\documentclass[a4paper,12pt]{article} 

\usepackage[top = 2.5cm, bottom = 2.5cm, left = 2.5cm, right = 2.5cm]{geometry} 

% packages
\usepackage{amsmath, amsfonts, amsthm} % basic math packages
\usepackage{tikz} % for making illustrations
\usetikzlibrary{shapes.arrows, arrows, decorations.markings, positioning, shadings}
\usetikzlibrary{calc,cd}
\tikzset{
	node split radius/.initial=1,
	node split color 1/.initial=red,
	node split color 2/.initial=yellow,
	node split color 3/.initial=cyan,
	node split color 4/.initial=green,
	node split half/.style={node split={#1,#1+180}},
	node split quarter/.style={node split={#1,#1+90, #1+180, #1+270}},
	node split/.style args={#1,#2}{
		path picture={
			\tikzset{
				x=($(path picture bounding box.east)-(path picture bounding box.center)$),
				y=($(path picture bounding box.north)-(path picture bounding box.center)$),
				radius=\pgfkeysvalueof{/tikz/node split radius}}
			\foreach \ang[count=\iAng, remember=\ang as \prevAng (initially #1)] in {#2,360+#1}
			\fill[line join=round, draw, fill=\pgfkeysvalueof{/tikz/node split color \iAng}]
			(path picture bounding box.center)
			--++(\prevAng:\pgfkeysvalueof{/tikz/node split radius})
			arc[start angle=\prevAng, end angle=\ang] --cycle;
} } }
\usetikzlibrary{3d}
\usepackage{graphicx} % for importing images
\usepackage{xcolor} % more color options
\usepackage{colortbl}
\usepackage{multicol} % for making two-column lists
\usepackage{hyperref} % for hyperlinking
%\hypersetup{colorlinks=true, urlcolor=cyan,}
\usepackage{mathabx}
\usepackage{cleveref}
\usepackage{subfig}
\usepackage{array}
\usepackage{wrapfig}
\usepackage{bbm}
\usepackage{fancyhdr}
\usepackage{algorithm, algorithmicx, algpseudocode}
\usepackage{stmaryrd}
\usepackage{adjustbox}
\usepackage{physics}
\usepackage{rotating}


% The following two packages - multirow and booktabs - are needed to create nice looking tables.
\usepackage{multirow} % Multirow is for tables with multiple rows within one cell.
\usepackage{booktabs} % For even nicer tables.

% As we usually want to include some plots (.pdf files) we need a package for that.
\usepackage{graphicx} 
\usepackage{longtable}

% The default setting of LaTeX is to indent new paragraphs. This is useful for articles. But not really nice for homework problem sets. The following command sets the indent to 0.
\usepackage{setspace}
\setlength{\parindent}{0in}

% Package to place figures where you want them.
\usepackage{float}

% The fancyhdr package let's us create nice headers.
\usepackage{fancyhdr}

% theorems, lemmas, examples, etc.
\newtheorem{theorem}{Theorem}[section]
% \newtheorem{corollary}{Corollary}[theorem]
% \newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{solution}{Solution}

\def\mydefb#1{\expandafter\def\csname bf#1\endcsname{\mathbf{#1}}}
\def\mydefallb#1{\ifx#1\mydefallb\else\mydefb#1\expandafter\mydefallb\fi}
\mydefallb aAbBcCdDeEfFgGhHiIjJkKlLmMnNoOpPqQrRsStTuUvVwWxXyYzZ\mydefallb

\def\mydefb#1{\expandafter\def\csname cal#1\endcsname{\mathcal{#1}}}
\def\mydefallb#1{\ifx#1\mydefallb\else\mydefb#1\expandafter\mydefallb\fi}
\mydefallb aAbBcCdDeEfFgGhHiIjJkKlLmMnNoOpPqQrRsStTuUvVwWxXyYzZ\mydefallb

%% Change this to just the normal N,Z,R,C,P,E
\def\mydefb#1{\expandafter\def\csname bb#1\endcsname{\mathbb{#1}}}
\def\mydefallb#1{\ifx#1\mydefallb\else\mydefb#1\expandafter\mydefallb\fi}
\mydefallb CEGIKNPQRST\mydefallb

\newcommand{\half}{\frac{1}{2}}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\newcommand{\matlab}{\textsc{Matlab}}

\makeatletter
\newcommand{\thickhline}{%
	\noalign {\ifnum 0=`}\fi \hrule height 1pt
	\futurelet \reserved@a \@xhline
}
\newcolumntype{"}{@{\hskip\tabcolsep\vrule width 1pt\hskip\tabcolsep}}
\makeatother



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3. Header (and Footer)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% To make our document nice we want a header and number the pages in the footer.

