Remove appendix

paper.tex

@@ -244,11 +244,10 @@ unknown distribution, the test we chose to verify this observation is a Mann
 Whitney U test. We found that $S _\text{L}$ is faster than $S_\text{R}$ with a
 significance of $p < 0.0001$, while no significant statement could be made
 about $S_\text{R}$ and $S_\text{B}$.
-
 Fig. \ref{fig:Behavior} shows the results of the behavioral measurement.
-During this part of the experiment, we also measured the time each participant
+% During this part of the experiment, we also measured the time each participant
-needed to fill up their bottle. Using the measured flowrates we calculated
+% needed to fill up their bottle. Using the measured flowrates we calculated
-the mean refill volume to be $\SI{673.92}{\milli\liter}$.
+% the mean refill volume to be $\SI{673.92}{\milli\liter}$.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -256,16 +255,16 @@ the mean refill volume to be $\SI{673.92}{\milli\liter}$.
 
 We can consider the water dispenser and students as comprising a queueing
 system, specifically an M/G/1 queue \cite{stewart_probability_2009}.
-The response time, i.e., the time spent waiting as well as
+The expected response time, i.e., the time spent waiting as well as
 the time dispensing water, is \cite[Section 14.3]{stewart_probability_2009}%
 \begin{align*}
     W = E\mleft\{ S \mright\} + \frac{\lambda E\mleft\{ S^2 \mright\}}{2\mleft( 1-\rho \mright)}
     ,%
 \end{align*}%
 where $S$ denotes the service time (i.e., the time spent refilling a bottle),
-$\lambda$ the mean arrival time, and $\rho$ the system utilization. Using our
+$\lambda$ the mean arrival time, and $\rho = \lambda \cdot E\mleft\{ S \mright\}$ the system utilization. Using our
-experimental data we can approximate all parameters and calculate
+experimental data we can approximate all parameters and obtain
-\todo{$W \approx 123$} (See the appendix for a complete derivation).
+\todo{$W \approx 123$}.
 % We examine the effects of the choice of hydration strategy. To
 % this end, we start by estimating the potential time savings possible by always
 % choosing the fastest strategy:%
@@ -338,69 +337,69 @@ academic performance of KIT students: always pressing the right button.
 
