FailureDistribution

FailureDistribution[bexpr,{{x1,dist1},{x2,dist2},}]
represents the failure distribution for a system with events having reliability distribution where the top event occurs when the Boolean expression bexpr is True and event has occurred when is True.

DetailsDetails

  • FailureDistribution[bexpr,] corresponds to a fault tree specification.
  • The Boolean expression bexpr is also known as the structure function for the system.
  • Typical structure functions include:
  • Or gate
    And gate
    TemplateBox[{BooleanCountingFunction, paclet:ref/BooleanCountingFunction}, RefLink, BaseStyle -> InlineFormula][{k,n},n]-out-of- system
    BooleanConsecutiveFunction[k,n]consecutive--out-of- system
  • The structure function bexpr can be any positive unate Boolean function.
  • UnateQ[bexpr] can be used to test whether a Boolean expression is positive unate.
  • The event reliability distributions need to be univariate with PDF[disti,t] zero for t0.
  • For FailureDistribution[bexpr,] with event indicators :
  • xiTrueindicates event has occurred
    xiFalseindicates event has not occurred
  • The CDF at time t for FailureDistribution[bexpr,{{x1,dist1},}] is given by Probability[bexpr/.{x1->t1t,},{t1dist1,}].
  • FailureDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.

ExamplesExamplesopen allclose all

Basic Examples  (3)Basic Examples  (3)

Define a system whose top event occurs if either of the two events occur:

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Compute the survival function:

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Define a system whose top event occurs if both of the underlying events occur:

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Compute the survival function:

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A structure with both an And and an Or gate:

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Distribution functions:

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Mean and median time to failure:

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The probability that the system fails before time :

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Introduced in 2012
(9.0)