Exact combinatorial reliability analysis of dynamic systems with sequence-dependent failures

Many real-life fault-tolerant systems are subjected to sequence-dependent failure behavior, in which the order in which the fault events occur is important to the system reliability. Such systems can be modeled by dynamic fault trees (DFT) with priority-AND (pAND) gates. Existing approaches for the reliability analysis of systems subjected to sequence-dependent failures are typically state-space-based, simulation-based or inclusion-exclusion-based methods. Those methods either suffer from the state-space explosion problem or require long computation time especially when results with high degree of accuracy are desired. In this paper, an analytical method based on sequential binary decision diagrams is proposed. The proposed approach can analyze the exact reliability of non-repairable dynamic systems subjected to the sequence-dependent failure behavior. Also, the proposed approach is combinatorial and is applicable for analyzing systems with any arbitrary component time-to-failure distributions. The application and advantages of the proposed approach are illustrated through analysis of several examples.     

Quantitative Analysis of Dynamic Fault Trees Based on the Structure Function

This paper presents a probabilistic model of dynamic gates which allows to perform the quantitative analysis of any dynamic fault tree (DFT) from its structure function. Both these probabilistic models and the quantitative analysis which can be performed thanks to them can accommodate any failure distribution of basic events. We illustrate our approach on a DFT example from the literature. Copyright © 2013 John Wiley & Sons, Ltd.     

Quantitative analysis of a fault tree with priority AND gates

A method for calculating the exact top event probability of a fault tree with priority AND gates and repeated basic events is proposed when the minimal cut sets are given. A priority AND gate is an AND gate where the input events must occur in a prescribed order for the occurrence of the output event. It is known that the top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, this method is not realistic for a complex system model because the number of states which should be considered in the Markov analysis increases explosively as the number of basic events increases. To overcome the shortcomings of the Markov model, we propose an alternative method to obtain the top event probability in this paper. We assume that the basic events occur independently, exponentially distributed, and the component whose failure corresponds to the occurrence of the basic event is non-repairable. First, we obtain the probability of occurrence of the output event of a single priority AND gate by Markov analysis. Then, the top event probability is given by a cut set approach and the inclusion–exclusion formula. An efficient procedure to obtain the probabilities corresponding to logical products in the inclusion–exclusion formula is proposed. The logical product which is composed of two or more priority AND gates having at least one common basic event as their inputs is transformed into the sum of disjoint events which are equivalent to a priority AND gate in the procedure. Numerical examples show that our method works well for complex systems.     

Quantification of Highly Coupled Dynamic Fault Tree Using IRVPM and SBDD

Dynamic fault tree (DFT) is a commonly used method to model systems having sequence‐dependent and function‐dependent failure behaviors. The failure structure function of a DFT can be expressed by logic OR of all minimal cut sequences, that is, minimal cut sequence set (MCSS). The occurrence probability to the top event of a DFT can be calculated using inclusion–exclusion (IE) principle based on enumerating the MCSS. However, the IE‐based approach would have exponential evaluation complexity. Then, a sequential binary decision diagram (SBDD)‐based method is proposed and successfully applied to analyze simple dynamic systems. This method is more efficient than IE‐based method in asymptotic analysis. But this method cannot handle complex systems modeled by different highly coupled dynamic gates. In this paper, we put forward using Independent Random Variable Probabilistic Model‐based plus SBDD‐based methods to quantify an MCSS to obtain the failure probability of a complex DFT. The results obtained by the proposed method are exactly matched with those obtained by the existing methods. In addition, this method enhances the analyzing ability of the original SBDD and retains the advantage of high computational efficiency. The application and advantage of our proposed method is demonstrated by a case study. Copyright © 2014 John Wiley & Sons, Ltd.     

Construction of event-tree/fault-tree models from a Markov approach to dynamic system reliability

While the event-tree (ET)/fault-tree (FT) methodology is the most popular approach to probability risk assessment (PRA), concerns have been raised in the literature regarding its potential limitations in the reliability modeling of dynamic systems. Markov reliability models have the ability to capture the statistical dependencies between failure events that can arise in complex dynamic systems. A methodology is presented that combines Markov modeling with the cell-to-cell mapping technique (CCMT) to construct dynamic ETs/FTs and addresses the concerns with the traditional ET/FT methodology. The approach is demonstrated using a simple water level control system. It is also shown how the generated ETs/FTs can be incorporated into an existing PRA so that only the (sub)systems requiring dynamic methods need to be analyzed using this approach while still leveraging the static model of the rest of the system.     

