# Crow-AMSAA Grouped Data Examples

### From ReliaWiki

*These examples appear in the Reliability Growth and Repairable System Analysis Reference book*.

## Grouped Data Example 1

Consider the grouped failure times data given in the following table. Solve for the Crow-AMSAA parameters using MLE.

Run Number | Cumulative Failures | End Time(hours) | ||||
---|---|---|---|---|---|---|

1 | 2 | 200 | 5.298 | 28.072 | 0.693 | 3.673 |

2 | 3 | 400 | 5.991 | 35.898 | 1.099 | 6.582 |

3 | 4 | 600 | 6.397 | 40.921 | 1.386 | 8.868 |

4 | 11 | 3000 | 8.006 | 64.102 | 2.398 | 19.198 |

Sum = | 25.693 | 168.992 | 5.576 | 38.321 |

**Solution**

Using RGA, the value of , which must be solved numerically, is 0.6315. Using this value, the estimator of is:

Therefore, the intensity function becomes:

## Grouped Data Example 2

A new helicopter system is under development. System failure data has been collected on five helicopters during the final test phase. The actual failure times cannot be determined since the failures are not discovered until after the helicopters are brought into the maintenance area. However, total flying hours are known when the helicopters are brought in for service, and every 2 weeks each helicopter undergoes a thorough inspection to uncover any failures that may have occurred since the last inspection. Therefore, the cumulative total number of flight hours and the cumulative total number of failures for the 5 helicopters are known for each 2-week period. The total number of flight hours from the test phase is 500, which was accrued over a period of 12 weeks (six 2-week intervals). For each 2-week interval, the total number of flight hours and total number of failures for the 5 helicopters were recorded. The grouped data set is displayed in the following table.

Interval | Interval Length | Failures in Interval |
---|---|---|

1 | 0 - 62 | 12 |

2 | 62 -100 | 6 |

3 | 100 - 187 | 15 |

4 | 187 - 210 | 3 |

5 | 210 - 350 | 18 |

6 | 350 - 500 | 16 |

Do the following:

- Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
- Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.

**Solution**

- Using RGA, the value of , must be solved numerically. Once has been estimated then the value of can be determined. The parameter values are displayed below:
- The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for hour are:
The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for hours are:

and:

The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.