# RiskAchievementImportance

RiskAchievementImportance[rdist,t]

gives the risk achievement importances for all components in the ReliabilityDistribution rdist at time t.

RiskAchievementImportance[fdist,t]

gives the risk achievement importances for all components in the FailureDistribution fdist at time t.

# Details

• RiskAchievementImportance is also known as risk achievement worth.
• The risk achievement importance at time t for component is given by , where is the probability that the system failed given 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

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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(19)

### ReliabilityDistribution Models(10)

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:

Compute the importance:

Component is most important, and and are equally important because of symmetry:

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

Show the importance:

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 more reliable component is more important:

ReliabilityDistribution can contain many different distributions:

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

Plot the importance over time:

### FailureDistribution Models(9)

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:

The basic event is most important:

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:

Compute the importance:

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

Any valid FailureDistribution can be used:

The more reliable component is more important:

FailureDistribution can contain many different distributions:

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

Plot the importance over time:

## Applications(3)

Find out which component has the highest potential to make the system unreliable for a mission time of 1 day:

Show the importance over time:

With a mission time of 1 day, component could worsen the system reliability the most if it failed:

The importance measure is the factor with which the system unreliability can be increased by worsening . The worst case is a failed , which results in an unreliability of 1:

Analyze what component has the highest risk achievement importance for 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 should be monitored to avoid a high unreliability increase:

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

The valve needs to be monitored and maintained most to avoid a high increase in unreliability:

## Properties & Relations(4)

RiskAchievementImportance can be defined in terms of Probability:

Compute the base reliability for the system:

The reliability for the system when the component is failed:

Divide the reliability with a failed component with the base reliability:

RiskAchievementImportance approaches 1 as :

The RiskAchievementImportance is the same for all components in a series system:

Irrelevant components have importance 1:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_riskachievementimportance, author="Wolfram Research", title="{RiskAchievementImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/RiskAchievementImportance.html}", note=[Accessed: 16-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_riskachievementimportance, organization={Wolfram Research}, title={RiskAchievementImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/RiskAchievementImportance.html}, note=[Accessed: 16-July-2024 ]}