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 ^(th) component is perfect and is the probability that the system is working given that the ^(th) component 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  (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 :

Introduced in 2012
 (9.0)