BarlowProschanImportance

BarlowProschanImportance[rdist]

gives the Barlow–Proschan importances for all components in the ReliabilityDistribution rdist.

BarlowProschanImportance[fdist]

gives the Barlow–Proschan importances for all components in the FailureDistribution fdist.

Details

  • The Barlow–Proschan importance for component is the probability that the failure of component coincides with the failure of the system.
  • The Barlow–Proschan importance for component is given by the expectation of Birnbaum importance for component using the ^(th) component lifetime distribution.
  • The results are returned in the component order given in the distribution list in rdist or fdist.

Examples

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Basic Examples  (3)

Two components connected in series, with different lifetime distributions:

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

Two components connected in parallel, with different lifetime distributions:

Use fault tree-based modeling to define the system:

Scope  (16)

ReliabilityDistribution Models  (8)

Two components connected in parallel, with identical lifetime distributions:

The importance is identical:

Two components connected in series, with identical lifetime distributions:

The importance is identical:

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

The importance is identical:

A simple mixed system with identical lifetime distributions:

Show the importance:

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

The component is critical to the system, and therefore most important:

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

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

Any valid ReliabilityDistribution can be used:

The less reliable component has a much higher risk of coinciding with the failure of the system:

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

The subsystem is more reliable, and therefore has a lower risk of coinciding with system failure:

FailureDistribution Models  (8)

Any of two basic events lead to the top event:

The importance is identical:

Only both basic events together lead to the top event:

The importance is identical:

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

The importance is identical:

A simple system with both And and Or gates:

Show the importance:

A simple system with both And and Or gates:

Show the importance:

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

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

Any valid FailureDistribution can be used:

The standby component is more important:

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

The subsystem is more important:

Applications  (2)

Analyze what component is most likely to have caused a failure at the launch of an aircraft. The hangar door can be opened electronically or manually:

Two fuel pumps require power to run:

Two more pumps run on reliable batteries, giving the following fuel transfer structure:

Also needed is de-icing of the aircraft and a fuel storage tank:

Define the lifetime distributions:

Compute the importance:

It is very likely that failure of a pump coincides with a failure to launch the aircraft:

Consider a water pumping system, with one valve and two redundant pumps. The reliability of the components are given as probabilities:

It is very likely that failure of the valve coincides with system failure:

Properties & Relations  (3)

BarlowProschanImportance is defined as an Expectation of BirnbaumImportance:

BarlowProschanImportance always sums to 1:

Irrelevant components have importance 0:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_barlowproschanimportance, author="Wolfram Research", title="{BarlowProschanImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BarlowProschanImportance.html}", note=[Accessed: 18-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_barlowproschanimportance, organization={Wolfram Research}, title={BarlowProschanImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/BarlowProschanImportance.html}, note=[Accessed: 18-April-2025 ]}