# CriticalityFailureImportance

CriticalityFailureImportance[rdist,t]

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

CriticalityFailureImportance[fdist,t]

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

# Details

• CriticalityFailureImportance is also known as failure criticality or criticality importance factor.
• The criticality failure importance for component is the probability that component has caused system failure, when the system has failed at time .
• The criticality failure importance at time for component is given by , where is the Birnbaum importance for component , is the probability that the component has failed, and is the probability that the system 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)

Two components connected in series, with different lifetime distributions:

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

Both parallel components are contributing to system failure:

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 have caused system failure:

Two components connected in series, with identical lifetime distributions:

Both components are equally likely to have caused system failure:

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

All components are equally likely to have caused system failure:

A simple mixed system with identical lifetime distributions:

Compute the importance:

Component is most likely to have caused system failure:

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

Calculate the importance:

For a failed system, component has always failed. The probability for its causing system failure is therefore 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 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:

The standby component is less likely to have caused system failure:

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 are equally likely to lead to the top event:

Only both basic events together lead to the top event:

When the top event has occurred, both and have the probability 1 of having contributed to it:

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

Identical events in a voting gate have the same probability of causing the top event to occur:

A simple system with both And and Or gates:

Calculate the criticality importance:

Event has always occurred when the top event has occurred:

A simple system with both And and Or gates:

Show the importance:

Event is most likely to have caused the top event:

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:

The less reliable event is more likely to have caused the top event than the standby modeled event:

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

Plot the importance over time:

## Applications(4)

An underwater dry maintenance cabin is used to repair pipelines underwater. The life support system to this cabin follows the distribution:

The typical mission time is 24 hours. Compute the criticality failure importances for such a mission:

Consider a special forces team consisting of one person in command, an intelligence officer, and two specialists in each of the areas of weapons, engineering, medical, and communications. Assume the team functions as long as one specialist in each area is available:

Compute the probability of success on a 10-hour mission:

Find out the role most likely to cause the mission to fail:

Sort the result by importance:

Improve the team by adding a secondary intelligence officer:

The survival probability goes up from 99.6 to 99.9%:

Analyze what component has the highest failure criticality in 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 deicing of the aircraft and a fuel storage tank:

Compute the importance:

The fuel storage and the power are most likely to have caused the launch to fail:

Find out which component is most important during a mission time of three hours:

Show the importance over time:

During a mission of three hours, component is most likely to have caused a failure:

## Properties & Relations(5)

Failure-based criticality importance can be defined in terms of Probability:

The BirnbaumImportance for all components:

Component weights as component unreliability divided by system unreliability:

The resulting criticality failure importance:

Compare with definition:

Failure-based criticality importance is related to RiskReductionImportance:

Compare to definition:

Failure-based criticality importance is related to ImprovementImportance:

Compute the ImprovementImportance:

Divide the improvement importance by the unreliability of the system:

Compare with the definition:

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

Irrelevant components have importance 0:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_criticalityfailureimportance, author="Wolfram Research", title="{CriticalityFailureImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CriticalityFailureImportance.html}", note=[Accessed: 09-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_criticalityfailureimportance, organization={Wolfram Research}, title={CriticalityFailureImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/CriticalityFailureImportance.html}, note=[Accessed: 09-August-2024 ]}