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@ -262,9 +262,9 @@ the time dispensing water, is \cite[Section 14.3]{stewart_probability_2009}%
,% ,%
\end{align*}% \end{align*}%
where $S$ denotes the service time (i.e., the time spent refilling a bottle), 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 % We examine the effects of the choice of hydration strategy. To
% this end, we start by estimating the potential time savings possible by always % this end, we start by estimating the potential time savings possible by always
% choosing the fastest strategy:% % choosing the fastest strategy:%
@ -337,69 +337,69 @@ academic performance of KIT students: always pressing the right button.
\printbibliography \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} \end{document}