paper.tex

@@ -244,10 +244,11 @@ 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 $\SI{673.92}{\milli\liter}$.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -255,16 +256,16 @@ Fig. \ref{fig:Behavior} shows the results of the behavioral measurement.
 
 We can consider the water dispenser and students as comprising a queueing
 system, specifically an M/G/1 queue \cite{stewart_probability_2009}.
-The expected response time, i.e., the time spent waiting as well as
+The 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 $\rho = \lambda \cdot E\mleft\{ S \mright\}$ the system utilization. Using our
+$\lambda$ the mean arrival time, and $\rho$ the system utilization. Using our
-experimental data we can approximate all parameters and obtain
+experimental data we can approximate all parameters and calculate
-\todo{$W \approx 123$}.
+\todo{$W \approx 123$} (See the appendix for a complete derivation).
 % 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:%
@@ -337,69 +338,69 @@ 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}