This paper reviews and classifies fault-tree analysis methods developed since 1960 for system safety and reliability. Fault-tree analysis is a useful analytic tool for the reliability and safety of complex systems. The literature on fault-tree analysis is, for the most part, scattered through conference proceedings and company reports. We have classified the literature according to system definition, fault-tree construction, qualitative evaluation, quantitative evaluation, and available computer codes for fault-tree analysis.

Fault Tree Analysis, Methods, and Applications ߝ A Review

This paper is a state-of-the-art review of the literature related to optimal maintenance models of systems subject to failure. The emphasis is on work appearing since the 1976 survey, “A Survey of Maintenance Models: The Control and Surveillance of Deteriorating Systems,” by W.P. Pierskalla and J.A. Voelker, published in this journal.

Optimal maintenance models for systems subject to failure–A Review

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Adaptive Quadrature—Revisited

Availability appears to be a more appropriate measure than reliability for measuring the effectiveness of maintained systems because it includes reliability as well as maintainability. This paper is a survey and a systematic classification of the literature relevant to availability. Emphasis in this paper is centered on the current state of the art on a variety of topics related to availability. Among the topics discussed are: the definition and concepts of the availability; the probability density functions (pdf) of failure times, the pdf of repair times, system configurations, and the various approaches employed to obtain the availability models; confidence intervals of availability; effect of preventive maintenance policies on availability; availability parameters in the model; and systems optimization.

Availability of Maintained Systems: A State-of-the-Art Survey

Quantitative analysis of a fault tree with priority AND gates

A formal method and an algorithm are presented to determine minimal cut sets (MCSs) from fault trees. The method is based upon segmenting the tree, constructing MCSs of subtrees and subsequent expanding into the MCS of the original tree. The major advantage of the method is the direct determination of MCSs up to any order required. More particularly, the algorithm DICOMICS allows straightforward analyses for all those cases (large trees, high order cut sets, repeated elements) where present combinatorial and Monte Carlo techniques are not effective. Compared with other structural algorithms now in use, DICOMICS provides a near optimal effective construction of MCSs. Thus, even when there are many redundant cut sets, computer implementations yield a substantial reduction of computing steps and effort. Hand calculation and interactive programming are also possible.

Efficient Construction of Minimal Cut Sets from Fault Trees

In reliability engineering and practice an important role is played by those units whose life characteristics change with time. The case is herewith considered where change underlies a nonhomogeneous Markov model. Simple repair processes then deal with a 2-state alternating policy resulting from the superposition of failure and restoration, both showing a continuous aging with time. Availability can be expressed through linear differential equations or by means of integral equations; the approaches are equivalent. A computer code is then described which calculates (i) availability, and (ii) transition densities for any continuous time dependence of failure and restoration rates. Numerical results are shown for a few examples.

Availability of Repairable Units When Failure and Restoration Rates Age in Real Time

A new adaptive algorithm for the integration of analytic functions is presented. The algorithm processes the integration interval by generating local subintervals whose length is controlled through a feedback loop. Control is performed by means of a relation derived on an analytical basis and valid for an arbitrary integration rule: two different estimates of an integral are used to compute the interval length necessary to obtain an integral estimate with accuracy within the assigned error bounds. The implied method for local generation of subintervals and an effective assumption of error partition among subintervals give rise to an adaptive algorithm provided with an accurate and very efficient integration process.

Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration

Repair Processes: Fundamentals and Computation. EUR 5232.

The paper introduces the concept of a cumulative stochastic process and derives the general mathematical expression of the distribution corresponding to such processes when they can be assumed to be Markovian. The behaviour of such a distribution in correspondence to accumulation functions of the type u(t) = atb and u(t) = l ln(l + t) is explored. It is shown how the exponential, Weibull, gamma, normal and lognormal distributions are particular cases of the general distribution. Next, the characteristics of the extreme values of n independent observations coming from such a general distribution are investigated. The central characteristics of the extreme values distributions are related to the hazard rate of the initial distribution. In particular, a simple method for relating the modal smallest value and the modal largest value to the sample size using the asymptotic expression of the hazard rate is given. The tail characteristics of the extreme values distributions are investigated numerically or analytically. The mathematical findings are applied to the volume effect on the failure probability of materials.

G. Reina

Papers

Efficient Construction of Minimal Cut Sets from Fault Trees

Availability of Repairable Units When Failure and Restoration Rates Age in Real Time

Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration

Repair Processes: Fundamentals and Computation. EUR 5232.

Extreme Value Characteristics of Distributions of Cumulative Processes