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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}
\#\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}
\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
$$FM(R) = \{R_o, R_s\}$$ 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 :-
$$\frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!} = 15$$
@ -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 $(3+1)$.
The number of combinations to check is thus 11 for this example and this can be verified
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{
\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
}
$$ \mathcal{P}_{2}(FG_cfg) = \{
\{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
\}
$$
And by inspection
$$ |\mathcal{P}_{2}(FG_cfg)| = 11 $$
The cardinality constrained powerset equation \ref{eqn:ccps} corrected for
unitary state failure modes can be
written as a general formula, 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 $\#C$
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 $|{C}|$
indicates the number of mutually exclusive fault modes each component has:-
%$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!} $$
\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}
\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.
%$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \big[ \frac{\#S!}{k!(\#S-k)!} - \sum_{j} (\#C_{j} \choose cc \big] $$

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{\secs}[1]{$#1^{\scriptsize\prime\prime}$} % Seconds symbol
\newcommand{\key}[1]{\fbox{\sc#1}} % Box for keys
\newcommand{\modulus}[1]{\ensuremathmode{|#1|}}
\newcommand{\?}{\_\hspace{0.115em}} % Proper spacing for
% underscore
\newcommand{\rev}{PA5}