A generic method for estimating system reliability using Bayesian networks

This study presents a holistic method for constructing a Bayesian network (BN) model for estimating system reliability. BN is a probabilistic approach that is used to model and predict the behavior of a system based on observed stochastic events. The BN model is a directed acyclic graph (DAG) where the nodes represent system components and arcs represent relationships among them. Although recent studies on using BN for estimating system reliability have been proposed, they are based on the assumption that a pre-built BN has been designed to represent the system. In these studies, the task of building the BN is typically left to a group of specialists who are BN and domain experts. The BN experts should learn about the domain before building the BN, which is generally very time consuming and may lead to incorrect deductions. As there are no existing studies to eliminate the need for a human expert in the process of system reliability estimation, this paper introduces a method that uses historical data about the system to be modeled as a BN and provides efficient techniques for automated construction of the BN model, and hence estimation of the system reliability. In this respect K2, a data mining algorithm, is used for finding associations between system components, and thus building the BN model. This algorithm uses a heuristic to provide efficient and accurate results while searching for associations. Moreover, no human intervention is necessary during the process of BN construction and reliability estimation. The paper provides a step-by-step illustration of the method and evaluation of the approach with literature case examples.     

Predictive maintenance of complex system with multi-level reliability structure

Onboard sensors, which constantly monitor the states of a system and its components, have made the predictive maintenance (PdM) of a complex system possible. To date, system reliability has been extensively studied with the assumption that systems are either single-component systems or they have a deterministic reliability structure. However, in many realistic problems, there are complex multi-component systems with uncertainties in the system reliability structure. This paper presents a PdM scheme for complex systems by employing discrete time Markov chain models for modelling multiple degradation processes of components and a Bayesian network (BN) model for predicting system reliability. The proposed method can be considered as a special type of dynamic Bayesian network because the same BN is repeatedly used over time for evaluating system reliability and the inter-time–slice connection of the same node is monitored by a sensor. This PdM scheme is able to make probabilistic inference at any system level, so PdM can be scheduled accordingly.     

Improving the analysis of dependable systems by mapping fault trees into Bayesian networks

Bayesian Networks (BN) provide a robust probabilistic method of reasoning under uncertainty. They have been successfully applied in a variety of real-world tasks but they have received little attention in the area of dependability. The present paper is aimed at exploring the capabilities of the BN formalism in the analysis of dependable systems. To this end, the paper compares BN with one of the most popular techniques for dependability analysis of large, safety critical systems, namely Fault Trees (FT). The paper shows that any FT can be directly mapped into a BN and that basic inference techniques on the latter may be used to obtain classical parameters computed from the former (i.e. reliability of the Top Event or of any sub-system, criticality of components, etc). Moreover, by using BN, some additional power can be obtained, both at the modeling and at the analysis level. At the modeling level, several restrictive assumptions implicit in the FT 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 the two methodologies is carried out by means of a running example, taken from the literature, that consists of a redundant multiprocessor system.     

Reliability assessment for systems suffering common cause failure based on Bayesian networks and proportional hazards model

The Bayesian network (BN) is an efficient tool for probabilistic modeling and causal inference, and it has gained considerable attentions in the field of reliability assessment. The common cause failure (CCF) is simultaneous failure of multiple elements in a system under a common cause, and it is a common phenomenon in engineering systems with dependent elements. Several models and methods have been proposed for modeling and assessment of complex systems with CCF. In this paper, a new reliability assessment method is proposed for the systems suffering from CCF in a dynamic environment. The CCF among components is characterized by a BN, which allows for bidirectional reasoning. A proportional hazards model is applied to capture the dynamic working environment of components and then the reliability function of the system is obtained. The proposed method is validated through an illustrative example, and some comparative studies are also presented.     

A Framework for Reliability Prediction During Product Development Process Incorporating Engineering Judgments

This paper presents a comprehensive framework for reliability prediction during the product development process. Early in the product development process, there is typically little or no quantitative evidence to predict the reliability of the new concept except indirect or qualitative information. The proposed framework addresses the issue of utilizing qualitative information in the reliability analysis. The framework is based on the Bayesian approach. The fuzzy logic theory is used to enhance the capability of the Bayesian approach to deal with qualitative information. This paper proposes to extract the information from various design tools and design review records and incorporate it into the Bayesian framework through a fuzzy inference system. The Weibull distribution is considered as failure/survival time distribution with the assumption of a known value of shape factor. Initial parameters of the Weibull distribution are estimated from warranty data of prior systems to estimate the initial Bayesian parameter ( λ_t). The applicability of the framework is illustrated via an example.     

