CriticalitySuccessImportance

CriticalitySuccessImportance[rdist,t]

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

CriticalitySuccessImportance[fdist,t]

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

Details

  • CriticalitySuccessImportance is also known as criticality importance factor.
  • The criticality success importance for component is the probability that component is the component that contributes to system success, given that the system is working.
  • The criticality success importance at time for component is given by , where is the Birnbaum importance for component , is the probability that the component is working, and is the probability that the system is working.
  • The results are returned in the component order given in the distribution list in rdist or fdist.

Examples

open allclose all

Basic Examples  (3)

Two components connected in parallel, with different lifetime distributions:

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

Both serial components are contributing to a working system:

Use fault tree-based modeling to define the system:

Scope  (17)

ReliabilityDistribution Models  (9)

Two components connected in parallel, with identical lifetime distributions:

Both components are equally likely to contribute to a working system:

Two components connected in series, with identical lifetime distributions:

For a working series system, both and are contributing to the success with probability 1:

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

All components are equally likely to contribute to the system's working:

A simple mixed system with identical lifetime distributions:

Compute the importance:

For a working system, component is always contributing by working:

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

Calculate the importance:

The component is most likely to contribute to a working system:

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

Compute the importance:

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

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

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

As machine-precision results:

As symbolic expressions:

Any valid ReliabilityDistribution can be used:

Plot the importance measures:

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

Plot the importance over time:

FailureDistribution Models  (8)

Either of two basic events leads to the top event:

Both events contribute to preventing the top event from occurring:

Only both basic events together lead to the top event:

Both events are equally likely to lead to the top event:

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

Identical events in a voting gate have the same probability of preventing failure:

A simple system with both And and Or gates:

Calculate the criticality importance:

Event is most likely to prevent the top event if it has not occurred yet:

A simple system with both And and Or gates:

Show the importance:

When the top event has not occurred, event is contributing to its not occurring with probability 1:

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

Compute the importance:

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

Any valid FailureDistribution can be used:

Plot the importance measures:

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

Plot the importance over time:

Applications  (3)

Find out which component is most likely to contribute to success with a mission time of three hours:

Show the importance over time:

With a mission time of three hours, component is most likely to contribute to system success:

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

The component always contributes to system success:

Assume the following system:

Compute the importance:

For all , it is true that the importances are in the order :

Properties & Relations  (3)

CriticalitySuccessImportance can be defined in terms of Probability:

The BirnbaumImportance for all components:

Component weights as component reliability divided by system reliability:

The resulting success-based criticality importance:

Compare with definition:

For a serial system, the success-based criticality importance is always 1 for all components:

Irrelevant components have importance 0:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_criticalitysuccessimportance, author="Wolfram Research", title="{CriticalitySuccessImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CriticalitySuccessImportance.html}", note=[Accessed: 23-September-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_criticalitysuccessimportance, organization={Wolfram Research}, title={CriticalitySuccessImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/CriticalitySuccessImportance.html}, note=[Accessed: 23-September-2023 ]}