Template:Growth rate for nominal idealized curve - Revision history

Richard House: Redirected page to Reliability Growth Planning#Growth Rate for Nominal Idealized Curve

2012-08-27T01:47:55Z

Redirected page to Reliability Growth Planning#Growth Rate for Nominal Idealized Curve

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	~~====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>~~

Nicolette Young: Created page with '====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]…'

2012-01-10T21:49:00Z

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]…'

New page

====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>

Template:Growth rate for nominal idealized curve - Revision history

Richard House: Redirected page to Reliability Growth Planning#Growth Rate for Nominal Idealized Curve

Nicolette Young: Created page with '====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]…'

Nicolette Young: Created page with '====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]…'