Template:Actual failure intensity function - Revision history

Richard House: Redirected page to Reliability Growth Planning#Actual Failure Intensity Function

2012-08-27T01:51:53Z

Redirected page to Reliability Growth Planning#Actual Failure Intensity Function

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	~~====Actual Failure Intensity Function====~~		#REDIRECT [[Reliability_Growth_Planning#Actual_Failure_Intensity_Function]]
	~~<br>~~
	Consider a test plan consisting of <math>i</math> phases. Taking into account the fix delay within each phase, we expect the actual failure intensity to be different (i.e. shifted) from the nominal failure intensity. This is because fixes are not incorporated instantaneously, thus growth is realized at a later time compared to the nominal case.
	~~Specifically, the actual failure intensity will be estimated as follows:~~


	~~'''Test Phase 1'''~~
	~~<br>~~
	~~For the first phase of a test plan, the actual idealized curve failure intensity, <math>{{r}_{AI}}(t)</math> , is <math>:</math>~~


	~~::<math>{{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left~~[ ~~\left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)t \right]}^{(\beta -1)}}\text{ for }0<t\le {{T}_{1}}</math>~~


	~~Note that the end time of Phase 1, <math>{{T}_{1}},</math> must be greater than <math>{{L}_{1}}+{{t}_{0}}</math> . That is, <math>{{T}_{1}}>{{L}_{1}}+{{t}_{0}}</math> .~~
	~~The actual idealized curve initialization time for Phase 1, <math>T_{0}^{AIC},</math> is calculated from:~~


	~~::<math>{{r}_{AI}}(T_{0}^{AIC})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}</math>~~


	~~Where <math>{{r}_{AI}}(T_{0}^{AIC})={{r}_{I}}.</math>~~
	~~Therefore, using Eqn. (lambdaqualsominal):~~


	~~::<math>{{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left~~[ ~~\left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right~~]~~}^{(\beta -1)}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}</math>~~


	~~Solving Eqn. (ActIhase1) for <math>T_{0}^{AIC}</math> we get:~~


	~~::<math>T_{0}^{AIC}=\frac{{{t}_{0}}}{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)}</math>~~




	~~'''Test Phase <math>i</math>'''~~
	~~<br>~~
	~~For any test phase <math>i</math> , the actual idealized curve failure intensity is given by:~~


	::<math>{{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right]~~}^{(\beta -1)}}</math>~~


	~~where <math>{{T}_{i-1}}\le t\le {{T}_{i}}</math> and <math>{{T}_{i}}</math> is the test time of each corresponding test phase.~~
	~~<br>~~
	~~The actual idealized curve MTBF is:~~


	~~::<math>{{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)}</math>~~

Nicolette Young: Created page with '====Actual Failure Intensity Function====
Consider a test plan consisting of $i$ phases. Taking into account the fix delay within each phase, we expect the actu…'

2012-01-10T21:52:36Z

Created page with '====Actual Failure Intensity Function==== Consider a test plan consisting of <math>i</math> phases. Taking into account the fix delay within each phase, we expect the actu…'

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====Actual Failure Intensity Function====
 
Consider a test plan consisting of <math>i</math> phases. Taking into account the fix delay within each phase, we expect the actual failure intensity to be different (i.e. shifted) from the nominal failure intensity. This is because fixes are not incorporated instantaneously, thus growth is realized at a later time compared to the nominal case.
Specifically, the actual failure intensity will be estimated as follows:

'''Test Phase 1'''
 
For the first phase of a test plan, the actual idealized curve failure intensity, <math>{{r}_{AI}}(t)</math> , is <math>:</math>

::<math>{{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)t \right]}^{(\beta -1)}}\text{ for }0<t\le {{T}_{1}}</math>

Note that the end time of Phase 1, <math>{{T}_{1}},</math> must be greater than <math>{{L}_{1}}+{{t}_{0}}</math> . That is, <math>{{T}_{1}}>{{L}_{1}}+{{t}_{0}}</math> .
The actual idealized curve initialization time for Phase 1, <math>T_{0}^{AIC},</math> is calculated from:

::<math>{{r}_{AI}}(T_{0}^{AIC})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}</math>

Where <math>{{r}_{AI}}(T_{0}^{AIC})={{r}_{I}}.</math>
Therefore, using Eqn. (lambdaqualsominal):

::<math>{{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}</math>

Solving Eqn. (ActIhase1) for <math>T_{0}^{AIC}</math> we get:

::<math>T_{0}^{AIC}=\frac{{{t}_{0}}}{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)}</math>

'''Test Phase <math>i</math>'''
 
For any test phase <math>i</math> , the actual idealized curve failure intensity is given by:

::<math>{{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right]}^{(\beta -1)}}</math>

where <math>{{T}_{i-1}}\le t\le {{T}_{i}}</math> and <math>{{T}_{i}}</math> is the test time of each corresponding test phase.
 
The actual idealized curve MTBF is:

::<math>{{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)}</math>

Template:Actual failure intensity function - Revision history

Richard House: Redirected page to Reliability Growth Planning#Actual Failure Intensity Function

Nicolette Young: Created page with '====Actual Failure Intensity Function==== Consider a test plan consisting of i phases. Taking into account the fix delay within each phase, we expect the actu…'

Nicolette Young: Created page with '====Actual Failure Intensity Function====
Consider a test plan consisting of $i$ phases. Taking into account the fix delay within each phase, we expect the actu…'