Fleet Analysis Example
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This example appears in the Reliability growth reference.
The following table presents data for a fleet of 27 systems. A cycle is a complete history from overhaul to overhaul. The failure history for the last completed cycle for each system is recorded. This is a random sample of data from the fleet. These systems are in the order in which they were selected. Suppose the intervals to group the current data are 10,000; 20,000; 30,000; 40,000 and the final interval is defined by the termination time. Conduct the fleet analysis.
| Sample Fleet Data | |||
| System | Cycle Time [math]\displaystyle{ {{T}_{j}}\,\! }[/math] | Number of failures [math]\displaystyle{ {{N}_{j}}\,\! }[/math] | Failure Time [math]\displaystyle{ {{X}_{ij}}\,\! }[/math] |
|---|---|---|---|
| 1 | 1396 | 1 | 1396 |
| 2 | 4497 | 1 | 4497 |
| 3 | 525 | 1 | 525 |
| 4 | 1232 | 1 | 1232 |
| 5 | 227 | 1 | 227 |
| 6 | 135 | 1 | 135 |
| 7 | 19 | 1 | 19 |
| 8 | 812 | 1 | 812 |
| 9 | 2024 | 1 | 2024 |
| 10 | 943 | 2 | 316, 943 |
| 11 | 60 | 1 | 60 |
| 12 | 4234 | 2 | 4233, 4234 |
| 13 | 2527 | 2 | 1877, 2527 |
| 14 | 2105 | 2 | 2074, 2105 |
| 15 | 5079 | 1 | 5079 |
| 16 | 577 | 2 | 546, 577 |
| 17 | 4085 | 2 | 453, 4085 |
| 18 | 1023 | 1 | 1023 |
| 19 | 161 | 1 | 161 |
| 20 | 4767 | 2 | 36, 4767 |
| 21 | 6228 | 3 | 3795, 4375, 6228 |
| 22 | 68 | 1 | 68 |
| 23 | 1830 | 1 | 1830 |
| 24 | 1241 | 1 | 1241 |
| 25 | 2573 | 2 | 871, 2573 |
| 26 | 3556 | 1 | 3556 |
| 27 | 186 | 1 | 186 |
| Total | 52110 | 37 | |
Solution
The sample fleet data set can be grouped into 10,000; 20,000; 30,000; 40,000 and 52,110 time intervals. The following table gives the grouped data.
| Grouped Data | |
| Time | Observed Failures |
|---|---|
| 10,000 | 8 |
| 20,000 | 16 |
| 30,000 | 22 |
| 40,000 | 27 |
| 52,110 | 37 |
Based on the above time intervals, the maximum likelihood estimates of [math]\displaystyle{ \widehat{\lambda }\,\! }[/math] and [math]\displaystyle{ \widehat{\beta }\,\! }[/math] for this data set are then given by:
- [math]\displaystyle{ \begin{matrix} \widehat{\lambda }=0.00147 \\ \widehat{\beta }=0.93328 \\ \end{matrix}\,\! }[/math]
The next figure shows the System Operation plot.