数字电路可测性设计的一种故障定位方法

在逻辑函数ReedMuller模式的电路可测性设计方面，文章采用AND门阵列和XOR门树结构来设计电路，提出了一种设计方案，可实现任意逻辑函数的功能，而且所得电路具有通用测试集和完全可故障定位的特点。给出了进行故障定位的方法，并可把它应用于其他相关电路的可测性设计。     

基于动态故障树的CTCS-3级ATP系统可靠性分析简

 针对传统的可靠性分析方法分析CTCS-3级ATP系统动态失效问题的不足,提出采用动态故障树分析其可靠性。首先,分析系统的结构和功能建立动态故障树模型;其次,采用深度优先最左遍历算法搜索动态故障树模型,得到独立的子树;最后,在引入可修系统可靠性指标基础上,采用解析法和马尔科夫矩阵迭代法求解子树,结合分层迭代方法对动态故障树分析法改进,以减小运算量,使得上述可靠性指标能用于CTCS-3级ATP系统的可靠性评估。计算所得可靠性指标与可靠性框图分析得到的结果对比表明：动态故障树能够更好地描述系统的冗余性和容错性等特点,提高了可靠性指标的精度。     

Assessment of fire vulnerability for nuclear power plant safety systems by combining fault-and success-oriented logic with physical models

The time behaviour of potential accident sequences may carry important information regarding nuclear power plant (NPP) safety operation and shutdown. In the case of external and environmental events, the ability of NPP components to operate correctly can be changed dramatically in a short time. In contrast to the failures caused by internal events, these two groups of undesirable events may lead to dynamic dependent failures among components of one or several systems. Such kinds of failure should be taken into account in the models of NPP behaviour. To evaluate how successfully the tasks of the safety systems will be carded out, logical models such as fault trees are usually used. The fault trees are not efficient at describing the short-term changes of the failure probabilities for system components. A method that has some advantages over the pure fault tree logic is proposed. The main features of the method are demonstrated by using examples.     

Comparison of Markov model and fault tree approach in determining initiating event frequency for systems with two train configurations

This study explores the use of Markov models in some areas of systems analysis in which time evolution of the system may be a significant factor in influencing the system reliability or availability. Comparisons are made between the Markov models and the time-averaged fault tree models for determining support system failure initiating event frequency in a nuclear power plant, for both power and shutdown conditions. Factors affecting consistency between the fault tree approach and the Markov model approach are studied for systems with common two train configurations. A correlation is developed to estimate the ratio between initiator frequencies through both approaches for a two parallel component system. Insights are developed as to when time averaged and simplified fault tree models support a good approximation to the more rigorous time-dependent Markov models.     

The robustness of markov reliability models

Markov models are an established part of current systems reliability and availability analysis. They are extensively used in various applications, including, in particular, electrical power supply systems. One of their advantages is that they considerably simplify availability evaluation so that the availability of very large and complex systems can be computed. It is generally assumed, with some justification, that the results obtained from such Markov reliability models are relatively robust. It has, however, been known for some time, that practical time to failure distributions are frequently non-exponential, particular attention being given in much reliability work to the Weibull family. Morover, recently additional doubt has been case on the validity of the Markov approach, both because of the work of Professor Kline and others on the non-exponentiality of practical repair time distribution, and because of the advantages to be obtained in terms of modelling visibility of the alternative simulation approach. In this paper we employ results on the ability of the k-out-of-n systems to span the coherent set to investigate the robustness of Markov reliability models based upon a simulation investigation of coherent systems of up to 10 identical components. We treat the case where adequate repair facilities are available for all components. The effects upon the conventional transient and steady-state measures of Weibull departures from exponentiality are considered. In general, the Markov models are found to be relatively robust, with alterations to failure distributions being more important than those to repair distributions, and decreasing hazard rates more critical than increasing hazard rates. Of the measures studied, the mean time to failure is most sensitive to variations in distributional shape.     

