An optimized technique for reliability analysis of safety‐critical systems: A case study of nuclear power plant

Stochastic models are extensively used in quantifying the reliability of safety critical systems. These models use the state‐space model for reliability quantification. Markov chain is comprehensively used in describing a sequence of possible events of any system in which the probability of each event depends only on the state attained in the previous event. Markov chains are convenient to model the software system of the SCS with the help of Petri Nets, a directed bipartite graph widely used for the verification and validation of real‐time systems. However, the stochastic model suffers from the state‐space explosion problem. In this paper, we proposed a technique for reliability analysis of safety critical systems, excavating into the coherent optimization of Markov chain. The approach has been validated on 17 safety critical systems of nuclear power plants.     

Comparative study of quantitative models for hardware,software and human reliability assessment

Various models which may be used for quantitative assessment of hardware, software and human reliability are compared in this paper. Important comparison criteria are the system life cycle phase in which the model is intended to be used, the failure category and reliability means considered in the model, model purpose, and model characteristic such as model construction approach, model output and model input. The main objective is to present limitations in the use of current models for reliability assessment of computer-based safety shutdown systems in the process industry and to provide recommendations on further model development. Main attention is given to presenting the overall concept of various models from a user's point of view rather than technical details of specific models. A new failure classification scheme is proposed which shows how hardware and software failures may be modelled in a common framework.     

Reliability performance of safety instrumented systems: A common approach for both low- and high-demand mode of operation

Safety instrumented systems (SISs) are usually divided into two modes of operation, low-demand and high-demand. Unfortunately, this classification is not easy to justify and the available formulas that are used to quantify the reliability performance in these two modes of operation are unable to capture combined effects of functional testing, spurious activations, and successful responses to demands. This article discusses some important modeling issues for SIS reliability performance quantification, and demonstrates their implementation in a Markov model. The accuracy of the Markov model for a simple case study of a pressure transmitter is verified through comparison with a scenario-based formula, and it is shown that the Markov approach gives a sufficiently accurate result for all demand rates, covering both low- and high-demand modes of operation.     

An analytical framework for reliability growth of one-shot systems

In this paper, we introduce a new reliability growth methodology for one-shot systems that is applicable to the case where all corrective actions are implemented at the end of the current test phase. The methodology consists of four model equations for assessing: expected reliability, the expected number of failure modes observed in testing, the expected probability of discovering new failure modes, and the expected portion of system unreliability associated with repeat failure modes. These model equations provide an analytical framework for which reliability practitioners can estimate reliability improvement, address goodness-of-fit concerns, quantify programmatic risk, and assess reliability maturity of one-shot systems. A numerical example is given to illustrate the value and utility of the presented approach. This methodology is useful to program managers and reliability practitioners interested in applying the techniques above in their reliability growth program.     

A new modeling approach to the safety evaluation of N-modular redundant computer systems in presence of imperfect maintenance

A large number of safety-critical control systems are based on N-modular redundant architectures, using majority voters on the outputs of independent computation units. In order to assess the compliance of these architectures with international safety standards, the frequency of hazardous failures must be analyzed by developing and solving proper formal models. Furthermore, the impact of maintenance faults has to be considered, since imperfect maintenance may degrade the safety integrity level of the system. In this paper, we present both a failure model for voting architectures based on Bayesian networks and a maintenance model based on continuous time Markov chains, and we propose to combine them according to a compositional multiformalism modeling approach in order to analyze the impact of imperfect maintenance on the system safety. We also show how the proposed approach promotes the reuse and the interchange of models as well the interchange of solving tools.     

Transient analysis of reliability with and without repair for K-out-of-N :G systems with M failure modes

This paper represents Markov models for transient analysis of reliability with and without repair for K-out-of-N :G systems subject to M failure modes. The reliability and the mean time between failures of repairable systems can be calculated as a result of numerical solution of simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability and the mean time to failure for nonrepairable systems.     

Transient analysis of reliability with and without repair for K-out-of-N:G systems with two failure modes

This paper presents Markov models for transient analysis of reliability with and without repair for K-out-of-N:G systems subject to two failure modes. The reliability of repairable systems can be calculated as a result of the numerical solution of a simultaneous set of linear differential equations. Closed form solutions of the transient probabilities are used to obtain the reliability for nonrepairable systems.     