\pagestyle{fancy} % With this command we can customize the header style.

\fancyhf{} % This makes sure we do not have other information in our header or footer.

\lhead{\footnotesize Math 190: Mathematics of Sustainability}% \lhead puts text in the top left corner. \footnotesize sets our font to a smaller size.

%\rhead works just like \lhead (you can also use \chead)
\rhead{\footnotesize  Project \#3} %<---- Fill in your lastnames.

% Similar commands work for the footer (\lfoot, \cfoot and \rfoot).
% We want to put our page number in the center.
\cfoot{\footnotesize \thepage} 

\begin{document}
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	% Title section of the document
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	% For the title section we want to reproduce the title section of the Problem Set and add your names.
	
	\thispagestyle{empty} % This command disables the header on the first page. 
	
	\begin{tabular}{p{15.5cm}} % This is a simple tabular environment to align your text nicely 
		{\large \sc Math 190:  Mathematics of Sustainability} \\
		Emory University \\ Spring 2026 \\ Prof. Scott \\
		\hline % \hline produces horizontal lines.
		\\
	\end{tabular} % Our tabular environment ends here.
	
	\vspace*{0.3cm} % Now we want to add some vertical space in between the line and our title.
	
	\begin{center} % Everything within the center environment is centered.
		{\Large \bf Project \# 3} % <---- Don't forget to put in the right number
		\vspace{2mm}
		
		% YOUR NAMES GO HERE
		{\bf Markov Chains in Public Transportation} % <---- Fill in your names here!
		
	\end{center}  
	
	\vspace{0.4cm}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	% Up until this point you only have to make minor changes for every week (Number of the homework). Your write up essentially starts here.
	
	
Markov Chains are all around us. In class, we looked at many environmental networks, complex graphs, physical states, etc. as Markov chains. However, for this assignment, we will merge environmentally friendly travel with Markov chains. \\


Each consulting group is hired by Metropolitan Atlanta Rapid Transit Authority (MARTA) to analyze their current train system and see if they can increase business by building an extension line to the MARTA system. Each consulting group will provide the below Markov Chain analysis using real world MARTA data, and present their findings in a 3-5 page perspective and 15 minute pitch to the MARTA Board of Trustees (your classmates in Math 190).

\begin{figure}
	\centering
	\begin{tikzpicture}[
		ry/.style={shape=circle, draw, minimum size=1cm,node split color 1=red, node split color 2=yellow, node split half=-45},
		bg/.style={shape=circle, draw, minimum size=1cm,node split color 1=green, node split color 2=cyan, node split half=-45},
		fP/.style={minimum size=1cm,
			node split color 1=red, node split color 2=yellow, node split color 3=cyan,  node split color 4=green, 
			node split={45,135,225,315}}
		]
		\node[ry] (airport) at (-0.5,-6) {\textbf{A}};
		\node[ry] (lakewood)       at (-1,-3.5) {\textbf{L}};
		\node[fP] (fivePoints) at (0,0) {\textbf{F}};
		\node[ry]   (lindberg) at (1,5)   {\textbf{G}};
		\node[circle, draw=black,fill=red,minimum size=1cm]        (northSpring)    at(1.5,10) {\textbf{N}};
		\node[circle, draw=black,fill=red,minimum size=1cm]        (windward)    at(3.5,12) {\textbf{W}};
		\node[circle, draw=black,fill=yellow, minimum size=1cm]        (doraville)       at(3.5,7.5) {\textbf{D}};
		\node[circle, draw=black,fill=yellow, minimum size=1cm]        (norcross)       at(5,9) {\textbf{C}};
		\node[circle, draw=black,fill=yellow, minimum size=1cm]        (grantPark)       at(2.5,-3.5) {\textbf{P}};
		
