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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.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Cold Standby and Perfect Switching  
Cold Standby and Imperfect Switching  
Warm Standby and Perfect Switching  
Warm Standby and Imperfect Switching  
Mixed Warm and Cold Standby Systems  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page

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.

Details

  • 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>a1∧p>u]+a3Boole[i3>a1+ a2Boole[i2>a1]∧p2>u]+⋯
    A1,{{I2,A2},{I3,A3},…},Sa1+a2 Boole[i2>a1∧s>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.

Examples

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Basic Examples  (3)

Define a cold standby system with perfect switching:

Compute its PDF:

Mean time to failure:

Compare to a non-standby system:

Define a cold standby system with imperfect switching:

Compute its PDF:

Mean time to failure:

Compare to a non-standby system:

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

Compute the mean time to failure:

Compare the survival functions:

Scope  (17)

Cold Standby and Perfect Switching  (3)

Define a cold standby system with three identical components:

Compute mean time to failure:

Define a cold standby system with two different components:

Compute the survival function:

Study a component with three identical components in standby:

Generate random variates:

Compare with the probability density function:

Cold Standby and Imperfect Switching  (4)

A cold standby system where the switch succeeds with probability p:

Find the mean time to failure:

Compare perfect switching to imperfect switching where the switch works half the time:

A cold standby system where the switch is modeled by a lifetime distribution:

Find the survival function:

Study the effect of having worse switches:

Cold standby system with three components and a switch modeled by a distribution:

Generate random variates:

Compare with the probability density function:

A switch modeled with a probability of success:

Compare with the probability density:

Warm Standby and Perfect Switching  (3)

Standby system where the component can fail while in standby:

Find the mean time to failure:

System with multiple components that can fail in standby:

Compare the survival function to a cold standby system:

Warm standby system with two components in standby:

Generate random variates:

Compare with the probability density function:

Warm Standby and Imperfect Switching  (4)

Warm standby system where the switch succeeds with a probability p:

Compute the mean time to failure:

Warm standby system where the switch has a lifetime distribution:

Compute the mean time to failure:

Warm standby system where the switch is modeled with a lifetime distribution:

Generate random variates:

Compare with the probability density function:

System where the switch succeeds with a probability:

Mixed Warm and Cold Standby Systems  (3)

Standby system where the second component can fail while in standby:

The system where the second and third component switch places:

Compare the survival functions:

A mixed cold and warm standby system, where the switch succeeds with probability 0.9`:

Find the hazard function:

Generate random numbers and compare with probability density:

Standby system where one component can fail while in standby, and a switch with a lifetime:

Compare the survival functions with different switch failure rates:

Applications  (2)

The lifetime of a component is exponentially distributed. To improve reliability, a second identical component is acquired. Find the most efficient use of this second component:

One alternative is a parallel configuration:

Another alternative is a standby configuration, with a switch that succeeds with probability p:

Plot the survival function of the two alternatives and compare with the original component, assuming perfect switching:

Simulate failure times for 30 standby systems and find the best configuration:

Check how bad a switch you can use while still being better than a parallel system:

The requirement on the switch to equal a parallel system gets lower with time:

Consider a computer server. It requires a power supply, hard drives, a network card, and a router to fulfill its intended function. The power supply is backed by a backup power outlet and a diesel generator in cold standby:

The hard drives are in a RAID configuration, which requires 2 out of 3 to work:

The network card has a second card in standby:

Two routers are connected in parallel:

The resulting survival function:

Plot it:

Compute the mean time to failure numerically:

Find the probability that the server survives for three months:

Define a consumer version that does not contain any redundancy:

Compare the survival functions:

Properties & Relations  (9)

Cold standby corresponds to the sum of component lifetimes:

Compare the survival functions:

Cold standby with identical exponentially distributed components is an ErlangDistribution:

Cold standby where component lifetimes follow the ExponentialDistribution corresponds to the HypoexponentialDistribution:

StandbyDistribution is a special case of TransformedDistribution:

Compare the survival functions:

StandbyDistribution is a special case of MixtureDistribution:

Compare the probability density function:

StandbyDistribution can be used in ReliabilityDistribution:

Compute the survival function:

ReliabilityDistribution can be used in StandbyDistribution:

Generate random numbers:

Compare with the probability density function:

StandbyDistribution can be used in FailureDistribution:

Compute the survival function:

FailureDistribution can be used in StandbyDistribution:

Generate random numbers:

Compare with the probability density function:

Possible Issues  (1)

Component distributions need to have a positive domain:

Use TruncatedDistribution to restrict the domain to positive values only:

Wolfram Research (2012), StandbyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StandbyDistribution.html.

Text

Wolfram Research (2012), StandbyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StandbyDistribution.html.

CMS

Wolfram Language. 2012. "StandbyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/StandbyDistribution.html.

APA

Wolfram Language. (2012). StandbyDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StandbyDistribution.html

BibTeX

@misc{reference.wolfram_2025_standbydistribution, author="Wolfram Research", title="{StandbyDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/StandbyDistribution.html}", note=[Accessed: 04-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_standbydistribution, organization={Wolfram Research}, title={StandbyDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/StandbyDistribution.html}, note=[Accessed: 04-February-2026]}