StandbyDistribution

StandbyDistribution[dist1,{dist2,,distn}]
represents a standby distribution with component lifetime distributions disti. When component i fails, component i+1 will become active.

StandbyDistribution[dist1,{dist2,,distn},p]
represents a standby distribution where switching from component i to component i+1 succeeds with probability p.

StandbyDistribution[dist1,{dist2,,distn},sdist]
represents a standby distribution where the switch component has lifetime distribution sdist.

StandbyDistribution[dist1,{,{disti,inactive,disti,active},},]
represents a standby distribution where the i^(th) component lifetime distribution follows disti,inactive in inactive mode and disti,active in active mode.

DetailsDetails

  • StandbyDistribution[,] represents a system with perfect switching where transitioning between components always succeeds.
  • StandbyDistribution[,,s] represents a system with imperfect switching. If s is a distribution, it represents that lifetime of the switch; otherwise it represents the probability of a successful transition between components.
  • StandbyDistribution[,{,Ai,},] represents a standby distribution where the i^(th) component follows a cold standby distribution Ai when it is active, and does not deteriorate when it is inactive.
  • StandbyDistribution[,{,{Ii,Ai},},] represents a standby distribution where the i^(th) component follows a warm standby distribution. The component deteriorates following distribution Ii when it is inactive and distribution Ai when it is active.
  • Any mix of cold and warm standby component distributions can be used.
  • The survival function and other properties for StandbyDistribution can be derived from the equivalent TransformedDistribution[expr,dists] with the distribution assumptions dists given by {a1A1,a2A2,,i2I2,i3I3,,sS,uUniformDistribution[{0,1}]}.
  • StandbyDistribution[]TransformedDistribution[,dists]
    A_(1),{A_(2),A_(3),...}a1+a2+a3+
    A1,{A2,A3,},pa1+ a2Boole[p>u]+a3Boole[p2>u]+
    A1,{A2,A3,},Sa1+a2Boole[s>a1]+a3Boole[s>a1+a2]+
    A1,{{I2,A2},{I3,A3},}a1+a2Boole[i2>a1]+a3Boole[i3>a1+a2Boole[i2>a1]]+
    A1,{{I2,A2},{I3,A3},},pa1+a2 Boole[i2>a1p>u]+a3Boole[i3>a1+ a2Boole[i2>a1]p2>u]+
    A1,{{I2,A2},{I3,A3},},Sa1+a2 Boole[i2>a1s>a1]+a3Boole[i3>a1+a2Boole[i2>a1]s>a1+a2Boole[i2>a1]]+
  • StandbyDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, ReliabilityDistribution, and RandomVariate.

ExamplesExamplesopen allclose all

Basic Examples  (3)Basic Examples  (3)

Define a cold standby system with perfect switching:

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Compute its PDF:

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Mean time to failure:

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Compare to a non-standby system:

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Define a cold standby system with imperfect switching:

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Compute its PDF:

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Mean time to failure:

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Compare to a non-standby system:

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Define cold and warm standby systems, with inactive failure rate half the active failure rate:

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Compute the mean time to failure:

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Compare the survival functions:

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Introduced in 2012
(9.0)
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