Repairable Systems Analysis Reference Example: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
 
(38 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Reference Example|Image=RGA_Reference_Examples_Banner.png|Link=RGA_Reference_Examples}}
{{Reference Example|{{Banner RGA Reference_Examples}}|Repairable Systems Analysis}}
 
This example validates the results for a repairable systems analysis in RGA.  
This example compares the results for a repairable systems analysis.  




Line 7: Line 6:


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.




{{Reference_Example_Heading2}}
{{Reference_Example_Heading2}}


{| {{table}}
The following table shows the data.
 
{| {{table|25%}}
!System 1
!System 1
!System 2
!System 2
Line 46: Line 49:
| ||190.8||
| ||190.8||
|-
|-
|+'''Simulated Data for 3 Systems with End Time = 200 hours'''
|}
|}




{{Reference_Example_Heading3}}
{{Reference_Example_Heading3}}
The book has the following results:


Beta = 0.615, Lambda = 0.461
Beta = 0.615, Lambda = 0.461




{{Reference_Example_Heading4|Software=RGA}}
{{Reference_Example_Heading4|RGA}}
 
Since <math>\,\!S_{1}=S_{2}=S_{3}=0</math> and <math>\,\!T_{1}=T_{2}=T_{3}=200</math>  then the maximum likelihood estimates of <math>\,\!\hat{\beta}</math> and <math>\,\!\hat{\lambda }</math> are given by:
 
 
::<math>\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>\begin{align}
\hat{\lambda }=&\frac{{\underset{q=1}{\overset{K}{\mathop \sum }}N_{q}}}{KT^{\hat{\beta }}}\\
\\
=&0.4605
\end{align}\,\!</math>
 


Since <math>\,\!S_{1}=S_{2}=S_{3}=0</math> and <math>\,\!T_{1}=T_{2}=T_{3}=200</math> then the maximum likelihood estimates of <math>\,\!\hat{\beta}</math> and <math>\,\!\hat{\lambda }</math> are given by:
The model parameters are:


::<math>\,\!\hat{\beta }=\frac{\sum_{q=1}^{K}N_{q}}{\sum_{q=1}^{K}\sum_{i=1}^{N_{q}}ln \left(\frac{T}{N_{iq}}\right)}</math>
[[image:Repairable SystemS SIAM_Results.png|center]]

Latest revision as of 18:26, 28 September 2015

RGA Reference Examples Banner.png


New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.

As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at RGA examples and RGA reference examples.



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
Retrieved from "https://www.reliawiki.com/index.php?title=Repairable_Systems_Analysis_Reference_Example&oldid=60842"