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StandbyDistribution
✖
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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Define a cold standby system with perfect switching:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ylr7e

Compute its PDF:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-6gojy
Out[2]=2

Mean time to failure:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-xows0
Out[3]=3

Compare to a non-standby system:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-inbwud
Out[4]=4

Define a cold standby system with imperfect switching:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-5nv0

Compute its PDF:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ba2h74
Out[2]=2

Mean time to failure:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-epsxfz
Out[3]=3

Compare to a non-standby system:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-kwmghc
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-koad2r
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-5dmgxi
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-d41rj4

Compute the mean time to failure:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-qn3cgl
Out[4]=4

Compare the survival functions:

In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-23gyd
Out[5]=5

Scope  (17)Survey of the scope of standard use cases

Cold Standby and Perfect Switching  (3)

Define a cold standby system with three identical components:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-666cni

Compute mean time to failure:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-idhtll
Out[2]=2

Define a cold standby system with two different components:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-uxvs60

Compute the survival function:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-9fylc3
Out[2]=2

Study a component with three identical components in standby:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ednm3c
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-8p420d

Generate random variates:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-swsfld

Compare with the probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-l7jr5j
Out[4]=4

Cold Standby and Imperfect Switching  (4)

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-n5v0hq

Find the mean time to failure:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-udmjrt
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ggq1pn
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-61un0v
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-un29z4

Find the survival function:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-z1gqxq
Out[2]=2

Study the effect of having worse switches:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-z69pqf
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-i4a3fw
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-0mf6m9

Generate random variates:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-46emkh

Compare with the probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-mzaj9s
Out[4]=4

A switch modeled with a probability of success:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-5o5ei5
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-3z6tqs
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-w3nfr1

Compare with the probability density:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-rv604u
Out[4]=4

Warm Standby and Perfect Switching  (3)

Standby system where the component can fail while in standby:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-25vjuq

Find the mean time to failure:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-x1pfc1
Out[2]=2

System with multiple components that can fail in standby:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-lt3u5p

Compare the survival function to a cold standby system:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-neca7s
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-3toxaj
Out[3]=3

Warm standby system with two components in standby:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-x4zheh
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-51yyyw

Generate random variates:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-iccnnh

Compare with the probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-qzkztr
Out[4]=4

Warm Standby and Imperfect Switching  (4)

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-g4ot6j

Compute the mean time to failure:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-7fy377
Out[2]=2

Warm standby system where the switch has a lifetime distribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-45v1gu

Compute the mean time to failure:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-i4x1on
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-zsdpj4
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-2fyei7

Generate random variates:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ngxct9

Compare with the probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-lb8c3j
Out[4]=4

System where the switch succeeds with a probability:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-tgq8et
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-20eqrd
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-oibb18
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-we52u8
Out[4]=4

Mixed Warm and Cold Standby Systems  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-6i5tdq
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-d2c442

The system where the second and third component switch places:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-6fx0k8

Compare the survival functions:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-vgmwy0
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-lidec8
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-swgnm

Find the hazard function:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-e5ui4r
Out[3]=3

Generate random numbers and compare with probability density:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-6s4skz
In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-d3cxmi
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-wnt63v
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-eud1g4

Compare the survival functions with different switch failure rates:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-hmy6z4
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ftj43p
Out[4]=4

Applications  (2)Sample problems that can be solved with this function

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:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-6aw57i

One alternative is a parallel configuration:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-2bswtw

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

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-8aqjv1

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

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ebpgrp
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-xl5ucd
In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-jn9ouu
Out[6]=6

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

In[7]:=7
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-bvyu33
Out[7]=7

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

In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-h7klo5
Out[6]=6
In[9]:=9
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-0z24ap
Out[9]=9

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:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-jp98o

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

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-v208y2

The network card has a second card in standby:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-p2yiil

Two routers are connected in parallel:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-on8eq7
In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-xy4wmi

The resulting survival function:

In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-coq0x2
Out[6]=6

Plot it:

In[7]:=7
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-yc26na
Out[7]=7

Compute the mean time to failure numerically:

In[8]:=8
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-q0g2k7
Out[8]=8

Find the probability that the server survives for three months:

In[9]:=9
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-43rtnc
Out[9]=9

Define a consumer version that does not contain any redundancy:

In[10]:=10
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-kw5aw2
In[11]:=11
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ft2o49
In[12]:=12
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-s92uvf
In[13]:=13
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-5pj4us
In[14]:=14
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-oukbue

Compare the survival functions:

In[15]:=15
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-kl0ug0
Out[15]=15

Properties & Relations  (9)Properties of the function, and connections to other functions

Cold standby corresponds to the sum of component lifetimes:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-giiezz
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-74kb4f
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-49vy3u

Compare the survival functions:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-wahyu1
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-78lr70
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ytn2n8
Out[6]=6

Cold standby with identical exponentially distributed components is an ErlangDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-s4ef1h
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-d5p92h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-7rltw8
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-0gua3z
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-1x9gob
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-s271x2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-kdjvbn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-l733ef
Out[4]=4

StandbyDistribution is a special case of TransformedDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-xrgq26
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ps4gh0
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ocvg8x

Compare the survival functions:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-vykdxf
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-qz587a
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-fg9i89
Out[6]=6

StandbyDistribution is a special case of MixtureDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-4nlrlf
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-y7aotg
In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ckgr4n

Compare the probability density function:

In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-5til8c
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-mefa7o
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-srg29x
Out[6]=6

StandbyDistribution can be used in ReliabilityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-hrmb92

Compute the survival function:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-bxz31p
Out[2]=2

ReliabilityDistribution can be used in StandbyDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-hhakiy

Generate random numbers:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-gb5vmv

Compare with the probability density function:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-s4z5hj
Out[3]=3

StandbyDistribution can be used in FailureDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-ek6btg

Compute the survival function:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-x7sl0r
Out[2]=2

FailureDistribution can be used in StandbyDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-t8i4t5

Generate random numbers:

In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-fqv8f1

Compare with the probability density function:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-7llkm5
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

Component distributions need to have a positive domain:

In[1]:=1
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-f7cake
In[2]:=2
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-dlbxq9
Out[2]=2

Use TruncatedDistribution to restrict the domain to positive values only:

In[3]:=3
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-xm6lhf
In[4]:=4
✖
https://wolfram.com/xid/0tp0cxgnlgcwpm-lvgk8t
Out[4]=4
Wolfram Research (2012), StandbyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StandbyDistribution.html.
✖
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.

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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.

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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

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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: 14-May-2025 ]}

✖
@misc{reference.wolfram_2025_standbydistribution, author="Wolfram Research", title="{StandbyDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/StandbyDistribution.html}", note=[Accessed: 14-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_standbydistribution, organization={Wolfram Research}, title={StandbyDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/StandbyDistribution.html}, note=[Accessed: 14-May-2025 ]}

✖
@online{reference.wolfram_2025_standbydistribution, organization={Wolfram Research}, title={StandbyDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/StandbyDistribution.html}, note=[Accessed: 14-May-2025 ]}