Crow-AMSAA Discrete Model Grouped Data Example
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This example appears in the Reliability Growth and Repairable System Analysis Reference book.
The table below 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 indicates that for an interval of 14 trials, 5 failures occurred.
| 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 the RGA software, the parameters of the Crow-AMSAA model are estimated as follows:
- [math]\displaystyle{ \widehat{\beta }=0.7950\,\! }[/math]
- and:
- [math]\displaystyle{ \widehat{\lambda }=0.5588\,\! }[/math]
As we have seen, the Crow-AMSAA instantaneous failure intensity, [math]\displaystyle{ {{\lambda }_{i}}(T)\,\! }[/math], is defined as:
- [math]\displaystyle{ \begin{align} {{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T\gt 0,\text{ }\lambda \gt 0\text{ and }\beta \gt 0 \end{align}\,\! }[/math]
Using the parameter estimates, we can calculate the instantaneous unreliability at the end of the test, or [math]\displaystyle{ T=68.\,\! }[/math]
- [math]\displaystyle{ {{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871\,\! }[/math]
This result that can be obtained from the Quick Calculation Pad (QCP), for [math]\displaystyle{ T=68,\,\! }[/math] as seen in the following picture.
The instantaneous reliability can then be calculated as:
- [math]\displaystyle{ \begin{align} {{R}_{inst}}=1-0.1871=0.8129 \end{align}\,\! }[/math]
The average unreliability is calculated as:
- [math]\displaystyle{ \text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\! }[/math]
and the average reliability is calculated as:
- [math]\displaystyle{ \text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\! }[/math]