# RiskReductionImportance

RiskReductionImportance[rdist,t]

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

RiskReductionImportance[fdist,t]

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

# Details • RiskReductionImportance is also known as risk reduction worth.
• The risk reduction importance for component is the factor by which the system unreliability would be decreased if component were perfect. As such, it shows the potential for increase in system reliability by making component better.
• The risk reduction importance at time t for component is given by , where is the probability that the system failed, given that the  component never fails, 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:

By improving a parallel component, the system reliability can be improved infinitely:

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:

Compute the importance:

Component is most important, but is not shown in the plot as it equals infinity:

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 less reliable component is more important:

ReliabilityDistribution can contain different distribution families:

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

Plot the importance over time:

### FailureDistribution Models(9)

Either of two basic events leads 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:

Both and are equally important because of symmetry:

A simple system with both And and Or gates:

Show the importance:

Study the effect of parameter variation 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 component is more important:

FailureDistribution can contain different distribution families:

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

Plot the importance over time:

## Applications(3)

Analyze what component has the best potential for improving the reliability of 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:

Improving the power would give a good reliability increase in the beginning, but for long-term reliability the various pumps should be improved:

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

Show the importance over time:

Component can improve system reliability the most, assuming a mission time of three hours:

The importance is the factor with which the system unreliability can be reduced by improving :

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

Find out which components can improve system reliability most:

## Properties & Relations(5)

RiskReductionImportance can be defined in terms of Probability:

Compute the base unreliability for the system:

The unreliability for when the component is always working:

The ratio of the base unreliability and the unreliability for the always working components:

Compare with definition:

RiskReductionImportance is related to CriticalityFailureImportance:

Compute the risk-reduction importance:

Compare with the definition:

A system with all components in parallel will have RiskReductionImportance equal to :

For all importances that are not , the RiskReductionImportance approaches 1 as :

Irrelevant components have importance 1: