Work on state explosion

fmmd_concept/System_safety_2011/state_exp.tex

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+\documentclass{beamer}
+
+\usepackage[utf8x]{inputenc}
+\usepackage{default}
+
+\begin{document}
+
+Consider the FMEA type methodologies
+where we look at all the failure modes in a system, and then 
+see how they can affect all other components within it,
+to determine its system level symptom or failure mode.
+We need to look at a large number of failure scenarios
+to do this completely (all failure modes against all components).
+This is represented in equation~\ref{eqn:fmea_state_exp},
+where $N$ is the total number of components in the system, and 
+$cfm$ is the number of failure modes per component.
+
+\begin{equation}
+  \label{eqn:fmea_state_exp}
+  N.(N-1).cfm % \\
+  %(N^2 - N).cfm 
+\end{equation}
+
+
+The FMMD methodology breaks the analysis down into small stages,
+by making the analyst choose functional groups, and then when analysed the groups
+are treated as components to be used for a higher stage.
+This is designed to address the state explosion (where $O$ is order
+of complexity) $O=N^2$ inherent in equation~\ref{eqn:fmea_state_exp}.
+
+\clearpage
+
+We can view the functional groups in FMMD as forming a hierarchy.
+If for the sake of example we consider each functional group to
+be three components, figure~\ref{fig:three_tree} shows
+how the levels work and converge to a top or system level.
+
+\begin{figure}
+ \centering
+ \includegraphics[width=300pt]{./three_tree.png}
+ % three_tree.png: 780x226 pixel, 72dpi, 27.52x7.97 cm, bb=0 0 780 226
+ \caption{Functional Group Tree example}
+ \label{fig:three_tree}
+\end{figure}
+
+\clearpage
+We can represent the number of failure scenarios to check in an FMMD hierarchy
+with equation~\ref{eqn:anscen}.
+
+\begin{equation}
+  \label{eqn:anscen}
+  \sum_{n=0}^{L} {fgn}^{n}.fgn.cfm.(fgn-1) 
+\end{equation}
+
+Where $fgn$ is the number of components in each functional group,
+and $cfm$ is the number of failure modes per component
+and L is the number of levels, the number of
+analysis scenarios to consider is show in equation~\ref{eqn:anscen}.
+
+
+So for a very simple analysis with three components forming a functional group where
+each component has three failure modes, we have only one level (zero'th).
+So to check every failure modes against the other components in the functional group
+requires 18 checks.
+
+\begin{equation}
+  \label{eqn:anscen2}
+  \sum_{n=0}^{0} {3}^{0}.3.3.(3-1) = 18 
+\end{equation}
+\clearpage
+
+
+
+In other words, we have three components in our functional group, 
+and nine failure modes to consider.
+So taking each failure mode and looking at how that could affect the functional group,
+we must compare each failure mode against the two other components (the `$fgn-1$' term).
+
+For the one `zero' level FMMD case we are doing the same thing as FMEA type analysis
+(but on a very simple small sub-system).
+We are looking at how each failure~mode can effect the system/top level.
+We can use equation~\ref{eqn:fmea_state_exp} to represent 
+the number of checks to rigorously perform FMEA, where $N$ is the total
+number of components in the system, and $cfm$ is the number of failures per component.
+
+
+
+Where $N=3$ and $cfm=3$ we can see that  the number of checks for this simple functional 
+group is the same for equation~\ref{eqn:fmea_state_exp}
+and equation~\ref{eqn:anscen}.
+\clearpage
+
+\section{Example}
+
+To see the effects of reducing `state~explosion' we need to look at a larger system.
+Let us take a system with 3 levels and apply these formulae.
+Having three levels (in addition to the top zero'th level)
+will require 81 base level components.
+
+$$
+%\begin{equation}
+  \label{eqn:fmea_state_exp}
+  81.(81-1).3 = 19440 % \\
+  %(N^2 - N).cfm 
+%\end{equation}
+$$
+
+$$
+%\begin{equation}
+ % \label{eqn:anscen}
+  \sum_{n=0}^{3} {3}^{n}.3.3.(2) = 720 
+%\end{equation}
+$$
+
+Thud for FMMD we needed to examine 720 failure mode scenarios, and for traditional FMEA
+type analysis methods 19440.
+% In practical example followed through, no more than 9 components have ever been required for a functional
+% group and the largest known number of failure modes has been 6.
+% If we take these numbers and double them (18 components per functional group
+% and 12 failure modes per component) and apply the formulas for a 4 level analysis
+% (i.e. 
+
+\clearpage
+Note that for double simultaneous failures the equation~\ref{eqn:fmea_state_exp} becomes
+equation~\ref{eqn:fmea_state_exp2} essentially making the order $N^3$.
+The FMMD case is cubic within the functional groups only, not all the components in the system.
+
+
+\begin{equation}
+  \label{eqn:fmea_state_exp2}
+  N.(N-1).(N-2).cfm % \\
+  %(N^2 - N).cfm 
+\end{equation}
+
+
+
+\end{document}