Test-Find-Test

Test-find-test is the case where all corrective actions are delayed until after the test. Therefore, there are no BC modes when analyzing test-find-test data. This scenario is also called the Crow-AMSAA Projection model, but for the purposes of RGA 7 it is simply a special case of the Crow Extended model.

Suppose a system is subjected to development testing for a period of time, Τ. The system can be considered as consisting of two types of failure modes: A modes and BD modes. It is assumed that all BD modes are in series and fail independently according to the exponential distribution. Also assume that the rate of occurrence of A modes follows an exponential distribution with failure intensity λA. The system MTBF is constant throughout the test phase since all of the corrective actions are delayed until after the completion of the test. After the delayed fixes have been implemented, the system MTBF will then jump to a higher value.

Let K denote the total number of BD modes in the system and let λi denote the failure intensity for the ith BD mode, such that i = 1, 2, ..., K. Then, at time equal to zero, the system failure intensity r(0) is:

where

During the test (0, T), a random number of M distinct BD modes will be observed, such that M ≤ K. Denote the effectiveness factor (EF) for the ith BD mode as di, i = 1, 2, ..., K. The effectiveness factor di is the percent decrease in λi after a corrective action has been made for the ith BD mode. That is, the corrective action for the ith BD mode removes 100 x di percent of the failure rate and 100 x (1 - di) percent remains. The failure intensity for the ith BD failure mode after a corrective action is (1 - di)λi. If corrective actions are taken on the M BD modes observed by time Τ, then the system failure intensity is reduced from r(0) to:

All M BD modes observed by test time Τ may not be fixed by time Τ so the actual failure intensity at time Τ may not be r(T). However, r(T) can be viewed as the achieved failure intensity at time Τ if all fixes were updated and incorporated into the system. All of the fixes for the BD modes found during the test are incorporated as delayed fixes at the end of the test phase. Therefore, the system failure intensity is constant at r(0) = λA + λBD through the test phase and will then jump to a lower value r(T) after the delayed fixes have been implemented. Let NA and NBD be the total number of A and BD failures observed during the test (0, T) and let N = NA + NBD. In addition, there are M distinct BD modes observed during the test. After implementing the M fixes, the failure intensity for the system at time Τ (after the jump) is given by the function r(T).

r(0) is actually the demonstrated failure intensity, which is based on actual system performance of the hardware tested and not of some future configuration. A demonstrated reliability value should be determined at the end of each test phase. The demonstrated failure intensity is:

(1)

The detailed procedure for estimating r(T) is given in Crow, L.H., "An Extended Reliability Growth Model for Managing and Assessing Corrective Actions" and is reviewed here.

Under realistic assumptions E[r(T)] also may be expressed as: MATH

where $\overline{d}$ is the mean effectiveness factor and h(T) is the instantaneous rate at which a new BD mode will occur at time T. The maximum likelihood estimate for the h(T) is:

And, $\overline{d}h(T)$ is the bias term, such that:

Estimation of Bias Term

Let X1 < X2 < ... < XM < Τ denote the cumulative test times for the first occurrences of BD modes. Then, the maximum likelihood estimates of λBD and βBD are:

In particular, the maximum likelihood estimate for the rate of occurrence for the distinct BD modes at time Τ is: MATH

The unbiased estimate of βBD is:

Thus the unbiased estimate of the bias term is given by:

Therefore, the projected failure intensity r(T) is then estimated at the end of the test phase by: MATH

(6)

Reliability Growth Potential

The failure intensity r(T) will depend on the management strategy that determines the classification of the A and BD failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, r(T) depends on h(t), which is the rate at which problem failure modes are being seen during testing. h(t) drives the opportunity to take corrective actions based on the seen failure modes and it is an important factor in the overall reliability growth rate. The reliability growth potential is the limiting value of r(T) as T increases. This limit is the maximum MTBF that can be attained with the current management strategy. The maximum MTBF will be attained when all K BD modes have been observed and fixed with EFs di. In terms of failure intensity, the growth potential is expressed by the following equation: MATH

In terms of the MTBF, the growth potential is given by:

The procedure for estimating the growth potential is as follows. Suppose that the system is tested for a period of time T and that N failures have been observed. According to the management strategy, NA of these failures are A modes and NBD of these failures are BD modes. For the BD modes, there will be M distinct fixes. As before, Ni is the total number of failures for the ith BD mode and di is the corresponding assigned EF. From this data, the growth potential failure intensity is estimated by: MATH

(7)

Crow Extended Example 1

Consider the data in Table 9.1. A system was tested for T = 400 hours. There were a total of N = 42 failures and all corrective actions will be delayed until after the end of the 400 hour test. Each failure has been designated as either an A failure mode (the cause will not receive a corrective action) or a BD mode (the cause will receive a corrective action). There are NA = 10 A mode failures and NBD = 32 BD mode failures. In addition, there are M = 16 distinct BD failure modes, which means 16 distinct corrective actions will be incorporated into the system at the end of test. The total number of failures for the ith observed distinct BD mode is denoted by Nj and the total number of BD failures during the test is

. These values and effectiveness factors are given in Table 9.2.

i	Xi	Mode	i	Xi	Mode
1	15	BD1	22	260.1	BD1
2	25.3	BD2	23	263.5	BD8
3	47.5	BD3	24	273.1	A
4	54	BD4	25	274.7	BD6
5	56.4	BD5	26	285	BD13
6	63.6	A	27	304	BD9
7	72.2	BD5	28	315.4	BD4
8	99.6	BD6	29	317.1	A
9	100.3	BD7	30	320.6	A
10	102.5	A	31	324.5	BD12
11	112	BD8	32	324.9	BD10
12	120.9	BD2	33	342	BD5
13	125.5	BD9	34	350.2	BD3
14	133.4	BD10	35	364.6	BD10
15	164.7	BD9	36	364.9	A
16	177.4	BD10	37	366.3	BD2
17	192.7	BD11	38	373	BD8
18	213	A	39	379.4	BD14
19	244.8	A	40	389	BD15
20	249	BD12	41	394.9	A
21	250.8	A	42	395.2	BD16

BD Mode	Number Nj	First Occurrence	EF di
1	2	15.0	.67
2	3	25.3	.72
3	2	47.5	.77
4	2	54.0	.77
5	3	54.0	.87
6	2	99.6	.92
7	1	100.3	.50
8	3	112.0	.85
9	3	125.5	.89
10	4	133.4	.74
11	1	192.7	.70
12	2	249.0	.63
13	1	285.0	.64
14	1	379.4	.72
15	1	389.0	.69
16	1	395.2	.46

Solution to Crow Extended Example 1

Based on the data in Table 9.2,

0.72125. Therefore,

. From Eqn. (6), the projected failure intensity due to incorporating the 16 corrective actions is: MATH

Figure 9.3 shows the demonstrated, projected and growth potential MTBF. Figure 9.4 shows the demonstrated, projected and growth potential failure intensity.