Template:Growth rate for nominal idealized curve: Difference between revisions

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(Created page with '====Growth Rate for Nominal Idealized Curve==== <br> The growth rate for the nominal idealized curve is defined in the same context as the growth rate for the Duane Postulate [8]…')
 
 
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====Growth Rate for Nominal Idealized Curve====
#REDIRECT [[Reliability_Growth_Planning#Growth_Rate_for_Nominal_Idealized_Curve]]
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The growth rate for the nominal idealized curve is defined in the same context as the growth rate for the Duane Postulate [8]. The nominal idealized curve has the same functional form for the growth rate as the Duane Postulate and the Crow-AMSAA (NHPP) model.
For both the Duane Postulate and the Crow-AMSAA (NHPP) model, the average failure intensity is given by:
 
 
::<math>C(t)=\frac{\lambda {{t}^{\beta }}}{t}=\lambda {{t}^{(\beta -1)}}</math>
 
 
Also, for both the Duane Postulate and the Crow-AMSAA (NHPP) model, the instantaneous failure intensity is given by:
 
 
::<math>r(t)=\lambda \beta {{t}^{(\beta -1)}}</math>
 
 
Taking the difference,  <math>D(t),</math>  between the average failure intensity,  <math>C(t)</math> , and the instantaneous failure intensity,  <math>r(t)</math> , yields:
 
 
::<math>D(t)=\lambda {{t}^{(\beta -1)}}-\lambda \beta {{t}^{(\beta -1)}}</math>
 
 
:Then:
 
 
::<math>D(t)=\lambda {{t}^{(\beta -1)}}[1-\beta ]</math>
 
 
For reliability growth to occur,  <math>D(t)</math>  must be decreasing.
The growth rate for both the Duane Postulate and the Crow-AMSAA (NHPP) model is the negative of the slope of  <math>\log (D(t))</math>  as a function of  <math>\log (t)</math> :
 
 
::<math>{{\log }_{e}}(D(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t)</math>
 
 
 
The slope is negative under reliability growth and equals:
 
 
::<math>\text{slope}=-(1-\beta )</math>
 
 
The growth rate for both the Duane Postulate and the Crow-AMSAA (NHPP) model is equal to the negative of this slope:
 
 
::<math>\text{Growth Rate}=(1-\beta )</math>
 
 
The instantaneous failure intensity for the nominal idealized curve is:
 
 
::<math>{{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{(t)}^{(\beta -1)}}</math>
 
 
The cumulative failure intensity for the nominal idealized curve is:
 
 
::<math>{{C}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda {{(t)}^{(\beta -1)}}</math>
 
 
:Therefore:
 
 
::<math>{{D}_{NI}}(t)=[{{C}_{NI}}(t)-{{r}_{NI}}(t)]=\lambda {{t}^{(\beta -1)}}[1-\beta ]</math>
 
 
:and:
 
 
::<math>{{\log }_{e}}({{D}_{NI}}(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t)</math>
 
 
Therefore, in accordance with the Duane Postulate and the Crow-AMSAA (NHPP) model,  <math>a=1-\beta </math>  is the growth rate for the reliability growth plan.
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Latest revision as of 01:47, 27 August 2012

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