How to Use the Minimal Cuts of Fault Tree Analysis in Markov Modelling

Abstract

This contribution has been stimulated by the NATO Conference at Sogesta and worked out there. It is an attempt to clarify the concepts of a fault tree and of minimal cuts in Markov-modelling of binary systems. These concepts which are currently used in reliability theory are still meaningful if we describe a binary system by a time evolutive process instead of a statical fault tree. So it seems to us worth-wile to demonstrate by a fairly simple example what can be done with these concepts in Markov modelling. Formal proofs and further examples will be given by the author elsewhere. ¹

The result will be that we can use the minimal cuts of a fault tree model to identify in a formal way the failed states relevant in the corresponding Markov model. This may be helpful for the evaluation of a Markov model related to a larger binary system by means of a computer code involving the following two problems:

• (i) Automatization of the construction of the transition matrix A of a Markov process.

• (ii) Calculation of the eigenvalues of the transition matrix A.

A lot of work has been done in numerical mathematics to solve problem (ii). By contrast not much work has been done to solve problem (i).² The main purpose of this paper is a tutorial one.