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Duane Model

Time-to-Failure Data

Discrete Data

General Examples Using the Duane Model

Duane Example 6

A prototype of a system was tested with design changes incorporated during the test. A total of 12 failures occurred. The data set is given in Table 4.5.

  1. Estimate the Duane parameters.
  2. Plot the cumulative and instantaneous MTBF curves.
  3. How many cumulative test and development hours are required to meet an instantaneous MTBF goal of 500 hours?
  4. How many cumulative test and development hours are required to meet a cumulative MTBF goal of 500 hours?

Table 4.5 - Developmental test data for Example 6

Failure Number

Cumulative Test Time (hr)

1

80

2

175

3

265

4

400

5

590

6

1100

7

1650

8

2010

9

2400

10

3380

11

5100

12

6400
Solution to Duane Example 6
  1. Figure 4.11 shows the data entered into RGA along with the estimated Duane parameters.
  2. Figure 4.12 shows the cumulative and instantaneous MTBF curves.
  3. Figure 4.13 shows the cumulative test and development hours needed for an instantaneous MTBF goal of 500 hours.
  4. Figure 4.14 shows the cumulative test and development hours needed for a cumulative MTBF goal of 500 hours.

Figure

Figure 4.11: Entered data and the estimated Duane parameters
 

Figure

Figure 4.12: The cumulative and instantaneous MTBF curves
 

Figure

Figure 4.13: Required test time for an instantaneous MTBF of 500 hours

Figure

Figure 4.14: Required test time for a cumulative MTBF of 500 hours
Duane Example 7

Two identical systems were tested. Any design changes made to improve the reliability of these systems were incorporated into both systems when any system failed. A total of 29 failures occurred. The data set is given in Table 4.6. Do the following:

  1. Estimate the Duane parameters.
  2. Assume both units are tested for an additional 100 hrs each. How many failures do you expect in that period?
  3. If testing/development were halted at this point, what would the reliability equation for this system be?

Table 4.6 - Developmental test data
Failure
Number		 Failed
Unit		 Test Time
Unit 1 (hr)		 Test Time
Unit 2 (hr)
1 1 0.2 2.0
2 2 1.7 2.9
3 2 4.5 5.2
4 2 5.8 9.1
5 2 17.3 9.2
6 2 29.3 24.1
7 1 36.5 61.1
8 2 46.3 69.6
9 1 63.6 78.1
10 2 64.4 85.4
11 1 74.3 93.6
12 1 106.6 103
13 2 195.2 117
14 2 235.1 134.3
15 1 248.7 150.2
16 2 256.8 164.6
17 2 261.1 174.3
18 2 299.4 193.2
19 1 305.3 234.2
20 1 326.9 257.3
21 1 339.2 290.2
22 1 366.1 293.1
23 2 466.4 316.4
24 1 504 373.2
25 1 510 375.1
26 2 543.2 386.1
27 2 635.4 453.3
28 1 641.2 485.8
29 2 755.8 573.6
Solution to Duane Example 7
  1. Figure 4.15 shows the data entered into RGA along with the estimated Duane parameters.
     

    Figure

    Figure 4.15: Entered data and the estimated parameters
     
  2. The current accumulated test time for both units is 1329.4 hr. If the process were to continue for an additional combined time of 200 hr, the expected cumulative number of failures at T = 1529.4 is 31.2695, as shown in Figure 4.16. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hr is 31.2695 - 29.2004 = 2.0691 ≈ 2.
  3. If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or λ = 0.0107 failures/hr. See Figure 4.17. An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:
 

MATH

Weibull++ can be utilized (from within RGA) to provide a Reliability vs. Time plot. This is shown in Figure 4.18.

Figure

Figure 4.16: The expected cumulative number of failures at T = 1529.4
 

Figure

Figure 4.17: Calculate the instantaneous failure intensity at the end of the test
 

Figure

Figure 4.18: Reliability vs. Time plot
Duane Example 8

Given the sequential success/failure data in the Table 4.7, do the following:

  1. Estimate the Duane parameters.
  2. What is the instantaneous MTBF at the end of the test?
  3. How many additional test runs with a one-sided 90% confidence level are required to meet an instantaneous MTBF goal of 5 hours?

Table 4.7 - Sequential data for Example 8
Run
Number		  
Result
1 F
2 F
3 S
4 S
5 S
6 F
7 S
8 F
9 F
10 S
11 S
12 S
13 F
14 S
15 S
16 S
17 S
18 S
19 S
20 S
Solution to Duane Example 8
  1. Figure 4.19 shows the data set entered into RGA along with the estimated Duane parameters.
  2. The MTBF at the end of the test is equal to 4.5904 hours. Note that this is the DMTBF that is shown in the Control Panel in Figure 4.20.
  3. Figure 4.20 shows the number of test runs with both one-sided confidence bounds at 90% confidence level to achieve an instantaneous MTBF of 5 hours. Therefore, the number of additional test runs required with a 90% confidence level is equal to 42.2481 - 20 = 22.2481 ≈ 23 test runs.

Figure

Figure 4.19: Entered data and the estimated parameters
 

Figure

Figure 4.20: Number of test runs with a one-sided 90% confidence
level required to meet an instantaneous MTBF goal of 5 hours