Template:Modified gompertz model

Modified Gompertz Model
Sometimes reliability growth data with an S-shaped trend cannot be described accurately by the Gompertz or Logistic (Chapter 8) curves. Since these two models have fixed values of reliability at the inflection points, only a few reliability growth data sets following an S-shaped reliability growth curve can be fitted to them. A modification of the Gompertz curve, which overcomes this shortcoming, is given next [5]. If we apply a shift in the vertical coordinate, then the Gompertz model is defined by:


 * $$R=d+a{{b}^}$$


 * where:


 * $$0<a+d\le 1$$
 * $$0<b<1,0<c<1,\text{and}T\ge 0$$


 * $$R$$ = system's reliability at development time $$T$$ or at launch number $$T$$, or stage number $$T$$.
 * $$d$$ = shift parameter.
 * $$d+a$$ = upper limit that the reliability approaches asymptotically as $$T\to\infty$$
 * $$d+ab$$ = initial reliability at $$T=0$$
 * $$c$$ = growth pattern indicator(small values of $$c$$ indicate rapid early reliability growth and large values of $$c$$ indicate slow reliability growth).

The Modified Gompertz model is more flexible than the original, especially when fitting growth data with S-shaped trends.