Modeling of system reliability Petri nets with aging tokens

The paper addresses the dynamic modeling of degrading and repairable complex systems. Emphasis is placed on the convenience of modeling for the end user, with special attention being paid to the modeling part of a problem, which is considered to be decoupled from the choice of solution algorithms. Depending on the nature of the problem, these solution algorithms can include discrete event simulation or numerical solution of the differential equations that govern underlying stochastic processes. Such modularity allows a focus on the needs of system reliability modeling and tailoring of the modeling formalism accordingly. To this end, several salient features are chosen from the multitude of existing extensions of Petri nets, and a new concept of aging tokens (tokens with memory) is introduced. The resulting framework provides for flexible and transparent graphical modeling with excellent representational power that is particularly suited for system reliability modeling with non-exponentially distributed firing times. The new framework is compared with existing Petri-net approaches and other system reliability modeling techniques such as reliability block diagrams and fault trees. The relative differences are emphasized and illustrated with several examples, including modeling of load sharing, imperfect repair of pooled items, multiphase missions, and damage-tolerant maintenance. Finally, a simple implementation of the framework using discrete event simulation is described.     

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.     

Non-stationary functional series modeling and analysis of hardware reliability series: a comparative study using rail vehicle interfailure times

A novel reliability modeling and analysis framework based upon the distinct class of non-stationary Functional Series (FS) models is introduced. This framework allows for non-stationary reliability modeling, evolution assessment, analysis (including non-stationarity assessment, dependency assessment, as well as cycle detection), and prediction. The Functional Series framework is used for the modeling and analysis of two rail vehicle reliability series (Times Between Failures, TBFs), while comparisons with alternative (ARIMA, adaptive RARMA–RML, and Bayesian) modeling approaches are also made. The results indicate the advantages and usefulness of the Functional Series framework, as the TBF modeling accuracy is improved, its non-stationarity and serial dependency are established, the presence of cyclic patterns is revealed, and reliability evolution is assessed. It is conjectured that the cycles revealed in the TBF series may be related to maintenance policies. Finally, reliability prediction is shown to be feasible, although the “larger” excursions in the TBF series are difficult to accurately predict.     

A simple approach to search for all d-MCs of a limited-flow network

Many real-world systems are multistate systems composed of multistate components in which the reliability can be computed in terms of the lower bound points of level d, called d-MCs. Such systems (electric power, transportation, etc.) may be regarded as flow networks whose arcs have independent, discrete, limited and multivalued random capacities. In this study, all MCs are assumed to be known in advance and we focused on how to find the entire d-MCs before calculating the reliability value of a network. Just based on the definition of d-MC, we develop an intuitive algorithm which is better than the best-known existing method. Analysis of our algorithm and comparison to existing algorithms shows that our proposed method is easier to understand and implement. Finally, the computational complexity of the proposed algorithm is analysed and compared with the existing methods.     

Bayesian updating of reliability of civil infrastructure facilities based on condition-state data and fault-tree model

This paper considers a difficult but practical circumstance of civil infrastructure management—deterioration/failure data of the infrastructure system are absent while only condition-state data of its components are available. The goal is to develop a framework for estimating time-varying reliabilities of civil infrastructure facilities under such a circumstance. A novel method of analyzing time-varying condition-state data that only reports operational/non-operational status of the components is proposed to update the reliabilities of civil infrastructure facilities. The proposed method assumes that the degradation arrivals can be modeled as a Poisson process with unknown time-varying arrival rate and damage impact and that the target system can be represented as a fault-tree model. To accommodate large uncertainties, a Bayesian algorithm is proposed, and the reliability of the infrastructure system can be quickly updated based on the condition-state data. Use of the new method is demonstrated with a real-world example of hydraulic spillway gate system.     

Incorporating organizational factors into probabilistic safety assessment of nuclear power plants through canonical probabilistic models

The ω-factor approach is a method that explicitly incorporates organizational factors into Probabilistic safety assessment of nuclear power plants. Bayesian networks (BNs) are the underlying formalism used in this approach. They have a structural part formed by a graph whose nodes represent organizational variables, and a parametric part that consists of conditional probabilities, each of them quantifying organizational influences between one variable and its parents in the graph. The aim of this paper is twofold. First, we discuss some important limitations of current procedures in the ω-factor approach for either assessing conditional probabilities from experts or estimating them from data. We illustrate the discussion with an example that uses data from Licensee Events Reports of nuclear power plants for the estimation task. Second, we introduce significant improvements in the way BNs for the ω-factor approach can be constructed, so that parameter acquisition becomes easier and more intuitive. The improvements are based on the use of noisy-OR gates as model of multicausal interaction between each BN node and its parents.     

