FussellVeselyImportance
✖
FussellVeselyImportance
gives the Fussell–Vesely importances for all components in the ReliabilityDistribution rdist at time t.
gives the Fussell–Vesely importances for all components in the FailureDistribution fdist at time t.
Details
- The Fussell–Vesely importance at time for component is given by where is the probability that at least one minimal cut set containing component has failed at time and is the probability that the system has failed at time . A minimal cut set is a minimal set of components which, if failed, causes the system to fail.
- The results are returned in the component order given in the distribution list in rdist or fdist.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Two components connected in series, with different lifetime distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-e66rb
The result is given in the same order as the distribution list in ReliabilityDistribution:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-9epgzw
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-4wtin8
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-buotvu
Two components connected in parallel, with different lifetime distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-yn943z
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-8e7d7o
Use fault tree-based modeling to define the system:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-28utav
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-fmkne5
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-0zv592
Scope (16)Survey of the scope of standard use cases
ReliabilityDistribution Models (8)
Two components connected in series, with identical lifetime distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-rwabaw
A change in reliability for either component will result in the same system reliability change:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-f1wvq8
A system where two out of three components need to work, with identical lifetime distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-mji74e
Components are equally important:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xc5531
A simple mixed system with identical lifetime distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-bs7f7v
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-jy1k7y
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-6vuxir
Changing the reliability of component x will impact the system reliability most:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-fn2idk
A system with a series connection in parallel with a component:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-8pfw08
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-mjkl5o
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-t1mcxq
Improving the x component has the biggest impact on system reliability:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xfjq50
Study the effect of a change in parameter in a simple mixed system:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-3daofy
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-lqd3ci
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-srbs5o
Show the changes in importance when worsening one of the parallel components, z:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xwt0s4
One component in parallel with two others, with different distributions:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-pr0k1j
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-bkyutz
Find the importance measures at one specific point in time as exact results:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-urftd7
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-vf85i0
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-4n9ka6
Any valid ReliabilityDistribution can be used:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-cjmrsj
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-6zfypk
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-6lxtrj
Later in the lifetime, changing the reliability of the standby component y will have more effect:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-iedm9b
Model the system in steps to get the importance measure for a subsystem:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-dcrxnl
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-2f3o3y
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-n8cikq
Plot the importance over time:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-kf0c4n
FailureDistribution Models (8)
Either of two basic events lead to the top event:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-l11ptk
A change in reliability for either event will result in the same top event reliability change:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-6nc5zk
Only both basic events together lead to the top event:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xmgcyf
FussellVeselyImportance will rank both events as equally important:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-nlxcpv
A voting gate with identical distributions on the basic events:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-natpec
A change in reliability for any of the events will result in the same top event reliability change:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xz9lrc
A simple system with both And and Or gates:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-4jsp53
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-3a7nr5
The basic event x is most important:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-wh2s2m
Changing the reliability of event x will impact the top event reliability most:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-tynywc
A simple system with both And and Or gates:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-fecls
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-sr2e4s
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-gbr67k
Improving event x has the biggest impact on preventing the top event:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-88i8fa
Study the effect of a change in parameter in a simple mixed system:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-ig97mn
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-xdjp3g
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-4smvi5
Show the changes in importance when worsening one of the basic events, z:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-pcd9ai
Any valid FailureDistribution can be used:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-34mp4o
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-i1m9g0
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-9qepae
Early in the lifetime, changing the reliability of the standby component y will have more effect:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-cmc5tk
Model the system in steps to get the importance measure for a subsystem:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-lxyr6j
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-mvuep3
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-okc513
Plot the importance over time:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-z93seu
Applications (3)Sample problems that can be solved with this function
Find out which component is most important in a system that has to last for three hours:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-q3cuud
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-wgnax0
Component v is most important according to the Fussell–Vesely importance:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-ud0au2
A problem at coal mines is bulldozers falling through bridged voids in coal piles. The bulldozer can be over a void intentionally or unintentionally:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-yg9xeu
To form a void, there has to be subsurface flow in the coal. This requires removal of coal from below on a conveyor belt, and an open feeder to that belt:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-tnnjgz
It is also required that no flow occurs on the surface. This can happen if the coal freezes:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-nltvy5
Compacted coal can also lead to a non-flowing surface:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-pj5htc
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-9nqf9x
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-j7r2i5
Assume the following distributions for the events:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-160mmq
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-67e78o
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-bm4ry1
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-zmzoc6
To determine what actions to take to avoid accidents, compute the importance of the basic events:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-g42h00
We can see that the events with importance 1 are the highest:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-847eq0
Find the basic events with importance 1:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-r7jngy
Consider a water pumping system with one valve and two redundant pumps. The reliability of the components are given as probabilities:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-dd2ary
Find out which components are most important:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-q9ocrf
Properties & Relations (6)Properties of the function, and connections to other functions
FussellVeselyImportance for serial connections can be defined in terms of probability:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-geskj
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-lxk16l
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-nsq607
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-89k2sp
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-j9a75f
All parallel systems have FussellVeselyImportance equal to 1:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-08flls
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-mkns50
Subsystems with parallel structure will have the same importance:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-tw18x1
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-12k5yu
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-4i2dr1
CriticalityFailureImportance is always less than or equal to Fussell–Vesely importance:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-epxbpg
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-k20k25
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-p058vt
CriticalityFailureImportance approaches Fussell–Vesely for highly reliable components:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-gfl5oq
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-j9a712
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-s72o2r
The difference is always when the failure rate approaches :
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-tc59nr
Irrelevant components have importance 0:
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-qx43x6
https://wolfram.com/xid/0bywougjhqp40xxqg07sluq-kvr29j
Wolfram Research (2012), FussellVeselyImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/FussellVeselyImportance.html.
Text
Wolfram Research (2012), FussellVeselyImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/FussellVeselyImportance.html.
Wolfram Research (2012), FussellVeselyImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/FussellVeselyImportance.html.
CMS
Wolfram Language. 2012. "FussellVeselyImportance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FussellVeselyImportance.html.
Wolfram Language. 2012. "FussellVeselyImportance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FussellVeselyImportance.html.
APA
Wolfram Language. (2012). FussellVeselyImportance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FussellVeselyImportance.html
Wolfram Language. (2012). FussellVeselyImportance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FussellVeselyImportance.html
BibTeX
@misc{reference.wolfram_2024_fussellveselyimportance, author="Wolfram Research", title="{FussellVeselyImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FussellVeselyImportance.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_fussellveselyimportance, organization={Wolfram Research}, title={FussellVeselyImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/FussellVeselyImportance.html}, note=[Accessed: 08-January-2025
]}