Power Law Model Example - Revision history

Lisa Hacker at 21:15, 18 September 2023

2023-09-18T21:15:05Z

← Older revision		Revision as of 21:15, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}{{Navigation box}}		<noinclude>{{Banner RGA Examples}}{{Navigation box}}
	''These examples appear in the [~~[Repairable_Systems_Analysis\|~~Reliability ~~Growth and Repairable System Analysis Reference book]~~]''.		''These examples appear in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

Kate Racaza at 19:50, 31 January 2014

2014-01-31T19:50:24Z

← Older revision		Revision as of 19:50, 31 January 2014
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	==Confidence Bounds ~~Example~~==		===Confidence Bounds===
	{{:Power Law Model Confidence Bounds Example}}		{{:Power Law Model Confidence Bounds Example}}


			==Auto Transmission Example==
			{{:Auto Transmission Example}}


			==Optimum Overhaul Example==
			{{:Optimum Overhaul Example}}

Kate Racaza at 19:01, 29 January 2014

2014-01-29T19:01:24Z

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 </noinclude>
-For the data in the following table, the starting time for each system is equal to 0 and the ending time for each system is 2,000 hours. Calculate the maximum likelihood estimates <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>.
+==Parameter Estimation Example==
-{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
+==Confidence Bounds Example==
-|-
+{{:Power Law Model Confidence Bounds Example}}

Kate Racaza: Created page with '{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book…'

2014-01-29T18:47:45Z

Created page with '<noinclude>{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book…'

New page

<noinclude>{{Banner RGA Examples}}{{Navigation box}}
''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

For the data in the following table, the starting time for each system is equal to 0 and the ending time for each system is 2,000 hours. Calculate the maximum likelihood estimates <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
|colspan="3" style="text-align:center"|'''Repairable system failure data'''
|-
!System 1 ( <math>{{X}_{i1}}\,\!</math> )
!System 2 ( <math>{{X}_{i2}}\,\!</math> )
!System 3 ( <math>{{X}_{i3}}\,\!</math> )
|-
|1.2|| 1.4|| 0.3
|-
|55.6|| 35.0|| 32.6
|-
|72.7|| 46.8|| 33.4
|-
|111.9|| 65.9|| 241.7
|-
|121.9|| 181.1|| 396.2
|-
|303.6|| 712.6|| 444.4
|-
|326.9|| 1005.7|| 480.8
|-
|1568.4|| 1029.9 ||588.9
|-
|1913.5|| 1675.7|| 1043.9
|-
| ||1787.5|| 1136.1
|-
| ||1867.0|| 1288.1
|-
| || ||1408.1
|-
| || ||1439.4
|-
| || ||1604.8
|-
|<math>{{N}_{1}}=9\,\!</math> || <math>{{N}_{2}}=11\,\!</math> ||<math>{{N}_{3}}=14\,\!</math>
|}

'''Solution'''

Because the starting time for each system is equal to zero and each system has an equivalent ending time, the general equations for <math>\widehat{\beta }\,\!</math> and <math>\widehat{\lambda }\,\!</math> reduce to the closed form equations. The maximum likelihood estimates of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> are then calculated as follows:

::<math>\widehat{\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 (\tfrac{T}{{{X}_{iq}}})} = 0.45300 </math>

::<math>\widehat{\lambda }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} = 0.36224 \,\!</math>

The system failure intensity function is then estimated by:

::<math>\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}},\text{ }t>0\,\!</math>

The next figure is a plot of <math>\widehat{u}(t)\,\!</math> over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.

[[File:Instantaneous Failure Intensity vs. Time plot.png|450px|thumb|center|Instantaneous Failure Intensity vs. Time plot]]