Editing Fault tree analysis (section)

==Basic mathematical foundation==
Events in a fault tree are associated with [[Statistics|statistical]] [[probability theory|probabilities]] or Poisson-Exponentially distributed constant rates. For example, component failures may 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:

<math> P = 1 - e^{- \lambda t} </math>

where:

<math> P \approx \lambda t </math> if <math> \lambda t < 0.001 </math>

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 numeral system|binary]] values of TRUE (1) or FALSE (0), the gates in a fault tree output probabilities related to the [[Algebra of sets|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 [[independence (probability theory)|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 theory|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.

Quite often, Poisson-Exponentially distributed rates<ref>Olofsson and Andersson, Probability, Statistics and Stochastic Processes,  John Wiley and Sons, 2011.</ref> are used to quantify a fault tree instead of probabilities. Rates are often modeled as constant in time while probability is a function of time. Poisson-Exponential events are modelled as infinitely short so no two events can overlap. An OR gate is the superposition (addition of rates) of the two input failure frequencies or failure rates which are modeled as [[Poisson point process]]es. The output of an AND gate is calculated using the unavailability (Q<sub>1</sub>) of one event thinning the Poisson point process of the other event (λ<sub>2</sub>). The unavailability (Q<sub>2</sub>) of the other event then thins the Poisson point process of the first event (λ<sub>1</sub>). The two resulting Poisson point processes are superimposed according to the following equations.

The output of an AND gate is the combination of independent input events 1 and 2 to the AND gate:
:Failure Frequency = λ<sub>1</sub>Q<sub>2</sub> + λ<sub>2</sub>Q<sub>1</sub>   where Q = 1 - e<sup>-λt</sup> ≈ λt  if λt < 0.001
:Failure Frequency ≈ λ<sub>1</sub>λ<sub>2</sub>t<sub>2</sub> + λ<sub>2</sub>λ<sub>1</sub>t<sub>1</sub>   if   λ<sub>1</sub>t<sub>1</sub> < 0.001 and λ<sub>2</sub>t<sub>2</sub> < 0.001

In a fault tree, unavailability (Q) may be defined as the unavailability of safe operation and may not refer to the unavailability of the system operation depending on how the fault tree was structured. The input terms to the fault tree must be carefully defined.