Bayesian and Multiple Bayesian Analysis of the Reliability Performances for Series System with Cold Standby Units

By using Bayesian and multiple Bayesian method, the failure probability, reliability and mean time to failure(MTTF) of series system with cold standby units are estimated. At last, we compare the two estimators by means of Monte_Carlo simulation.

Fully Bayesian reliability assessment of multi-state systems with overlapping data

The failure data at the system level are often limited, resulting in high uncertainty to system reliability assessment. Integrating data drawn from various structural levels of the target system (e.g. the system, subsystems, assemblies and components), i.e. the multi-level data, through Bayesian analysis can improve the precision of system reliability assessment. However, if the multi-level data are overlapping, it is challenging for Bayesian integration to develop the likelihood function. Especially for multi-state systems (MSS), the Bayesian integration with overlapping data is even more difficult. The major disadvantage of previous approaches is the intensive computation for the development of the likelihood function caused by the workload to opt the appropriate combinations of the vectors of component states consistent with the overlapping data. An improved fully Bayesian integration approach from a geometric perspective is proposed for the reliability assessment of MSS with overlapping data. In this method, a specific combination of component states is regarded as a state vector, which leads to a specific system state of the MSS, and all state vectors generate a system state space. The overlapping data are regarded as the constraints which create hyperplanes in the system state space. And a point in a hyperplane corresponds to a particular combination of the state vectors. In the light of the features of the constraints, the proposed approach introduces space partition and hyperplane segmentation, which reduces the selection workload significantly and simplifies the likelihood function for overlapping data. Two examples demonstrate the feasibility and efficiency of the proposed approach. © 1990-2011 Beijing Institute of Aerospace Information.

Reliability analysis for wireless communication networks via dynamic Bayesian network

The dynamic wireless communication network is a complex network that needs to consider various influence factors including communication devices,radio propagation,network topology,and dynamic behaviors.Existing works focus on suggesting simplified reliability analysis methods for these dynamic networks.As one of the most popular modeling methodologies,the dynamic Bayesian network(DBN)is proposed.However,it is insufficient for the wireless communication network which contains temporal and non-temporal events.To this end,we present a modeling methodology for a generalized continuous time Bayesian network(CTBN)with a 2-state conditional probability table(CPT).Moreover,a comprehensive reliability analysis method for communication devices and radio propagation is suggested.The proposed methodology is verified by a reliability analysis of a real wireless communication network.

Reliability analysis of monotone coherent multi-state systems based on Bayesian networks

The Bayesian networks (BNs) provide a robust probabilistic method of reasoning under uncertainty and have been successfully applied to a variety of real-world tasks. Aiming to explore the capabilities of the BN formalism in reliability analysis of monotone coherent multi-state systems, the BNs are compared with a popular tool for reliability analysis of monotone coherent multi-state systems, namely the multi-state fault trees (MFTs). It is shown that any MFT can be directly mapped into BN and the basic inference techniques on the latter may be used to obtain classical parameters computed from the former (i.e. probability distribution of top variable, minimal upper vectors and maximum lower vectors for any performance level, importance measures of components). Furthermore, some additional information can be obtained by using BN, both at the modeling and analysis level. At the modeling level, several restrictive assumptions implicit in the MFT methodology can be removed and various kinds of dependencies among components can be accommodated. At the analysis level, a general diagnostic analysis can be performed. The comparison of these methods is illustrated by an example of the water supply system. © 2016 Beijing Institute of Aerospace Information.

Reliability Analysis of Lithography Wafer Stage Based on Fuzzy Bayesian Networks

Bayesian network( BN) is a powerful tool of uncertainty reasoning. Considering the insufficient information,incorporating fuzzy probability into BN is an effective method. Fuzzy BN was used to solve this problem. In this paper,fuzzy BN was applied in wafer stage system,which was an important part of lithography. BN of wafer stage was transferred from fault tree( FT). The quantitative assessment based on fuzzy BN was carried out. The Birnbaum importance factors of basic events were calculated. Therefore,the system failure probability and the vulnerable components could be gotten.

