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At present, the fault big data of open source software are opened as the open data set. In particular, the fault detection phenomenon depends on various situation of operation in OSS. Actually, various software reliability growth models have been actively proposed by several researchers in the past. This paper applies the deep learning approach to the OSS fault big data. Then, we propose several reliability assessment measures based on the deep learning. As an approach, the range of estimate expands by the Wiener process embedded for the data preprocessing. Furthermore, this paper proposes the performability as novel reliability assessment measure from the proposed deep learning model. In particular, we develop the prototype of 3D reliability assessment tool. Several illustration examples based on the developed prototype of 3D reliability assessment tool by using the actual fault big data sets are shown in this paper.

There is a problem of poor predictive performance when predicting the remaining service life of products. Therefore, this paper proposes a finite data product remaining service life prediction method based on Gaussian stochastic processes. Analyze the types of product failure and degradation under limited data, and determine the factors affecting the degradation performance of product life through constant stress accelerated degradation tests, step stress accelerated degradation tests, and sequential stress accelerated aging tests; using the Wiener process to analyze the performance degradation characteristics of products, setting product failure thresholds, defining the remaining life of the product based on the degradation process, and using equal interval partitioning methods to determine the product health index, and clarifying the product failure process under limited data; by calculating the mathematical expected value, the numerical characteristics of the remaining life of the product are defined. Variance and moments are used to determine the predicted second-order central moment and origin moment. Based on the monotonic increasing characteristics of the inverse Gaussian process, the probability distribution of the random variable of product failure is determined after the Gaussian process. Under limited data, the actual working time of the product and the degradation amount at the time scale are clarified, and a preset threshold is set for the degradation amount. Build a residual life prediction model for product degradation and output the prediction results. The experimental results show that this method can accurately predict the remaining service life of the product and has good predictive performance.

We present a mathematical model for stock market volatility flocking. Our proposed model consists of geometric Brownian motions with time-varying volatilities coupled with Cucker–Smale (C–S) flocking and regime switching mechanisms. For all-to-all interactions, we assume that all assets' volatilities are coupled to each other with a constant interaction weight, and we show that the common volatility emerges asymptotically and discuss its financial applications. We also provide several numerical simulations and compare them to existing analytical results.

There are always some difficulties in storage reliability evaluation of high-reliability, long-life, and high-value products, such as the test sample being small, degradation speed being slow, and failure data being inadequate. Temperature–humidity step-stress accelerated degradation test (THSS-ADT) is an effective method to evaluate the reliability of this type of products, but the test data processing is an extremely complex work. The motivation of this paper is to provide a clear, effective, and convenient method to evaluate the reliability on the basis of THSS-ADT data. Considering the stochastic volatility in degradation process, Wiener process is used to modeling the accelerated degradation process. The methods to estimate the parameters of Peck accelerated model and degradation model are discussed under temperature–humidity step-stress. As ordinary optimization algorithms (such as Newton Iteration Method and impelling function method) find it difficult to get the solutions, particle swarm optimization (PSO) method is used to solve the problem of maximum-likelihood estimation. Finally, the proposed methods are demonstrated for two examples, in which one is a numerical simulation, and another is an engineering practice of a microwave power amplifier.

This paper focuses on the sustainability based on the effort by using the fault big data of open source software (OSS). The fault detection phenomenon depends on the maintenance effort, because the number of software fault is influenced by the effort expenditure. Actually, the software reliability growth models with testing-effort have been proposed in the past. In this paper, we apply the deep learning approach to the OSS fault big data. Also, we propose the reliability assessment measure of sustainability. Then, we show several sustainability assessment measure based on the deep learning. Moreover, several numerical illustrations based on the proposed deep learning model are shown in this paper.

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Lévy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesàro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise.

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup P_t; further, we identify its infinitesimal generator in the space L² (H; ν).

The production of natural resource-based commodities is frequently affected by environmental uncertainty and the strategic response of producers to uncertainty. We ask when uncertainty induces cooperation. Using a model of Cournot competition under environmentally induced price uncertainty, we consider the conditions under which cooperation is favored over defection. We find that the probability of cooperation depends on the length of the time period over which production levels are adjusted to price shocks. When uncertainty is low, the probability of cooperation is monotonically increasing in the length of this adjustment period. When uncertainty is high, the probability of cooperation initially increases as the adjustment period lengthens, but beyond some threshold starts to fall. In this case, the expected outcome is defection. We consider the broader implications of these findings.

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

We introduce the discrete distribution of a Wiener process range (DDWPR). Rather than finding some basic distributional properties including hazard rate function, moments, stress-strength parameter and order statistics of this distribution, this paper studies some basic properties of the truncated version of this distribution. The effectiveness of this distribution is established using a data set.

The integration of renewable energy sources into the power systems and the growth of electricity consumption leads to a considerable increase in the power fluctuations. In the first part of this study, the control of the rotor angle stability of single machine infinite bus system in the presence of Wiener type power fluctuations has been achieved by minimizing the corresponding stochastic sensitivity function. In the second part, the power fluctuations have been modeled by alpha-stable Levy processes and since stochastic sensitivity function is not available for alpha-stable Levy processes, then the control of the rotor angle stability has been numerically achieved by minimizing the corresponding rotor angle dispersion for the first time in the literature.

We prove the existence and uniqueness of solutions to Wiener–Poisson type multivalued stochastic evolution equations in abstract spaces. We also prove that the solution has the Markov property. Moreover, applications to stochastic ordinary differential equations and stochastic partial differential equations are presented.

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [13], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in $p$th-moment. An example is provided to illustrate the abstract results.

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the $L^2$-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere $\mathcal{ℳ}$ in a Hilbert space $L^2$.

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.

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov’s equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planckequation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

With the popularization of accelerated test, how to get reliability and lifetime information accurately with limited samples has been more and more important. In this paper, a cyclic iterative algorithm of data and parameters to get more accurate parameters of accelerated degradation test (ADT) with a few samples is proposed. The method uses the parameters of acceleration model which is estimated by raw data to transform the data of stress at higher level to the data of stress at this level to increase the number of data in this level. Then, using the new data to estimate the new parameters until the parameters are not changed or changed negligibly. The maximum likelihood estimation (MLE) method is used to get more accurate parameters in each step of the cycle. Particle Swarm Optimization (PSO) algorithm is used to hunt for the optimum solution of the maximum likelihood function. In order to verify the validity of the method, a simulation model is built with MATLAB and an experiment of electric energy meter has been done. The proposed method is compared with the conventional methods in terms of their accuracy of parameters, which can fully demonstrate the effectiveness of the proposed method.

Over the past few decades, the method of reliability assessment based on performance degradation data has attracted many researchers because of its effectiveness and practical significance for complex systems of small-sample, highly reliable and longevity. A degradation model was presented in this paper for system with linear failure threshold under dynamic environment based on Wiener process, and the parameter of the influence function determined by dynamic environment in the degradation model was estimated by Bayesian method. Then a method of reliability assessment was presented for system with linear failure threshold in terms of the distribution of first passage time. The models and method are appropriate in applications and engineering which is meaningful and will play an important role in the remaining useful life estimation and condition-based maintenance for complex systems.