		%% East -West
		\node[circle, draw=black, fill=cyan, minimum size=1cm]       (holmes) at (-5,0)   {\textbf{H}};
		\node[bg]     (ashby) at (-2.5,0)   {\textbf{S}};
		\node[bg]      (reynoldstown) at (2.5,0)   {\textbf{R}};
		\node[circle, draw=black, fill=cyan, minimum size=1cm]       (avondale) at (5.0,0)   {\textbf{V}};
		\node[circle, draw=black, fill=cyan, minimum size=1cm]       (indianCreek) at (6.5,0)   {\textbf{I}};
		\node[circle, draw=black, fill=green, minimum size=1cm]       (bankhead) at (-4,1.5)   {\textbf{B}};
		\node[circle, draw=black, fill=green, minimum size=1cm]       (stoneMountain) at (8.5,2.5)   {\textbf{M}};
		\node[circle, draw=black, fill=green, minimum size=1cm]       (bolton) at (-5,4)   {\textbf{T}};
		
		\draw[transform canvas={xshift=-2.5pt}, red, very thick] (airport.north) -- (lakewood.south);
		\draw[transform canvas={xshift=+2.5pt}, yellow, very thick] (airport.north) -- (lakewood.south);
		\draw[transform canvas={xshift=-2pt}, red, very thick] (lakewood.north) -- (fivePoints.south);
		\draw[transform canvas={xshift=+2pt}, yellow, very thick] (lakewood.north) -- (fivePoints.south);
		\draw[transform canvas={xshift=-1.5pt}, red, very thick] (fivePoints.north) -- (lindberg.south);
		\draw[transform canvas={xshift=+1.5pt}, yellow, very thick] (fivePoints.north) -- (lindberg.south);
		\draw[red, very thick] (lindberg.north) -- (northSpring.south);
		\draw[red,  ultra thick,dashed] (northSpring.north east) -- (windward.south west);		
		\draw[yellow, very thick] (lindberg.north east) -- (doraville.south west);
		\draw[yellow, ultra thick,dashed] (doraville.north east) -- (norcross.south west);
		\draw[yellow, ultra thick,dashed] (lakewood.east) -- (grantPark.west);
		%% East-West
		\draw[cyan, very thick] (holmes.east) -- (ashby.west);
		\draw[transform canvas={yshift=+1.5pt}, green, very thick] (ashby.east) -- (fivePoints.west);
		\draw[transform canvas={yshift=-1.5pt}, cyan, very thick] (ashby.east) -- (fivePoints.west);
		\draw[transform canvas={yshift=+1.5pt}, green, very thick] (fivePoints.east) -- (reynoldstown.west);
		\draw[transform canvas={yshift=-1.5pt}, cyan, very thick] (fivePoints.east) -- (reynoldstown.west);
		\draw[transform canvas={yshift=-1.5pt}, cyan, very thick](reynoldstown.east) -- (avondale.west);
		\draw[transform canvas={yshift=+1.5pt}, green, ultra thick, dashed](reynoldstown.east) -- (avondale.west);
		\draw[cyan, very thick] (avondale.east) -- (indianCreek.west);
		\draw[green, very thick] (bankhead.south east) -- (ashby.north west);
		\draw[green, ultra thick, dashed](avondale.north east) -- (stoneMountain.south west);
		\draw[green, ultra thick, dashed](bankhead.north west) -- (bolton.south);
	\end{tikzpicture}
	\caption{Depiction of the simplified MARTA system we will use as our project.}
	\label{fig:marta}
\end{figure}
\begin{table}[!h]
	\centering
	\begin{tabular}{|cl|cl|}
		\hline
		Name& Station & Name  & Station  \\
		\hline\hline
		\textbf{A}& Airport &\textbf{S}  & Ashby  \\
		\textbf{L}&  Lakewood- Ft. McPherson&  \textbf{H}& Hamilton E. Holmes  \\
		\textbf{F}&  Five Points& \textbf{B} & Bankhead  \\
		\textbf{G}&  Lindbergh&  \textbf{R}& Inman Park - Reynoldstown  \\
		\textbf{N}& North Springs  & \textbf{V} & Avondale  \\
		\textbf{D}& Doraville &  \textbf{I}&  Indian Creek\\
		\textbf{C}& Chamblee &  \textbf{M}& Stone Mountain  \\
		\textbf{T}&  Bolton&\textbf{P}& Panthersville\\
		\textbf{W}&  Windward Pkwy & &\\
		\hline
	\end{tabular}
\end{table}
	

