Bayesian networks for multilevel system reliability

Bayesian networks have recently found many applications in systems reliability; however, the focus has been on binary outcomes. In this paper we extend their use to multilevel discrete data and discuss how to make joint inference about all of the nodes in the network. These methods are applicable when system structures are too complex to be represented by fault trees. The methods are illustrated through four examples that are structured to clarify the scope of the problem.     

Bayesian networks in reliability

Over the last decade, Bayesian networks (BNs) have become a popular tool for modelling many kinds of statistical problems. We have also seen a growing interest for using BNs in the reliability analysis community. In this paper we will discuss the properties of the modelling framework that make BNs particularly well suited for reliability applications, and point to ongoing research that is relevant for practitioners in reliability.     

Matrix-based system reliability method and applications to bridge networks

Using a matrix-based system reliability (MSR) method, one can estimate the probabilities of complex system events by simple matrix calculations. Unlike existing system reliability methods whose complexity depends highly on that of the system event, the MSR method describes any general system event in a simple matrix form and therefore provides a more convenient way of handling the system event and estimating its probability. Even in the case where one has incomplete information on the component probabilities and/or the statistical dependence thereof, the matrix-based framework enables us to estimate the narrowest bounds on the system failure probability by linear programming. This paper presents the MSR method and applies it to a transportation network consisting of bridge structures. The seismic failure probabilities of bridges are estimated by use of the predictive fragility curves developed by a Bayesian methodology based on experimental data and existing deterministic models of the seismic capacity and demand. Using the MSR method, the probability of disconnection between each city/county and a critical facility is estimated. The probability mass function of the number of failed bridges is computed as well. In order to quantify the relative importance of bridges, the MSR method is used to compute the conditional probabilities of bridge failures given that there is at least one city disconnected from the critical facility. The bounds on the probability of disconnection are also obtained for cases with incomplete information.     

System reliability and sensitivity factors via the MPPSS method

A procedure is presented for obtaining the reliability of series, parallel, and mixed systems. The method, called Most Probable Point System Simulation (MPPSS) is simple, and numerically-based. We do not make any claims of analytic advance. Nevertheless, the procedure is more accurate than current analytically-based bounding methods (e.g. bi-modal) and it is computationally efficient. In addition, the method can be used to obtain system sensitivity factors, that is, the importance of each random variable to the system reliability. The procedure is demonstrated through three example illustrations.     

An efficient analytical Bayesian method for reliability and system response updating based on Laplace and inverse first-order reliability computations

This paper presents an efficient analytical Bayesian method for reliability and system response updating without using simulations. The method includes additional information such as measurement data via Bayesian modeling to reduce estimation uncertainties. Laplace approximation method is used to evaluate Bayesian posterior distributions analytically. An efficient algorithm based on inverse first-order reliability method is developed to evaluate system responses given a reliability index or confidence interval. Since the proposed method involves no simulations such as Monte Carlo or Markov chain Monte Carlo simulations, the overall computational efficiency improves significantly, particularly for problems with complicated performance functions. A practical fatigue crack propagation problem with experimental data, and a structural scale example are presented for methodology demonstration. The accuracy and computational efficiency of the proposed method are compared with traditional simulation-based methods.     

A discrete-time Bayesian network reliability modeling and analysis framework

Dependability tools are becoming an indispensable tool for modeling and analyzing (critical) systems. However the growing complexity of such systems calls for increasing sophistication of these tools. Dependability tools need to not only capture the complex dynamic behavior of the system components, but they must be also easy to use, intuitive, and computationally efficient. In general, current tools have a number of shortcomings including lack of modeling power, incapacity to efficiently handle general component failure distributions, and ineffectiveness in solving large models that exhibit complex dependencies between their components. We propose a novel reliability modeling and analysis framework based on the Bayesian network (BN) formalism. The overall approach is to investigate timed Bayesian networks and to find a suitable reliability framework for dynamic systems. We have applied our methodology to two example systems and preliminary results are promising. We have defined a discrete-time BN reliability formalism and demonstrated its capabilities from a modeling and analysis point of view. This research shows that a BN based reliability formalism is a powerful potential solution to modeling and analyzing various kinds of system components behaviors and interactions. Moreover, being based on the BN formalism, the framework is easy to use and intuitive for non-experts, and provides a basis for more advanced and useful analyses such as system diagnosis.     

