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==Crow Extended Model==
#REDIRECT [[Reliability_Growth_Planning#Crow_Extended_Model]]
===Introduction===
The Crow Extended model for reliability growth planning is a revised and improved version of the MIL-HDBK-189 growth curve [13]. MIL-HDBK-189 can be considered as the growth curve based on the Crow-AMSAA (NHPP) model. Using MIL-HDBK-189 for reliability growth planning assumes that the corrective actions for the observed failure modes are incorporated during the test and at the specific time of failure. However, in actual practice, fixes may be delayed until after the completion of the test or some fixes may be implemented during the test while others are delayed and some are not fixed at all. The Crow Extended model for reliability growth planning provides additional input to be able to account for a specific management strategy and delayed fixes with specified effectiveness factors.
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===Definitions===
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{{management strategy ratio crow extended}}
 
{{effectiveness factor crow extended}}
 
{{mtbf goal crow extended}}
 
{{growth potential crow extended}}
 
{{growth potential design margin crow extended}}
 
{{nominal idealized growth curve}}
 
===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====
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Fix delays reflect how long it takes from the time a problem failure mode is discovered in testing, to the time the corrective action is incorporated into the system and reliability growth is realized. The consideration of the fix delay is often in terms of how much calendar time it takes to incorporate a corrective action fix after the problem is first seem. However, the impact of the delay on reliability growth is reflected in the average test time it takes between finding a problem failure mode and incorporating a corrective action. The fix delay is reflected in the actual idealized growth curve in terms of test time.
In other words, the average fix delay is calendar time converted to test hours. For example, say that we expect an average fix delay of two weeks: if in two weeks the total test time is 1000 hours, the average fix delay is 1000 hours. If in the same two weeks the total test time is 2000 hours (maybe there are more units available or more shifts) then the average fix delay is 2000 hours.
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There can be a constant fix delay across all test phases or, as a practical matter, each test phase can have a different fix delay time. In practice, the fix delay will generally be constant over a particular test phase.  <math>{{L}_{i}}</math>  denotes the fix delay for phase  <math>i=1,...,P,</math>  where  <math>P</math>  is the total number of phases in the test. RGA 7 allows for a maximum of seven test phases.
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====Actual Failure Intensity Function====
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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'''
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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>'''
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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.
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The actual idealized curve MTBF is:
 
 
::<math>{{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)}</math>
 
 
====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 01:40, 27 August 2012

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