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| ==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).
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| <br>
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| ===Model Development===
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| Suppose system development is represented by <math>i</math> configurations. This corresponds to <math>i-1</math> configuration changes, unless fixes are applied at the end of the test phase, in which case there would be <math>i</math> configuration changes. Let <math>{{N}_{i}}</math> be the number of trials during configuration <math>i</math> and let <math>{{M}_{i}}</math> be the number of failures during configuration <math>i</math> . Then the cumulative number of trials through configuration <math>i</math> , namely <math>{{T}_{i}}</math> , is the sum of the <math>{{N}_{i}}</math> for all <math>i</math> , or:
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| ::<math>{{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}}</math>
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| And the cumulative number of failures through configuration <math>i</math> , namely <math>{{K}_{i}}</math> , is the sum of the <math>{{M}_{i}}</math> for all <math>i</math> , or:
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| ::<math>{{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}}</math>
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| The expected value of <math>{{K}_{i}}</math> can be expressed as <math>E[{{K}_{i}}]</math> and defined as the expected number of failures by the end of configuration <math>i</math> . Applying the learning curve property to <math>E[{{K}_{i}}]</math> implies:
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| ::<math>E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta }</math>
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| Denote <math>{{f}_{1}}</math> as the probability of failure for configuration 1 and use it to develop a generalized equation for <math>{{f}_{i}}</math> in terms of the <math>{{T}_{i}}</math> and <math>{{N}_{i}}</math> . From Eqn. (expectedn), the expected number of failures by the end of configuration 1 is:
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| ::<math>E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}}</math>
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| ::<math>\therefore {{f}_{1}}=\frac{\lambda T_{1}^{\beta }}{{{N}_{1}}}</math>
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| Applying Eqn. (expectedn) again and noting that the expected number of failures by the end of configuration 2 is the sum of the expected number of failures in configuration 1 and the expected number of failures in configuration 2:
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| ::<math>\begin{align}
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| & E\left[ {{K}_{2}} \right]= & \lambda T_{2}^{\beta } \\
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| & = & {{f}_{1}}{{N}_{1}}+{{f}_{2}}{{N}_{2}} \\
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| & = & \lambda T_{1}^{\beta }+{{f}_{2}}{{N}_{2}}
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| \end{align}</math>
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| ::<math>\therefore {{f}_{2}}=\frac{\lambda T_{2}^{\beta }-\lambda T_{1}^{\beta }}{{{N}_{2}}}</math>
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| By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, <math>{{f}_{i}}</math> , is obtained, such that:
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| ::<math>{{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}}</math>
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| For the special case where <math>{{N}_{i}}=1</math> for all <math>i</math> , Eqn. (dfi) becomes a smooth curve, <math>{{g}_{i}}</math> , that represents the probability of failure for trial by trial data, or:
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| ::<math>{{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }}</math>
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| In Eqn. (dfi1), <math>i</math> represents the trial number. Thus using Eqn. (dfi), an equation for the reliability (probability of success) for the <math>{{i}^{th}}</math> configuration is obtained:
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| ::<math>{{R}_{i}}=1-{{f}_{i}}</math>
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| And using Eqn. (dfi1), the equation for the reliability for the <math>{{i}^{th}}</math> trial is:
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| ::<math>{{R}_{i}}=1-{{g}_{i}}</math>
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| ===Maximum Likelihood Estimators===
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| 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:
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| ::<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>
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| And the probability of success (reliability) for each configuration <math>i</math> is equal to:
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| ::<math>{{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}</math>
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| The likelihood function is:
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| <br>
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| ::<math>\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}
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| {{N}_{i}} \\
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| {{M}_{i}} \\
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| \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>
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| Taking the natural log on both sides yields:
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| ::<math>\begin{align}
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| & \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}
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| {{N}_{i}} \\
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| {{M}_{i}} \\
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| \end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\
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| & & +\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]
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| \end{align}</math>
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| 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:
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| ::<math>\begin{align}
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| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\
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| & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0
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| \end{align}</math>
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| :where:
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| ::<math>\begin{align}
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| & {{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] \\
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| & {{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]} \\
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| & {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\,
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| \end{align}</math>
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| '''Example 8'''
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| <br>
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| 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.
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| <br>
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| • Configuration 1 experienced 5 failures,
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| <br>
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| • Configuration 2 experienced 3 failures,
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| <br>
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| • Configuration 3 experienced 4 failures and
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| <br>
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| • Configuration 4 experienced 4 failures.
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| <br>
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| <br>
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| :1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
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| :2) Estimate the unreliability and reliability by configuration.
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| <br>
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| <br>
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| '''Solution'''
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| <br>
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| :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.
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| :2) Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).
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| Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration.
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| <br>
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| <center>Table 5.6 - Estimated failure probability and reliability by configuration</center>
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| {|style= align="center" border="2"
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| !Configuration(<math>i</math>)
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| !Estimated Failure Probability
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| !Estimated Reliability
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| |-
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| |1|| 0.333|| 0.667
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| |-
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| |2|| 0.234|| 0.766
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| |-
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| |3|| 0.206|| 0.794
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| |-
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| |4|| 0.190|| 0.810
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| |}
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| <br>
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| [[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
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| <br>
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| [[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]
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| <br>
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| ===Mixed Data===
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| <br>
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| 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.
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| <br>
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| '''Example 9'''
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| <br>
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| Table 5.7 shows the number of fai
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| <math>\widehat{\beta }=0.7950</math>
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| 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.
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| <br>
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| {|style= align="center" border="2"
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| |+'''Table 5.7 - Mixed data for Example 9'''
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| !Failures in Interval
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| !Cumulative Trials
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| |-
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| |5|| 14
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| |-
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| |3|| 33
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| |-
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| |4|| 48
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| |-
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| |0|| 52
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| |-
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| |1|| 53
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| |-
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| |0|| 57
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| |-
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| |1|| 58
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| |-
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| |0|| 62
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| |-
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| |1|| 63
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| |-
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| |0|| 67
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| |-
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| |1|| 68
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| |}
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| <br>
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| Using RGA 7, the parameters of the Crow-AMSAA model are estimated as follows:
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| :and:
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| ::<math>\widehat{\lambda }=0.5588</math>
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| As we have seen, the Crow-AMSAA instantaneous failure intensity, <math>{{\lambda }_{i}}(T)</math> , is defined as:
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| ::<math>{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0</math>
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| Using the above parameter estimates, we can calculate the or instantaneous unreliability at the end of the test, or <math>T=68.</math>
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| ::<math>{{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871</math>
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| This result that can be obtained from the Quick Calculation Pad (QCP), for <math>T=68,</math> as seen in Figure Mixednst.FI.
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| <br>
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| [[Image:rga5.18.png|thumb|center|400px|Instantaneous unreliability at the end of the test.]]
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| <br>
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| The instantaneous reliability can then be calculated as:
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| ::<math>{{R}_{inst}}=1-0.1871=0.8129</math>
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| The average unreliability is calculated as:
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| ::<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}</math>
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| and the average reliability is calculated as:
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| ::<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}</math>
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| ====Bounds on Average Failure Probability for Mixed Data====
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| The process to calculate the average unreliability confidence bounds for mixed data is as follows:
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| <br>
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| :1) Calculate the average failure probability .
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| :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 .
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| <br>
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| <br>
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| ====Bounds on Average Reliability for Mixed Data====
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| The process to calculate the average reliability confidence bounds for mixed data is as follows:
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| <br>
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| :1) Calculate confidence bounds for average unreliability as described above.
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| :2) The confidence bounds for reliability are 1 minus these confidence bounds for average unreliability.
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