 \printbibliography
 
-\appendix
+% \appendix
-
+%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Derivation of Service Time}
+% \section{Derivation of Service Time}
-\label{sec:Derivation of Service Time}
+% \label{sec:Derivation of Service Time}
-
+%
-
+%
-We want to compute the response time of our queueing system, i.e.,
+% We want to compute the response time of our queueing system, i.e.,
-\cite[Section 14.3]{stewart_probability_2009}
+% \cite[Section 14.3]{stewart_probability_2009}
-\begin{align*}
+% \begin{align*}
-    W = E\mleft\{ S \mright\} + \frac{\lambda E\mleft\{ S^2 \mright\}}{2\mleft( 1-\rho \mright)}
+%     W = E\mleft\{ S \mright\} + \frac{\lambda E\mleft\{ S^2 \mright\}}{2\mleft( 1-\rho \mright)}
-    .%
+%     .%
-\end{align*}%
+% \end{align*}%
-We start by modelling the service time and subsequently calculate $\lambda$
+% We start by modelling the service time and subsequently calculate $\lambda$
-and $\rho$.
+% and $\rho$.
-
+%
-Let $S, V$ and $R$ be random variables denoting the service time, refill volume
+% Let $S, V$ and $R$ be random variables denoting the service time, refill volume
-and refill rate, respectively. Assuming that $V$ and $R$ are independent, we 
+% and refill rate, respectively. Assuming that $V$ and $R$ are independent, we 
-have
+% have
-\begin{gather*}
+% \begin{gather*}
-    S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm}
+%     S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm}
-    P_R(r) = \left\{\begin{array}{rl}
+%     P_R(r) = \left\{\begin{array}{rl}
-        P(S_\text{L}), & r = r_{S_\text{L}} \\
+%         P(S_\text{L}), & r = r_{S_\text{L}} \\
-        1-P(S_\text{L}), & r = r_{S_\text{R}}
+%         1-P(S_\text{L}), & r = r_{S_\text{R}}
-    \end{array}\right.
+%     \end{array}\right.
-\end{gather*}%
+% \end{gather*}%
-\begin{align*}
+% \begin{align*}
-    E\mleft\{ S^2 \mright\} &= E\mleft\{ V^2 \mright\}E\mleft\{ 1 / R^2  \mright\} \\
+%     E\mleft\{ S^2 \mright\} &= E\mleft\{ V^2 \mright\}E\mleft\{ 1 / R^2  \mright\} \\
-    & = E\mleft\{ V^2 \mright\} \mleft( P(S_\text{L})\frac{1}{r^2_{S_\text{L}}} + \mleft( 1 - P(S_\text{L}) \mright)\frac{1}{r^2_{S_\text{R}}}  \mright)
+%     & = E\mleft\{ V^2 \mright\} \mleft( P(S_\text{L})\frac{1}{r^2_{S_\text{L}}} + \mleft( 1 - P(S_\text{L}) \mright)\frac{1}{r^2_{S_\text{R}}}  \mright)
-    .%
+%     .%
-\end{align*}
+% \end{align*}
-We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the
+% We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the
-measured fill times and flow rates) to compute
+% measured fill times and flow rates) to compute
-\begin{align*}
+% \begin{align*}
-    \left.
+%     \left.
-    \begin{array}{r}
+%     \begin{array}{r}
-        E\mleft\{ V^2 \mright\} \approx \frac{1}{N+1} \sum_{n=1}^{N} v_n^2 = \todo{15}\\
+%         E\mleft\{ V^2 \mright\} \approx \frac{1}{N+1} \sum_{n=1}^{N} v_n^2 = \todo{15}\\
-        P(S_\text{L}) \approx \frac{\text{\# times $S_\text{L}$ was chosen}}{N} = \todo{123} \\
+%         P(S_\text{L}) \approx \frac{\text{\# times $S_\text{L}$ was chosen}}{N} = \todo{123} \\
-        r^2_{S_\text{L}} \approx \todo{125} \\
+%         r^2_{S_\text{L}} \approx \todo{125} \\
-        r^2_{S_\text{R}} \approx \todo{250}
+%         r^2_{S_\text{R}} \approx \todo{250}
-    \end{array}
+%     \end{array}
-    \right\} \Rightarrow
+%     \right\} \Rightarrow
-    \left\{
+%     \left\{
-    \begin{array}{l}
+%     \begin{array}{l}
-        E\mleft\{ S \mright\} \approx \todo{678} \\
+%         E\mleft\{ S \mright\} \approx \todo{678} \\
-        E\mleft\{ S^2 \mright\} \approx \todo{123}
+%         E\mleft\{ S^2 \mright\} \approx \todo{123}
-    \end{array}
+%     \end{array}
-    \right.
+%     \right.
-    .%
+%     .%
-\end{align*}
+% \end{align*}
-
+%
-$\lambda$ is the mean arrival time.
+% $\lambda$ is the mean arrival time.
-
+%
-\todo{
+% \todo{
-    \textbf{TODOs:}
+%     \textbf{TODOs:}
-    \begin{itemize}
+%     \begin{itemize}
-        \item Complete text describing / obtaining $\rho$ and $\lambda$
+%         \item Complete text describing / obtaining $\rho$ and $\lambda$
-        \item Move model derivation to method section
+%         \item Move model derivation to method section
-        \item Move calculations with model to results section
+%         \item Move calculations with model to results section
-        \item Add grade gain derivation
+%         \item Add grade gain derivation
-        \item Idea: Make the whole thing 2 pages and print on A3
+%         \item Idea: Make the whole thing 2 pages and print on A3
-    \end{itemize}
+%     \end{itemize}
-}
+% }
 
 
 \end{document}