Management controls the resources for corrective actions. Consequently, the effectiveness factors are part of the management strategy. For the BD mode failure intensity that has been seen during development testing, 100 d percent will be removed and 100(1 - d) percent will remain in the system. Therefore, after the corrective actions have been made, the current system instantaneous failure intensity consists of the failure intensity due to the A modes plus the failure intensity for the BC unseen plus the failure intensity for the unseen BD modes, plus the failure intensity for the BD modes that have been seen. Figure 9.5 shows how the system's instantaneous failure intensity can be broken down into its individual pieces based on the current failure mode strategy.
Figure 9.5: System failure intensity after the corrective actions
Keep in mind that the individual components of the system's instantaneous failure intensity will depend on the classifications defined in the data. For example, if BC modes are not present within the data then the BC mode MTBF will not be a part of the overall system MTBF. The individual pieces of the pie, as shown in the above plot, are calculated using the following equations. Let
where T is the test time and
and 
are
the maximum likelihood estimates of the Crow-AMSAA model for all of the
data. 
is the biased estimate of 
. Therefore:
where N is the total number
of failures and Xi
is the ith
time-to-failure. Let the successive failures 0 < X1 < X2
< ... < X3 < XN be partitioned
in the A mode failures (NA),
BC first occurrence failures (NBCF),
BC remaining failures (NBCR),
BD first occurrence failure (NBDF)
and the BD remaining failures (NBDR). For continuos
data, each portion of the pie chart due to each of the modes is calculated
as follows:
A modes
BC modes unseen
BC modes seen
BD modes unseen
BD modes seen
BD modes remain
BD modes removed
For grouped data, from the Crow-AMSAA
(N.H.P.P.) chapter the maximum likelihood estimates of 
and 
are calculated
such that the following equations are satisfied:
where K is the number of groups
and 
.
A modes
BC modes unseen
BC modes seen
BD modes unseen
BD modes seen
BD modes remain
BD modes removed
Consider the data in Table 9.3. There were 56 total failures and T = 400. The effectiveness factors of the BD modes are given in Table 9.2. Determine the following:
Calculate the demonstrated MTBF and failure intensity.
Calculate the projected MTBF and failure intensity.
What is the rate at which unique BD modes are being generated during this test?
If the test continues for an additional 50 hours, what is the minimum number of new unique BD modes expected to be generated?
Table 9.3 - Test-fix-find-test data
|
In order to obtain 
, use the traditional
Crow-AMSAA model for test-fix-test
to fit all 56 data points, regardless of the failure mode classification
to get:
Thus the achieved or demonstrated failure intensity is estimated by
The achieved or demonstrated MTBF, MCA, is the system reliability attained at the end of test, T = 400, and is estimated by
For this data set, M = 16 and T = 400.
Calculate 
and 
of the BD modes using Eqns. 3 and 4:
Then:
Therefore:
The Crow Extended model projected MTBF is:
Consequently, based on the Crow Extended model and the data in Tables 9.2 and 9.3, the MTBF grew to 7.85 as a result of the corrective actions for the BC failure modes during the test. The MTBF then jumped to 11.29 after the test as a result of the delayed corrective actions for the BD failure modes. The management strategy can be summarized by the Failure Mode Strategy plot shown in Figure 9.6.
Figure 9.6: Failure Mode Strategy plot
Figure 9.6 shows that 9.48% of the system's failure intensity has been left in (A modes), 31.81% of the failure intensity due to the BC modes has not been seen yet and 13.40% was removed during the test (BC modes - seen). In addition, 33.23% of the failure intensity due to the BD modes has not been seen yet, 3.37% will remain in the system since the corrective actions will not be completely effective at eliminating the identified failure modes and 8.72% will be removed after the delayed corrective actions.
The rate at which unique BD modes are being generated is equal to h(T|BD)-1, where
Unique BD modes are being generated every 33.4605 hours. If the test continues for another 50 hours, then at least one new unique BD mode would be expected to be seen from this additional testing.
As shown in Figure 9.7, the MTBF of each individual failure mode can be plotted and the failure modes with the lowest MTBF can be identified. These are the failure modes that cause the majority of the system failures.
Figure 9.7: Individual Mode MTBF
See
Also:
Crow Extended
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