An algorithm of discrete-time Bayesian network for reliability analysis of multilevel system with warm spare gate

Discrete-time Bayesian network (DTBN) is a popular tool for the reliability analysis of dynamic systems, which, however, is insufficient in analyzing the reliability of multilevel system (MLS) with warm spare (WSP) gates. Additionally, conventional DTBNs are not able to consider the situation that dormant components and primary components may fail during the same time interval. To this end, this paper analyzes the dynamic reliability characteristics of dormant systems with WSP gates by utilizing DTBNs. Moreover, an algorithm of modeling the conditional probability table (CPT) for WSP gates together with a new schedule of constructing dynamic Bayesian networks is put forward. The validation of the proposed techniques is implemented by Monte Carlo simulation (MCS) and reliability analysis of an actual communication station system.     

Inference in hybrid Bayesian networks

Since the 1980s, Bayesian networks (BNs) have become increasingly popular for building statistical models of complex systems. This is particularly true for boolean systems, where BNs often prove to be a more efficient modelling framework than traditional reliability techniques (like fault trees and reliability block diagrams). However, limitations in the BNs’ calculation engine have prevented BNs from becoming equally popular for domains containing mixtures of both discrete and continuous variables (the so-called hybrid domains). In this paper we focus on these difficulties, and summarize some of the last decade's research on inference in hybrid Bayesian networks. The discussions are linked to an example model for estimating human reliability.     

Probability of infancy problems for space launch vehicles

This paper addresses the treatment of ‘infancy problems’ in the reliability analysis of space launch systems. To that effect, we analyze the probability of failure of launch vehicles in their first five launches. We present methods and results based on a combination of Bayesian probability and frequentist statistics designed to estimate the system's reliability before the realization of a large number of launches. We show that while both approaches are beneficial, the Bayesian method is particularly useful when the experience base is small (i.e. for a new rocket). We define reliability as the probability of success based on a binary failure/no failure event. We conclude that the mean failure rates appear to be higher in the first and second flights (≈1/3 and 1/4, respectively) than in subsequent ones (third, fourth and fifth), and Bayesian methods do suggest that there is indeed some difference in launch risk over the first five launches. Yet, based on a classical frequentist analysis, we find that for these first few flights, the differences in the mean failure rates over successive launches or over successive generations of vehicles, are not statistically significant (i.e. do not meet a 95% confidence level). This is true because the frequentist analysis is based on a fixed confidence level (here: 95%), whereas the Bayesian one allows more flexibility in the conclusions based on a full probability density distribution of the failure rate and therefore, permits better interpretation of the information contained in a small sample. The approach also gives more insight into the considerable uncertainty in failure rate estimates based on small sample sizes.     

Bayes approach in RDT using accelerated and long-term life data

A common problem of reliability demonstration testing (RDT) is the magnitude of total time on test required to demonstrate reliability to the consumer’s satisfaction, particularly in the case of high reliability components. One solution is the use of accelerated life testing (ALT) techniques. Another is to incorporate prior beliefs, engineering experience, or previous data into the testing framework. This may have the effect of reducing the amount of testing required in the RDT in order to reach a decision regarding conformance to the reliability specification. It is in this spirit that the use of a Bayesian approach can, in many cases, significantly reduce the amount of testing required.We demonstrate the use of this approach to estimate the acceleration factor in the Arrhenius reliability model based on long-term data given by a manufacturer of electronic components (EC). Using the Bayes approach we consider failure rate and acceleration factor to vary randomly according to some prior distributions. Bayes approach enables for a given type of technology the optimal choice of test plan for RDT under accelerated conditions when exacting reliability requirements must be met. These requirements are given by a hypothetical consumer by two different ways. The calculation of posterior consumer’s risk is demonstrated in both cases.The test plans are optimum in that they take into account Var{λ|data}, posterior risk, E{λ|data}, Median λ or other percentiles of λ at data observed at the accelerated conditions. The test setup assumes testing of units with time censoring.     

