Template:Model development camsaa-cd

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


 * $${{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}}$$

And the cumulative number of failures through configuration $$i$$, namely  $${{K}_{i}}$$ , is the sum of the  $${{M}_{i}}$$  for all  $$i$$ , or:


 * $${{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}}$$

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


 * $$E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta }$$

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


 * $$E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}}$$


 * $$\therefore {{f}_{1}}=\frac{\lambda T_{1}^{\beta }}$$

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:


 * $$\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}$$


 * $$\therefore {{f}_{2}}=\frac{\lambda T_{2}^{\beta }-\lambda T_{1}^{\beta }}$$

By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, $${{f}_{i}}$$, is obtained, such that:


 * $${{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}$$

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


 * $${{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }}$$

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


 * $${{R}_{i}}=1-{{f}_{i}}$$

And using Eqn. (dfi1), the equation for the reliability for the $${{i}^{th}}$$  trial is:


 * $${{R}_{i}}=1-{{g}_{i}}$$