Slope reliability and back analysis of failure with geotechnical parameters estimated using Bayesian inference

A Bayesian approach is proposed for the inference of the geotechnical parameters used in slope design.The methodology involves the construction of posterior probability distributions that combine prior information on the parameter values with typical data from laboratory tests and site investigations used in design.The posterior distributions are often complex,multidimensional functions whose analysis requires the use of Markov chain Monte Carlo(MCMC)methods.These procedures are used to draw representative samples of the parameters investigated,providing information on their best estimate values,variability and correlations.The paper describes the methodology to define the posterior distributions of the input parameters for slope design and the use of these results for evaluation of the reliability of a slope with the first order reliability method(FORM).The reliability analysis corresponds to a forward stability analysis of the slope where the factor of safety(FS)is calculated with a surrogate model from the more likely values of the input parameters.The Bayesian model is also used to update the estimation of the input parameters based on the back analysis of slope failure.In this case,the condition FS?1 is treated as a data point that is compared with the model prediction of FS.The analysis requires a sufficient number of observations of failure to outbalance the effect of the initial input parameters.The parameters are updated according to their uncertainty,which is determined by the amount of data supporting them.The methodology is illustrated with an example of a rock slope characterised with a Hoek-Brown rock mass strength.The example is used to highlight the advantages of using Bayesian methods for the slope reliability analysis and to show the effects of data support on the results of the updating process from back analysis of failure.

Simulation Evaluation Study of System Reliability Based on Improved BayesBootstrap Method

For the disadvantages of analyzing system reliability using common Bayes method,an evaluation method based on improved Bayes-Bootstrap method is presented.Firstly,the data obliteration problem existing in traditional Bayes method is analyzed,then an improved sampling arithmetic is proposed on the foundation of an experience function.Finally,effectiveness of this method is proved by an example.The simulating result shows that this method is more accurate than common Bayes method.

Assessment Method of Heavy NC Machine Reliability Based on Bayes Theory

It is difficult to collect the prior information for small-sample machinery products when their reliability is assessed by using Bayes method. In this study, an improved Bayes method with gradient reliability(GR) results as prior information was proposed to solve the problem. A certain type of heavy NC boring and milling machine was considered as the research subject, and its reliability model was established on the basis of its functional and structural characteristics and working principle. According to the stress-intensity interference theory and the reliability model theory, the GR results of the host machine and its key components were obtained. Then the GR results were deemed as prior information to estimate the probabilistic reliability(PR) of the spindle box, the column and the host machine in the present method. The comparative studies demonstrated that the improved Bayes method was applicable in the reliability assessment of heavy NC machine tools.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available,this study proposes a Bayesianmethod to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data.According to the given interval ranges of uncertainties,we determine the initial characteristic parameters of a multi-ellipsoid convex set.Moreover,to update the plausibility of characteristic parameters,a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed.Then,an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved.The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function.Several numerical examples are presented to validate the proposed Bayesian updating method.

One-Sample Bayesian Predictive Analyses for an Exponential Non-Homogeneous Poisson Process in Software Reliability

The Goel-Okumoto software reliability model, also known as the Exponential Nonhomogeneous Poisson Process,is one of the earliest software reliability models to be proposed. From literature, it is evident that most of the study that has been done on the Goel-Okumoto software reliability model is parameter estimation using the MLE method and model fit. It is widely known that predictive analysis is very useful for modifying, debugging and determining when to terminate software development testing process. However, there is a conspicuous absence of literature on both the classical and Bayesian predictive analyses on the model. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model. Driven by the requirement of highly reliable software used in computers embedded in automotive, mechanical and safety control systems, industrial and quality process control, real-time sensor networks, aircrafts, nuclear reactors among others, we address four issues in single-sample prediction associated closely with software development process. We have adopted Bayesian methods based on non-informative priors to develop explicit solutions to these problems. An example with real data in the form of time between software failures will be used to illustrate the developed methodologies.

High-speed Railway External Power Supply Reliability Evaluation of Bayesian Network

Reliability evaluation is important in high speed railway external power supply design, based on probability reasoning bayesian network applied in high-speed railway external power supply reliability evaluation, establish the minimum cut and the minimum path of bayesian network model, quantitative calculation external power supply system in each element posterior probability, and the example analysis verified the feasibility and correctness of the above method. Using bayesian network bidirection reasoning technology, quantitative calculation the posterior probability of each element in external power supply system, realized the identification of weak link in external power supply. The research methods and the results of the study can be used in the scheme optimization design of high speed railway external power supply.

