# CriticalitySuccessImportance

CriticalitySuccessImportance[rdist,t]

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

CriticalitySuccessImportance[fdist,t]

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

# Details • CriticalitySuccessImportance is also known as criticality importance factor.
• The criticality success importance for component is the probability that component is the component that contributes to system success, given that the system is working.
• The criticality success importance at time for component is given by , where is the Birnbaum importance for component , is the probability that the component is working, and is the probability that the system is working.
• 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 parallel, with different lifetime distributions:

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

Both serial components are contributing to a working system:

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 contribute to a working system:

Two components connected in series, with identical lifetime distributions:

For a working series system, both and are contributing to the success with probability 1:

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

All components are equally likely to contribute to the system's working:

A simple mixed system with identical lifetime distributions:

Compute the importance:

For a working system, component is always contributing by working:

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

Calculate the importance:

The component is most likely to contribute to a working system:

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:

Plot the importance measures:

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 contribute to preventing the top event from occurring:

Only both basic events together lead to the top event:

Both events are equally likely to lead to the top event:

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

Identical events in a voting gate have the same probability of preventing failure:

A simple system with both And and Or gates:

Calculate the criticality importance:

Event is most likely to prevent the top event if it has not occurred yet:

A simple system with both And and Or gates:

Show the importance:

When the top event has not occurred, event is contributing to its not occurring with probability 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 basic events, :

Any valid FailureDistribution can be used:

Plot the importance measures:

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

Plot the importance over time:

## Applications(3)

Find out which component is most likely to contribute to success with a mission time of three hours:

Show the importance over time:

With a mission time of three hours, component is most likely to contribute to system success:

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

The component always contributes to system success:

Assume the following system:

Compute the importance:

For all , it is true that the importances are in the order :

## Properties & Relations(3)

CriticalitySuccessImportance can be defined in terms of Probability:

The BirnbaumImportance for all components:

Component weights as component reliability divided by system reliability:

The resulting success-based criticality importance:

Compare with definition:

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

Irrelevant components have importance 0: