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General Examples Using the Crow-AMSAA Model

Crow Discrete Reliability Growth Model

The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).

Model Development

Suppose system development is represented by configurations. This corresponds to configuration changes, unless fixes are applied at the end of the test phase, in which case there would be configuration changes. Let $N_{i}$ be the number of trials during configuration and let $M_{i}$ be the number of failures during configuration . Then the cumulative number of trials through configuration , namely $T_{i}$ , is the sum of the $N_{i}$ for all , or:

And the cumulative number of failures through configuration , namely $K_{i}$ , is the sum of the $M_{i}$ for all , or:

The expected value of $K_{i}$ can be expressed as $E[K_{i}]$ and defined as the expected number of failures by the end of configuration . Applying the learning curve property to $E[K_{i}]$ implies:

(59)

Denote $f_{1}$ as the probability of failure for configuration 1 and use it to develop a generalized equation for $f_{i}$ in terms of the $T_{i}$ and $N_{i}$ . From Eqn. (59), the expected number of failures by the end of configuration 1 is:

MATH

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:

MATH MATH

By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, $f_{i}$ , is obtained, such that: MATH (60)

For the special case where $N_{i}=1$ for all , Eqn. (60) becomes a smooth curve, $g_{i}$ , that represents the probability of failure for trial by trial data, or: (61)

In Eqn. (61), represents the trial number. Thus using Eqn. (60), an equation for the reliability (probability of success) for the $i^{th}$ configuration is obtained:

And using Eqn. (61), the equation for the reliability for the $i^{th}$ trial is:

Maximum Likelihood Estimators

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, $\lambda$ and $\beta$ . The MLEs for $\lambda$ and $\beta$ allow for point estimates for the probability of failure, given by: MATH (62)

And the probability of success (reliability) for each configuration is equal to:

(63)

The likelihood function is: MATH

Taking the natural log on both sides yields: MATH

Taking the derivative with respect to $\lambda$ and $\beta$ respectively, exact MLEs for $\lambda$ and $\beta$ are values satisfying the following two equations:

MATH (64)
MATH (65)

where: MATH

Crow-AMSAA Example 8

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.

Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
Estimate the unreliability and reliability by configuration.

Solution to Crow-AMSAA Example 8

The solution of Eqns. (64) and (65) provides for $\lambda$ and $\beta$ corresponding to 0.5954 and 0.7801, respectively.
Table 5.6 displays the results of Eqns. (62) and (63).

Figures 5.16 and 5.17 show plots of the estimated unreliability and reliability by configuration.

Table 5.6 - Estimated failure probability and reliability by configuration

Configuration ()	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

Figure 5.16: Estimated unreliability by configuration

Figure 5.17: Estimated reliability by configuration

Mixed Data

In the RGA software, the Discrete Data > Mixed Data option gives a data sheet that can have input data that is either configuration in groups or individual trial by trial, or a mixed combination of individual trials and configurations of more than one trial. The calculations use the same mathematical methods described in Grouped Data for the Crow-AMSAA Model for the Crow-AMSAA grouped data.

Crow-AMSAA Example 9

Table 5.7 shows the number of failures of each interval of trials and the cumulative number of trials in each interval for a reliability growth test. For example, the first row of Table 5.7 indicates that for an interval of 14 trials, 5 failures occurred.

Table 5.7 - Mixed data for Example 9

Failures in Interval	Cumulative Trials
5	14
3	33
4	48
0	52
1	53
0	57
1	58
0	62
1	63
0	67
1	68

Solution to Crow-AMSAA Example 9

Using RGA 7, the parameters of the Crow-AMSAA model are estimated as follows:

and:

As we have seen, the Crow-AMSAA instantaneous failure intensity, $\lambda _{i}(T)$ , is defined as:

Using the above parameter estimates, we can calculate the instantaneous unreliability at the end of the test, or

This result that can be obtained from the Quick Calculation Pad (QCP), for as seen in Figure 5.18.

rga7_ch_amsaa(5)_inst_failure_intensity.wmf

Figure 5.18: Instantaneous unreliability at the end of the test

The instantaneous reliability can then be calculated as:

The average unreliability is calculated as:

MATH

and the average reliability is calculated as:

MATH

Bounds on Average Failure Probability for Mixed Data

The process to calculate the average failure probability confidence bounds for mixed data is as follows:

Calculate the average failure probability $(t_{1},\ t_{2})$ .
There will exist a $t^{\ast }$ between $t_{1}$ and $t_{2}$ such that the instantaneous failure probability at $t^{\ast }$ equals the average failure probability $(t_{1},\ t_{2})$ . The confidence intervals for the instantaneous failure probability at $t^{\ast }$ are the confidence intervals for the average failure probability $(t_{1},\ t_{2})$ .

Bounds on Average Reliability for Mixed Data

The process to calculate the average reliability confidence bounds for mixed data is as follows:

Calculate confidence bounds for average failure probability $(t_{1},\ t_{2})$ as described above.
The confidence bounds for reliability are 1 minus these confidence bounds for average failure probability.