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expanded final equation and gave reason why it is useful

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Robin 2010-06-04 16:29:50 +01:00
parent 05dd07ce83
commit dddb0da334
2 changed files with 47 additions and 23 deletions
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69
component_failure_modes_definition/component_failure_modes_definition.tex
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@ -358,7 +358,7 @@ from $1$ to $cc$ thus
% %
\begin{equation} \begin{equation}
\#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!} |{\mathcal{P}_{cc}S}| = \sum^{k}_{1..cc} \frac{|{S}|!}{ k! ( |{S}| - k)!}
\label{eqn:ccps} \label{eqn:ccps}
\end{equation} \end{equation}
@ -377,7 +377,11 @@ from the cardinality constrain powerset number.
Thus were we to have a simple functional group with two components R and T, of which Thus were we to have a simple functional group with two components R and T, of which
$$FM(R) = \{R_o, R_s\}$$ and $$FM(T) = \{T_o, T_s, T_h\}$$. $$FM(R) = \{R_o, R_s\}$$ and $$FM(T) = \{T_o, T_s, T_h\}$$.
For a cardinality constrained powerset of 2, because there are 5 error modes
This means that a functional~group $FG=\{R,T\}$ will have a component failure modes set % $FM_{cfg} $
of $FM_{cfg} = \{R_o, R_s, T_o, T_s, T_h\}$
For a cardinality constrained powerset of 2, because there are 5 error modes ( $|{FG_{cfg}}|=5$),
applying equation \ref{eqn:ccps} gives :- applying equation \ref{eqn:ccps} gives :-
$$\frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!} = 15$$ $$\frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!} = 15$$
@ -392,41 +396,60 @@ $R_o \wedge R_s$. As a combination ${2 \choose 2} = 1$ . For $T$ the component w
Thus for $cc == 2$ we must subtract $(3+1)$. Thus for $cc == 2$ we must subtract $(3+1)$.
The number of combinations to check is thus 11 for this example and this can be verified The number of combinations to check is thus 11 for this example and this can be verified
by listing all the required combinations: by listing all the required combinations:
%
%\vbox{
%\subsubsection{All Eleven Cardinality Constrained \\ Powerset of 2 Elements Listed}
%%\tiny
%\begin{enumerate}
%\item $\{R_o T_o\}$
%\item $\{R_o T_s\}$
%\item $\{R_o T_h\}$
%\item $\{R_s T_o\}$
%\item $\{R_s T_s\}$
%\item $\{R_s T_h\}$
%\item $\{R_o \}$
%\item $\{R_s \}$
%\item $\{T_o \}$
%\item $\{T_s \}$
%\item $\{T_h \}$
%\end{enumerate}
%%\normalsize
%}
%
\vbox{ $$ \mathcal{P}_{2}(FG_cfg) = \{
\subsubsection{All Eleven Cardinality Constrained \\ Powerset of 2 Elements Listed} \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
%\tiny \}
\begin{enumerate} $$
\item $\{R_o T_o\}$
\item $\{R_o T_s\}$
\item $\{R_o T_h\}$
\item $\{R_s T_o\}$
\item $\{R_s T_s\}$
\item $\{R_s T_h\}$
\item $\{R_o \}$
\item $\{R_s \}$
\item $\{T_o \}$
\item $\{T_s \}$
\item $\{T_h \}$
\end{enumerate}
%\normalsize
}
And by inspection
$$ |\mathcal{P}_{2}(FG_cfg)| = 11 $$
The cardinality constrained powerset equation \ref{eqn:ccps} corrected for The cardinality constrained powerset equation \ref{eqn:ccps} corrected for
unitary state failure modes can be unitary state failure modes can be
written as a general formula, where C is a set of the components (indexed by j where J written as a general formula (see equation \ref{eqn:correctedccps}), where C is a set of the components (indexed by j where J
is the set of components in the functional~group under analyis) and $\#C$ is the set of components in the functional~group under analyis) and $|{C}|$
indicates the number of mutually exclusive fault modes each component has:- indicates the number of mutually exclusive fault modes each component has:-
%$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!} $$ %$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!} $$
\begin{equation} \begin{equation}
\#\mathcal{P}_{cc} S = {\sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!}} - {\sum^{j}_{j \in J} {\#C_{j} \choose cc}} |{\mathcal{P}_{cc}S}| = {\sum^{k}_{1..cc} \frac{|{S}|!}{k!(|{S}| - k)!}} - {\sum^{j}_{j \in J} {|{C_{j}}| \choose cc}}
\label{eqn:correctedccps} \label{eqn:correctedccps}
\end{equation} \end{equation}
Expanding the combination in equation \ref{eqn:correctedccps}
\begin{equation}
|{\mathcal{P}_{cc}S}| = {\sum^{k}_{1..cc} \frac{|{S}|!}{k!(|{S}| - k)!}} - {\sum^{j}_{j \in J} \frac{|{C_j}|!}{cc!(|{C_j}| - cc)!}}
\label{eqn:correctedccps2}
\end{equation}
The equation \ref{eqn:correctedccps2} is now useful for an automated tool that
would verify that a `N' simultaneous failures model had been completly covered
by knowing how many test case should be covered.
%$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \big[ \frac{\#S!}{k!(\#S-k)!} - \sum_{j} (\#C_{j} \choose cc \big] $$ %$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \big[ \frac{\#S!}{k!(\#S-k)!} - \sum_{j} (\#C_{j} \choose cc \big] $$

1
component_failure_modes_definition/style.tex
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@ -57,6 +57,7 @@ depth -0.5ex\hfill}\newcommand{\innerhead}[1]{\def\lp@innerhead{#1}}
\newcommand{\mins}[1]{$#1^{\scriptsize\prime}$} % Minutes symbol \newcommand{\mins}[1]{$#1^{\scriptsize\prime}$} % Minutes symbol
\newcommand{\secs}[1]{$#1^{\scriptsize\prime\prime}$} % Seconds symbol \newcommand{\secs}[1]{$#1^{\scriptsize\prime\prime}$} % Seconds symbol
\newcommand{\key}[1]{\fbox{\sc#1}} % Box for keys \newcommand{\key}[1]{\fbox{\sc#1}} % Box for keys
\newcommand{\modulus}[1]{\ensuremathmode{|#1|}}
\newcommand{\?}{\_\hspace{0.115em}} % Proper spacing for \newcommand{\?}{\_\hspace{0.115em}} % Proper spacing for
% underscore % underscore
\newcommand{\rev}{PA5} \newcommand{\rev}{PA5}