A simple minimal path method for estimating the weighted multi-commodity multistate unreliable networks reliability

The weighted multicommodity multistate unreliable network (WMMUN) is a novel network composed of multistate unreliable components (arcs and nodes) capable of transmitting different types of commodities in which capacity weight varies with components. It is an extension of the multistate network. The current method for evaluating the directed WMMUN reliability has been derived from minimal cut (MC) based algorithm. The existing best-known method needed extensive comparison and verification, and failed to find the real directed WMMUN reliability. A very simple algorithm based on minimal paths (MPs) is developed for the WMMUN reliability problem. The correctness and computational complexity of the proposed algorithm will be analyzed and proven. An example is given to illustrate how the WMMUN reliability is evaluated using the proposed algorithm. The relationships among all different versions of MPs are also clarified.     

Integrated system identification and reliability evaluation of stochastic building structures

System identification and reliability evaluation play a significant role in structural health monitoring to ensure the serviceability and safety of existing structures. Although the development of system identification methods has attained much attention and some degree of maturity, reliability evaluation of existing structures still remains a challenging problem especially when uncertainties in measurement data and inherent randomness, which are inevitably involved in civil structures, are considered. In this regard, this paper presents a framework for integrated system identification and reliability evaluation of stochastic building structures. Two algorithms are proposed to respectively evaluate component reliability and system reliability of stochastic building structures by combining a statistical moment-based system identification method and a probability density evolution equation-based reliability evaluation method. System identification is embedded in the procedure of reliability evaluation of a stochastic building structure. The uncertainties in both the structure and the external excitation are considered. Numerical examples show that the structural component and system reliabilities of a three-story shear building structure with three damage scenarios can be effectively evaluated by the proposed methods.     

A better method for verifying production reliability

Various schemes have been created for verifying that reliability is not degraded during production. These include the periodic performance of reliability tests during production, three versions of an all-equipment reliability test plan and Bayesian approaches. Each method has its drawbacks. The purpose of all of these is to verify that the production process is continuing to produce products of acceptable reliability, for which the long-existing tools of statistical process control are directly applicable and advantageous. A method of verifying production reliability based on the use of a control chart for failure rate is proposed as a better way than the current standards and alternatives discussed in this paper.     

A model for Bayesian software reliability analysis

Two problems which are of great interest in relation to software reliability are the prediction of future times to failure and the calculation of the optimal release time. An important assumption in software reliability analysis is that the reliability grows whenever bugs are found and removed. In this paper we present a model for software reliability analysis using the Bayesian statistical approach in order to incorporate in the analysis prior assumptions such as the (decreasing) ordering in the assumed constant failure rates of prescribed intervals. We use as prior model the product of gamma functions for each pair of subsequent interval constant failure rates, considering as the location parameter of the first interval the failure rate of the following interval. In this way we include the failure rate ordering information. Using this approach sequentially, we predict the time to failure for the next failure using the previous information obtained. Using also the relevant predictive distributions obtained, we calculate the optimal release time for two different requirements of interest: (a) the probability of an in‐service failure in a prescribed time t; (b) the cost associated with a single or more failures in a prescribed time t. Finally a numerical example is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Bayesian system reliability assessment under fuzzy environments

The Bayesian system reliability assessment under fuzzy environments is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. The (conventional) Bayes estimation method will be used to create the fuzzy Bayes point estimator of system reliability by invoking the well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. On the other hand, we also provide the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability. In order to achieve this purpose, we transform the original problem into a nonlinear programming problem. This nonlinear programming problem is then divided into four subproblems for the purpose of simplifying computation. Finally, the subproblems can be solved by using any commercial optimizers, e.g. GAMS or LINGO.     

A new reliability evaluation method for networks with imperfect vertices using BDD

As an efficient data structure for representation and manipulation of Boolean functions, binary decision diagrams (BDDs) have been applied to network reliability analysis. However, most of the existing BDD methods on network reliability analysis have assumed perfectly reliable vertices, which is often not true for real‐world networks where the vertices can fail because of factors such as limited resources (eg, power and memory) or harsh operating environments. Extensions have been made to the existing BDD methods (particularly, edge expansion diagram and boundary set–based methods) to address imperfect vertices. But these extended methods have various constraints leading to problems in accuracy or space efficiency. To overcome these constraints, in this paper, we propose a new BDD‐based algorithm called ordered BDD dependency test for K‐terminal network reliability analysis considering both edge and vertex failures. Based on a newly defined concept “dependency set”, the proposed algorithm can accurately compute the reliability of networks with imperfect vertices. In addition, the proposed algorithm has no restrictions on the starting vertex for the BDD model construction. Comprehensive examples and experiments are provided to show effectiveness of the proposed approach.     

