General Examples Using the Crow-AMSAA Model

Crow-AMSAA Example 9

Six systems were subjected to a reliability growth test and a total of 81 failures were observed. The start and end times, along with the times-to-failure for each system, are presented in Table 5.7. Do the following:

Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
How many additional failures would be generated if testing continued until 3000 hours?

Table 5.7 - Multiple Systems (Unknown Equivalent Times) data for Example 9

	1	2	3	4	5	6
Start Time	0	0	0	0	0	0
End Time	504	541	454	474	436	500
Times-to-Failure	21	83	26	36	23	7
	29	83	26	306	46	13
	43	83	57	306	127	13
	43	169	64	334	166	31
	43	213	169	354	169	31
	66	299	213	395	213	82
	115	375	231	403	213	109
	159	431	231	448	255	137
	199		231	456	369	166
	202		231	461	374	200
	222		304		380	210
	248		383		415	220
	248					422
	255					437
	286					469
	286					469
	304
	320
	348
	364
	404
	410
	429

Solution to Crow-AMSAA Example 9

The parameters estimated using RGA are shown in Figure 5.18.
The number of failures can be estimated using the Quick Calculation Pad as shown in Figure 5.19. The estimated number of failures at 3000 hours is equal to . Therefore, the number of additional failures generated if testing continued until 3000 hours is equal to .

Figure 5.18: Estimated parameters of the Crow-AMSAA model

Figure 5.19: Estimate the number of failures at 3000 hours

Crow-AMSAA Example 10

A prototype of a system was tested at the end of one of its design stages. The test was run for a total of 300 hours and 27 failures were observed. The collected data is shown in Table 5.8. The prototype has a design specification of an MTBF equal to 10 hours with a 90% confidence level at 300 hours. Do the following:

Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
Does the prototype meet the specified goal?

Table 5.8 - Failure Times data for Example 10

2.6	56.5	98.1	190.7
16.5	63.1	101.1	193
16.5	70.6	132	198.7
17	73	142.2	251.9
21.4	77.7	147.7	282.5
29.1	93.9	149	286.1
33.3	95.5	167.2

Solution to Crow-AMSAA Example 10

The parameters estimated using RGA are shown in Figure 5.20.
Calculate the instantaneous MTBF with one-sided 90% confidence bounds using the Quick Calculation Pad (QCP) as shown in Figure 5.21. From the QCP, it is estimated that the lower limit on the MTBF at 300 hours with a 90% confidence level is equal to 10.8170. Therefore, the prototype has met the specified goal.

Figure 5.20: Estimated parameters of the Crow-AMSAA model

Figure 5.21: Instantaneous MTBF with one-sided 90% confidence bounds