Component Importance Reference Example - Revision history

Richard House at 14:39, 8 August 2016

2016-08-08T14:39:56Z

← Older revision		Revision as of 14:39, 8 August 2016
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	~~Brinbaum’s~~ importance measures for each computer can be formulated via Equation 4.59 on page 100 as:		Birnbaum’s importance measures for each computer can be formulated via Equation 4.59 on page 100 as:

	:<math>I_{B}(i\|t)= \frac{\partial R(t)}{\partial R_{i}(t)}\,\!</math>		:<math>I_{B}(i\|t)= \frac{\partial R(t)}{\partial R_{i}(t)}\,\!</math>

Kate Racaza at 17:30, 20 August 2015

2015-08-20T17:30:51Z

← Older revision		Revision as of 17:30, 20 August 2015
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	A computing system with four computers is configured as given in the BlockSim flowchart diagram below. (Please see Figure 4.29 on page 100 in the reference book for the original configuration.) Each computer follows an exponentially distributed failure rate with the given failure rate parameters in the table below. The Birnbaum’s importance measures of each computer in the system at 4000 and 8000 hours of operation are sought in this example. The system reliability equation, the reliabilities of each computer at the given times and the reliability importance at different times are also scrutinized.		A computing system with four computers is configured as given in the BlockSim flowchart diagram below. (Please see Figure 4.29 on page 100 in the reference book for the original configuration.) Each computer follows an exponentially distributed failure rate with the given failure rate parameters in the table below. The Birnbaum’s importance measures of each computer in the system at 4000 and 8000 hours of operation are sought in this example. The system reliability equation, the reliabilities of each computer at the given times and the reliability importance at different times are also scrutinized.


	~~{{Reference_Example_Heading3}}~~

	{\| {{Table}}		{\| {{Table}}
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	\|}		\|}


			{{Reference_Example_Heading3}}

	The system reliability equation is computed from Equation 4.40 on page 87 as:		The system reliability equation is computed from Equation 4.40 on page 87 as:
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	{{Reference_Example_Heading4\|BlockSim}}		{{Reference_Example_Heading4\|BlockSim}}
	~~<!--~~

	In BlockSim, the computer system RBD is configured as shown below.		In BlockSim, the computer system RBD is configured as shown below.
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	[[Image:importance_qcp1.png\|center\|500px]]		[[Image:importance_qcp1.png\|center\|500px\|Reliability at 4000 hours.]]


	~~[[Image:importance_qcp2.png\|center\|500px]]~~

			[[Image:importance_qcp2.png\|center\|500px\|Reliability at 8000 hours.]]

	~~The QCP report shows the reliabilities of each computer at the given times (4000 and 8000 hours).~~

	~~[[Image:importance_qcpreport1~~.~~png\|center]]~~		The QCP report (accessed by clicking the '''Report''' button in the QCP window) shows the reliabilities of each computer at the given times (4000 and 8000 hours).

			[[Image:importance_qcpreport.png\|center\|600px]]

	~~[[Image:importance_qcpreport2.png\|center]]~~

			The static reliability importance for each computer at 4000 hours is presented below in graphical format, with the values highlighted in red. The color of each bar represents the computer’s reliability.

	~~The static reliability~~ importance ~~for each computer~~ at 4000 ~~and 8000~~ hours ~~is presented below in graphical format, with the values highlighted in red. The color of each bar represents the computer’s reliability~~.		[[Image:importance_plot1.png\|center\|500px\|Reliability importance at 4000 hours.]]

	~~[[Image:importance_plot1.png\|center\|500px]]~~

			The static reliability importance for each computer at 8000 hours is shown next.

	[[Image:importance_plot2.png\|center\|500px]]		[[Image:importance_plot2.png\|center\|500px\|Reliability importance at 8000 hours.]]


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	[[Image:importance_plot3.png\|center\|500px]]		[[Image:importance_plot3.png\|center\|500px]]

