Multi-Phase - Mixed Data - Revision history

Lisa Hacker at 20:53, 18 September 2023

2023-09-18T20:53:38Z

← Older revision		Revision as of 20:53, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow_Extended_-_Continuous_Evaluation\|~~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 17:19, 30 May 2014

2014-05-30T17:19:46Z

← Older revision		Revision as of 17:19, 30 May 2014
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	<li>The next figure shows the data entered in the RGA software.		<li>The next figure shows the data entered in the RGA software.

	[[File:multi-Phase Discrete Data Entered in RGA.png\|~~400px\|thumb~~\|center~~\|Multi-Phase Discrete Data Entered~~]]		[[File:multi-Phase Discrete Data Entered in RGA.png\|600px\|center]]

	The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]] chapter):		The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]] chapter):
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	The next figure shows the plot for the discovery failure intensity function.		The next figure shows the plot for the discovery failure intensity function.

	[[Image:rga10.16.png\|center\|~~400px\|Discovery failure intensity function.~~]]		[[Image:rga10.16.png\|center\|450px]]

	</li>		</li>
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	The next figure shows the plot for the instantaneous demonstrated, projected and growth potential reliability.		The next figure shows the plot for the instantaneous demonstrated, projected and growth potential reliability.

	[[Image:rga10.17.png\|center\|~~400px\|Instantaneous demonstrated projected and growth potential reliability.~~]]		[[Image:rga10.17.png\|center\|450px]]
	</li>		</li>
	</ol>		</ol>

David J. Groebel at 00:03, 23 April 2014

2014-04-23T00:03:49Z

← Older revision		Revision as of 00:03, 23 April 2014
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	<li>The next figure shows the data entered in the RGA software.		<li>The next figure shows the data entered in the RGA software.

	[[File:multi-Phase Discrete Data Entered in RGA.png\|~~300px~~\|thumb\|center\|Multi-Phase Discrete Data Entered]]		[[File:multi-Phase Discrete Data Entered in RGA.png\|400px\|thumb\|center\|Multi-Phase Discrete Data Entered]]

	The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]] chapter):		The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]] chapter):
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	:<math>{{\widehat{\lambda }}_{BD}}=0.6920\,\!</math>		:<math>{{\widehat{\lambda }}_{BD}}=0.6920\,\!</math>


	So the discovery failure intensity function at time <math>T=20\,\!</math> is:		So the discovery failure intensity function at time <math>T=20\,\!</math> is:

David J. Groebel at 00:02, 23 April 2014

2014-04-23T00:02:53Z

Show changes

Chris Kahn at 20:21, 6 February 2014

2014-02-06T20:21:43Z

← Older revision		Revision as of 20:21, 6 February 2014
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	The table also gives the classifications of the failure modes. There are 5 BD modes. Of these 5 modes, 2 are corrected during the test (BD3 and BD4) and 3 have not been corrected by time <math>T=20\,\!</math> (BD1, BD2 and BD5). Do the following:		The table also gives the classifications of the failure modes. There are 5 BD modes. Of these 5 modes, 2 are corrected during the test (BD3 and BD4) and 3 have not been corrected by time <math>T=20\,\!</math> (BD1, BD2 and BD5). Do the following:

	~~:1)~~ Calculate the parameters of the Crow Extended - Continuous Evaluation model.		#Calculate the parameters of the Crow Extended - Continuous Evaluation model.
	~~:2)~~ Calculated the demonstrated reliability at the end of the test.		#Calculated the demonstrated reliability at the end of the test.
	~~:3)~~ Calculate parameter <math>p\,\!</math>.		#Calculate parameter <math>p\,\!</math>.
	~~:4)~~ Calculate the unfixed BD mode failure probability.		#Calculate the unfixed BD mode failure probability.
	~~:5)~~ Calculate the nominal growth potential factor.		#Calculate the nominal growth potential factor.
	~~:6)~~ Calculate the nominal average effectiveness factor.		#Calculate the nominal average effectiveness factor.
	~~:7)~~ Calculate the discovery failure intensity function at the end of the test.		#Calculate the discovery failure intensity function at the end of the test.
	~~:8)~~ Calculate the nominal projected reliability at the end of the test.		#Calculate the nominal projected reliability at the end of the test.
	~~:9)~~ Calculate the nominal growth potential reliability at the end of the test.		#Calculate the nominal growth potential reliability at the end of the test.


