Template:Actual idealized growth curve: Difference between revisions

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===Actual Idealized Growth Curve===
#REDIRECT [[Reliability_Growth_Planning#Actual_Idealized_Growth_Curve]]
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The actual idealized growth curve differs from the nominal idealized curve in that it takes into account the average fix delay that might occur in each test phase. The actual idealized growth curve is continuous and goes through each of the test phase target MTBFs.
 
{{fix delays and test phase target mtbf}}
 
{{actual failure intensity function}}
 
====Actual Time to Reach Goal====
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The actual time to reach the target MTBF or failure intensity goal,  <math>{{t}_{AC,G}},</math>  can be found by solving Eqn. (ActIhase):
 
 
::<math>\begin{align}
  & {{r}_{AI}}({{t}_{AC,G}})= & {{\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}_{AC,G}}-{{T}_{i-1}}) \right]}^{(\beta -1)}} 
\end{align}</math>
 
 
Since the actual idealized growth curve depends on the phase durations and average fix delays, there are three different cases that need to be treated differently in order to determine the actual time to reach the MTBF goal. The cases depend on when the actual MTBF that can be reached within the specific phase durations and fix delays becomes equal to the MTBF goal. This can be determined by solving Eqn. (ActIhase) for phases  <math>1</math>  through  <math>i</math> , then solving in terms of actual MTBF using Eqn. (MTBFctualq) for each phase and finding the phase during which the actual MTBF becomes equal to the goal MTBF. The three cases are presented next.
 
''Case 1: MTBF goal is met during the last phase''
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If  <math>{{T}_{F}}</math>  indicates the cumulative end phase time for the last phase and  <math>{{L}_{F}}</math>  indicates the fix delay for the last phase, then we have:
 
 
::<math>\begin{align}
  & {{r}_{G}}= & {{\lambda }_{A}}+(1-d){{\lambda }_{B}} \\
&  & +d\lambda \beta {{\left[ {{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{F-1}}) \right]}^{(\beta -1)}} 
\end{align}</math>
 
 
Starting to solve for  <math>{{t}_{AC,G}}</math>  yields:
 
 
::<math>{{\left[ \frac{{{r}_{G}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\lambda \beta } \right]}^{\tfrac{1}{\beta -1}}}={{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{i}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{F-1}})</math>
 
 
We can substitute the left term by using Eqn. (tgoalominal), thus we have:
 
 
::<math>{{t}_{N,G}}={{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}})</math>
 
 
:Therefore:
 
 
::<math>{{t}_{AC,G}}=\frac{{{t}_{N,G}}-{{T}_{F-1}}+{{L}_{F-1}}}{\left( \tfrac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)}+{{T}_{F-1}}</math>
 
 
''Case 2: MTBF goal is met before the last phase''
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Eqn. (toalctual) still applies, but in this case  <math>{{T}_{F}}</math>  and  <math>{{L}_{F}}</math>  are the time and fix delay of the phase during which the goal is met.
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''Case 3: MTBF goal is met after the final phase''
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If the goal MTBF,  <math>{{M}_{G}},</math>  is met after the final test phase, then the actual time to reach the goal is not calculated, since additional phases have to be added with specific duration and fix delays. The reliability growth program needs to be re-evaluated with the following options:
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:• Add more phase(s) to the program.
:• Re-examine the phase duration of the existing phases.
:• Investigate whether there are potential process improvements in the program that can <br>
reduce the average fix delay for the phases.
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Other alternative routes for consideration would be to investigate the rest of the inputs in the model:
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:• Change the management strategy.
:• Consider if further program risk can be acceptable, and if so, reduce the growth potential design margin.
:• Consider if it is feasible to increase the effectiveness factors of the delayed fixes by using more robust engineering redesign methods.
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Note that each change of input variables into the model can significantly influence the results. <br>
With that in mind, any alteration in the input parameters should be justified by actionable decisions that will influence the reliability growth program. For example, increasing the average effectiveness factor value should be done only when there is proof that the program will pursue a different, more effective path in terms of addressing fixes.
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Latest revision as of 01:49, 27 August 2012

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