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BirnbaumImportance
✖
BirnbaumImportance

✖

gives the Birnbaum importances for all components in the ReliabilityDistribution rdist at time t.

✖

gives the Birnbaum importances for all components in the FailureDistribution fdist at time t.

Details

  • BirnbaumImportance is also known as reliability importance.
  • The Birnbaum importance is the improvement in the reliability that would be gained by replacing a failed component with a perfect component .
  • The Birnbaum importance at time for component is given by where is the probability that the system is working given that the ^(th) component is perfect and is the probability that the system is working given that the ^(th) component has failed.
  • The results are returned in the component order given in the distribution list in rdist or fdist.

Examples

open allclose all

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

Two components connected in series, with different lifetime distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-e66rb

The result is given in the same order as the distribution list in ReliabilityDistribution:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-4wtin8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-d43v3q
Out[3]=3

Two components connected in parallel, with different lifetime distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-yn943z
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-8e7d7o
Out[2]=2

Use fault tree-based modeling to define the system:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-28utav
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-fmkne5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-0zv592
Out[3]=3

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

ReliabilityDistribution Models  (9)

Two components connected in parallel, with identical lifetime distributions:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-08flls

A change in reliability for either component will result in the same system reliability change:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-mkns50
Out[4]=4

Two components connected in series, with identical lifetime distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-rwabaw

A change in reliability for either component will result in the same system reliability change:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-f1wvq8
Out[2]=2

A system where two out of three components need to work, with identical lifetime distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-mji74e

Components are equally important:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xc5531
Out[2]=2

A simple mixed system with identical lifetime distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-bs7f7v
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-jy1k7y

Compute the importance:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-6vuxir
Out[3]=3

Changing the reliability of component will impact the system reliability most:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-fn2idk
Out[4]=4

A system with a series connection in parallel with a component:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-8pfw08
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-mjkl5o

Compute the importance:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-t1mcxq
Out[3]=3

Improving the component has the biggest impact on system reliability:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xfjq50
Out[4]=4

Study the effect of a change in parameter in a simple mixed system:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-3daofy
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-lqd3ci

Compute the importance:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-srbs5o
Out[3]=3

Show the changes in importance when worsening one of the parallel components, :

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xwt0s4
Out[4]=4

One component in parallel with two others, with different distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-pr0k1j
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-bkyutz

Find the importance measures at one specific point in time as exact results:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-urftd7
Out[3]=3

As machine precision results:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-vf85i0
Out[4]=4

As symbolic expressions:

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-4n9ka6
Out[5]=5

Any valid ReliabilityDistribution can be used:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-cjmrsj
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-6zfypk
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-6lxtrj
Out[3]=3

Early in the lifetime, changing the reliability of the standby component will have more effect:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-iedm9b
Out[4]=4

Model the system in steps to get the importance measure for a subsystem:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-dcrxnl
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-2f3o3y
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-n8cikq
Out[3]=3

Plot the importance over time:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-kf0c4n
Out[4]=4

FailureDistribution Models  (8)

Any of two basic events lead to the top event:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-l11ptk

A change in reliability for either event will result in the same top event reliability change:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-6nc5zk
Out[2]=2

Only both basic events together lead to the top event:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xmgcyf

A change in reliability for either event will result in the same top event reliability change:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-nlxcpv
Out[2]=2

A voting gate, with identical distributions on the basic events:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-natpec

A change in reliability for any of the events will result in the same top event reliability change:

In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xz9lrc
Out[2]=2

A simple system with both And and Or gates:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-4jsp53
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-3a7nr5

The basic event is most important:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-wh2s2m
Out[3]=3

Changing the reliability of event will impact the top event reliability most:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-tynywc
Out[4]=4

A simple system with both And and Or gates:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-fecls
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-sr2e4s

Compute the importance:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-gbr67k
Out[3]=3

Improving event has the biggest impact on preventing the top event:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-88i8fa
Out[4]=4

Study the effect of a change in parameter in a simple mixed system:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-ig97mn
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-xdjp3g

Compute the importance:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-4smvi5
Out[3]=3

Show the changes in importance when worsening one of the basic events, :

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-pcd9ai
Out[4]=4

Any valid FailureDistribution can be used:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-34mp4o
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-i1m9g0
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-9qepae
Out[3]=3