Negating a Generalized Cut Sequence: Bridging the Gap Between Dynamic Fault Trees Quantification and Sum of Disjoint Products Methods

Dynamic fault trees (DFTs) are powerful tools to model industrial systems having dynamic failure mechanisms, such as sequence‐ and function‐dependent failure behaviors. Yet for large and complex DFTs, their quantitative analyses are still of great challenges. Up to now, many researchers have presented several approaches to deal with this problem, and among which, the sum of disjoint products (SDP) methods, such as dynamic binary decision tree, sequential binary decision diagram (SBDD), and improved SBDD, have proven to be an efficient way. In SDP methods, negating a generalized cut sequence is an unavoidable task. Yet, for a complex cut sequence expression where normal, cold and warm spares basic events coexist, its negating operation is still difficult and needs to be further studied. In this paper, based on De Morgan theorem, improved explicit formulas for negating a generalized cut sequences are presented. The new concept of universal set of basic event and its operating rules are proposed to deduce the simplified expressions of general enforcing occurring cut sequences and warm spares occurring cut sequences. To validate the presented approaches, a typical system DFT is analyzed. The results indicate the reasonability and effectiveness of the improved negating formulas. Copyright © 2016 John Wiley & Sons, Ltd.     

Identification of independent modules in fault trees which contain dependent basic events

The reliability performance of a system is frequently a function of component failures of which some are independent whilst others are interdependent. It is possible to represent the system failure logic in a fault tree diagram, however only the sections containing independent events can be assessed using the conventional fault tree analysis methodology. The analysis of the dependent sections will require a Markov analysis. Since the efficiency of the Markov analysis largely depends on the size of the established Markov model, the key is to extract from the fault tree the smallest sections which contain dependencies. This paper proposes a method aimed at establishing the smallest Markov model for the dependencies contained within the fault tree.     

A methodology of alarm filtering using dynamic fault tree

This article presents a new approach for filtering the faults, thanks to the defined dynamic fault tree (DFT). The proposed methodology includes the dependencies between fault events in the models. Two problems must thus be solved: they relate to the filtering of false alarms, and the reduction of the size of the ambiguity of fault isolation related to the occurrence of a failure. In response to the expressed need for diagnosis, as well as for the need for filtering and localization of the failures, it is necessary to introduce new dynamic gates, making it possible to translate new dependencies, relationships. Based on previous techniques, the approach presented in this paper is based on four peculiar powerful features. First, the concept of the precedence between events is taken into account in order to resort to an adapted configuration for the fault isolation. Second, another relevant data to establish a diagnosis is to take into account the concepts of redundancies between various sets. The appearance of the same phenomenon on various sets can make it possible to refine the fault isolation. The knowledge of the character of the failures is a third important concept; indeed according to the character of the identified breakdowns, one will be able for example to refine the localization or to filter certain events considered non-representative of the character of the breakdown. Fourth, the time duration of the alarm is a more interesting resource to be exploited. The proposed DFT model can be modularized and each module translated into a High Level Petri Net (HLPN). Translation of DFT modules into HLPN has proved to be very flexible and various kinds of new dependencies can be easily accommodated. In order to exploit this flexibility a new representation, called the event diagram, is introduced.     

Quantification of sequential failure logic for fault tree analysis

Fault tree analysis (FTA) is generally accepted as an efficient method for analyzing system failures. It is well known that a fault tree (FT) is equivalent to a minimal cut set fault tree with all minimal cut-AND structures. The minimal cut-AND structure is an AND conjunction of an output and all inputs that compose a minimal cut set. For the structure, the failed state of the output becomes true when all failed states of inputs exist simultaneously. There are cases where the output of the minimal cut-AND structure depends not only on all failed states of inputs but also on the sequence of occurrences of those failures. This sequential failure logic (SFL) is equivalently expressed with Priority-AND gates in FTA, where inputs to the gates have constant failure and repair rates. A probabilistic model for analysis of SFL was proposed and equations with multiple integration for arbitrary number of inputs were derived from the model. However, it is usually difficult to solve the multiple integration when the number of inputs exceeds a certain range. This paper presents analytical solutions of the probability that the output is in a failed state at time t and the statistically expected number of failures of the output per unit time at time t for the special case where inputs are characterized by common failure and repair rates. In addition, the analysis of FT involving SFL is demonstrated by means of software Mathematica.     

