, which confirm theoretical properties. [ABSTRACT FROM AUTHOR]

In the field of reliability analysis, the constant-stress accelerated degradation test is one of the most commonly used methods to evaluate a product's reliability as degradation data are provided. In this paper, a constant-stress accelerated degradation test model of the Wiener process with random effects is proposed. First, the generalized confidence intervals of the model parameters are developed by constructing generalized pivotal quantities. Second, utilizing the substitution method, the generalized confidence intervals for the reliability function of lifetime, mean time to failure and the generalized prediction intervals for the degradation characteristic at the normal operating condition are also developed. Simulation studies are conducted to investigate the performances of the proposed generalized confidence intervals and prediction intervals. The simulation results reveal that the proposed generalized confidence intervals and prediction intervals work well in terms of the coverage percentage. In particular, a comparative analysis is made with the traditional bootstrap confidence intervals. At last, the proposed procedures are used for a real data analysis. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of stochastic FitzHugh–Nagumo equation (SFNE) has been obtained by Chebyshev spectral collocation. Semi-implicit Euler–Maruyama scheme has been used for temporal variable to discretize the stochastic FitzHugh–Nagumo equation. A detailed stability analysis for stochastic FitzHugh–Nagumo equation has also been discussed. Graphical representations of obtained results of the stochastic FitzHugh–Nagumo equation have been discussed to provide a clear idea about the behavior of the solutions. [ABSTRACT FROM AUTHOR]

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state. [ABSTRACT FROM AUTHOR]

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]