Template:Model development camsaa-cd: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(Created page with '===Model Development=== Suppose system development is represented by <math>i</math> configurations. This corresponds to <math>i-1</math> configuration changes, unless fixes a…')
 
 
Line 1: Line 1:
===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:
 
::<math>{{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}}</math>
 
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:
 
::<math>{{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}}</math>
 
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:
 
::<math>E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta }</math>
 
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:
 
::<math>E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}}</math>
 
::<math>\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>\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>\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>{{f}_{i}}</math> , is obtained, such that:
 
::<math>{{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}}</math>
 
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:
 
::<math>{{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }}</math>
 
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:
 
::<math>{{R}_{i}}=1-{{f}_{i}}</math>
 
And using Eqn. (dfi1), the equation for the reliability for the  <math>{{i}^{th}}</math>  trial is:
 
::<math>{{R}_{i}}=1-{{g}_{i}}</math>

Latest revision as of 23:06, 23 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Model_development_camsaa-cd&oldid=33632"