Early in the lifetime, changing the reliability of the standby component will have more effect:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-cmc5tk
Out[4]=4

Model the system in steps to get the importance measure for a subsystem:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-lxyr6j
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-mvuep3
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-okc513
Out[3]=3

Plot the importance over time:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-5ll2fu
Out[4]=4

Applications  (5)Sample problems that can be solved with this function

Find out which component is best to improve in a system that has a mission time of 3 hours:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-bqrgiq
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-wgnax0
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-ud0au2
Out[3]=3

Show the importance over time:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-7l5tir
Out[4]=4

With a mission time of 3 hours, improvement of component x will lead to the largest system improvement:

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-k1mgn7
Out[5]=5

Study a simple system with one component in series and two components in parallel. Determine which component is the most important according to the Birnbaum importance measure:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-px940y
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-r7aaoz
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-z0lrx3
In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-vh5o5o
Out[4]=4

Find the cumulative Birnbaum importance over the entire lifetime:

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-o42yzk
Out[5]=5

The cumulative Birnbaum importance gives the difference of mean time to failure:

In[6]:=6
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-exmlhu
In[7]:=7
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-b1at50
In[8]:=8
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-cytpg1
Out[8]=8

Assume the following system:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-wdz6a0
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-oqtwlm

The cost for increasing the reliability of any component is given as :

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-mef539

Find out which component is best to improve for highest gain with least cost at time :

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-evcow9
Out[4]=4

Component is best to improve. It is more cost effective to improve component than component :

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-elyj2o
Out[5]=5

Two points in a city are connected through a network of water pipes . Find the pipes most critical to maintain the supply of water:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-d08rnu
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-r3su1o
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-kmt3or
In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-ji6eio
In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-uemvn8
Out[5]=5

An oil pipeline system with five pumps works if no more than two consecutive pumps fail. Find the most important pumps:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-2qi6zi
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-qxfyu
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-5uxb2l
Out[3]=3

Properties & Relations  (6)Properties of the function, and connections to other functions

BirnbaumImportance can be defined in terms of probability:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-h5mvwn
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-gdczdm
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-lac9wy
Out[3]=3

It is the difference in system reliability with a perfect and a failed component:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-crj1mx
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-8wr007
Out[5]=5

CriticalityFailureImportance is related to BirnbaumImportance:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-hwswhu
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-rdk742

Component weights as component unreliability divided by system unreliability:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-b29d79
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-b3nojy
Out[4]=4

Compare with definition of CriticalityFailureImportance:

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-kt65u7
Out[5]=5

ImprovementImportance is related to BirnbaumImportance:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-2uwhqu
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-5hmaxc

The unreliability of the components:

In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-hqhhkl
Out[3]=3

The improvement importance is the component unreliability multiplied by Birnbaum importance:

In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-0rzj3t
Out[4]=4

Compare with definition of ImprovementImportance:

In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-caz8bu
Out[5]=5

The Birnbaum importance for a component is independent of its own lifetime distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-zn9d03
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-m1491r
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-g2591
Out[3]=3

Irrelevant components have importance 0:

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-qx43x6
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-kvr29j
Out[2]=2

StructuralImportance is Birnbaum importance with component reliabilities :

In[1]:=1
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-bjyuku
In[2]:=2
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-k1oaw5
In[3]:=3
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-q7t7kn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-cwsfzm
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsxqckh9twewpyxsy-hj41uj
Out[5]=5
Wolfram Research (2012), BirnbaumImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/BirnbaumImportance.html.
✖
Wolfram Research (2012), BirnbaumImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/BirnbaumImportance.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_birnbaumimportance, author="Wolfram Research", title="{BirnbaumImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BirnbaumImportance.html}", note=[Accessed: 22-May-2025 ]}

✖
@misc{reference.wolfram_2025_birnbaumimportance, author="Wolfram Research", title="{BirnbaumImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BirnbaumImportance.html}", note=[Accessed: 22-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_birnbaumimportance, organization={Wolfram Research}, title={BirnbaumImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/BirnbaumImportance.html}, note=[Accessed: 22-May-2025 ]}

✖
@online{reference.wolfram_2025_birnbaumimportance, organization={Wolfram Research}, title={BirnbaumImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/BirnbaumImportance.html}, note=[Accessed: 22-May-2025 ]}