ReliabilityDistribution

ReliabilityDistribution[bexpr,{{x1,dist1},{x2,dist2},…}]

represents the reliability distribution for a system with components xi having reliability distribution disti, where the whole system is working when the Boolean expression bexpr is True, and component xi is working when xi is True.

Details

  • ReliabilityDistribution[bexpr,…] corresponds to a reliability block diagram specification.
  • The Boolean expression bexpr is also known as the structure function for the system.
  • Typical structure functions include:
  • x_(1)∧...∧x_(n)series system
    x_(1)∨...∨x_(n)parallel system
    TemplateBox[{BooleanCountingFunction, paclet:ref/BooleanCountingFunction}, RefLink, BaseStyle -> {2ColumnTableMod}][{k,n},n]k-out-of-n system
    BooleanConsecutiveFunction[k,n]consecutive-k-out-of-n 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 component reliability distributions disti need to be univariate with PDF[disti,t] zero for t≤0.
  • For ReliabilityDistribution[bexpr,…] with component indicator variables xi:
  • xiTrueindicates component xi working
    xiFalseindicates component xi failed
  • The survival function at time t for ReliabilityDistribution[bexpr,{{x1,dist1},…}] is given by Probability[bexpr/.{x1->t1>t,…},{t1disti,…}].
  • ReliabilityDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.

Examples

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Basic Examples  (3)

A series-connected system:

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Mean time to failure:

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A parallel-connected system:

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Mean time to failure:

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One component in series with two parallel components:

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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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Scope  (26)

Applications  (12)

Properties & Relations  (14)

Possible Issues  (3)

Neat Examples  (2)

Introduced in 2012
(9.0)