Software reliability analysis for safety-critical and control systems

The transition from analog to digital safety-critical instrumentation and control (I&C) systems has introduced new challenges for software experts to deliver increased software reliability. Since the 1970s, researchers are continuing to propose software reliability models for reliability estimation of software. However, these approaches rely on the failure history for the assessment of reliability. Due to insufficient failure data, these models fail to predict the reliability of safety critical systems. This paper utilizes the Bayesian update methodology and proposes a framework for the reliability assessment of the safety-critical systems (SCSs). The proposed methodology is validated using experiments performed on real data of 12 safety-critical control systems of nuclear power plants.     

The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system

The aim of this paper is to improve evaluation of the reliability of probabilistic and non-probabilistic hybrid structural system. Based on the probabilistic reliability model and interval arithmetic, a new model of interval estimation for reliability of the hybrid structural system was proposed. Adequately considering all uncertainties affecting the hybrid structural system, the lower and upper bounds of reliability for the hybrid structural system were obtained through the probabilistic and non-probabilistic analysis. In the process of non-probabilistic analysis, the interval truncation method was used. In addition, a recognition method of the main failure modes in the hybrid structural system was presented. A five-bar statically indeterminate truss structure and an intermediate complexity wing structure were used to demonstrate the new model is more suitable for analysis and design of these structural systems in comparison with the probabilistic model. The results also show that the method of recognition of main failure modes is effective. In addition, range obtained through interval estimation is shown to be more credible than certain results of other reliability models.     

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.     

Optimal Bayesian maintenance policy for a gearbox subject to two dependent failure modes

Most maintenance optimization models of gear systems have considered single failure mode. There have been very few papers dealing with multiple failure modes, considering mostly independent failure modes. In this paper, we present an optimal Bayesian control scheme for early fault detection of the gear system with dependent competing risks. The system failures include degradation failure and catastrophic failure. A three‐state continuous‐time–homogeneous hidden Markov model (HMM), namely the model with unobservable healthy and unhealthy states, and an observable failure state, describes the deterioration process of the gear system. The condition monitoring information as well as the age of the system are considered in the proposed optimal Bayesian maintenance policy. The objective is to maximize the long‐run expected average system availability per unit time. The maintenance optimization model is formulated and solved in a semi‐Markov decision process (SMDP) framework. The posterior probability that the system is in the warning state is used for the residual life estimation and Bayesian control chart development. The prediction results show that the mean residual lives obtained in this paper are much closer to the actual values than previously published results. A comparison with the Bayesian control chart based on the previously published HMM and the age‐based replacement policy is given to illustrate the superiority of the proposed approach. The results demonstrate that the Bayesian control scheme with two dependent failure modes can detect the gear fault earlier and improve the availability of the system.     

A Bayesian framework for accelerated reliability growth testing with multiple sources of uncertainty

Reliability growth tests are often used for achieving a target reliability for complex systems via multiple test‐fix stages with limited testing resources. Such tests can be sped up via accelerated life testing (ALT) where test units are exposed to harsher‐than‐normal conditions. In this paper, a Bayesian framework is proposed to analyze ALT data in reliability growth. In particular, a complex system with components that have multiple competing failure modes is considered, and the time to failure of each failure mode is assumed to follow a Weibull distribution. We also assume that the accelerated condition has a fixed time scaling effect on each of the failure modes. In addition, a corrective action with fixed ineffectiveness can be performed at the end of each stage to reduce the occurrence of each failure mode. Under the Bayesian framework, a general model is developed to handle uncertainty on all model parameters, and several special cases with some parameters being known are also studied. A simulation study is conducted to assess the performance of the proposed models in estimating the final reliability of the system and to study the effects of unbiased and biased prior knowledge on the system‐level reliability estimates.     

Reliability importance analysis of Markovian systems at steady state using perturbation analysis

Sensitivity analysis has been primarily defined for static systems, i.e. systems described by combinatorial reliability models (fault or event trees). Several structural and probabilistic measures have been proposed to assess the components importance. For dynamic systems including inter-component and functional dependencies (cold spare, shared load, shared resources, etc.), and described by Markov models or, more generally, by discrete events dynamic systems models, the problem of sensitivity analysis remains widely open. In this paper, the perturbation method is used to estimate an importance factor, called multi-directional sensitivity measure, in the framework of Markovian systems. Some numerical examples are introduced to show why this method offers a promising tool for steady-state sensitivity analysis of Markov processes in reliability studies.     

