Template:Actual failure intensity function: Difference between revisions

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(Created page with '====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 actu…')
 
 
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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>

Latest revision as of 01:51, 27 August 2012

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