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| ===Model Development===
| | #REDIRECT [[Crow-AMSAA - NHPP]] |
| 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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