Template:Expected failure time plots - Revision history

Kate Racaza: Redirected page to Reliability Test Design

2012-06-29T08:05:28Z

Redirected page to Reliability Test Design

← Older revision		Revision as of 08:05, 29 June 2012
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	==Test Design ~~Using Expected Failure Time Plots==~~		#REDIRECT [[Reliability Test Design]]
	Test duration is one of the key factors that should be considered in designing a test. If the expected test duration can be estimated prior to the test, test resources can be better allocated. In this section, we will explain how to estimate the expected test time based on test sample size and the assumed underlying failure distribution.


	~~The binomial equation used in non-parametric demonstration test design is the base for predicting expected failure times. The equation is:~~

	~~<center><math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{n!}{i!\cdot (n-i)!}\cdot {{(1-{{R}_{TEST}})}^{i}}\cdot R_{TEST}^{(n-i)}</math></center>~~

	~~where:~~

	~~::<math>\begin{align}~~
	~~& CL= \text{the required confidence level} \\~~
	~~& r= \text{the number of failures} \\~~
	~~& n= \text{the total number of units on test} \\~~
	~~& {{R}_{TEST}}= \text{the reliability on test}~~
	~~\end{align}</math>~~


	If ''CL'', ''r'' and ''n'' are given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probability of failure whose value is 1-''R'') is the so called '''median rank''' for the corresponding failure. Please see [[Median Ranks]].


	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. This means, at the time when the second failure occurs, the estimated system probability of failure is 0.385728. The median rank can be calculated in Weibull++ using the '''Quick Statistical Reference''', as shown below:

	~~[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|250px\| ]]~~


	Similarly, if we set ''r'' = 3 for the above example, we can get the probability of failure at the time when the third failure occurs. Using the estimated median rank for each failure and the assumed underlying failure distribution, we can calculate the expected time for each failure. Assume the failure distribution is Weibull, then we know:

	~~<center> <math>Q=1-{{e}^{{{\left( \frac{t}{\eta } \right)}^{\beta }}}}</math> </center>~~

	~~where:~~
	~~: <math>\beta </math> is the shape parameter~~
	~~: <math>\eta</math> is the scale parameter~~


	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'' we can get the upper bound of the time for each failure. The calculated ''Q'' is given in the next figure:

	~~[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|250px\| ]]~~


	~~If we set ''CL''=0.1, from the calculated ''Q'' we can get the lower bound of the time for each failure. The calculated ''Q'' is given in the figure below:~~

	~~[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|250px\| ]]~~


	~~'''Example 6:'''~~
	~~{{Example: Test Design Using Expected Failure Times Plot}}~~

Nicolette Young: /* Test Design Using Expected Failure Time Plots */

2012-04-25T23:21:20Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 23:21, 25 April 2012
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	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. This means, at the time when the second failure occurs, the estimated system probability of failure is 0.385728. The median rank can be calculated in Weibull++ using the '''Quick Statistical Reference''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. This means, at the time when the second failure occurs, the estimated system probability of failure is 0.385728. The median rank can be calculated in Weibull++ using the '''Quick Statistical Reference''', as shown below:

	[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|~~400px~~\| ]]		[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|250px\| ]]


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	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'' we can get the upper bound of the time for each failure. The calculated ''Q'' is given in the next figure:		Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'' we can get the upper bound of the time for each failure. The calculated ''Q'' is given in the next figure:

	[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|~~500px~~\| ]]		[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|250px\| ]]


	If we set ''CL''=0.1, from the calculated ''Q'' we can get the lower bound of the time for each failure. The calculated ''Q'' is given in the figure below:		If we set ''CL''=0.1, from the calculated ''Q'' we can get the lower bound of the time for each failure. The calculated ''Q'' is given in the figure below:

	[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|~~500px~~\| ]]		[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|250px\| ]]


	'''Example 6:'''		'''Example 6:'''
	{{Example: Test Design Using Expected Failure Times Plot}}		{{Example: Test Design Using Expected Failure Times Plot}}

Lisa Hacker: /* Test Design Using Expected Failure Time Plots */

2012-03-30T00:14:22Z

Test Design Using Expected Failure Time Plots

Chris Kahn: /* Test Design Using Expected Failure Time Plots */

2012-03-08T18:55:49Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 18:55, 8 March 2012
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	The Binomial equation used in the non-~~parameteric~~ demonstration test design is the base for predicting expected failure times. The equation is:		The Binomial equation used in the non-parametric demonstration test design is the base for predicting expected failure times. The equation is:

	::<math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{n!}{i!\cdot (n-i)!}\cdot {{(1-{{R}_{TEST}})}^{i}}\cdot R_{TEST}^{(n-i)}</math>		::<math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{n!}{i!\cdot (n-i)!}\cdot {{(1-{{R}_{TEST}})}^{i}}\cdot R_{TEST}^{(n-i)}</math>

Chris Kahn: /* Test Design Using Expected Failure Time Plots */

2012-03-08T18:48:52Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 18:48, 8 March 2012
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	If ''CL'', ''r'', and ''n'' is given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the ~~probabilty~~ of failure whose value is 1-''R'') is the so called '''Median Rank''' for the corresponding failure. Please see [[Median Ranks]].		If ''CL'', ''r'', and ''n'' is given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probability of failure whose value is 1-''R'') is the so called '''Median Rank''' for the corresponding failure. Please see [[Median Ranks]].