Building Bayesian Network(BN)-Based System Reliability Model by Dual Genetic Algorithm(DGA)

A system reliability model based on Bayesian network(BN)is built via an evolutionary strategy called dual genetic algorithm(DGA).BN is a probabilistic approach to analyze relationships between stochastic events.In contrast with traditional methods where BN model is built by professionals,DGA is proposed for the automatic analysis of historical data and construction of BN for the estimation of system reliability.The whole solution space of BN structures is searched by DGA and a more accurate BN model is obtained.Efficacy of the proposed method is shown by some literature examples.

Two-Sample Bayesian Predictive Analyses for an Exponential Non-Homogeneous Poisson Process in Software Reliability

The Goel-Okumoto software reliability model is one of the earliest attempts to use a non-homogeneous Poisson process to model failure times observed during software test interval. The model is known as exponential NHPP model as it describes exponential software failure curve. Parameter estimation, model fit and predictive analyses based on one sample have been conducted on the Goel-Okumoto software reliability model. However, predictive analyses based on two samples have not been conducted on the model. In two-sample prediction, the parameters and characteristics of the first sample are used to analyze and to make predictions for the second sample. This helps in saving time and resources during the software development process. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model based on two samples. We have addressed three issues in two-sample prediction associated closely with software development testing process. Bayesian methods based on non-informative priors have been adopted to develop solutions to these issues. The developed methodologies have been illustrated by two sets of software failure data simulated from the Goel-Okumoto software reliability model.

Component-Oriented Reliability Analysis Based on Hierarchical Bayesian Model for an Open Source Software

The successful experience of adopting distributed development models in such open source projects includes GNU/Linux operating system, Apache HTTP server, Android, BusyBox, and so on. The open source project contains special features so-called software composition by which several geographically-dispersed compo-nents are developed in all parts of the world. We propose a method of component-oriented reliability as-sessment based on hierarchical Bayesian model and Markov chain Monte Carlo methods. Especially, we fo-cus on the fault-detection rate for each component reported to the bug tracking system. We can assess the reliability for the whole open source software system by using the confidence interval for each component. Also, we analyze actual software fault-count data to show numerical examples of reliability assessment for OSS.

Reliability Analysis for Retaining Pile in Foundation Pit Based on Bayesian Principle

Moso bamboo has the advantages of high short-term strength and reproducibility,appropriating for temporary supporting structure of shallow foundation pit.According to the displacement of the pile top from an indoor model test,the reliability of the supporting effect of the moso bamboo pile was analyzed.First,the calculation formula of reliability index was deduced based on themean-value first-order second-moment(MVFOSM)method and probability theory under ultimate limit state and serviceability limit state.Then,the dimensionless bias factor(the ratio of the measured value to the calculated value)was introduced to normalize the displacement.The mathematical characteristics of the displacement were estimated and optimized based on Bayesian theory.Finally,taking 2.5 as the design reliability index,the effect of safety factor,tolerable limit displacement,and the ratio of the ultimate limit displacement to the tolerable on reliability index was analyzed.The results show that the safety level of the supporting pile can be increased by 1–2 levels when the safety factor increases by 0.5.When the coefficient of variation of tolerable limit displacement is less than 0.3,the safety factor can be 2–2.5.And the ratio of the ultimate limit displacement to the tolerable has a great influence on the reliability index,when the soil conditions is well,the ratio can be 1.2–1.3.

The Effectiveness of the Squared Error and Higgins-Tsokos Loss Functions on the Bayesian Reliability Analysis of Software Failure Times under the Power Law Process

Reliability analysis is the key to evaluate software’s quality. Since the early 1970s, the Power Law Process, among others, has been used to assess the rate of change of software reliability as time-varying function by using its intensity function. The Bayesian analysis applicability to the Power Law Process is justified using real software failure times. The choice of a loss function is an important entity of the Bayesian settings. The analytical estimate of likelihood-based Bayesian reliability estimates of the Power Law Process under the squared error and Higgins-Tsokos loss functions were obtained for different prior knowledge of its key parameter. As a result of a simulation analysis and using real data, the Bayesian reliability estimate under the Higgins-Tsokos loss function not only is robust as the Bayesian reliability estimate under the squared error loss function but also performed better, where both are superior to the maximum likelihood reliability estimate. A sensitivity analysis resulted in the Bayesian estimate of the reliability function being sensitive to the prior, whether parametric or non-parametric, and to the loss function. An interactive user interface application was additionally developed using Wolfram language to compute and visualize the Bayesian and maximum likelihood estimates of the intensity and reliability functions of the Power Law Process for a given data.

