Template:Gompertz models (standard and modified)

=Gompertz Models (Standard and Modified)=

Example 4
A new design is put through a reliability growth test. The requirement is that after the ninth stage the design will exhibit an 85% reliability with a 90% confidence level. Given the data in Table 7.5, do the following:
 * 1)	Estimate the parameters of the Standard Gompertz model.
 * 2)	What is the initial reliability at $$T=0$$ ?
 * 3)	Determine the reliability at the end of the ninth stage and check to see if the goal has been met.

Solution to Example 4

 * 1)	The data is entered in cumulative format and the estimated Standard Gompertz parameters are shown in Figure Gompex4a.


 * 2)	The initial reliability at $$T=0$$  is equal to:


 * $$\begin{align}

& {{R}_{T=0}}= & a\cdot b \\ & = & 0.9497\cdot 0.5249 \\ & = & 0.4985 \end{align}$$


 * 3)	The reliability at the ninth stage can be calculated using the Quick Calculation Pad (QCP) as shown in Figure Gompex4b.

The estimated reliability at the end of the ninth stage is equal to 91.92%. However, the lower limit at the 90% 1-sided confidence bound is equal to 82.15%. Therefore, the required goal of 85% reliability at a 90% confidence level has not been met.

Example 5
Using the data in Table 7.6, determine whether the Standard Gompertz or Modified Gompertz would be better suited for analyzing the given data.

Solution to Example 5
The Standard Gompertz Reliability vs. Time plot is shown in Figure Ex5Std. The Standard Gompertz seems to do a fairly good job of modeling the data. However, it appears that it is having difficulty modeling the S-shape of the data. The Modified Gompertz Reliability vs. Time plot is shown in Figure Ex5Mod.

The Modified Gompertz, as expected, does a much better job of handling the S-shape presented by the data and provides a better fit for this data.