	'''Solution'''		'''Solution'''
			<ol>
	~~:1)~~ The next figure shows the data entered in the RGA software.		<li>The next figure shows the data entered in the RGA software.


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	::<math>\widehat{\lambda }=0.4602\,\!</math>		::<math>\widehat{\lambda }=0.4602\,\!</math>

			</li>
	~~:2)~~ The corresponding demonstrated unreliability is calculated as:		<li>The corresponding demonstrated unreliability is calculated as:

	::<math>\begin{align}		::<math>\begin{align}
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	:Note that in discrete data, we calculate the reliability and not MTBF because we are dealing with the number of trials and not test time.		:Note that in discrete data, we calculate the reliability and not MTBF because we are dealing with the number of trials and not test time.

	~~:3)~~ Assume that the following effectiveness factors are assigned to the unfixed BD modes:		</li>
			<li>Assume that the following effectiveness factors are assigned to the unfixed BD modes:

	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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	~~:4)~~ The unfixed BD mode failure probability at time <math>T\,\!</math> is the total number of unfixed BD failures (classified at time <math>T\,\!</math> ) divided by the total trials. Based on the table at the beginning of the example, we have:		</li>
			<li>The unfixed BD mode failure probability at time <math>T\,\!</math> is the total number of unfixed BD failures (classified at time <math>T\,\!</math> ) divided by the total trials. Based on the table at the beginning of the example, we have:

	::<math>{{f}_{BD, unfixed}}=\frac{4}{20}=0.2\,\!</math>		::<math>{{f}_{BD, unfixed}}=\frac{4}{20}=0.2\,\!</math>


	~~:5)~~ The nominal growth potential factor is:		</li>
			<li>The nominal growth potential factor is:

	::<math>\begin{align}		::<math>\begin{align}
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	~~:6)~~ The nominal average effectiveness factor is:		</li>
			<li>The nominal average effectiveness factor is:

	::<math>\begin{align}		::<math>\begin{align}
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	~~:7)~~ The discovery function at time <math>T\,\!</math> is calculated using all the first occurrences of all the BD modes, both fixed and unfixed. In our example, we calculate <math>{{\widehat{\beta }}_{BD}}\,\!</math> and <math>{{\widehat{\lambda }}_{BD}}\,\!</math> using only the unfixed BD modes and excluding the second occurrence of BD2, which results in the following:		</li>
			<li>The discovery function at time <math>T\,\!</math> is calculated using all the first occurrences of all the BD modes, both fixed and unfixed. In our example, we calculate <math>{{\widehat{\beta }}_{BD}}\,\!</math> and <math>{{\widehat{\lambda }}_{BD}}\,\!</math> using only the unfixed BD modes and excluding the second occurrence of BD2, which results in the following:

	::<math>{{\widehat{\beta }}_{BD}}=0.6602\,\!</math>		::<math>{{\widehat{\beta }}_{BD}}=0.6602\,\!</math>
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	~~:8)~~ The nominal projected failure probability at time <math>T=20\,\!</math> is:		</li>
			<li>The nominal projected failure probability at time <math>T=20\,\!</math> is:

	::<math>\begin{align}		::<math>\begin{align}
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	~~:9)~~ The nominal growth potential unreliability is:		</li>
			<li>The nominal growth potential unreliability is:

	::<math>{{f}_{NGP}} = {{f}_{D}} - {{f}_{BD unfixed}} + {{\lambda}_{NGPFactor}} + {{d}_{N}}\cdot p \cdot h(T) - {{d}_{N}}h(T)\,\!</math>		::<math>{{f}_{NGP}} = {{f}_{D}} - {{f}_{BD unfixed}} + {{\lambda}_{NGPFactor}} + {{d}_{N}}\cdot p \cdot h(T) - {{d}_{N}}h(T)\,\!</math>
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	[[Image:rga10.17.png\|center\|300px\|Instantaneous demonstrated projected and growth potential reliability.]]		[[Image:rga10.17.png\|center\|300px\|Instantaneous demonstrated projected and growth potential reliability.]]
			</li>
			</ol>

Chris Kahn at 21:45, 22 January 2014

2014-01-22T21:45:37Z

← Older revision		Revision as of 21:45, 22 January 2014
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	[[File:multi-Phase Discrete Data Entered in RGA.png\|300px\|thumb\|center\|Multi-Phase Discrete Data Entered]]		[[File:multi-Phase Discrete Data Entered in RGA.png\|300px\|thumb\|center\|Multi-Phase Discrete Data Entered]]

