Tetsushi Yuge

Quantitative analysis of a fault tree with priority and gates

Abstract 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.

Calculating top event probability of a fault tree with many repeated events

Purpose – Calculating the exact top event probability of fault trees is an important analysis in quantitative risk assessments. However, it is a difficult problem for the trees with complex structure. Therefore, the paper aims to provide an efficient calculation method to obtain an exact top event probability of a fault tree with many repeated events when the minimal cut sets of the tree model are given.Design/methodology/approach – The method is based on the inclusion‐exclusion method. Generally, the inclusion‐exclusion method tends to get into computational difficulties for a large‐scale fault tree. The computation time has been reduced by enumerating only non‐canceling terms.Findings – The method enables the calculation of the probability more quickly than the conventional method. The effect increases as the number of repeated events increases, namely the tree structure becomes complex. This method also can be applied to obtain the lower and upper bounds of the top event probability easily.Originality/...

Reliability of a k-out-of-n System with Common-cause Failures Using Multivariate Exponential Distribution

In recent years, numerous papers dealing with extensions of Marshall-Olkin distributions have appeared. However, the Marshall- Olkin model is not yet a commonly used mathematical model in the field of risk analysis, even though it has been considered to be suitable for common cause analysis in the field of statistics. We consider the reliability of a k-out-of-n system subjected to Marshall-Olkin type shocks. All combinations of components in the system are assumed to be shock sources in the analysis. We formulate the system reliability and numerically compare the results with those obtained using the conventional α-factor model.

Minimal cut sequences and top event probability of dynamic fault tree

Purpose – This paper aims to present a method for calculating the top event probability of a fault tree with priority AND gates.Design/methodology/approach – The paper makes use of Merles temporal operators for obtaining the minimal cut sequence set of a dynamic fault tree. Although Merles expression is based on the occurrence time of an event sequence, the paper treats the expression as an event containing the order of events. This enables the authors to treat the minimal cut sequence set by using the static fault tree techniques. The proposed method is based on the sum of disjoint products. The method for a static FT is extended to a more applicable one that can deal with the order operators proposed by Merle et al.Findings – First, an algorithm to obtain the minimal cut sequence set of dynamic fault trees is proposed. This algorithm enables the authors to analyze reasonably large scale dynamic fault trees. Second, the proposed method of obtaining the top event probability of a dynamic fault tree is e...

An approximation to the steady state probabilities of a multi-echelon repair model for a series system

Abstract A multi-echelon repair system is often employed for the repair of a complex device. There have been many papers on a multi-echelon repair system. Most of them, however, treat the case that i) the repair system has one central repair station(two-echelon repair model), ii)the device is a single item system. The purpose of this paper is to extend a two-echelon repair model with single item systems to a multi-echelon repair model with multi-item systems. This extension makes the repair model more realistic. Then an approximation method to obtain the steady state probabilities of the repair model is presented. This method obtains the probabilities rapidly and accurately. The reliability measure considered in this model is the system availability. It is obtained as a byproduct of the steady state probabilities.

Fault tree analysis considering sequence dependence and repairable input events

Purpose – This paper aims to present two methods for calculating the steady state probability of a repairable fault tree with priority AND gates and repeated basic events when the minimal cut sets are given.Design/methodology/approach – The authors consider a situation that the occurrence of an operational demand and its disappearance occur alternately. We assume that both the occurrence and the restoration of the basic event are statistically independent and exponentially distributed. Here, restoration means the disappearance of the occurring event as a result of a restoration action. First, we obtain the steady state probability of an output event of a single‐priority AND gate by Markov analysis. Then, we propose two methods of obtaining the top event probability based on an Inclusion‐Exclusion method and by considering the sum of disjoint probabilities.Findings – The closed form expression of steady state probability of a priority AND gate is derived. The proposed methods for obtaining the top event pr...

Reliability of a 2-Dimensional Lattice System Subject to Dependent Component Failure

In this paper an analysis of component and system reliability for lattice systems is proposed when component failures are not statistically independent. We deal the case that the failure rate of a component depends on the number of the adjacent failed components. And we discuss the maintainability of the system when a failed component is replaced by a spare component. At first we discuss the approximated reliability of each component. Then we estimate the mean number of failed components. Furthermore, the system reliability is approximated by using the component reliability.

Repairable Fault Tree Analysis Using Renewal Intensities

Abstract Fault tree (FT) is one of powerful tools for reliability analysis. The conventional FT usually considers only failure occurrence as an input event. However, considering repair in FT improves the analysis capability of FT. This FT is referred to as a repairable FT (RFT). This paper deals with RFTs, including dynamic FTs. In RFTs, it is necessary to consider both the occurrence of a basic event and its disappearance. This means that the analysis of the RFT needs not only the information about whether basic/intermediate events occur, but also the one about in which states the event outputs are. This forces us into considering the state transition of a system with time domain. Markov analysis is usually adopted for the state transition analysis. The difficulty of this method is mainly due to the explosion of the number of states to be considered in the analysis. We try to handle the state transitions in the RFT analysis as equivalent event occurrences by using the concept of renewal process, and introduce a new approach to obtain the steady state top event probability. That is, the state transition of a gate output is regarded as an alternating renewal process. The renewal intensities of the process are derived applying the limit theorem of a renewal process with the up-time-ratio analysis, the mean up time and the down time analyses. Starting from the gate located at the bottom of an FT, the top event probability is calculated by a bottom up procedure.

Estimation of the Change Point for Failure-Censored Data via Bayesian Information Criterion

This paper considers an estimation problem of the change point for failure-censored data. The data follow an exponential distribution and its parameter changes with progress of time. The time point when the parameter changes is referred to as a change point. We apply Bayesian information criterion (BIC) to the estimation of the change point of the parameter. Simulation analysis is conducted to investigate the accuracy of the estimate.