|
|
| Line 1: |
Line 1: |
| ====Growth Rate for Nominal Idealized Curve====
| | #REDIRECT [[Reliability_Growth_Planning#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]. 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.
| |
| <br>
| |