Template:Growth rate for nominal idealized curve

Growth Rate for Nominal Idealized Curve
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:


 * $$C(t)=\frac{\lambda {{t}^{\beta }}}{t}=\lambda {{t}^{(\beta -1)}}$$

Also, for both the Duane Postulate and the Crow-AMSAA (NHPP) model, the instantaneous failure intensity is given by:


 * $$r(t)=\lambda \beta {{t}^{(\beta -1)}}$$

Taking the difference, $$D(t),$$  between the average failure intensity,  $$C(t)$$, and the instantaneous failure intensity,  $$r(t)$$ , yields:


 * $$D(t)=\lambda {{t}^{(\beta -1)}}-\lambda \beta {{t}^{(\beta -1)}}$$


 * Then:


 * $$D(t)=\lambda {{t}^{(\beta -1)}}[1-\beta ]$$

For reliability growth to occur, $$D(t)$$  must be decreasing. The growth rate for both the Duane Postulate and the Crow-AMSAA (NHPP) model is the negative of the slope of $$\log (D(t))$$  as a function of  $$\log (t)$$ :


 * $${{\log }_{e}}(D(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t)$$

The slope is negative under reliability growth and equals:


 * $$\text{slope}=-(1-\beta )$$

The growth rate for both the Duane Postulate and the Crow-AMSAA (NHPP) model is equal to the negative of this slope:


 * $$\text{Growth Rate}=(1-\beta )$$

The instantaneous failure intensity for the nominal idealized curve is:


 * $${{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{(t)}^{(\beta -1)}}$$

The cumulative failure intensity for the nominal idealized curve is:


 * $${{C}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda {{(t)}^{(\beta -1)}}$$


 * Therefore:


 * $${{D}_{NI}}(t)=[{{C}_{NI}}(t)-{{r}_{NI}}(t)]=\lambda {{t}^{(\beta -1)}}[1-\beta ]$$


 * and:


 * $${{\log }_{e}}({{D}_{NI}}(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t)$$

Therefore, in accordance with the Duane Postulate and the Crow-AMSAA (NHPP) model, $$a=1-\beta $$  is the growth rate for the reliability growth plan.