	~~-->~~

Kate Racaza at 22:46, 14 August 2015

2015-08-14T22:46:22Z

@@ Line 1: / Line 1: @@
 {{Reference Example|{{Banner BlockSim Reference Examples}}}}
-This example validates the component importance results for BlockSim's analytical and simulation diagrams.
+This example validates the component importance results for BlockSim diagrams.
@@ Line 28: / Line 28: @@
 |}
 The system reliability equation is computed from Equation 4.40 on page 87 as:
@@ Line 52: / Line 50: @@
 :<math>\begin{align}
 R_{C1}(8000)&=0.9570; R_{C2}(8000) = 0.5945\\
-R_{C3}(8000)&=0.7089; R_{C2}(8000) = 0.9712\\
+R_{C3}(8000)&=0.7089; R_{C2}(8000) = 0.9433\\
 \end{align}\,\!</math>
@@ Line 67: / Line 65: @@
 :<math>I_{B}(C1|t)= R_{2}(1-R_{3}); I_{B}(C2|t) = R_{1}+R_{3}-R_{1}R_{3}-R_{3}R_{4}\,\!</math>
-:<math>I_{B}(C3|t)= R_{2}+ R_{4}- R_{1}R_{2} - R_{2}R_{4}; I_{B}(C4|t) = R_{3}(1 - R_{2}\,\!</math>
+:<math>I_{B}(C3|t)= R_{2}+ R_{4}- R_{1}R_{2} - R_{2}R_{4}; I_{B}(C4|t) = R_{3}(1 - R_{2})\,\!</math>

Kate Racaza: Created page with '{{Reference Example|{{Banner BlockSim Reference Examples}}}} This example validates the component importance results for BlockSim's analytical and simulation diagrams. {{Refer…'

2015-08-14T22:06:54Z

Created page with '{{Reference Example|{{Banner BlockSim Reference Examples}}}} This example validates the component importance results for BlockSim's analytical and simulation diagrams. {{Refer…'

New page

{{Reference Example|{{Banner BlockSim Reference Examples}}}}
This example validates the component importance results for BlockSim's analytical and simulation diagrams.

{{Reference_Example_Heading1}}

The data set is from example 4.14 on page 100 in the book ''Life Cycle Reliability Engineering'' by Dr. Guangbin Yang, John Wiley & Sons, 2007.

{{Reference_Example_Heading2}}

A computing system with four computers is configured as given in the BlockSim flowchart diagram below. (Please see Figure 4.29 on page 100 in the reference book for the original configuration.) Each computer follows an exponentially distributed failure rate with the given failure rate parameters in the table below. The Birnbaum’s importance measures of each computer in the system at 4000 and 8000 hours of operation are sought in this example. The system reliability equation, the reliabilities of each computer at the given times and the reliability importance at different times are also scrutinized.

{{Reference_Example_Heading3}}

{| {{Table}}
!Computers of the Computing System
!Number of Failures, <math>\lambda</math> (per hour)
|-
|Computer 1||5.5 x 10-6
|-
|Computer 2||6.5 x 10-5
|-
|Computer 3||4.3 x 10-5
|-
|Computer4||7.3 x 10-6
|}

{{Reference_Example_Heading4|BlockSim}}

The system reliability equation is computed from Equation 4.40 on page 87 as:

:<math>\begin{align}
R&=Pr(system good|A)Pr(A)+Pr(system good|\bar{A})Pr(\bar{A})\\
R(t)&=[1-(1-R_{C1})(1-R_{C3})]R_{C2}+R_{C3}R_{C4}(1-R_{C2})\\
R(t)&=R_{C1}R_{C2}+R_{C2}R_{C3}+R_{C3}R_{C4}-R_{C1}R_{C2}R_{C3}-R_{C2}R_{C3}R_{C4}\\
\end{align}\,\!</math>

Since the times to failure of the computers are exponentially distributed and the reliability can be calculated via <math>R_{i} (t)=e^{-\lambda_{i} t}\,\!</math> for ''i'' = C1, C2, C3, and C4, the reliabilities of each computer at 4000 and 8000 hours are calculated as:

:<math>\begin{align}
R_{C1}(4000)&=0.9782; R_{C2}(4000) = 0.7711\\
R_{C3}(4000)&=0.8420; R_{C2}(4000) = 0.9712\\
\end{align}\,\!</math>

and

:<math>\begin{align}
R_{C1}(8000)&=0.9570; R_{C2}(8000) = 0.5945\\
R_{C3}(8000)&=0.7089; R_{C2}(8000) = 0.9712\\
\end{align}\,\!</math>

The system reliabilities at the specified times are calculated as:

:<math>R(4000)=0.9556; R(8000) = 0.8582\,\!</math>

Brinbaum’s importance measures for each computer can be formulated via Equation 4.59 on page 100 as:

:<math>I_{B}(i|t)= \frac{\partial R(t)}{\partial R_{i}(t)}\,\!</math>

:<math>I_{B}(C1|t)= R_{2}(1-R_{3}); I_{B}(C2|t) = R_{1}+R_{3}-R_{1}R_{3}-R_{3}R_{4}\,\!</math>

:<math>I_{B}(C3|t)= R_{2}+ R_{4}- R_{1}R_{2} - R_{2}R_{4}; I_{B}(C4|t) = R_{3}(1 - R_{2}\,\!</math>