	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system ~~proability~~ of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system probability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:

	[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]		[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]
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	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above ~~calcualtion~~ gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:		Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:

	[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]		[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]

Harry Guo: /* Test Design Using Expected Failure Time Plots */

2012-02-24T18:02:47Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 18:02, 24 February 2012
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	'''Example 6:'''		'''Example 6:'''
	{{Example: Test Design Using Expected Failure Times Plot}}		{{Example: Test Design Using Expected Failure Times Plot}}

	~~The EFTC tool works by first calculating the probability of failure ''F'' as a function of the sample size ''n'' and number of failures ''r''.~~
	~~This function is given by the equation <math> f\left(n,r,CL\right)=\frac{1}{1+\frac{n-r+1}{rF_{2r,2\left(n-r+1\right),1-CL}}} </math> where ''CL'' is the confidence level.~~

	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the median rank for ''F'' is <math> f\left(4,2,0.5\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.5}}}=0.385728 </math> The 80% 2-sided confidence interval on the probability of failure F is bounded by <math> f\left(4,2,0.1\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.9}}}=0.142559 </math> and <math> f\left(4,2,0.9\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.1}}}=0.679539 </math> With the probabilities of failure known, and assuming the failure distribution is exponentially distributed with parameter \(\lambda\,\!\) = 1, the expected times to failure can be calculated.

	For example, the median time to the second failure is calculated as <math> t=-\frac{ln\left(1-F\right)}{\lambda}=-\frac{1-0.385728}{1}=0.48732 </math> Similarly, the 10th percentile and 90th percentiles of time to failure are calculated as <math> t=-\frac{1-0.142559}{1}=0.153803 </math> and <math> t=-\frac{1-0.679539}{1}=1.137995 </math> The above results can also be seen in Weibull++ as shown below.

	~~[[Image:Eftc2.jpg\|thumb\|center\|500px\| ]]~~

Harry Guo: /* Test Design Using Expected Failure Time Plots */

2012-02-24T17:39:38Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 17:39, 24 February 2012
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	'''Example 6:'''		'''Example 6:'''
			{{Example: Test Design Using Expected Failure Times Plot}}

	The EFTC tool works by first calculating the probability of failure ''F'' as a function of the sample size ''n'' and number of failures ''r''.		The EFTC tool works by first calculating the probability of failure ''F'' as a function of the sample size ''n'' and number of failures ''r''.

Harry Guo at 17:37, 24 February 2012

2012-02-24T17:37:07Z

← Older revision		Revision as of 17:37, 24 February 2012
Line 22:		Line 22:
	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:

	[[Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]		[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]


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	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calcualtion gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:		Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calcualtion gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:

	[[Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]		[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]


	If we set ''CL''=0.1, from the calculated ''Q'', we can get the lower bound of the failure time for each failure. The calculated ''Q'' is given in the Figure below:		If we set ''CL''=0.1, from the calculated ''Q'', we can get the lower bound of the failure time for each failure. The calculated ''Q'' is given in the Figure below:

	[[Expected Failure Plot 0.1 Rank.png\|thumb\|center\|500px\| ]]		[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|500px\| ]]

Harry Guo: /* Test Design Using Expected Failure Time Plots */

2012-02-24T17:36:37Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 17:36, 24 February 2012
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	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:

			[[Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]


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	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calcualtion gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:		Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calcualtion gives the '''Median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the failure time for each failure. The calculated ''Q'' is given in the next Figure:

			[[Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]


	If we set ''CL''=0.1, from the calculated ''Q'', we can get the lower bound of the failure time for each failure. The calculated ''Q'' is given in the Figure below:		If we set ''CL''=0.1, from the calculated ''Q'', we can get the lower bound of the failure time for each failure. The calculated ''Q'' is given in the Figure below:

			[[Expected Failure Plot 0.1 Rank.png\|thumb\|center\|500px\| ]]


	'''Example 6:'''		'''Example 6:'''

Harry Guo: /* Test Design Using Expected Failure Time Plots */

2012-02-24T16:55:30Z

Test Design Using Expected Failure Time Plots

← Older revision		Revision as of 16:55, 24 February 2012
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	If ''CL'', ''r'', and ''n'' is given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probabilty of failure whose value is 1-''R'') is the so called '''Median Rank''' for the corresponding failure. Please see [[Median Ranks]].		If ''CL'', ''r'', and ''n'' is given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probabilty of failure whose value is 1-''R'') is the so called '''Median Rank''' for the corresponding failure. Please see [[Median Ranks]].