	The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]]chapter):		The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]] chapter):

	::<math>\widehat{\beta }=0.8572\,\!</math>		::<math>\widehat{\beta }=0.8572\,\!</math>

Chris Kahn at 21:45, 22 January 2014

2014-01-22T21:45:14Z

← Older revision		Revision as of 21:45, 22 January 2014
Line 69:		Line 69:
	[[File:multi-Phase Discrete Data Entered in RGA.png\|300px\|thumb\|center\|Multi-Phase Discrete Data Entered]]		[[File:multi-Phase Discrete Data Entered in RGA.png\|300px\|thumb\|center\|Multi-Phase Discrete Data Entered]]

	The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-~~AMSAA_-_NHPP~~#Parameter_Estimation\|Crow-AMSAA (NHPP)]]chapter):		The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA (NHPP)#Parameter_Estimation\|Crow-AMSAA (NHPP)]]chapter):

	::<math>\widehat{\beta }=0.8572\,\!</math>		::<math>\widehat{\beta }=0.8572\,\!</math>

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''.
2014-01-21T18:43:44Z

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

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

A one-shot system underwent reliability growth development for a total of 20 trials. The test was performed as a combination of configuration in groups and individual trial-by-trial. The following table shows the obtained data set. The '''Failures in Interval''' column specifies the number of failures that occurred in each interval, and the '''Cumulative Trials''' column specifies the cumulative number of trials at the end of that interval.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
|colspan="5" style="text-align:center"|'''Mixed Data'''
|-
!Event
!Failures in Interval
!Cumulative Trials
!Classification
!Mode
|-
|F||1||8||BD||1
|-
|F||1||8||BD||2
|-
|F||1||8||BD||3
|-
|F||0||10||A||
|-
|F||0||11||A||
|-
|F||0||12||A||
|-
|F||1||13||BD||2
|-
|F||0||14||A||
|-
|F||0||15||A||
|-
|I||0||15||BD||3
|-
|F||1||16||BD||4
|-
|F||0||17||A||
|-
|I||0||17||BD||4
|-
|F||0||18||A||
|-
|F||0||18||A||
|-
|F||1||20||BD||5
|}

The table also gives the classifications of the failure modes. There are 5 BD modes. Of these 5 modes, 2 are corrected during the test (BD3 and BD4) and 3 have not been corrected by time <math>T=20\,\!</math> (BD1, BD2 and BD5). Do the following:

:1) Calculate the parameters of the Crow Extended - Continuous Evaluation model.
:2) Calculated the demonstrated reliability at the end of the test.
:3) Calculate parameter <math>p\,\!</math>.
:4) Calculate the unfixed BD mode failure probability.
:5) Calculate the nominal growth potential factor.
:6) Calculate the nominal average effectiveness factor.
:7) Calculate the discovery failure intensity function at the end of the test.
:8) Calculate the nominal projected reliability at the end of the test.
:9) Calculate the nominal growth potential reliability at the end of the test.

'''Solution'''

:1) The next figure shows the data entered in the RGA software.

[[File:multi-Phase Discrete Data Entered in RGA.png|300px|thumb|center|Multi-Phase Discrete Data Entered]]

The parameters <math>\beta \,\!</math> and <math>\lambda \,\!</math> are calculated as follows (the calculations for these parameters are presented in detail in the [[Crow-AMSAA_-_NHPP#Parameter_Estimation|Crow-AMSAA (NHPP)]]chapter):

::<math>\widehat{\beta }=0.8572\,\!</math>

:and:

::<math>\widehat{\lambda }=0.4602\,\!</math>

:2) The corresponding demonstrated unreliability is calculated as:

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

:Using the parameter estimates given above, the instantaneous unreliability at the end of the test (or <math>T=20\,\!</math>) is equal to:

::<math>{{f}_{D}}(20)=0.8572\cdot 0.4602\cdot {{20}^{0.8572-1}}=0.2572\,\!</math>

:The demonstrated reliability is:

::<math>\begin{align}
{{R}_{D}}= & 1-{{f}_{D}} \\
= & 1-0.2572=0.7428
\end{align}\,\!</math>

:Note that in discrete data, we calculate the reliability and not MTBF because we are dealing with the number of trials and not test time.

