BarlowProschanImportance

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`BarlowProschanImportance`

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BarlowProschanImportance[rdist]

gives the Barlow–Proschan importances for all components in the ReliabilityDistribution rdist.

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BarlowProschanImportance[fdist]

gives the Barlow–Proschan importances for all components in the FailureDistribution fdist.

Details

The Barlow–Proschan importance for component is the probability that the failure of component coincides with the failure of the system.
The Barlow–Proschan importance for component is given by the expectation of Birnbaum importance for component using the component lifetime distribution.
The results are returned in the component order given in the distribution list in rdist or fdist.

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Two components connected in series, with different lifetime distributions:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-e66rb

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

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-4wtin8

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-sp1s0r

Out[3]=3

Two components connected in parallel, with different lifetime distributions:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-yn943z

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-8e7d7o

Out[2]=2

Use fault tree-based modeling to define the system:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-28utav

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-fmkne5

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-0zv592

Out[3]=3

Scope (16)Survey of the scope of standard use cases

ReliabilityDistribution Models (8)

Two components connected in parallel, with identical lifetime distributions:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-08flls

The importance is identical:

In[4]:=4

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-mkns50

Out[4]=4

Two components connected in series, with identical lifetime distributions:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-rwabaw

The importance is identical:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-f1wvq8

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-mji74e

The importance is identical:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-xc5531

Out[2]=2

A simple mixed system with identical lifetime distributions:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-bs7f7v

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-jy1k7y

Show the importance:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-fn2idk

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-8pfw08

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-mjkl5o

The component is critical to the system, and therefore most important:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-kx9gi2

Out[3]=3

Study the effect of a change in parameter in a simple mixed system:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-3daofy

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-lqd3ci

Show the changes in importance when worsening one of the parallel components:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-xwt0s4

Out[3]=3

Any valid ReliabilityDistribution can be used:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-cjmrsj

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-6zfypk

The less reliable component has a much higher risk of coinciding with the failure of the system:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-6lxtrj

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-dcrxnl

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-2f3o3y

The subsystem is more reliable, and therefore has a lower risk of coinciding with system failure:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-n8cikq

Out[3]=3

FailureDistribution Models (8)

Any of two basic events lead to the top event:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-l11ptk

The importance is identical:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-6nc5zk

Out[2]=2

Only both basic events together lead to the top event:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-xmgcyf

The importance is identical:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-qcmglo

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-natpec

The importance is identical:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-xz9lrc

Out[2]=2

A simple system with both And and Or gates:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-4jsp53

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-3a7nr5

Show the importance:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-vi4s0n

Out[3]=3

A simple system with both And and Or gates:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-fecls

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-sr2e4s

Show the importance:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-fhectw

Out[3]=3

Study the effect of a change in parameter in a simple mixed system:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-q6ngyg

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-mjw0g1

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

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-olq15f

Out[3]=3

Any valid FailureDistribution can be used:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-34mp4o

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-i1m9g0

The standby component is more important:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-cmc5tk

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-lxyr6j

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-mvuep3

The subsystem is more important:

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-okc513

Out[3]=3

Applications (2)Sample problems that can be solved with this function

Analyze what component is most likely to have caused a failure at the launch of an aircraft. The hangar door can be opened electronically or manually:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-z9j4b1

Two fuel pumps require power to run:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-nefezh

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-l66sqs

Two more pumps run on reliable batteries, giving the following fuel transfer structure:

In[4]:=4

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-lbq8mc

Also needed is de-icing of the aircraft and a fuel storage tank:

In[5]:=5

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-qek6nb

Define the lifetime distributions:

In[6]:=6

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-nkxog6

In[7]:=7

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-2q97ci

In[8]:=8

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-ifr45q

Compute the importance:

In[9]:=9

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-yz5yqc

It is very likely that failure of a pump coincides with a failure to launch the aircraft:

In[10]:=10

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-uzze9r

Out[10]=10

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

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-dd2ary

It is very likely that failure of the valve coincides with system failure:

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-q9ocrf

Out[2]=2

Properties & Relations (3)Properties of the function, and connections to other functions

BarlowProschanImportance is defined as an Expectation of BirnbaumImportance:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-9k73s6

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-nrxrrh

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-76yuw6

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-tp4w65

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-lzpbbu

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-vr9vex

Out[6]=6

BarlowProschanImportance always sums to 1:

In[1]:=1

✖

https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-3xo9cx

In[2]:=2

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-uf80xb

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-ycld4p

Out[3]=3

Irrelevant components have importance 0:

In[1]:=1

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https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-qx43x6

In[2]:=2

✖

https://wolfram.com/xid/0dnlbjuc3fynz7nuamcwlgi-kvr29j

Out[2]=2

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