Fault Tree Analysis - Basic Mathematical Foundation

Basic Mathematical Foundation

Events in a fault tree are associated with statistical probabilities. For example, component failures typically occur at some constant failure rate λ (a constant hazard function). In this simplest case, failure probability depends on the rate λ and the exposure time t:

P = 1 - exp(-λt)

P ≈ λt, λt < 0.1

A fault tree is often normalized to a given time interval, such as a flight hour or an average mission time. Event probabilities depend on the relationship of the event hazard function to this interval.

Unlike conventional logic gate diagrams in which inputs and outputs hold the binary values of TRUE (1) or FALSE (0), the gates in a fault tree output probabilities related to the set operations of Boolean logic. The probability of a gate's output event depends on the input event probabilities.

An AND gate represents a combination of independent events. That is, the probability of any input event to an AND gate is unaffected by any other input event to the same gate. In set theoretic terms, this is equivalent to the intersection of the input event sets, and the probability of the and gate output is given by:

P(A and B) = P(A ∩ B) = P(A) P(B)

An OR gate, on the other hand, corresponds to set union:

P(A or B) = P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Since failure probabilities on fault trees tend to be small (less than .01), P(A ∩ B) usually becomes a very small error term, and the output of an OR gate may be conservatively approximated by using an assumption that the inputs are mutually exclusive events:

P(A or B) ≈ P(A) + P(B), P(A ∩ B) ≈ 0

An exclusive OR gate with two inputs represents the probability that one or the other input, but not both, occurs:

P(A xor B) = P(A) + P(B) - 2P(A ∩ B)

Again, since P(A ∩ B) usually becomes a very small error term, the exclusive OR gate has limited value in a fault tree.