	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the 2nd failure occurs, the estimated system proability of failure is 0.385728. The Median Rank can be calculated using the '''Quick Statistical Calculation''', as shown below:

← Older revision		Revision as of 00:14, 30 March 2012
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	==Test Design Using Expected Failure Time Plots==		==Test Design Using Expected Failure Time Plots==
	Test duration is one of the key factors that should be considered in designing a test. If the expected test duration can be estimated ~~ahead of~~ the test, test resources can be better allocated. In this section, we will explain how to ~~estimated~~ the expected test time based on test sample size and the assumed underlying failure distribution.		Test duration is one of the key factors that should be considered in designing a test. If the expected test duration can be estimated prior to the test, test resources can be better allocated. In this section, we will explain how to estimate the expected test time based on test sample size and the assumed underlying failure distribution.


	The ~~Binomial~~ equation used in ~~the~~ non-parametric demonstration test design is the base for predicting expected failure times. The equation is:		The binomial equation used in non-parametric demonstration test design is the base for predicting expected failure times. The equation is:

	::<math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{n!}{i!\cdot (n-i)!}\cdot {{(1-{{R}_{TEST}})}^{i}}\cdot R_{TEST}^{(n-i)}</math>		<center><math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{n!}{i!\cdot (n-i)!}\cdot {{(1-{{R}_{TEST}})}^{i}}\cdot R_{TEST}^{(n-i)}</math></center>

	where:		where:
Line 17:		Line 17:


	If ''CL'', ''r'', and ''n'' is given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probability of failure whose value is 1-''R'') is the so called '''~~Median Rank~~''' for the corresponding failure. Please see [[Median Ranks]].		If ''CL'', ''r'' and ''n'' are given, the ''R'' value can be solved from the above equation. When ''CL''=0.5, the solved ''R'' (or ''Q'', the probability of failure whose value is 1-''R'') is the so called '''median rank''' for the corresponding failure. Please see [[Median Ranks]].


	For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. It means, at the time when the ~~2nd~~ failure occurs, the estimated system probability of failure is 0.385728. The ~~Median Rank~~ can be calculated using the '''Quick Statistical ~~Calculation~~''', as shown below:		For example, given ''n'' = 4, ''r'' = 2 and ''CL'' = 0.5, the calculated ''Q'' is 0.385728. This means, at the time when the second failure occurs, the estimated system probability of failure is 0.385728. The median rank can be calculated in Weibull++ using the '''Quick Statistical Reference''', as shown below:

	[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]		[[Image: Expected Failure Plot Median Rank.png\|thumb\|center\|400px\| ]]


	Similarly, ~~for the above example~~ if we set ''r'' = 3, we can get the probability of failure at the time when the ~~3rd~~ failure occurs. Using the estimated median rank for each failure and the assumed underlying failure distribution, we can calculate the expected ~~failure~~ time for each failure. Assume the failure distribution is Weibull, then we know:		Similarly, if we set ''r'' = 3 for the above example, we can get the probability of failure at the time when the third failure occurs. Using the estimated median rank for each failure and the assumed underlying failure distribution, we can calculate the expected time for each failure. Assume the failure distribution is Weibull, then we know:

	<center> <math>Q=1-{{e}^{{{\left( \frac{t}{\eta } \right)}^{\beta }}}}</math> </center>		<center> <math>Q=1-{{e}^{{{\left( \frac{t}{\eta } \right)}^{\beta }}}}</math> </center>

	where:		where:
	: <math>\beta </math> is the shape parameter,		: <math>\beta </math> is the shape parameter
	: <math>\eta</math> is the scale parameter,		: <math>\eta</math> is the scale parameter


	Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''~~Median~~''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'', we can get the upper bound of the ~~failure~~ time for each failure. The calculated ''Q'' is given in the next ~~Figure~~:		Using the above equation, for a given ''Q'', we can get the corresponding time ''t''. The above calculation gives the '''median''' of each failure time for ''CL'' = 0.5. If we set ''CL'' at different values, the confidence bounds of each failure time can be obtained. For the above example, if we set ''CL''=0.9, from the calculated ''Q'' we can get the upper bound of the time for each failure. The calculated ''Q'' is given in the next figure:

	[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]		[[Image: Expected Failure Plot 0.9 Rank.png\|thumb\|center\|500px\| ]]


	If we set ''CL''=0.1, from the calculated ''Q'', we can get the lower bound of the ~~failure~~ time for each failure. The calculated ''Q'' is given in the ~~Figure~~ below:		If we set ''CL''=0.1, from the calculated ''Q'' we can get the lower bound of the time for each failure. The calculated ''Q'' is given in the figure below:

	[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|500px\| ]]		[[Image: Expected Failure Plot 0.1 Rank.png\|thumb\|center\|500px\| ]]