Bayesian Analysis of System Reliability using the Burr Distribution

On the mean remaining strength at the system level for some bivariate survival models based on exponential distribution

Bayesian approaches have been demonstrated as effective methods for reliability analysis of complex systems with small-amount data, which integrate prior information and sample data using Bayes’ theorem. However, there is an assumption that precise prior probability distributions are available for unknown parameters, yet these prior distributions are sometimes unavailable in practical engineering. A possible way to avoiding this assumption is to generalize Bayesian reliability analysis approach by using imprecise probability theory. In this paper, we adopt a set of imprecise Dirichlet distributions as priors to quantify uncertainty of unknown parameters and extend traditional Bayesian reliability analysis approach by introducing an imprecise Dirichlet model (IDM). When the prior information is rare, the result of imprecise Bayesian analysis method is too rough to support engineering decision-making, so we proposed an optimization model to reduce the imprecision of the new method. Spindles are crucial for machine tools and reliability data related to spindles of new-developed machine tools are often rare. We can then use the imprecise Bayesian reliability analysis method to assess its reliability. In this paper, we mainly investigate the reliability assessment of a motorized spindle to illustrate the effectiveness of the proposed method.

Bayesian reliability analysis (BRA) technique has been actively used in reliability assessment for engineered systems. However, there are two key controversies surrounding the BRA: the reasonableness of the prior and the consistency among all data sets. These issues have been debated in Bayesian analysis for many years. As we observed, they have not been resolved satisfactorily. These controversies have seriously hindered the applications of BRA as a useful reliability analysis tool to support engineering design. In this paper, a Bayesian reliability analysis methodology with a prior and data validation and adjustment scheme (PDVAS) is developed to address these issues. As the part of the PDVAS development, a consistency measure is first defined that judges the level of consistency among all data sets including the prior. The consistency measure is then used to adjust either the prior or the data or both to the extent that the prior and the data are statistically consistent. This prior and data validation and adjustment scheme is developed for Binomial sampling with Beta prior, called Beta-Binomial Bayesian model. The properties of the scheme are presented and discussed that provides some insights of PDVAS. Various forms of the adjustment formulas are shown, and a selection framework of a specific formula, based on engineering design and analysis knowledge, is established. Several illustrative examples are presented, which show the reasonableness, effectiveness, and usefulness of PDVAS. General discussion of the scheme is offered to enhance the Bayesian reliability analysis in engineering design for reliability assessment.

Prior information and elicitation are the prerequisite in Bayesian reliability inference. Multiple sources for priors such as probability fitting based on historical data and expert judgment are often available when estimating the reliability of complex systems. This paper investigates the integration of multiple priors in Bayesian reliability analysis. Specifically, methods for multiple priors' integration based on Bayesian Melding are investigated. The performance of the studied methods with different prior information integration algorithms such as the arithmetic and geometric averaging is investigated. The impacts of the prior misspecification and the pooling parameter selection for prior integration algorithms are also studied. In numerical examples, simulation methods are applied for posterior reliability inference under the proposed prior integration methods and the performance of the two methods are compared.

Burr distribution is one of the most important types of distribution in Burr system and has gained special attention. It has an important role in various disciplines, such as reliability analysis, life testing, survival analysis, actuarial science, economics, forestry, hydrology and meteorology. Thus, the parameter estimation for Burr distribution becomes an important thing to do. The frequentist approach using the maximum likelihood method is the most commonly used way to estimate the parameters of a distribution. In this paper we considered using the Bayesian method to estimate the shape parameter k of Burr distribution using gamma prior which is a conjugate prior. The Bayes estimate for the shape parameter k is obtained under the squared-error loss function (SELF) which is one of the symmetric loss function and the precautionary loss function (PLF) which is one of the asymmetric loss function. Through a simulation study, the comparison was made on the performance of the Bayes estimate for the shape parameter k under these two loss functions with respect to the mean-squared error (MSE) and the posterior risk.

A better understanding of the travel time distribution shape or pattern could improve the decision made by the transport operator to estimate the travel time required for the vehicle to travel from one place to another. Finding the most appropriate distribution to represent the day-to-day travel time variation of an individual link of a bus route is the main purpose of this study. Klang Valley, Malaysia is the study area for the research. A consecutive of 7 months ten bus routes automatic vehicle location (AVL) data are used to examine the distribution performance. The leading distribution proposed for the research is the Burr distribution. Both symmetrical and asymmetrical distributions that have been proposed in existing studies are also used for comparison purposes. Maximum likelihood estimation is applied for parameter estimation while loglikelihood value, Akaike information criterion (AIC) and Bayesian information criterion (BIC) are applied for performance assessment of the distributions. Promising results are obtained by the leading model in all different kinds of operating environment and could be treated as the preliminary preparation for further reliability analysis.

