Add proper queuing theory based model; Add TODOs

paper.bib

@@ -221,3 +221,32 @@ Publisher: Select Press},
 	pages = {19--38},
 	file = {JSTOR Full Text PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/H58YHZEJ/Rau and Durand - 2000 - The Academic Ethic and College Grades Does Hard Work Help Students to Make the Grade.pdf:application/pdf},
 }
+
+@book{gross_fundamentals_2008,
+	address = {Hoboken, NJ},
+	edition = {4. ed},
+	series = {Wiley series in probability and statistics},
+	title = {Fundamentals of queueing theory},
+	isbn = {978-0-471-79127-0},
+	language = {eng},
+	publisher = {Wiley},
+	author = {Gross, Donald and Shortle, John F. and Thompson, James M. and Harris, Carl M.},
+	year = {2008},
+	keywords = {/unread, Queuing theory, Warteschlangentheorie},
+	file = {Ebook:/home/andreas/workspace/work/hiwi/Zotero/storage/ZGBENDHH/Gross et al. - 2008 - Fundamentals of queueing theory.epub:application/epub+zip},
+}
+
+@book{stewart_probability_2009,
+	title = {Probability, {Markov} {Chains}, {Queues}, and {Simulation}: {The} {Mathematical} {Basis} of {Performance} {Modeling}},
+	isbn = {978-1-4008-3281-1},
+	shorttitle = {Probability, {Markov} {Chains}, {Queues}, and {Simulation}},
+	abstract = {Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics.The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation.Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available (to professors only).Numerous examples illuminate the mathematical theoriesCarefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approachEach chapter concludes with an extensive set of exercises},
+	language = {en},
+	publisher = {Princeton University Press},
+	author = {Stewart, William J.},
+	month = jul,
+	year = {2009},
+	note = {Google-Books-ID: ZfRyBS1WbAQC},
+	keywords = {Mathematics / Applied, /unread, Computers / Data Science / Data Modeling \& Design, Mathematics / Probability \& Statistics / General, Technology \& Engineering / Engineering (General)},
+	file = {PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/L2FEI8HG/Stewart - 2009 - Probability, Markov Chains, Queues, and Simulation The Mathematical Basis of Performance Modeling.pdf:application/pdf},
+}

paper.tex

@@ -140,7 +140,7 @@ into the productivity of their employees that found that ``workers with the
 most patents often shared lunch or breakfast with a Bell Labs electrical
 engineer named Harry Nyquist'' \cite{gertner_idea_2012}, and we presume that
 they also paired their food with something to drink. We can see that
-intellectual achievement and hydration are related even for the most
+intellectual achievement and fluid consumption are related even for the most
 prestigious research institutions.
 
 In this work, we quantify this relationship in the context of studying at the
@@ -159,10 +159,13 @@ of the choice of hydration strategy of the participants: $S_\text{L}$ denotes
 pressing the left button of the water dispenser, $S_\text{R}$ the right one,
 and $S_\text{B}$ pressing both buttons.
 
-As is always the case with measurements, care must be taken not to alter
-quantities by measuring them. To this end, we made sure only to take system
-measurements in the absence of participants and to only record data on the
-behaviour of participants discreetly.
+For the system measurement $10$ datapoints were recorded for each strategy,
+for the behavioral measurement $113$ in total.
+
+% As is always the case with measurements, care must be taken not to alter
+% quantities by measuring them. To this end, we made sure only to take system
+% measurements in the absence of participants and to only record data on the
+% behaviour of participants discreetly.
 
 % TODO: Describe the actual measurement setup? (e.g., filling up a 0.7l bottle
 % and timing with a standard smartphone timer)
@@ -177,8 +180,8 @@ behaviour of participants discreetly.
 
     \begin{tikzpicture}
         \begin{axis}[
-            width=0.85\columnwidth,
-            height=0.4\columnwidth,
+            width=0.8\columnwidth,
+            height=0.35\columnwidth,
             boxplot/draw direction = x,
             grid,
             ytick = {1, 2, 3},
@@ -199,11 +202,13 @@ behaviour of participants discreetly.
         \end{axis}
     \end{tikzpicture}
 
-    \caption{Flow rate of the water dispenser depending on the button pressed.}
+    \vspace*{-3mm}
+
+    \caption{Flow rate of the water dispenser depending on the hydration strategy.}
     \label{fig:System}
 \end{figure}
 
