General Examples Using the Gompertz Models

Gompertz Models 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:

Estimate the parameters of the Standard Gompertz model.
What is the initial reliability at T = 0?
Determine the reliability at the end of the ninth stage and check to see if the goal has been met.

Table 7.5 - Grouped per configuration data for Example 4

Stage	Number of Units	Number of Failures
1	10	5
2	8	3
3	9	3
4	9	2
5	10	2
6	10	1
7	10	1
8	10	1
9	10	1

Solution to Gompertz Models Example 4

The data is entered in cumulative format and the estimated Standard Gompertz parameters are shown in Figure 7.6.

Figure 7.6: Entered data and the estimated Standard Gompertz parameters

The initial reliability at T = 0 is equal to:

MATH

The reliability at the ninth stage can be calculated using the Quick Calculation Pad (QCP) as shown in Figure 7.7.

Figure 7.7: Calculate the reliability at the end of the ninth stage with
90% confidence bounds

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.

Gompertz Models 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.

Table 7.6 - Reliability data for Example 5

Stage	Reliability (%)
0	36
1	38
2	46
3	58
4	71
5	80
6	86
7	88
8	90
9	91

Solution to Gompertz Models Example 5

The Standard Gompertz Reliability vs. Time plot is shown in Figure 7.8.

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 7.9.

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.

Figure 7.8: Standard Gompertz Reliability vs. Time plot

Figure 7.9: Modified Gompertz Reliability vs. Time plot