Template:Model development camsaa-cd

Model Development

Suppose system development is represented by [math]\displaystyle{ i }[/math] configurations. This corresponds to [math]\displaystyle{ i-1 }[/math] configuration changes, unless fixes are applied at the end of the test phase, in which case there would be [math]\displaystyle{ i }[/math] configuration changes. Let [math]\displaystyle{ {{N}_{i}} }[/math] be the number of trials during configuration [math]\displaystyle{ i }[/math] and let [math]\displaystyle{ {{M}_{i}} }[/math] be the number of failures during configuration [math]\displaystyle{ i }[/math] . Then the cumulative number of trials through configuration [math]\displaystyle{ i }[/math] , namely [math]\displaystyle{ {{T}_{i}} }[/math] , is the sum of the [math]\displaystyle{ {{N}_{i}} }[/math] for all [math]\displaystyle{ i }[/math] , or:

[math]\displaystyle{ {{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}} }[/math]

And the cumulative number of failures through configuration [math]\displaystyle{ i }[/math] , namely [math]\displaystyle{ {{K}_{i}} }[/math] , is the sum of the [math]\displaystyle{ {{M}_{i}} }[/math] for all [math]\displaystyle{ i }[/math] , or:

[math]\displaystyle{ {{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}} }[/math]

The expected value of [math]\displaystyle{ {{K}_{i}} }[/math] can be expressed as [math]\displaystyle{ E[{{K}_{i}}] }[/math] and defined as the expected number of failures by the end of configuration [math]\displaystyle{ i }[/math] . Applying the learning curve property to [math]\displaystyle{ E[{{K}_{i}}] }[/math] implies:

[math]\displaystyle{ E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta } }[/math]

Denote [math]\displaystyle{ {{f}_{1}} }[/math] as the probability of failure for configuration 1 and use it to develop a generalized equation for [math]\displaystyle{ {{f}_{i}} }[/math] in terms of the [math]\displaystyle{ {{T}_{i}} }[/math] and [math]\displaystyle{ {{N}_{i}} }[/math] . From Eqn. (expectedn), the expected number of failures by the end of configuration 1 is:

[math]\displaystyle{ E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}} }[/math]

[math]\displaystyle{ \therefore {{f}_{1}}=\frac{\lambda T_{1}^{\beta }}{{{N}_{1}}} }[/math]

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:

[math]\displaystyle{ \begin{align} & E\left[ {{K}_{2}} \right]= & \lambda T_{2}^{\beta } \\ & = & {{f}_{1}}{{N}_{1}}+{{f}_{2}}{{N}_{2}} \\ & = & \lambda T_{1}^{\beta }+{{f}_{2}}{{N}_{2}} \end{align} }[/math]

[math]\displaystyle{ \therefore {{f}_{2}}=\frac{\lambda T_{2}^{\beta }-\lambda T_{1}^{\beta }}{{{N}_{2}}} }[/math]

By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, [math]\displaystyle{ {{f}_{i}} }[/math] , is obtained, such that:

[math]\displaystyle{ {{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}} }[/math]

For the special case where [math]\displaystyle{ {{N}_{i}}=1 }[/math] for all [math]\displaystyle{ i }[/math] , Eqn. (dfi) becomes a smooth curve, [math]\displaystyle{ {{g}_{i}} }[/math] , that represents the probability of failure for trial by trial data, or:

[math]\displaystyle{ {{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }} }[/math]

In Eqn. (dfi1), [math]\displaystyle{ i }[/math] represents the trial number. Thus using Eqn. (dfi), an equation for the reliability (probability of success) for the [math]\displaystyle{ {{i}^{th}} }[/math] configuration is obtained:

[math]\displaystyle{ {{R}_{i}}=1-{{f}_{i}} }[/math]

And using Eqn. (dfi1), the equation for the reliability for the [math]\displaystyle{ {{i}^{th}} }[/math] trial is:

[math]\displaystyle{ {{R}_{i}}=1-{{g}_{i}} }[/math]