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

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 be the number of trials during configuration and let be the number of failures during configuration . Then the cumulative number of trials through configuration , namely , is the sum of the for all , or:

And the cumulative number of failures through configuration , namely , is the sum of the for all , or:

The expected value of can be expressed as and defined as the expected number of failures by the end of configuration . Applying the learning curve property to implies:

(59)

Denote as the probability of failure for configuration 1 and use it to develop a generalized equation for in terms of the and . 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, , is obtained, such that: (60)

For the special case where for all , Eqn. (60) becomes a smooth curve, , 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 configuration is obtained:

And using Eqn. (61), the equation for the reliability for the 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 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.

- Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
- 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.
- 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 |

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.

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 |

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, , 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.

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:

and the average reliability is calculated as:

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

- Calculate the average failure probability .
- There will exist a between and such that the instantaneous failure probability at equals the average failure probability . The confidence intervals for the instantaneous failure probability at are the confidence intervals for the average failure probability .

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

- Calculate confidence bounds for average failure probability as described above.
- The confidence bounds for reliability are 1 minus these confidence bounds for average failure probability.