-\begin{figure}[H]
+\begin{figure}
     \centering
 
     \begin{tikzpicture}
@@ -211,57 +216,92 @@ behaviour of participants discreetly.
             ybar,
             bar width=15mm,
             width=\columnwidth,
-            height=0.4\columnwidth,
+            height=0.35\columnwidth,
             area style,
             xtick = {0, 1, 2},
             grid,
             ymin = 0,
             enlarge x limits=0.3,
-            xticklabels = {Left button, Right button, Both buttons},
-            ylabel = {No. of presses},
+            xticklabels = {\footnotesize{$S_\text{L}$ (Left button)}, \footnotesize{$S_\text{R}$  (Right button)}, \footnotesize{$S_\text{B}$}  (Both buttons)},
+            ylabel = {No. chosen},
         ]
             \addplot+[ybar,mark=no,fill=scol1] table[skip first n=1, col sep=comma, x=button, y=count]
                 {res/left_right_distribution.csv};
         \end{axis}
     \end{tikzpicture}
 
+    \vspace*{-3mm}
+
     \caption{Distribution of the choice of hydration strategy.}
     \label{fig:Behavior}
 \end{figure}
 
-Fig. \ref{fig:System} indicates that $S_\text{L}$ is the slowest
-strategy, while $S_\text{R}$ and $S_\text{B}$ are similar.
-Due to the small sample size ($N=10$) and the 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:System} shows the results of the system measurement.
+We observe that $S_\text{L}$ is the slowest strategy, while $S_\text{R}$
+and $S_\text{B}$ are similar. Due to the small sample size and the
+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
+needed to fill up their bottle. Using the measured flowrates we calculated
+the mean refill volume to be $\SI{673.92}{\milli\liter}$.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Discussion}
 
-
-We examine the effects of the choice of hydration strategy. To
-this end, we first estimate the amount of time saved by choosing a certain
-strategy and relate that to a possible gain in academic performance, i.e.,
-grades.%
-%
-\todo{
-\begin{itemize}
-    \item ``We measured the average bottle size''
-    \item Quantify relationship: Compute average time saving by using right
-        button $\rightarrow$ translate into grade gain
-    \item People using the left button slow down the entire queue
-        behind them, not only themselves
-\end{itemize}
-}%
+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 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
+experimental data we can approximate all parameters and calculate
+\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:%
+% %
+% % We can model the time needed for one person to refill their bottle as a random
+% % variable (RV) $T_1 = V/R$ and the time saved by choosing the fastest strategy
+% % as $\Delta T_1 = T_1 - V/\max r$, where $V$ and $R$ are RVs representing the
+% % bottle volume and flowrate. The potential time saving for the last person in a
+% % queue of $N$ people is thus $\Delta T_N = N\cdot\Delta T_1$. We can then model
+% % the total time savings as $\Delta T_\text{tot} = \sum_{n=1}^{N} \Delta T_n$,
+% % where N is an RV describing the queue length. Assuming the independence of all
+% % RVs we can compute the mean total time savings as
+% %
+% \begin{gather*}
+%     T_1 = V/R, \hspace{3mm} \Delta T_1 = T_1 - V/\max R, \hspace{3mm} \Delta T_n = n \cdot \Delta T_1 \\
+%     \Delta T_\text{tot} = \sum_{n=1}^{N} \Delta T_\text{n} = \sum_{n=1}^{N} n \cdot \Delta T_1 = \Delta T_1 \frac{N\mleft( N+1 \mright)}{2} \\
+%     \Delta t := E\mleft\{ \Delta T_\text{tot} \mright\} = E\mleft\{ \Delta T_1 \mright\} \cdot \mleft[ E\mleft\{ N^2 \mright\} + E\mleft\{ N \mright\} \mright]/2
+%     ,%
+% \end{gather*}
+% %
+% where $V$ and $R$ are random variables (RVs) representing the volume of a
+% bottle and the flowrate, $\Delta T_n$ describes the time the last of $n$
+% people saves, $\Delta T_\text{tot}$ the total time savings and $N$ the length
+% of the queue. It is plausible to assume independence of $R,V$ and $N$.
+% Using our experimental measurements we estimate $\todo{\Delta t = \SI{20}{\second}}$
 
 Many attempts have been made in the literature to relate the time spent
 studying to academic achievement -  see, e.g.
 \cite{schuman_effort_1985, zulauf_use_1999, michaels_academic_1989, dickinson_effect_1990}.
 The overwhelming consensus is that there is a significant relationship,
 though it is a weak one.
+%
+\todo{
+\begin{itemize}
+    \item Compute possible grade gain
+\end{itemize}}
 %Many of the studies were only performed over
 % a period of one week or even day, so we believe care should be taken when
 % generlizing these results. Nevertheless, the overwhelming consensus in the
@@ -273,20 +313,20 @@ though it is a weak one.
 