An integrated methodology for the dynamic performance and reliability evaluation of fault-tolerant systems

We propose an integrated methodology for the reliability and dynamic performance analysis of fault-tolerant systems. This methodology uses a behavioral model of the system dynamics, similar to the ones used by control engineers to design the control system, but also incorporates artifacts to model the failure behavior of each component. These artifacts include component failure modes (and associated failure rates) and how those failure modes affect the dynamic behavior of the component. The methodology bases the system evaluation on the analysis of the dynamics of the different configurations the system can reach after component failures occur. For each of the possible system configurations, a performance evaluation of its dynamic behavior is carried out to check whether its properties, e.g., accuracy, overshoot, or settling time, which are called performance metrics, meet system requirements. Markov chains are used to model the stochastic process associated with the different configurations that a system can adopt when failures occur. This methodology not only enables an integrated framework for evaluating dynamic performance and reliability of fault-tolerant systems, but also enables a method for guiding the system design process, and further optimization. To illustrate the methodology, we present a case-study of a lateral-directional flight control system for a fighter aircraft.     

A novel framework for the reliability modelling of repairable multistate complex mechanical systems considering propagation relationships

Due to the propagation, amplification, and concatenation in a failure process, the reliabilities of repairable multistate complex mechanical systems (RMCMSs) may be affected by a significant fluctuation due to a small exception associated with a reliability indicator. Focused on the problems arising from the lack of propagation relationships among fault modes, functional components, and failure causes in conventional reliability models, a novel framework for reliability modelling is proposed to comprehensively analyse the reliabilities of RMCMSs. First, the reliability models are abstracted as weighted and directed networks with five layers. Second, an improved failure mode and effects analysis (IFMEA) method combined with the D‐number method and VIKOR approach is presented to determine the importance of reliability nodes. Third, a cut set of the reliability model is generated by any exception of a reliability indicator by considering the propagation relationships, and the reliability sensibility index is defined to characterize the fluctuations in system reliability. The effectiveness of the proposed framework is demonstrated in an actual reliability modelling application. As an intuitive method, the proposed framework inherits the advantages of conventional models but overcomes the drawbacks of these existing methods. Therefore, this method can be flexibly and efficiently used in the reliability modelling of RMCMSs. Moreover, the approach provides a foundation for comprehensive and dynamic reliability analysis and the failure mechanism mining of RMCMSs, and it can be used in other engineering applications.     

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.     

Modeling the reliability of threshold weighted voting systems

In many applications, ranging from target detection to safety monitoring systems, we are interested in determining whether or not to accept a hypothesis based on the information available. In this paper we model the reliability of threshold weighted voting systems (WVS) with multi-failure-modes, where a general recursive reliability function of the WVS is presented. We also develop approximation formulas for calculating the reliability of WVS based on a large number of units. We also develop reliability functions of time-dependent threshold weighted voting systems, where each unit is a function of time. Finally, the optimal stopping time that minimizes the total cost of the systems subject to a reliability constraint is discussed.     

Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.     

A framework for capturing and analyzing the failures due to system/component interactions

To keep up with the speed of globalization and growing customer demands for more technology‐oriented products, modern systems are becoming increasingly more complex. This complexity gives rise to unpredictable failure patterns. While there are a number of well‐established failure analysis (physics‐of‐failure) models for individual components, these models do not hold good for complex systems as their failure behaviors may be totally different. Failure analysis of individual components does consider the environmental interactions but is unable to capture the system interaction effects on failure behavior. These models are based on the assumption of independent failure mechanisms. Dependency relationships and interactions of components in a complex system might give rise to some new types of failures that are not considered during the individual failure analysis of that component. This paper presents a general framework for failure modes and effects analysis (FMEA) to capture and analyze component interaction failures. The advantage of the proposed methodology is that it identifies and analyzes the system failure modes due to the interaction between the components. An example is presented to demonstrate the application of the proposed framework for a specific product architecture (PA) that captures interaction failures between different modules. However, the proposed framework is generic and can also be used in other types of PA. Copyright © 2007 John Wiley & Sons, Ltd.