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 6 it is simply a special case of the Crow Extended model. Suppose a system is subjected to development testing for a period of time, T. 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 occurrence of A modes follows an exponential distribution with failure intensity . 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 is 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 . 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 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 . If corrective actions are taken on the M BD modes observed by time T, then the system failure intensity is reduced from r(0) to:

where

• is the failure intensity for the M modes after the corrective actions

• is the remaining failure intensity for all unseen BD modes

All M BD modes observed by test time T may not be fixed by time T so the actual failure intensity at time T may not be r(T). However, r(T) can be viewed as the achieved failure intensity at time T 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 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 T (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 demonstrated MTBF is given by:

(2)

The detailed procedure for estimating r(T) is given in reference [20] and is reviewed here.

Let E[•] denote the expected value.

It is also noted that under realistic assumptions E[r(T)] may be expressed as:

where is the mean effectiveness factor and h(T) is the instantaneous rate at which a new BD mode will occur at time T. The ML estimate for the h(T) is:

And, is the bias term, such that

Estimation of Bias Term

Let X1 < X2 < ... < XM < T denote the cumulative test times for the first occurrence of BD modes. Then, the ML estimates of and are:

(3)

(4)
 

The intensity function h(T) for t > 0 is estimated by:

In particular, the ML estimate for the rate of occurrence for the distinct BD modes at time T is:

Furthermore, the ML estimate of the bias term B(T) is given by:

The unbiased estimate of is:

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

The mean is given by:

(5)

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

(6)

The projected MTBF is:

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 determine 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.

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:

(7)

The growth potential MTBF is estimated by:

(8)

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 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 jth 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.

Table 9.1 - Test-find-test data

Table 9.2 - Effectiveness factors for each unique BD mode

1. Determine the projected MTBF and failure intensity.

2. Determine the growth potential MTBF and failure intensity.

3. Determine the demonstrated MTBF and failure intensity.

Solution to Crow Extended Example 1

1. From Eqns. 3 and 4, the ML estimates of and are determined to be:

The unbiased estimate of is:

Based on the data in Table 9.2, . Therefore, . From Eqn. 6, the projected failure intensity due to incorporating the 16 corrective actions is:

The projected MTBF is:

2. To estimate the maximum reliability that can be attained with this management strategy, use the following calculations.

From Eqn. 7, the growth potential failure intensity is estimated by:

The growth potential MTBF is estimated by:

3. From Eqn. 1, the demonstrated failure intensity and MTBF are estimated by:

   
    
   

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

Figure 9.3: Demonstrated, projected and growth potential MTBF

Figure 9.4: Demonstrated, projected and growth potential failure intensity

See Also:
Crow Extended