\section*{Assignment}
\begin{enumerate}
	\item 	{\bf Analyze the current MARTA train lines to figure out yearly revenue}
	\begin{enumerate}
		\item Turn the ridership data in Table \ref{tab:ridership} into a stochastic matrix. Then using \matlab's \texttt{dtmc} command to visualize the Markov Chain.
		\item Determine the stationary distribution for the data by using the \matlab \  Live Script notes, if it is possible, and explain what condition it violates if it's not possible.
		\item\label{ques:1c}	What is the stationary distribution of trains showing up on time based on the data in Table \ref{tab:onTime}? 
		\item  When the train is late, riders, many of whom have somewhere to get to, tend to leave the station, not paying for the ride. Below is the data:
		
		\begin{table}[!h]
			\centering
			\begin{tabular}{|c|c|}
				\hline
				Train Timing & \% of passengers leaving\\
				\hline
				\hline
				On-time & 0 \\
				3-5 min late & 0.5\%\\
				5-10 min late & 2\%\\
				10+ min late & 5\%\\
				\hline
			\end{tabular}
			\caption{Later trains tend to lead people to leave the station and find another way to where they are going.}
			\label{tab:upsetRider}
		\end{table}

		Using your answer in part \ref{ques:1c}, how much money is lost due to the trains being late? (Assume that everyone traveling pays the MARTA fare of \$2.50 for a one-way trip.)
	\end{enumerate}
	\item 	{\bf Analyze the proposed extension of MARTA train lines for your group to figure out projected yearly revenue}
	\begin{enumerate}
		\item Based on the new extension data, make a modified stochastic matrix, and create a visualization of your Markov Chain.
		\item Does the new MARTA system change the structure of the Markov Chain, i.e. is it periodic, connected, transient, etc?
		\item Determine the stationary distribution for the data, if possible, and explain what condition it violates if it's not possible. How many people will use the new station.
		\item Does this extension bring more people into the MARTA system, or will it just eat into its own profits? Use discussion from previous questions.
		\item Using the data above, does this extension make fiscal sense? What about when we add in environmental sense?
	\end{enumerate}
	\item {\bf Write a 3-5 page prospectus on your project with your detailed work included.}
	\item {\bf Give a \textbf{20 minute} presentation on your data. Treat it like a business meeting where your group has been asked to determine if the extension should happen. Does it generate profit, will it be work the hastle, etc?}
\end{enumerate}

\section*{Data}
\begin{table}[h!]
	\centering
	\begin{tabular}{|c||cccc|}
		\hline
		Train Timeliness& On-time & 2-5 minutes  & 5-10 minutes & 10+ minutes \\
		\hline\hline
		On-time& 0.7 &0.24	&0.06	&	0\\
		2-5 minutes &0.09 &0.72 & 0.12 &0.01 \\
		5-10 minutes &0.005 &0.24 & 0.65&0.105 \\
		10+ minutes &0 & 0 & 0.15 &0.85 \\
		\hline
	\end{tabular}
	\caption{Data for on-time MARTA trains from 9-27-2024.}
	\label{tab:onTime}
\end{table}

\begin{sidewaystable}[!h]
	\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}	
		\hline
		Start $\backslash$ Stop& \textbf{A} & \textbf{L}  & \textbf{F} & \textbf{G} & \textbf{N} &\textbf{D} & \textbf{S} & \textbf{H} & \textbf{B} & \textbf{R} & \textbf{V} &\textbf{I}  \\
		\hline\hline
		\textbf{A} & & 1581918 & & & & & & & & &&\\\hline
		\textbf{L} & 66810 & & 179432 & & & & & && & &\\\hline
		\textbf{F} & & 311814 & & 469009 & & &  113054 &  && 214147 && \\\hline
		\textbf{G} & & & 836490 & & 121884&109804 & & & & & &\\\hline
		\textbf{N} & & & & 606238 & & & & & & & &\\\hline
		\textbf{D} & & & & 688018 && & & & & & &\\\hline
		\textbf{S} && & 75613 & & &  & & 11097 & 2607 && & \\\hline
		\textbf{H} & & &  & & & & 553278 & & & & &\\\hline
		\textbf{B} & & & & & & & 90242 & & & &  &\\\hline
		\textbf{R}& & & 247961 & & & & & & & & 59526& \\\hline
		\textbf{V} & & & & & & & & & & 280633& &43390 \\\hline
		\textbf{I} &&&&&&& & && & 431706 & \\\hline
		
	\end{tabular}
	\caption{2024 MARTA ridership data}
	\label{tab:ridership}
\end{sidewaystable}






\section*{Acknowledgements}
The data presented in this project is based on 2024 real ridership data provided by MARTA. Special thanks to Ryan VanSickle who provided the ridership and station data, and Paul Lopes, who provided the on-time train data.

