Repairable Systems Analysis Reference Example: Difference between revisions

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{{Reference Example|{{Banner RGA Reference_Examples}}}}
{{Reference Example|{{Banner RGA Reference_Examples}}|Repairable Systems Analysis}}
This example compares the results for a repairable systems analysis.  
This example validates the results for a repairable systems analysis in RGA.  




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Crow, L.H., ''Reliability Analysis for Complex Repairable Systems'', Reliability and Biometry: Statistical Analysis of Lifelength, pg. 385, 1974.
Crow, L.H., ''Reliability Analysis for Complex Repairable Systems'', Reliability and Biometry: Statistical Analysis of Lifelength, pg. 385, 1974.
For this example, the Power Law model parameters will be calculated.




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::<math>\begin{align}
::<math>\begin{align}
\hat{\beta }=&\frac{\sum_{q=1}^{K}N_{q}}{\sum_{q=1}^{K}\sum_{i=1}^{N_{q}}ln \left(\frac{T}{X_{iq}}\right)}\\
\hat{\beta} =&\frac{\underset{q=1}{\overset{K}{\mathop \sum }}N_{q}}{\underset{q=1}{\overset{K}{\mathop \sum }}\,\underset{i=1}{\overset{N_{q}}{\mathop \sum }}\ln \left ( \frac{T}{X_{iq}} \right )}\\
\\
\\
=&0.6153
=&0.6153
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::<math>\begin{align}
::<math>\begin{align}
\hat{\lambda }=&\frac{{\underset{i=1}{\overset{N}{\mathop \sum }}N_{q}}}{KT^{\hat{\beta }}}\\
\hat{\lambda }=&\frac{{\underset{q=1}{\overset{K}{\mathop \sum }}N_{q}}}{KT^{\hat{\beta }}}\\
\\
\\
=&0.4605
=&0.4605
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The model parameters are:


[[image:Repairable SystemS SIAM_Results.png|center]]
[[image:Repairable SystemS SIAM_Results.png|center]]

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Repairable Systems Analysis

This example validates the results for a repairable systems analysis in RGA.


Reference Case

Crow, L.H., Reliability Analysis for Complex Repairable Systems, Reliability and Biometry: Statistical Analysis of Lifelength, pg. 385, 1974.

For this example, the Power Law model parameters will be calculated.


Data

The following table shows the data.

System 1 System 2 System 3
4.3 0.1 8.4
4.4 5.6 32.4
10.2 18.6 44.7
23.5 19.5 48.4
23.8 24.2 50.6
26.4 26.7 73.6
74 45.1 98.7
77.1 45.8 112.2
92.1 72.7 129.8
197.2 75.7 136
98.6 195.8
120.1
161.8
180.6
190.8
Simulated Data for 3 Systems with End Time = 200 hours


Result

The book has the following results:

Beta = 0.615, Lambda = 0.461


Results in RGA

Since [math]\displaystyle{ \,\!S_{1}=S_{2}=S_{3}=0 }[/math] and [math]\displaystyle{ \,\!T_{1}=T_{2}=T_{3}=200 }[/math] then the maximum likelihood estimates of [math]\displaystyle{ \,\!\hat{\beta} }[/math] and [math]\displaystyle{ \,\!\hat{\lambda } }[/math] are given by:


[math]\displaystyle{ \begin{align} \hat{\beta} =&\frac{\underset{q=1}{\overset{K}{\mathop \sum }}N_{q}}{\underset{q=1}{\overset{K}{\mathop \sum }}\,\underset{i=1}{\overset{N_{q}}{\mathop \sum }}\ln \left ( \frac{T}{X_{iq}} \right )}\\ \\ =&0.6153 \end{align}\,\! }[/math]


[math]\displaystyle{ \begin{align} \hat{\lambda }=&\frac{{\underset{q=1}{\overset{K}{\mathop \sum }}N_{q}}}{KT^{\hat{\beta }}}\\ \\ =&0.4605 \end{align}\,\! }[/math]


The model parameters are:

Repairable SystemS SIAM Results.png
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