Reliability‑Based Analysis of a Caisson Breakwater with the Application of Bayesian Inference

Caisson breakwaters are mainly constructed in deep waters to protect an area against waves.These breakwaters are con-ventionally designed based on the concept of the safety factor.However,the wave loads and resistance of structures have epistemic or aleatory uncertainties.Furthermore,sliding failure is one of the most important failure modes of caisson breakwaters.In most previous studies,for assessment purposes,uncertainties,such as wave and wave period variation,were ignored.Therefore,in this study,Bayesian reliability analysis is implemented to assess the failure probability of the sliding of Tombak port breakwater in the Persian Gulf.The mean and standard deviations were taken as random variables to consider dismissed uncertainties.For this purpose,the frst-order reliability method(FORM)and the frst principal curvature cor-rection in FORM are used to calculate the reliability index.The performances of these methods are verifed by importance sampling through Monte Carlo simulation(MCS).In addition,the reliability index sensitivities of each random variable are calculated to evaluate the importance of diferent random variables while calculating the caisson sliding.The results show that the reliability index is most sensitive to the coefcients of friction,wave height,and caisson weight(or concrete density).The sensitivity of the failure probability of each of the random variables and their uncertainties are calculated by the derivative method.Finally,the Bayesian regression is implemented to predict the statistical properties of breakwater sliding with non-informative priors,which are compared to Goda’s formulation,used in breakwater design standards.The analysis shows that the model posterior for the sliding of a caisson breakwater has a mean and standard deviation of 0.039 and 0.022,respectively.A normal quantile analysis and residual analysis are also performed to evaluate the correctness of the model responses.

Reliability-based design in rock engineering: Application of Bayesian regression methods to rock strength data

Reliability-based design (RBD) is being adopted by geotechnical design codes worldwide, and it is therefore necessary that rock engineering practice evolves to embrace RBD. This paper examines the Hoek-Brown (H-B) strength criterion within the RBD framework, and presents three distinct analyses using a Bayesian approach. Firstly, a compilation of intact compressive strength test data for six rock types is used to examine uncertainty and variability in the estimated H-B parameters m and σc, and corresponding predicted axial strength. The results suggest that within- and between-rock type variabilities are so large that these parameters need to be determined from rock testing campaigns, rather than reference values being used. The second analysis uses an extensive set of compressive and tensile (both direct and indirect) strength data for a granodiorite, together with a new Bayesian regression model, to develop joint probability distributions of m and σc suitable for use in RBD. This analysis also shows how compressive and indirect tensile strength data may be robustly used to fit an H-B criterion. The third analysis uses the granodiorite data to investigate the important matter of developing characteristic strength criteria. Using definitions from Eurocode 7, a formal Bayesian interpretation of characteristic strength is proposed and used to analyse strength data to generate a characteristic criterion. These criteria are presented in terms of characteristic parameters mk and σck, the values of which are shown to depend on the testing regime used to obtain the strength data. The paper confirms that careful use of appropriate Bayesian statistical analysis allows the H-B criterion to be brought within the framework of RBD. It also reveals that testing guidelines such as the International Society for Rock Mechanics and Rock Engineering (ISRM) suggested methods will require modification in order to support RBD. Importantly, the need to fully understand the implications of uncertainty in nonlinear strength criteria is identified.

Reliability Evaluation of CRH_(3C) Brake Pads Based on Bayes Method

In the reliability life evaluation of CRH_(3C) brake pads,the evaluation model of reliability life is put forward based on the Bayes method in the small sample. The correctness of evaluation model is validated by comparing and analyzing with the evaluation results based on Bootstrap simulation. Also by comparing the result with the semi-empirical method,the life evaluation results of the brake pads which are based on the Bayes method are more actual. The results which are based on the Bayes method can provide the theoretical basis and guidance for the repair and replacement of brake pads.

E-Bayesian Estimation of the Products Reliability when Testing Reveals no Failure

This paper develops a new method, named E-Bayesian estimation method, to estimate the reliability parameters. The E-Bayesian estimation method of the reliability are derived for the zero-failure data from the product with Binomial distribution. Firstly, for the product reliability, the definitions of E-Bayesian estimation were given, and on the base, expressions of the E-Bayesian estimation and hierarchical Bayesian estimation of the products reliability was given. Secondly, discuss properties of the E-Bayesian estimation. Finally, the new method is applied to a real zero-failure data set, and as can be seen, it is both efficient and easy to operate.