Crow-AMSAA Discrete Model Grouped Data Example - Revision history

Lisa Hacker at 21:22, 18 September 2023

2023-09-18T21:22:41Z

← Older revision		Revision as of 21:22, 18 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow-AMSAA (NHPP)\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 18:32, 30 May 2014

2014-05-30T18:32:43Z

← Older revision		Revision as of 18:32, 30 May 2014
Line 54:		Line 54:
	This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.		This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.

	[[Image:rga5.18.png\|center\|~~400px\|Instantaneous unreliability at the end of the test.~~]]		[[Image:rga5.18.png\|center\|450px]]

	The instantaneous reliability can then be calculated as:		The instantaneous reliability can then be calculated as:

Chris Kahn at 16:46, 14 May 2014

2014-05-14T16:46:30Z

← Older revision		Revision as of 16:46, 14 May 2014
Line 6:		Line 6:

	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Mixed ~~data~~'''		\|+'''Mixed Data'''
	!Failures in Interval		!Failures in Interval
	!Cumulative Trials		!Cumulative Trials

David J. Groebel at 00:11, 23 April 2014

2014-04-23T00:11:35Z

← Older revision		Revision as of 00:11, 23 April 2014
Line 1:		Line 1:
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

David J. Groebel at 01:24, 31 January 2014

2014-01-31T01:24:42Z

← Older revision		Revision as of 01:24, 31 January 2014
Line 61:		Line 61:
	{{R}_{inst}}=1-0.1871=0.8129		{{R}_{inst}}=1-0.1871=0.8129
	\end{align}\,\!</math>		\end{align}\,\!</math>

	~~The average unreliability is calculated as:~~

	~~:<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>~~

	~~and the average reliability is calculated as:~~

	~~:<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>~~

David J. Groebel at 00:26, 31 January 2014

2014-01-31T00:26:13Z

← Older revision		Revision as of 00:26, 31 January 2014
Line 36:		Line 36:
	Using the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:		Using the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:

	:<math>\~~widehat~~{\beta }=0.7950\,\!</math>		:<math>\hat{\beta }=0.7950\,\!</math>

	and:		and:

	:<math>\~~widehat~~{\lambda }=0.5588\,\!</math>		:<math>\hat{\lambda }=0.5588\,\!</math>

	As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)\,\!</math>, is defined as:		As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)\,\!</math>, is defined as:

David J. Groebel at 17:44, 27 January 2014

2014-01-27T17:44:07Z

← Older revision		Revision as of 17:44, 27 January 2014
Line 38:		Line 38:
	:<math>\widehat{\beta }=0.7950\,\!</math>		:<math>\widehat{\beta }=0.7950\,\!</math>

	:and:		and:

	:<math>\widehat{\lambda }=0.5588\,\!</math>		:<math>\widehat{\lambda }=0.5588\,\!</math>

David J. Groebel at 17:43, 27 January 2014

2014-01-27T17:43:56Z

← Older revision		Revision as of 17:43, 27 January 2014
Line 36:		Line 36:
	Using the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:		Using the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:

	::<math>\widehat{\beta }=0.7950\,\!</math>		:<math>\widehat{\beta }=0.7950\,\!</math>

	:and:		:and:

	::<math>\widehat{\lambda }=0.5588\,\!</math>		:<math>\widehat{\lambda }=0.5588\,\!</math>


	As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)\,\!</math>, is defined as:		As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)\,\!</math>, is defined as:

	::<math>\begin{align}		:<math>\begin{align}
	{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0		{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Using the parameter estimates, we can calculate the instantaneous unreliability at the end of the test, or <math>T=68.\,\!</math>		Using the parameter estimates, we can calculate the instantaneous unreliability at the end of the test, or <math>T=68.\,\!</math>

	::<math>{{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871\,\!</math>		:<math>{{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871\,\!</math>


	This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.		This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.
Line 61:		Line 58:
	The instantaneous reliability can then be calculated as:		The instantaneous reliability can then be calculated as:

	::<math>\begin{align}		:<math>\begin{align}
	{{R}_{inst}}=1-0.1871=0.8129		{{R}_{inst}}=1-0.1871=0.8129
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The average unreliability is calculated as:		The average unreliability is calculated as:

	::<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>		:<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>


	and the average reliability is calculated as:		and the average reliability is calculated as:

	::<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>		:<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>

Chris Kahn at 21:52, 22 January 2014

2014-01-22T21:52:37Z

← Older revision		Revision as of 21:52, 22 January 2014
Line 1:		Line 1:
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow-~~AMSAA_-_NHPP~~\|Reliability Growth and Repairable System Analysis Reference book]]''.		''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference book]]''.
	</noinclude>		</noinclude>

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. The table…'

2013-11-21T23:36:16Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> The table…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Crow-AMSAA_-_NHPP|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

The table below shows the number of failures of each interval of trials and the cumulative number of trials in each interval for a reliability growth test. For example, the first row indicates that for an interval of 14 trials, 5 failures occurred.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Mixed data'''
!Failures in Interval
!Cumulative Trials
|-
|5|| 14
|-
|3|| 33
|-
|4|| 48
|-
|0|| 52
|-
|1|| 53
|-
|0|| 57
|-
|1|| 58
|-
|0|| 62
|-
|1|| 63
|-
|0|| 67
|-
|1|| 68
|}

Using the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:

::<math>\widehat{\beta }=0.7950\,\!</math>

:and:

::<math>\widehat{\lambda }=0.5588\,\!</math>

As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)\,\!</math>, is defined as:

::<math>\begin{align}
{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0
\end{align}\,\!</math>

Using the parameter estimates, we can calculate the instantaneous unreliability at the end of the test, or <math>T=68.\,\!</math>

::<math>{{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871\,\!</math>

This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.

[[Image:rga5.18.png|center|400px|Instantaneous unreliability at the end of the test.]]

The instantaneous reliability can then be calculated as:

::<math>\begin{align}
{{R}_{inst}}=1-0.1871=0.8129
\end{align}\,\!</math>

The average unreliability is calculated as:

::<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>

and the average reliability is calculated as:

::<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>