:3) Assume that the following effectiveness factors are assigned to the unfixed BD modes:

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
!Classification
!Mode
!Effectiveness Factor
!Implemented at End of Phase
|-
|BD||1||0.65||Phase 1
|-
|BD||2||0.70||Phase 1
|-
|BD||5||0.75||Phase 1
|}

:The parameter <math>p\,\!</math> is the total number of distinct unfixed BD modes at time <math>T\,\!</math> divided by the total number of distinct BD (fixed and unfixed) modes.

:In this example:

::<math>p=\frac{3}{5}=0.6\,\!</math>

:4) The unfixed BD mode failure probability at time <math>T\,\!</math> is the total number of unfixed BD failures (classified at time <math>T\,\!</math> ) divided by the total trials. Based on the table at the beginning of the example, we have:

::<math>{{f}_{BD, unfixed}}=\frac{4}{20}=0.2\,\!</math>

:5) The nominal growth potential factor is:

::<math>\begin{align}
{{\lambda }_{NGPFactor}}= & \underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{i}} \right)\frac{{{N}_{i}}}{T} \\
= & \left( 1-{{d}_{1}} \right)\frac{{{N}_{1}}}{T}+\left( 1-{{d}_{2}} \right)\frac{{{N}_{2}}}{T}+\left( 1-{{d}_{5}} \right)\frac{{{N}_{5}}}{T} \\
= & \left( 1-0.65 \right)\frac{1}{20}+\left( 1-0.70 \right)\frac{2}{20}+\left( 1-0.75 \right)\frac{1}{20} \\
= & 0.06
\end{align}\,\!</math>

:6) The nominal average effectiveness factor is:

::<math>\begin{align}
{{d}_{N}}= & \frac{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{Ni}}}{M} \\
= & \frac{0.65+0.70+0.75}{3} \\
= & 0.70
\end{align}\,\!</math>

:7) The discovery function at time <math>T\,\!</math> is calculated using all the first occurrences of all the BD modes, both fixed and unfixed. In our example, we calculate <math>{{\widehat{\beta }}_{BD}}\,\!</math> and <math>{{\widehat{\lambda }}_{BD}}\,\!</math> using only the unfixed BD modes and excluding the second occurrence of BD2, which results in the following:

::<math>{{\widehat{\beta }}_{BD}}=0.6602\,\!</math>

:and:

::<math>{{\widehat{\lambda }}_{BD}}=0.6920\,\!</math>

:So the discovery failure intensity function at time <math>T=20\,\!</math> is:

::<math>\begin{align}
\widehat{h}(T|BD)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\
= & 0.6920\cdot 0.6602\cdot {{20}^{0.6602-1}} \\
= & 0.16507
\end{align}\,\!</math>

:The next figure shows the plot for the discovery failure intensity function.

[[Image:rga10.16.png|center|300px|Discovery failure intensity function.]]

:8) The nominal projected failure probability at time <math>T=20\,\!</math> is:

::<math>\begin{align}
{{f}_{NP}}= & {{f}_{NGP}}+{{d}_{N}}h(T) \\
= & 0.0701+0.7\cdot 0.16507 \\
= & 0.1865
\end{align}\,\!</math>

:Therefore, the nominal projected reliability is:

::<math>\begin{align}
{{R}_{P}}= & 1-0.1856= \\
= & 0.8135
\end{align}\,\!</math>

:9) The nominal growth potential unreliability is:

::<math>{{f}_{NGP}} = {{f}_{D}} - {{f}_{BD unfixed}} + {{\lambda}_{NGPFactor}} + {{d}_{N}}\cdot p \cdot h(T) - {{d}_{N}}h(T)\,\!</math>

:Based on the previous calculation for this example, we have:

::<math>\begin{align}
{{f}_{NGP}}= & 0.2572-0.2+0.06+0.7\cdot 0.6\cdot 0.16507-0.7\cdot 0.16507 \\
= & 0.0709
\end{align}\,\!</math>

:So the nominal growth potential reliability is:

::<math>\begin{align}
{{R}_{NGP}}= & 1-0.0709 \\
= & 0.9291
\end{align}\,\!</math>

The next figure shows the plot for the instantaneous demonstrated, projected and growth potential reliability.

[[Image:rga10.17.png|center|300px|Instantaneous demonstrated projected and growth potential reliability.]]