Gap Analysis Example - Revision history

Lisa Hacker at 20:38, 18 September 2023

2023-09-18T20:38:21Z

← Older revision		Revision as of 20:38, 18 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	~~''This example appears in the article [[Gap Analysis]]''.~~
	</noinclude>		</noinclude>

David J. Groebel at 18:36, 30 May 2014

2014-05-30T18:36:09Z

← Older revision		Revision as of 18:36, 30 May 2014
Line 73:		Line 73:
	The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.		The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.

	[[Image:rga5.14.png\|center\|~~400px\|Observed and estimated number of failures for the full data set.~~]]		[[Image:rga5.14.png\|center\|450px]]

	Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.
Line 86:		Line 86:
	The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.		The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

	[[Image:rga5.15.png\|center\|~~400px\|Observed and estimated number of failures for the gap data set.~~]]</li>		[[Image:rga5.15.png\|center\|450px]]</li>
	</ol>		</ol>

Chris Kahn at 20:17, 15 May 2014

2014-05-15T20:17:42Z

← Older revision		Revision as of 20:17, 15 May 2014
Line 3:		Line 3:
	</noinclude>		</noinclude>

	Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=~~1000~~\,\!</math> hours of testing.		Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=1,000\,\!</math> hours of testing.

	<center><math>{{X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>		<center><math>{{X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>

David J. Groebel at 00:14, 23 April 2014

2014-04-23T00:14:09Z

← Older revision		Revision as of 00:14, 23 April 2014
Line 73:		Line 73:
	The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.		The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.

	[[Image:rga5.14.png\|center\|~~500px~~\|Observed and estimated number of failures for the full data set.]]		[[Image:rga5.14.png\|center\|400px\|Observed and estimated number of failures for the full data set.]]

	Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.
Line 86:		Line 86:
	The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.		The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

	[[Image:rga5.15.png\|center\|~~500px~~\|Observed and estimated number of failures for the gap data set.]]</li>		[[Image:rga5.15.png\|center\|400px\|Observed and estimated number of failures for the gap data set.]]</li>
	</ol>		</ol>

Chris Kahn at 23:22, 31 January 2014

2014-01-31T23:22:04Z

← Older revision		Revision as of 23:22, 31 January 2014
Line 54:		Line 54:
	\|}		\|}

	1) Determine the maximum likelihood estimators for the Crow-AMSAA model.		#Determine the maximum likelihood estimators for the Crow-AMSAA model.
			#Evaluate the goodness-of-fit for the model.
	2) Evaluate the goodness-of-fit for the model.		#Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.

	3) Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.

	'''Solution'''		'''Solution'''
			<ol>
	1) For the time terminated test:		<li>For the time terminated test:

	:<math>\begin{align}		:<math>\begin{align}
Line 68:		Line 66:
	& \widehat{\lambda }= & 0.4521		& \widehat{\lambda }= & 0.4521
	\end{align}\,\!</math>		\end{align}\,\!</math>
			</li>
	2) The Cramér-von Mises goodness-of-fit test for this data set yields:		<li>The Cramér-von Mises goodness-of-fit test for this data set yields:

	:<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>		:<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>
Line 78:		Line 76:

	Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.
			</li>
	3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA (NHPP)#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:		<li>Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA (NHPP)#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:

	:<math>\begin{align}		:<math>\begin{align}
Line 88:		Line 86:
	The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.		The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

	[[Image:rga5.15.png\|center\|500px\|Observed and estimated number of failures for the gap data set.]]		[[Image:rga5.15.png\|center\|500px\|Observed and estimated number of failures for the gap data set.]]</li>
			</ol>

Chris Kahn at 22:09, 29 January 2014

2014-01-29T22:09:15Z

← Older revision		Revision as of 22:09, 29 January 2014
Line 75:		Line 75:
	The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.		The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.

	[[Image:rga5.14.png\|center\|~~400px~~\|Observed and estimated number of failures for the full data set.]]		[[Image:rga5.14.png\|center\|500px\|Observed and estimated number of failures for the full data set.]]

	Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.

Chris Kahn at 22:08, 29 January 2014

2014-01-29T22:08:49Z

← Older revision		Revision as of 22:08, 29 January 2014
Line 88:		Line 88:
	The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.		The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

	[[Image:rga5.15.png\|center\|~~400px~~\|Observed and estimated number of failures for the gap data set.]]		[[Image:rga5.15.png\|center\|500px\|Observed and estimated number of failures for the gap data set.]]

David J. Groebel at 01:28, 25 January 2014

2014-01-25T01:28:16Z

← Older revision		Revision as of 01:28, 25 January 2014
Line 4:		Line 4:

	Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=1000\,\!</math> hours of testing.		Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=1000\,\!</math> hours of testing.


	<center><math>{{X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>		<center><math>{{X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>
Line 54:		Line 53:
	\|7\|\| 875-1000\|\|1\|\| 86		\|7\|\| 875-1000\|\|1\|\| 86
	\|}		\|}


	1) Determine the maximum likelihood estimators for the Crow-AMSAA model.		1) Determine the maximum likelihood estimators for the Crow-AMSAA model.

David J. Groebel at 01:21, 25 January 2014

2014-01-25T01:21:23Z

← Older revision		Revision as of 01:21, 25 January 2014
Line 56:		Line 56:


	:1) Determine the maximum likelihood estimators for the Crow-AMSAA model.		1) Determine the maximum likelihood estimators for the Crow-AMSAA model.
	:2) Evaluate the goodness-of-fit for the model.
	:3) Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.		2) Evaluate the goodness-of-fit for the model.

			3) Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.

	'''Solution'''		'''Solution'''

	:1) For the time terminated test:		1) For the time terminated test:

	::<math>\begin{align}		:<math>\begin{align}
	& \widehat{\beta }= & 0.7597 \\		& \widehat{\beta }= & 0.7597 \\
	& \widehat{\lambda }= & 0.4521		& \widehat{\lambda }= & 0.4521
	\end{align}\,\!</math>		\end{align}\,\!</math>

	:2) The Cramér-von Mises goodness-of-fit test for this data set yields:		2) The Cramér-von Mises goodness-of-fit test for this data set yields:

	~~::<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>~~

	:The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\~~ln N(t)~~\,\~~!</math> versus <math>~~\~~ln t~~\,\~~!</math> with the fitted line <math>~~\~~ln \hat~~{~~\lambda~~ }+\~~hat~~{\beta }\~~ln t\,\!</math>, where <math>\widehat~~{\~~lambda~~ }~~=0.4521\,\!</math> and <math>\widehat~~{~~\beta~~ }=0.~~7597~~\,\!</math> ~~are the maximum likelihood estimates.~~		:<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>

			The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.

	[[Image:rga5.14.png\|center\|400px\|Observed and estimated number of failures for the full data set.]]		[[Image:rga5.14.png\|center\|400px\|Observed and estimated number of failures for the full data set.]]

	:Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.

	:3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA (NHPP)#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:		3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA (NHPP)#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:

	::<math>\begin{align}		:<math>\begin{align}
	& \widehat{\beta }= & 0.5596 \\		& \widehat{\beta }= & 0.5596 \\
	& \widehat{\lambda }= & 1.1052		& \widehat{\lambda }= & 1.1052
	\end{align}\,\!</math>		\end{align}\,\!</math>

			The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.
	:The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

	[[Image:rga5.15.png\|center\|400px\|Observed and estimated number of failures for the gap data set.]]		[[Image:rga5.15.png\|center\|400px\|Observed and estimated number of failures for the gap data set.]]

Chris Kahn at 21:55, 22 January 2014

2014-01-22T21:55:12Z

← Older revision		Revision as of 21:55, 22 January 2014
Line 80:		Line 80:
	:Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.		:Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.

	:3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-~~AMSAA_-_NHPP~~#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:		:3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA (NHPP)#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:

	::<math>\begin{align}		::<math>\begin{align}