New insights on multi-state component criticality and importance

In this paper, new importance measures for multi-state systems with multi-state components are introduced and evaluated. These new measures complement and enhance current work done in the area of multi-state reliability. In general, importance measures are used to evaluate and rank the criticality of component or component states with respect to system reliability. The focus of the study is to provide intuitive and clear importance measures that can be used to enhance system reliability from two perspectives: (1) how a specific component affects multi-state system reliability and (2) how a particular component state or set of states affects multi-state system reliability. The first measure unsatisfied demand index, provides insight regarding a component or component state contribution to unsatisfied demand. The second measure multi-state failure frequency index, elaborates on an approach that quantifies the contribution of a particular component or component state to system failure. Finally, the multi-state redundancy importance identifies where to allocate component redundancy as to improve system reliability. The findings of this study indicate that both perspectives can be used to complement each other and as an effective tool to assess component criticality. Examples illustrate and compare the proposed measures with previous multi-state importance measures.     

A fuzzy logic based approach to reliability improvement estimation during product development

During early stages of product development process, a vast amount of knowledge and information is generated. However, most of it is subjective (imprecise) in nature and remains unutilized. This paper presents a formal structure for capturing this information and knowledge and utilizing it in reliability improvement estimation. The information is extracted as improvement indices from various design tools, experiments, and design review records and treated as fuzzy numbers or linguistic variables. Fuzzy reasoning method is used to combine and quantify the subjective information to map their impact on product reliability. The crisp output of the fuzzy reasoning process is treated as new evidence and incorporated into a Bayesian framework to update the reliability estimates. A case example is presented to demonstrate the proposed approach.     

Bayesian network approach to change propagation analysis

The prediction of change propagation is one of the important issues in engineering change management. The aim of this article is to explore the capability of Bayesian network (BN), which is an emerging tool for a wide range of risk management, in modeling and analysis of change propagation. To this end, we compare the BN with change prediction method (CPM), which is the most established probabilistic methods for predicting change propagation. This paper shows that a CPM-based model can be converted into an equivalent BN, and the probabilistic inference technique on the latter results in the same change prediction result obtained from the former. Then, this paper shows that several improvements can be obtained at various levels using the BN. At the modeling level, complex relationship between components such as combined effect of simultaneous changes or multistate relationship can be naturally represented with the BN. At the analysis level, various change propagation scenarios can be analyzed using probabilistic inference on the BN. Finally, BN provides a robust framework for learning change propagation probabilities from empirical data. The case study is conducted to show the feasibility of the model.     

Application of Bayesian Methods for Age‐Dependent Reliability Analysis

In this article, the authors present a general methodology for age‐dependent reliability analysis of degrading or ageing components, structures and systems. The methodology is based on Bayesian methods and inference—its ability to incorporate prior information and on ideas that ageing can be thought of as age‐dependent change of beliefs about reliability parameters (mainly failure rate), when change of belief occurs not only because new failure data or other information becomes available with time but also because it continuously changes due to the flow of time and the evolution of beliefs. The main objective of this article is to present a clear way of how practitioners can apply Bayesian methods to deal with risk and reliability analysis considering ageing phenomena. The methodology describes step‐by‐step failure rate analysis of ageing components: from the Bayesian model building to its verification and generalization with Bayesian model averaging, which as the authors suggest in this article, could serve as an alternative for various goodness‐of‐fit assessment tools and as a universal tool to cope with various sources of uncertainty. The proposed methodology is able to deal with sparse and rare failure events, as is the case in electrical components, piping systems and various other systems with high reliability. In a case study of electrical instrumentation and control components, the proposed methodology was applied to analyse age‐dependent failure rates together with the treatment of uncertainty due to age‐dependent model selection. Copyright © 2013 John Wiley & Sons, Ltd.     

A new algorithm for generating minimal cut sets in k-out-of-n networks

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. A k-out-of-n network is a special network in that some nodes must receive at least k (>1) flows from all of their input edges (n). In real-life cases, many networks such as computer and telecommunications include k-out-of-n nodes for redundancy. The minimal-cut-node-set (MCN) is the major and fundamental tools for evaluating the k-out-of-n network reliability. In this study, a very simple algorithm based on some intuitive theorems that characterize the structure of the MCN is developed to solve the k-out-of-n network reliability. Compared to the existing algorithms, the proposed algorithm generates all k-out-of-n MCs without duplication based on fewer MCNs and fewer (k-out-of-n MC) candidates. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithms. The correctness of the proposed algorithm will be analyzed and proven. One example is illustrated to show how all k-out-of-n MCs are generated, verified, and implemented to evaluate the k-out-of-n network reliability using the proposed algorithm.