|
|
| (2 intermediate revisions by one other user not shown) |
| Line 1: |
Line 1: |
| ==Crow Discrete Reliability Growth Model==
| | #REDIRECT [[Crow-AMSAA - NHPP]] |
| The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
| |
| <br>
| |
| {{model development camsaa-cd}}
| |
| | |
| ===Maximum Likelihood Estimators===
| |
| 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, <math>\lambda </math> and <math>\beta </math> . The MLEs for <math>\lambda </math> and <math>\beta </math> allow for point estimates for the probability of failure, given by:
| |
| | |
| ::<math>{{\hat{f}}_{i}}=\frac{\hat{\lambda }T_{i}^{{\hat{\beta }}}-\hat{\lambda }T_{i-1}^{{\hat{\beta }}}}{{{N}_{i}}}=\frac{\hat{\lambda }\left( T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}} \right)}{{{N}_{i}}}</math>
| |
| | |
| And the probability of success (reliability) for each configuration <math>i</math> is equal to:
| |
| | |
| ::<math>{{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}</math>
| |
| | |
| The likelihood function is:
| |
| <br>
| |
| ::<math>\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}
| |
| {{N}_{i}} \\
| |
| {{M}_{i}} \\
| |
| \end{matrix} \right){{\left( \frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{M}_{i}}}}{{\left( \frac{{{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{N}_{i}}-{{M}_{i}}}}</math>
| |
| | |
| Taking the natural log on both sides yields:
| |
| | |
| ::<math>\begin{align}
| |
| & \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}
| |
| {{N}_{i}} \\
| |
| {{M}_{i}} \\
| |
| \end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\
| |
| & & +\underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ ({{N}_{i}}-{{M}_{i}})\left[ \ln ({{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right]
| |
| \end{align}</math>
| |
| | |
| Taking the derivative with respect to <math>\lambda </math> and <math>\beta </math> respectively, exact MLEs for <math>\lambda </math> and <math>\beta </math> are values satisfying the following two equations:
| |
| | |
| ::<math>\begin{align}
| |
| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\
| |
| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0
| |
| \end{align}</math>
| |
| | |
| :where:
| |
| | |
| ::<math>\begin{align}
| |
| & {{H}_{i}}= & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ T_{i}^{\beta }\ln {{T}_{i}}-T_{i-1}^{\beta }\ln {{T}_{i-1}} \right] \\
| |
| & {{S}_{i}}= & \frac{{{M}_{i}}}{\left[ \lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta } \right]}-\frac{{{N}_{i}}-{{M}_{i}}}{\left[ {{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta } \right]} \\
| |
| & {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\,
| |
| \end{align}</math>
| |
| | |
| '''Example 8'''
| |
| <br>
| |
| 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.
| |
| <br>
| |
| • Configuration 1 experienced 5 failures,
| |
| <br>
| |
| • Configuration 2 experienced 3 failures,
| |
| <br>
| |
| • Configuration 3 experienced 4 failures and
| |
| <br>
| |
| • Configuration 4 experienced 4 failures.
| |
| <br>
| |
| <br>
| |
| :1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
| |
| :2) Estimate the unreliability and reliability by configuration.
| |
| <br>
| |
| <br>
| |
| '''Solution'''
| |
| <br>
| |
| :1) The solution of Eqns. (solution1) and (solution2) provides for <math>\lambda </math> and <math>\beta </math> corresponding to 0.5954 and 0.7801, respectively.
| |
| :2) Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).
| |
| Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration.
| |
| <br>
| |
| <center>Table 5.6 - Estimated failure probability and reliability by configuration</center>
| |
| {|style= align="center" border="2"
| |
| !Configuration(<math>i</math>)
| |
| !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
| |
| |}
| |
| <br>
| |
| [[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
| |
| <br>
| |
| [[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]] | |
| <br>
| |
| | |
| ===Mixed Data===
| |
| <br>
| |
| 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 section 5.3 for the Crow-AMSAA grouped data.
| |
| <br>
| |
| '''Example 9'''
| |
| <br>
| |
| Table 5.7 shows the number of fai
| |
| <math>\widehat{\beta }=0.7950</math>
| |
| lures 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.
| |
| <br>
| |
| {|style= align="center" border="2"
| |
| |+'''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
| |
| |}
| |
| | |
| <br>
| |
| Using RGA 7, the parameters of the Crow-AMSAA model are estimated as follows:
| |
| | |
| :and:
| |
| | |
| ::<math>\widehat{\lambda }=0.5588</math>
| |
| | |
| As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)</math> , is defined as:
| |
| | |
| ::<math>{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0</math>
| |
| | |
| Using the above parameter estimates, we can calculate the or instantaneous unreliability at the end of the test, or <math>T=68.</math>
| |
| | |
| ::<math>{{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>T=68,</math> as seen in Figure Mixednst.FI.
| |
| <br>
| |
| [[Image:rga5.18.png|thumb|center|400px|Instantaneous unreliability at the end of the test.]]
| |
| <br>
| |
| The instantaneous reliability can then be calculated as:
| |
| | |
| ::<math>{{R}_{inst}}=1-0.1871=0.8129</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>
| |
| | |
| ====Bounds on Average Failure Probability for Mixed Data====
| |
| The process to calculate the average unreliability confidence bounds for mixed data is as follows:
| |
| <br>
| |
| :1) Calculate the average failure probability .
| |
| :2) There will exist a <math>{{t}^{*}}</math> between <math>{{t}_{1}}</math> and <math>{{t}_{2}}</math> such that the instantaneous unreliability at <math>{{t}^{*}}</math> equals the average unreliability . The confidence intervals for the instantaneous unreliability at <math>{{t}^{*}}</math> are the confidence intervals for the average unreliability .
| |
| <br>
| |
| <br>
| |
| ====Bounds on Average Reliability for Mixed Data====
| |
| The process to calculate the average reliability confidence bounds for mixed data is as follows:
| |
| <br>
| |
| :1) Calculate confidence bounds for average unreliability as described above.
| |
| :2) The confidence bounds for reliability are 1 minus these confidence bounds for average unreliability.
| |