Fault tree. Time aspects

This article on fault trees (ADDs) follows on from [SE 4052] , which describes the history of ADDs, their Boolean context and their links with other approaches used in the field of operational safety. The article [SE 4052] describes in detail the use of ADDs as a qualitative analysis tool, as well as the mathematical basis for probabilistic calculations related to the failure of the modeled system. The ADD represents a static logic function (i.e. one in which time does not intervene) which essentially calculates, with constant probabilities, the probability of failure of the system modeled as a function of the probabilities of failure of its components.

However, when components evolve independently of each other over time, it is still possible, to a certain extent, to take some temporal aspects into account with this approach. This is the subject of this article, which explains how ADD makes it easy to calculate the unavailability, U(t), of the modeled system as a function of the unavailabilities of its components, U _i (t). It also explains how ADD can be used, albeit with more complicated calculations, to obtain the failure frequency, w(t), and the defiabilty, F(t), of the system under study (probability of observing a failure over the duration [0, t]). ADD calculation of the defiabilty, F(t), is generally not possible without approximation.

For a long time, these calculations were limited by the size of the ADDs considered, the power of the computers available and the low efficiency of the algorithms used in the presence of elements repeated several times in the models. These limitations have been overcome since the implementation of binary decision diagrams (BDD). This was first put to good use in the static case [SE 4052] , then for the introduction of temporal aspects and the...