An Efficient Approximate Markov Chain Method in Dynamic Fault Tree Analysis

Approximate Markov chain method for dynamic fault tree analysis is suggested for both reparable and non‐reparable systems. The approximation is based on truncation, aggregation and elimination of Markov chain states during the process of dynamic fault tree transformation to corresponding Markov chain. The method is valid for small probabilities. For reparable systems, it is true if mean time to repair is much less than mean time to failure. Several examples are studied. Additional simplification is considered in case the system is in a steady state. Copyright © 2015 John Wiley & Sons, Ltd.     

Methodology for the optimal component selection of electronic devices under reliability and cost constraints

The objective of this paper is to present an efficient computational methodology for the reliability optimization of electronic devices under cost constraints. The system modeling for calculating the reliability indices of the electronic devices is based on Bayesian networks using the fault tree approach, in order to overcome the limitations of the series–parallel topology of the reliability block diagrams. Furthermore, the Bayesian network modeling for the reliability analysis provides greater flexibility for representing multiple failure modes and dependent failure events, and simplifies fault diagnosis and reliability allocation. The optimal selection of components is obtained using the simulated annealing algorithm, which has proved to be highly efficient in complex optimization problems where gradient‐based methods can not be applied. The reliability modeling and optimization methodology was implemented into a computer program in Matlab using a Bayesian network toolbox. The methodology was applied for the optimal selection of components for an electrical switch of power installations under reliability and cost constraints. The full enumeration of the solution space was calculated in order to demonstrate the efficiency of the proposed optimization algorithm. The results obtained are excellent since a near optimum solution was found in a small fraction of the time needed for the complete enumeration (3%). All the optimum solutions found during consecutive runs of the optimization algorithm lay in the top 0.3% of the solutions that satisfy the reliability and cost constraints. Copyright © 2007 John Wiley & Sons, Ltd.     

Gastric esophageal surgery risk analysis with a fault tree and Markov integrated model

Reliability methods have been widely used in risk analysis of medical surgeries. In this study, the authors combine a fault tree with Markov models to assess time independent- and dependent factors together. Dynamics are integrated in the traditional fault tree, and meanwhile the processes of solving Markov are simplified with the modular approach. Continuous time Markov chains are adopted in evaluating the failure probability of a gastric esophageal surgery after categorizing basic events in the fault tree, and a certain time dependent variables, such as failure rate of medical equipment, surgery frequency, and rescue timeliness are involved into risk analysis. A case is studied with data collected from a general hospital, to illustrate the operational process of the proposed method. Results based on the inputs show that taking rescue actions into consideration can reduce the gap between the result of fault tree analysis and the reality. Sensitivity analysis for measuring the impacts of the above time relevant variables is conducted, as well as limitations of the Markov model are discussed.     

Fault trees analysis using expert opinion based on fuzzy‐bathtub failure rates

The main objective of fault tree analysis method is to estimate the “Top Event occurrence probability”. This requires determination of failure time distribution functions also known as “Bathtub Curves” for each of the system elements/events. This paper introduces a novel method to determine the failure time distribution functions using possibility theory. For this purpose, fuzzy‐bathtub distributions using expert opinions are generated for basic events and fuzzy formulas are derived for static and dynamic gates fault tree constructions. This process completed by proposed fuzzy Monte Carlo simulation throughout the preferred operational time and uses the actual time‐to‐failure data. Accordingly, the Top Event failure curve and the reliability profile of the system are depicted based on the defuzzificated basic‐events' bathtub‐failure‐rates. The results show that the proposed method not only is feasible and powerful but can also be accurate more than the other probabilistic and possibilistic techniques because of the component failure rates follow the real failure distributions.     

Mission Reliability Assessment of Space TT&C System by Discrete Event System Simulation

Based on the analysis of system characteristics and mission process, space tracking, telemetry and command (TT&C) system can be viewed as a phased‐mission system (PMS). A general methodology using discrete event system simulation is proposed to quantitatively assess mission reliability of space TT&C system, because the traditional method is difficult to solve such complex problem. By dividing the time sequence of TT&C mission profile into several phases, the fault tree model of PMS is built to represent the system logical structure in each phase. In order to efficiently build simulation models, unified modeling language static class diagram is used to describe simulation model architecture. Extensible markup language is adopted to represent the mission reliability model in standard format for simulation input. By randomly generating the failure and repair events of the system components, the changes of the system state are simulated. The logic structure function of fault tree and observation data of the system state change jointly determines the mission reliability. A case study is given to illustrate the approach and validate its effectiveness. Copyright © 2013 John Wiley & Sons, Ltd.