 
 In this study, we investigated how the choice of hydration strategy affects
-the average academic performance of a student. We found that always choosing to
+the average academic performance. We found that always choosing to
 press the right button leads to an average time gain of \todo{\SI{10}{\second}}
 per day, which translates into a grade improvement of $\todo{0.001}$ levels.
 We thus propose a novel and broadly applicable strategy to boost the average
 academic performance of KIT students: always pressing the right button.
 
-Further research is needed to develop a better model of how the choice of
-hydration strategy is related to academic performance. We
-suspect that there is a compounding effect that leads to $S_\text{L}$ being an
-even worse choice of hydration strategy: When the queue is long, students are
-less likely to refill their empty water bottles, leading to reduced mental
-ability. Nevertheless, we believe that with this work we have laid a solid
-foundation and hope that our results will find widespread acceptance among the
-local student population.
+% Further research is needed to develop a better model of how the choice of
+% hydration strategy is related to academic performance. We
+% suspect that there is a compounding effect that leads to $S_\text{L}$ being an
+% even worse choice of hydration strategy: When the queue is long, students are
+% less likely to refill their empty water bottles, leading to reduced mental
+% ability. Nevertheless, we believe that with this work we have laid a solid
+% foundation and hope that our results will find widespread acceptance among the
+% local student population.
 
 
 %
@@ -298,6 +338,70 @@ local student population.
 
 \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}
+}
+
 
 \end{document}
 

res/full_participant_measurement.csv

@@ -0,0 +1,114 @@
+time,button
+28,left
+22,left
+17,left
+40,left
+24,left
+41,left
+11,left
+11,left
+26.56,left
+37,left
+30,left
+30,left
+8,left
+21,left
+20,left
+19,left
+28,left
+20,left
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+16.43,left
+16,left
+29,left
+20,left
+24,left
+22,left
+15,left
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+40,left
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+35.38,left
+21,left
+16.3,left
+29.3,left
+30.3,left
+30.2,left
+25,left
+14,left
+14.1,left
+40,left
+24.4,left
+5.2,left
+50,left
+29.7,left
+39,left
+17,left
+40.7,left
+27.3,left
+19.8,left
+7.55,right
+14,right
+9,right
+13,right
+5,right
+13,right
+13.58,right
+15.58,right
+25,right
+20,right
+14,right
+13,right
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+13.3,right
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+13,right
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+15,right
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+22.9,right
+10,right
+20,right
+12,right
+19,right
+39.8,right
+20,both
+20,both
+15,both
+19,both
+13,both
+7,both
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+17.3,both
+12,both
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+12,both
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+12.35,both

scripts/calculate_mean_bottle_size.py

@@ -0,0 +1,45 @@
+import numpy as np
+import pandas as pd
+
+
+filename_participants = "res/full_participant_measurement.csv"
+
+filename_left = "res/flowrate_left.csv"
+filename_right = "res/flowrate_right.csv"
+filename_both = "res/flowrate_both.csv"
+
+
+def main():
+    # Get bottle fillup times
+
+    df_part = pd.read_csv(filename_participants)
+
+    times_left = np.array(df_part[df_part["button"] == "left"]["time"])
+    times_right = np.array(df_part[df_part["button"] == "right"]["time"])
+    times_both = np.array(df_part[df_part["button"] == "both"]["time"])
+
+    # Get mean flowrates
+
+    df_left = pd.read_csv(filename_left)
+    df_right = pd.read_csv(filename_right)
+    df_both = pd.read_csv(filename_both)
+
+    flowrate_left = np.mean(np.array(df_left["flowrate"]))
+    flowrate_right = np.mean(np.array(df_right["flowrate"]))
+    flowrate_both = np.mean(np.array(df_both["flowrate"]))
+
+    # Calculate mean bottle size
+
+    sizes_left = times_left * flowrate_left
+    sizes_right = times_right * flowrate_right
+    sizes_both = times_both * flowrate_both
+
+    sizes = np.concatenate([sizes_left, sizes_right, sizes_both])
+
+    mean_size = np.mean(sizes)
+
+    print(f"Mean bottle size: {mean_size}")
+
+
+if __name__ == "__main__":
+    main()

scripts/perform_hypothesis_tests.py

@@ -16,7 +16,7 @@ def main():
     flowrate_right = np.array(df_right["flowrate"])
 
     df_both = pd.read_csv(filename_both)
-    flowrate_both = np.array(df_right["flowrate"])
+    flowrate_both = np.array(df_both["flowrate"])
     
     U_lr, p_lr = mannwhitneyu(flowrate_left, flowrate_both, method="exact")
     U_rb, p_rb = mannwhitneyu(flowrate_right, flowrate_both, method="exact")