Optimizing gas turbine maintenance can reduce total operating costs but raises the risk of unit failures. A large pipeline fleet requires constant up time, but experiences variation in unit-to-unit reliability and availability. There is an ongoing desire to optimize maintenance budget to maximize fleet availability. This can be done using detailed maintenance records; however, analyzing these in-depth for the entire fleet can be cumbersome. In this study, a Bayesian reliability and availability analysis is used to predict failure risks for both the fleet and individual units using only high level summary information including MTBF, MTTR, and asset service factor. Bayesian statistics enable transformation of these summary metrics into predictive distributions for reliability and availability accounting for seasonal and annual variations. The study considers a fleet of gas turbines and compares individual units to the fleet to identify outliers. This provides the first step in prioritizing maintenance decisions.

Hardware systems are present in many fields of human activity. Markov models are sometimes used in hardware reliability, availability, and maintainability (RAM) modeling. They are especially useful in situations in which the system we want to analyze may be modeled with several states through which the system evolves, some of them corresponding to ON states, the rest to OFF states. We provide here RAM analyses of such systems within a Bayesian framework, addressing both short-term and long-term performance.

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.

The recent proliferation of Markov Chain Monte Carlo (MCMC) approaches has led to the use of the Bayesian inference in a wide variety of fields, including reliability engineering. With the current (and future) proliferation of new products, old problems continue to hamper us, while new challenges keep appearing. In Bayesian reliability, these include but are not limited to: (1) achieving and making use of prior information; (2) applying small data sets or system operating/environmental (SOE) data with big and complex data; and (3) making posterior inferences from high-dimensional numerical integration. To deal with old problems while meeting new challenges, this paper proposes an improved procedure for Bayesian reliability inference with MCMC, discussing modern reliability data and noting some applications where the Bayesian reliability approach with MCMC can be used. It also explores opportunities to use Bayesian reliability models to create stronger statistical methods from prior to posterior. Finally, it outlines some practical concerns and remaining challenges for future research.

The rapid growth in the use of wind energy calls for a more and more careful representation of wind speed probability distribution, both for identification and estimation purpose. In particular, a key point of the above identification and estimation aspects is the one of representing the values of wind speed probability distributions, which in the static case, as defined in the paper, may be expressed in terms of extreme upper This topic has indeed brought about an increasing number of studies in the last years, both for wind energy production assessment and also in risk and reliability analysis. Concerning the aspect of energy production, it is well known that a great sensitivity exists in the relationship between wind speed upper quantiles and the corresponding wind energy quantiles. Concerning risk and reliability analysis, the wind speed values characterization is useful for a proper understanding of the destructive wind forces which may affect mechanical tower reliability, and consequently a proper choice of the “cut off” wind speed value. The above modeling is however difficult due to uncertainty in wind speed probability distributions. For this purpose, the paper proposes a novel Bayes approach for the estimation of the probability that wind speed is lower than a prefixed value. The basic feature of the method is its being not dependent on the underlying wind speed model, be it the classical Weibull distribution, or alternative models such as the recently adopted Log-logistic or Burr distributions. A large set of numerical simulations are performed in the last part of the paper, in order to illustrate the feasibility and efficiency of the above method of estimation, especially when compared to the classical Maximum Likelihood method.

Applied mathematics in reliability engineering: Bayesian methods, Bayesian reliability

Describes how sampling distribution of the estimates of the reliability functions for various failure time distributions is either hard to obtain or is quite complicated and, as such, it becomes difficult to draw inferences about the reliability of a system. To overcome this situation, presents some distribution‐free inference techniques on system reliability and gives Bayesian reliability analysis using conjugate priors.

A Bayesian approach is used to make inferences given a random sample of observations from a Burr distribution. Complete and type-2 censored samples are considered and inferences are made on the unknown parameters and the reliability function. In the case of a type-2 censored sample prediction bounds are derived for the unobserved sample values.

In Reliability analysis, there are four main steps to perform. These steps are selection of the mathematical model that represents system states, selection of the mathematical representation that represents the operation of the system, quantitative and qualitative reliability analysis and to conclude the obtained results. In Faculty of Management Science and Informatics, there is a course called Reliability engineering in which reliability analysis is taught. During the course, the students are taught about the basic of the reliability engineering and reliability analysis. During this process, there are various topics that can be more challenging for a student, such as computation of the Importance measures. Thanks to this, we developed a software tool for such computation and this tool is also usable for the research activities in our faculty. By implementing this software tool in the teaching process in subject of the reliability engineering the students can better understand, how these measures can be computed and how can they be used for localizing weak spots in the system. Results from this tool can be then used in term projects and assignments by students of in research activities that uses these measures in reliability analysis.