# BirnbaumImportance

BirnbaumImportance[rdist,t]

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

BirnbaumImportance[fdist,t]

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

# Details

• BirnbaumImportance is also known as reliability importance.
• The Birnbaum importance is the improvement in the reliability that would be gained by replacing a failed component with a perfect component .
• The Birnbaum importance at time for component is given by where is the probability that the system is working given that the component is perfect and is the probability that the system is working given that the component 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:

Two components connected in parallel, with different lifetime distributions:

Use fault tree-based modeling to define the system:

## Scope(17)

### ReliabilityDistribution Models(9)

Two components connected in parallel, with identical lifetime distributions:

A change in reliability for either component will result in the same system reliability change:

Two components connected in series, with identical lifetime distributions:

A change in reliability for either component will result in the same system reliability change:

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

Components are equally important:

A simple mixed system with identical lifetime distributions:

Compute the importance:

Changing the reliability of component will impact the system reliability most:

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

Compute the importance:

Improving the component has the biggest impact on system reliability:

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:

Early in the lifetime, changing the reliability of the standby component will have more effect:

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

Plot the importance over time:

### FailureDistribution Models(8)

Any of two basic events lead to the top event:

A change in reliability for either event will result in the same top event reliability change:

Only both basic events together lead to the top event:

A change in reliability for either event will result in the same top event reliability change:

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

A change in reliability for any of the events will result in the same top event reliability change:

A simple system with both And and Or gates:

The basic event is most important:

Changing the reliability of event will impact the top event reliability most:

A simple system with both And and Or gates:

Compute the importance:

Improving event has the biggest impact on preventing 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:

Early in the lifetime, changing the reliability of the standby component will have more effect:

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

Plot the importance over time:

## Applications(5)

Find out which component is best to improve in a system that has a mission time of 3 hours:

Show the importance over time:

With a mission time of 3 hours, improvement of component x will lead to the largest system improvement:

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

Find the cumulative Birnbaum importance over the entire lifetime:

The cumulative Birnbaum importance gives the difference of mean time to failure:

Assume the following system:

The cost for increasing the reliability of any component is given as :

Find out which component is best to improve for highest gain with least cost at time :

Component is best to improve. It is more cost effective to improve component than component :

Two points in a city are connected through a network of water pipes . Find the pipes most critical to maintain the supply of water:

An oil pipeline system with five pumps works if no more than two consecutive pumps fail. Find the most important pumps:

## Properties & Relations(6)

BirnbaumImportance can be defined in terms of probability:

It is the difference in system reliability with a perfect and a failed component:

CriticalityFailureImportance is related to BirnbaumImportance:

Component weights as component unreliability divided by system unreliability:

Compare with definition of CriticalityFailureImportance:

ImprovementImportance is related to BirnbaumImportance:

The unreliability of the components:

The improvement importance is the component unreliability multiplied by Birnbaum importance:

Compare with definition of ImprovementImportance:

The Birnbaum importance for a component is independent of its own lifetime distribution:

Irrelevant components have importance 0:

StructuralImportance is Birnbaum importance with component reliabilities :

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_birnbaumimportance, author="Wolfram Research", title="{BirnbaumImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BirnbaumImportance.html}", note=[Accessed: 06-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_birnbaumimportance, organization={Wolfram Research}, title={BirnbaumImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/BirnbaumImportance.html}, note=[Accessed: 06-August-2024 ]}