Integration of computation and testing for reliability estimation

This paper develops a methodology to integrate reliability testing and computational reliability analysis for product development. The presence of information uncertainty such as statistical uncertainty and modeling error is incorporated. The integration of testing and computation leads to a more cost-efficient estimation of failure probability and life distribution than the tests-only approach currently followed by the industry. A Bayesian procedure is proposed to quantify the modeling uncertainty using random parameters, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. An adaptive method is developed to determine the number of tests needed to achieve a desired confidence level in the reliability estimates, by combining prior computational prediction and test data. Two kinds of tests — failure probability estimation and life estimation — are considered. The prior distribution and confidence interval of failure probability in both cases are estimated using computational reliability methods, and are updated using the results of tests performed during the product development phase.     

Efficient surrogate models for reliability analysis of systems with multiple failure modes

Despite many advances in the field of computational reliability analysis, the efficient estimation of the reliability of a system with multiple failure modes remains a persistent challenge. Various sampling and analytical methods are available, but they typically require accepting a tradeoff between accuracy and computational efficiency. In this work, a surrogate-based approach is presented that simultaneously addresses the issues of accuracy, efficiency, and unimportant failure modes. The method is based on the creation of Gaussian process surrogate models that are required to be locally accurate only in the regions of the component limit states that contribute to system failure. This approach to constructing surrogate models is demonstrated to be both an efficient and accurate method for system-level reliability analysis.     

Reliability graph with general gates: an intuitive and practical method for system reliability analysis

In this paper, we propose an intuitive and practical method for system reliability analysis. Among the existing methods for system reliability analysis, reliability graph is particularly attractive due to its intuitiveness, even though it is not widely used for system reliability analysis. We provide an explanation for why it is not widely used, and propose a new method, named reliability graph with general gates, which is an extension of the conventional reliability graph. An evaluation method utilizing existing commercial or free software tools are also provided. We conclude that the proposed method is intuitive, easy-to-use, and practical while as powerful as fault tree analysis, which is currently the most widely used method for system reliability analysis.     

Combined hardware and software aspects of reliability

This paper presents the similarities and differences between hardware, software and system reliability. Relative contributions to system failures are shown for software and hardware and failure and recovery propensities are also discussed. Reliability, availability and maintainability (RAM) concepts have been broadly developed for software reliability than hardware reliability. Extending these software concepts to hardware and system reliability helps in examining the reliability of complex systems. The paper concludes with assurance techniques for defending against faults. Most of the techniques discussed originate in software reliability but apply to all aspects of a system. Also, the effects of redundancy on overall system availability are shown.     

Building prior distributions to support Bayesian reliability growth modelling using expert judgement

A sound methodology for the elicitation of subjective expert judgement is a pre-requisite for specifying prior distributions for the parameters of reliability growth models. In this paper, we describe an elicitation process that is developed to ensure valid data are collected by suggesting how possible bias might be identified and managed. As well as discussing the theory underpinning the elicitation process, the paper gives practical guidance concerning its implementation during reliability growth testing. The collection of subjective data using the proposed elicitation process is embedded within a Bayesian reliability growth modelling framework and reflections upon its practical use are described.     

A response surface method based on weighted regression for structural reliability analysis

Approximation methods such as the response surface method (RSM) are widely used to alleviate the computational burden of engineering analyses. For reliability analysis, the common approach in the RSM is to use regression methods based on least square methods. However, for structural reliability problems, RSMs should approximate the performance function around the design point where its value is close to zero. Therefore, in this study, a new response surface called ADAPRES is proposed, in which a weighted regression method is applied in place of normal regression. The experimental points are also selected from the region where the design point is most likely to exist. Examples are given to demonstrate the benefit of the proposed method for both numerical and implicit performance functions.     

A new directional simulation method for system reliability. Part II: application of neural networks

A challenge in directional importance sampling is in identifying the location and the shape of the importance sampling density function when a realistic limit state for a structural system is considered in a finite element-supported reliability analysis. Deterministic point refinement schemes, previously studied in place of directional importance sampling, can be improved by prior knowledge of the limit state. This paper introduces two types of neural networks that identify the location and shape of the limit state quickly and thus facilitate directional simulation-based reliability assessment using the deterministic Fekete point sets introduced in the companion paper. A set of limit states composed of linear functions are used to test the efficiency and possible directional preference of the networks. These networks are shown in the tests and examples to reduce the simulation effort in finite element-based reliability assessment.     

A management view of software reliability

Projects with a high software content are frequently completed late, with overrun costs and inadequate performance. Much of this is due to inadequate management, mainly caused by a lack of availability or knowledge of the criteria, methods and tools on which effective management should be based. Examples are given of the most frequent problems met by project managers. A plea is made for a vigorous effort to be made in developing management strategies which will allow the tools and techniques now being researched and developed to be used effectively for producing the high quality software needed for systems.