Suppose system development is represented by i configurations. This corresponds to i - 1 configuration changes, unless fixes are applied at the end of the test phase, in which case there would be i configuration changes. Let Ni be the number of trials during configuration i and let Mi be the number of failures during configuration i. Then the cumulative number of trials through configuration i, namely Ti, is the sum of the Ni for all i, or:
And the cumulative number of failures through configuration i, namely Ki, is the sum of the Mi for all i, or:
The expected value of Ki can be expressed as E[Ki] and defined as the expected number of failures by the end of configuration i. Applying the learning curve property to E[Ki] implies:
(59)
Denote f1 as the probability of failure for configuration 1 and use it to develop a generalized equation for fi in terms of the Ti and Ni. From Eqn. 59, the expected number of failures by the end of configuration 1 is:
Applying Eqn. 59 again and noting that the expected number of failures by the end of configuration 2 is the sum of the expected number of failures in configuration 1 and the expected number of failures in configuration 2:
By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, fi, is obtained, such that:
(60)
For the special case where Ni = 1 for all i, Eqn. 60 becomes a smooth curve, gi, that represents the probability of failure for the option for trial by trial data, or:
(61)
In Eqn. 61, i represents the trial number. Thus using Eqn. 60, an equation for the reliability (probability of success) for the ith configuration is obtained:
And using Eqn. 61, the equation for the reliability for the ith trial is:
This section describes procedures for estimating the parameters of the
Crow-AMSAA model for Success/Failure
data. An example is presented illustrating these concepts. The estimation
procedures described below provide maximum likelihood estimates (MLEs)
for the model's two parameters, 
and 
. The MLEs for 
and 
allow for point estimates
for the probability of failure , given by:
(62)
And the probability of success (reliability) for each configuration i is equal to:
(63)
The likelihood function is:
Taking the natural log on both sides yields:
Taking the derivative with respect to 
and 
respectively, exact MLEs
for 
and 
are values satisfying the following two equations:
(64)
(65)
where:
A one-shot system underwent reliability growth development testing for a total of 68 trials. Delayed corrective actions were incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68, the configuration was not changed. Configuration 1 experienced 5 failures, configuration 2 experienced 3 failures, configuration 3 experienced 4 failures and configuration 4 experienced 4 failures.
Table 5.6 - Estimated failure probability and reliability by configuration
|
Configuration (i) |
Estimated Failure Probability |
Estimated Reliability |
|
1 |
0.333 |
0.667 |
|
2 |
0.234 |
0.766 |
|
3 |
0.206 |
0.794 |
|
4 |
0.190 |
0.810 |
Determine the maximum likelihood estimators.
Estimate the unreliability and reliability by configuration.
The solution of Eqns.
64 and 65 provides for 
and 
corresponding to 0.5954 and 0.7801, respectively.
Using Eqns. 62 and 63 results in the following table.
Figures 5.16 and 5.17 show plots of the estimated unreliability and reliability by configuration.
Figure 5.16: Estimated unreliability by configuration
Figure 5.17: Estimated reliability by configuration
See
Also:
Crow-AMSAA (N.H.P.P.)
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