\newpage
\section*{Rubric}
The rubric is a direct reflection of the learning objective for this course:
\begin{itemize}
	\item Breaking down problems into smaller, tractable steps
	\item Interpreting and analyzing real world data
	\item Communicating technical details to a general audience
	\item Working on a project with people in an equitable and responsible fashion.
\end{itemize}

The last learning objective will be measured by your peers on how the work was split, performed, and communicated.

\begin{table}
	\centering
	\begin{tabular}{|c|p{.2\textwidth}p{.2\textwidth} p{.2\textwidth}|c|}
		\hline
		\textbf{Category}& \textbf{0: Improving} & \textbf{1: Good} & \textbf{2: Excellent} & \textbf{Total} \\
		\hline\hline
		Communication & Communicating with this student was difficult. It was hard to get ahold of them, and when talking to them, it was sparse and disconnected. & Communicating with this student was overall good. They were somewhat repsonsive to messaging about meeting. They were receptive to chatting. & This student was great about communicating where they were, what they needed to do, and helped make the project succeed. &\\
		\hline
		Content & This student doesn't have mastery of the content, and it slowed the project down & This student has a good grasp on the content and made accurate and correct contributions to the project & This student has mastery of the material. They clearly know what is going on, and everything they added to the project was correct and beneficial &\\\hline
		Workload&  This student contributed very little to the content of this project. They did not pull their fair share.& This student overall did their fair share, but it did require check-ins and reminders. & This student did their fair share or more. They were willing to add more content and go above and beyond.& \\ 
		\hline
		Presentation &  This student did a poor job presenting the material, either from reading off the slides, not knowing the material, etc.& This student presented well. They knew what they were talking about and were engaging. & This student showed mastery in presenting. They laid out a wonderful narrative arc and were very engaging.& \\
		\hline
		Attitude & This student had a poor attitude about the project. It made working with them uncomfortable.  & This student had a positive attitude and made this project fun.  & This student had a wonderful attitude. They were really excited, interested, and motivated to help. &\\
		\hline
		\textbf{\textit{Peer Review Points}} & & & & \\\hline
	\end{tabular}
\end{table}

Below is the rubric that the professor will use to evaluate.

\begin{longtable}{| p{.80\textwidth} | p{.10\textwidth} |} 
	\hline
	\textbf{Presentation} & \textbf{Points}\\
	\hline\hline
	\textit{Delivery} (13 points) & \\ \hline
	Topical Knowledge (10 points) & \\\hline
	Time Limit (3 points) & \\\hline
	\textit{Content} (12 points) & \\ \hline
	Titles/ Headings (2 points) & \\ \hline
	Informative (7 points) & \\ \hline
	Cohesive Story (3 points) & \\\hline
	\textbf{\textit{Total Presentation Points}} & \\\hline
	\hline
	\textbf{Report} & \textbf{Points}\\
	\hline\hline
	\textit{Content} (50 points)& \\	\hline
	Stochastic Matrix (4 points) & \\	\hline
	Stationary Distribution - Passengers (4 points) & \\	\hline
	Stationary Distribution - Trains (4 points) & \\	\hline
	Revenue Computation (10 points) & \\	\hline
	Extension Stochastic Matrix (6 points) & \\	\hline
	Extension Stationary Distribution - Passengers (4 points) & \\	\hline
	Revenue Computation (18 points) & \\	\hline
	\textit{Clarity} (10 points) & \\	\hline
	Technical Communication (5 points) & \\	\hline
	Supporting Data (5 points) & \\	\hline
	\textit{Formatting} (5 points) & \\ 	\hline
	Font (1 point) & \\\hline
	Executive Summary (1 point) & \\ 	\hline
	Page Number (2 points) & \\ 	\hline
	Correct Margins (1 point) & \\ \hline
	\textbf{\textit{Total Report Points}} & \\\hline

	\textbf{Presentation} (25 points) & \\
	\textbf{Report} (65 points) & \\
	\textbf{Peer Review} (10 points) & \\
	\textbf{\textit{TOTAL PROJECT POINTS}} & \\\hline
\end{longtable}
	
\end{document}
