Crow-AMSAA Discrete Model Grouped Data Example: Difference between revisions

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<noinclude>{{Banner RGA Examples}}
<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Crow-AMSAA (NHPP)|Reliability Growth and Repairable System Analysis Reference book]]''.
''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
</noinclude>
</noinclude>


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{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Mixed data'''
|+'''Mixed Data'''
!Failures in Interval
!Failures in Interval
!Cumulative Trials
!Cumulative Trials
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This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.
This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,\,\!</math> as seen in the following picture.


[[Image:rga5.18.png|center|400px|Instantaneous unreliability at the end of the test.]]
[[Image:rga5.18.png|center|450px]]


The instantaneous reliability can then be calculated as:
The instantaneous reliability can then be calculated as:
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{{R}_{inst}}=1-0.1871=0.8129
{{R}_{inst}}=1-0.1871=0.8129
\end{align}\,\!</math>
\end{align}\,\!</math>
The average unreliability is calculated as:
:<math>\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>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}\,\!</math>

Latest revision as of 21:22, 18 September 2023

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This example appears in the Reliability growth reference.


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.

Mixed Data
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{ \hat{\beta }=0.7950\,\! }[/math]

and:

[math]\displaystyle{ \hat{\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.

Rga5.18.png

The instantaneous reliability can then be calculated as:

[math]\displaystyle{ \begin{align} {{R}_{inst}}=1-0.1871=0.8129 \end{align}\,\! }[/math]