General Examples Using the Lloyd-Lipow Model

Lloyd-Lipow Example 4

A 15-stage reliability development test program was performed. The grouped per configuration data set that was obtained is given in Table 6.3. Do the following:

Fit the Lloyd-Lipow model to the data using MLE.
What is the maximum reliability attained as the number of test stages approaches infinity?
What is the maximum achievable reliability with a 90% confidence level?

Table 6.3 - Grouped per Configuration data for Example 4

Stage, k	Number of Tests (nk)	Number of Successes (Sk)
1	10	3
2	10	3
3	10	4
4	10	5
5	10	5
6	12	6
7	12	5
8	12	7
9	14	8
10	14	8
11	14	10
12	14	12
13	14	11
14	14	12
15	14	12

Solution to Lloyd-Lipow Example 4

Figure 6.4 displays the entered data and the estimated Lloyd-Lipow parameters.
The maximum achievable reliability as the number of test stages approaches infinity is equal to the value of R. Therefore, R = 0.7157.
The maximum achievable reliability with a 90% confidence level can be estimated by viewing the confidence bounds on the parameters in the QCP, as shown in Figure 6.5. The lower bound on the value of R is equal to 0.6691.

Figure 6.4: Estimated Lloyd-Lipow parameters using MLE

Figure 6.5: Confidence bounds on the parameters

Lloyd-Lipow Example 5

Given the reliability data in Table 6.4, do the following:

Fit the Lloyd-Lipow model to the data using least squares analysis.
Plot the Lloyd-Lipow reliability with 90% 2-sided confidence bounds.
Determine how many months of testing are required to achieve a reliability goal of 90%.
Determine what is the attainable reliability if the maximum duration of testing is 30 months.

Table 6.4 - Reliability data

Time (months)	Reliability (%)
1	33.35
2	42.50
3	58.02
4	68.50
5	74.20
6	80.00
7	82.30
8	89.50
9	91.00
10	92.10

Solution to Lloyd-Lipow Example 5

Figure 6.6 displays the estimated parameters.
Figure 6.7 displays Reliability vs. Time plot with 90% 2-sided confidence bounds.
Figure 6.8 shows the number of months of testing required to achieve a reliability goal of 90%.
Figure 6.9 displays the reliability achieved after 30 months of testing.

Figure 6.6: Estimated Lloyd-Lipow parameters using least squares

Figure 6.7: Reliability vs. Time plot with 90% 2-sided confidence bounds

Figure 6.8: Number of months required to achieve a reliability of 90%

Figure 6.9: Maximum attainable reliability for a testing duration of 30 months

Lloyd-Lipow Example 6

Find the Lloyd-Lipow model that represents the data in Table 6.5 using MLE and plot it along with 95% 2-sided confidence bounds. Does the model follow the data?

Table 6.5 - Sequential data

Run Number	Result	Run Number	Result
1	F	11	S
2	F	12	S
3	S	13	F
4	S	14	S
5	S	15	S
6	F	16	S
7	S	17	S
8	F	18	S
9	F	19	S
10	S	20	S

Solution to Lloyd-Lipow Example 6

Figures 6.10 and 6.11 demonstrate the solution. As it can be seen from the plot in Figure 6.11, the model does not seem to follow the data. You may want to consider another model for this data set.

Figure 6.10: Estimated Lloyd-Lipow parameters using MLE

Figure 6.11: Reliability vs. Time plot with 95% 2-sided confidence bounds

Lloyd-Lipow Example 7

Find the Lloyd-Lipow model that represents the data in Table 6.6 using least squares. This data set includes information about the failure mode that was responsible for each failure, so that the probability of each failure mode recurring is taken into account in the analysis.

Table 6.6 - Sequential with mode data

Run Number	Result	Run Number	Result
1	F	11	S
2	F	12	S
3	S	13	F
4	S	14	S
5	S	15	S
6	F	16	S
7	S	17	S
8	F	18	S
9	F	19	S
10	S	20	S

Solution to Lloyd-Lipow Example 7

Figure 6.12 shows the analysis.

Figure 6.12: Estimated Lloyd-Lipow parameters using least squares