Concurrent Operating Times - Crow-AMSAA (NHPP) Example

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This example appears in the Reliability Growth and Repairable System Analysis Reference book.

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

1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.

2) How many additional failures would be generated if testing continues until 3,000 hours?

Multiple systems (concurrent operating times) Data

System	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

1) The next figure shows the parameters estimated using RGA.

Estimated parameters of the Crow-AMSAA model.

2) The number of failures can be estimated using the Quick Calculation Pad, as shown next. The estimated number of failures at 3,000 hours is equal to 83.2451 and 81 failures were observed during testing. Therefore, the number of additional failures generated if testing continues until 3,000 hours is equal to [math]\displaystyle{ 83.2451-81=2.2451\approx 3\,\! }[/math].

Expected number of failures at 3000 hours.