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_2025_fussellveselyimportance, author="Wolfram Research", title="{FussellVeselyImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FussellVeselyImportance.html}", note=[Accessed: 11-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_fussellveselyimportance, organization={Wolfram Research}, title={FussellVeselyImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/FussellVeselyImportance.html}, note=[Accessed: 11-July-2025
]}