From dac52007a7a85ea2f179b1fb4f326fef6b816441 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Thu, 29 May 2025 00:50:47 -0400 Subject: [PATCH] Remove appendix --- paper.tex | 132 +++++++++++++++++++++++++++--------------------------- 1 file changed, 66 insertions(+), 66 deletions(-) diff --git a/paper.tex b/paper.tex index dd37e4e..3aa1bbc 100644 --- a/paper.tex +++ b/paper.tex @@ -262,9 +262,9 @@ the time dispensing water, is \cite[Section 14.3]{stewart_probability_2009}% ,% \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 -experimental data we can approximate all parameters and calculate -\todo{$W \approx 123$} (See the appendix for a complete derivation). +$\lambda$ 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 obtain +\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:% @@ -337,69 +337,69 @@ academic performance of KIT students: always pressing the right button. \printbibliography -\appendix - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Derivation of Service Time} -\label{sec:Derivation of Service Time} - - -We want to compute the response time of our queueing system, i.e., -\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*}% -We start by modelling the service time and subsequently calculate $\lambda$ -and $\rho$. - -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 -have -\begin{gather*} - S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm} - P_R(r) = \left\{\begin{array}{rl} - P(S_\text{L}), & r = r_{S_\text{L}} \\ - 1-P(S_\text{L}), & r = r_{S_\text{R}} - \end{array}\right. -\end{gather*}% -\begin{align*} - 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) - .% -\end{align*} -We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the -measured fill times and flow rates) to compute -\begin{align*} - \left. - \begin{array}{r} - 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} \\ - r^2_{S_\text{L}} \approx \todo{125} \\ - r^2_{S_\text{R}} \approx \todo{250} - \end{array} - \right\} \Rightarrow - \left\{ - \begin{array}{l} - E\mleft\{ S \mright\} \approx \todo{678} \\ - E\mleft\{ S^2 \mright\} \approx \todo{123} - \end{array} - \right. - .% -\end{align*} - -$\lambda$ is the mean arrival time. - -\todo{ - \textbf{TODOs:} - \begin{itemize} - \item Complete text describing / obtaining $\rho$ and $\lambda$ - \item Move model derivation to method section - \item Move calculations with model to results section - \item Add grade gain derivation - \item Idea: Make the whole thing 2 pages and print on A3 - \end{itemize} -} +% \appendix +% +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% \section{Derivation of Service Time} +% \label{sec:Derivation of Service Time} +% +% +% We want to compute the response time of our queueing system, i.e., +% \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*}% +% We start by modelling the service time and subsequently calculate $\lambda$ +% and $\rho$. +% +% 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 +% have +% \begin{gather*} +% S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm} +% P_R(r) = \left\{\begin{array}{rl} +% P(S_\text{L}), & r = r_{S_\text{L}} \\ +% 1-P(S_\text{L}), & r = r_{S_\text{R}} +% \end{array}\right. +% \end{gather*}% +% \begin{align*} +% 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) +% .% +% \end{align*} +% We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the +% measured fill times and flow rates) to compute +% \begin{align*} +% \left. +% \begin{array}{r} +% 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} \\ +% r^2_{S_\text{L}} \approx \todo{125} \\ +% r^2_{S_\text{R}} \approx \todo{250} +% \end{array} +% \right\} \Rightarrow +% \left\{ +% \begin{array}{l} +% E\mleft\{ S \mright\} \approx \todo{678} \\ +% E\mleft\{ S^2 \mright\} \approx \todo{123} +% \end{array} +% \right. +% .% +% \end{align*} +% +% $\lambda$ is the mean arrival time. +% +% \todo{ +% \textbf{TODOs:} +% \begin{itemize} +% \item Complete text describing / obtaining $\rho$ and $\lambda$ +% \item Move model derivation to method section +% \item Move calculations with model to results section +% \item Add grade gain derivation +% \item Idea: Make the whole thing 2 pages and print on A3 +% \end{itemize} +% } \end{document}