Headway statistics of public transport in Mexican cities
Milan Krbalek ${ }^{1,3}$ and Petr Seba ${ }^{2,3}$
${ }^1$ \ Department of Mathematics
Faculty of Nuclear Sciences
and Physical Engineering
Trojanova 13
Prague
Czech Republic
${ }^2$ \ Department of Physics
V\'{\i}ta Nejedl\'eho 573
Hradec Kr\'alov\'e
Czech Academy of Sciences
Cukrovarnick\'a 10
We present a cellular automaton simulating the behavior of public bus transport in several Mexican cities. The headway statistics obtained from the model is compared to the measured time intervals between subsequent bus arrivals to a given bus stop and to a spacing distribution resulting from a random matrix theory. \vspace*{3mm} \noindent KEYWORDS: transport, cellular automata, random matrix theory\\ %\pacs{PACS Nos: *************************}
Article.tex
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\include{incpic}
\emptyplace{3.3in \special{psfile="Figure1.eps" hscale=50 vscale=50 hoffset=-30
voffset=-300}} {3.1in} {\footnotesize \noindent FIG.1. Time headway of the bus
transport in Puebla.\\ The curve represents the Poisson distribution
(\ref{Poisson}). Plus signs display the time headway distribution of buses in
Puebla and bars are taken from the CA model with $a=0.$ }

\emptyplace{3.3in \special{psfile="Figure2.eps" hscale=50 vscale=50 hoffset=-30
voffset=-300}} {3.1in} {\footnotesize \noindent FIG.2. Time headway of
 bus transport in Cuernavaca and Mexico City.\\ The curve represents
the Wigner formula (\ref{Wigner}.) Plus signs display the time headway
distribution of buses in Cuernavaca and bars represent the results of the CA
model with $a=1/36.$ The results obtained for buses in Mexico City are
presented on the inset.}

\emptyplace{3.3in \special{psfile="Figure3.eps" hscale=50 vscale=50 hoffset=-30
voffset=-300}} {3.1in} {\footnotesize \noindent FIG.3. Number variance of the
public transport in some Mexican cities.\\ Curve and line represent the
prediction of the RMT (\ref{NV_GUE}) and (\ref{NV_Poisson}), respectively. Plus
signs and stars display number variance obtained from the public bus transport
in Cuernavaca and Mexico City respectively. Circles are the results of the CA
model for $a=1/36.$ The number variance obtained from the transport in  Puebla
is displayed on the insert (plus signs).The circles show the result of the CA
model obtained for $a=0.$}

\title{Headway statistics of public transport in Mexican cities}

\author{Milan Krbalek ${ }^{1,3}$ and  Petr Seba ${ }^{2,3}$\\
${ }^1$ \ Department of Mathematics, Faculty of Nuclear Sciences\\
and Physical Engineering,\\ Trojanova 13, Prague, Czech Republic\\
${ }^2$ \ Department of Physics, University  of Hradec
Kr\'alov\'e,\\ V\'{\i}ta Nejedl\'eho 573, Hradec Kr\'alov\'e,
Czech Republic\\ ${ }^3$ \ Institute of Physics, Czech Academy of
Sciences,\\ Cukrovarnick\'a 10, Prague, Czech Republic\\}

\maketitle

\begin{abstract}

We present a cellular automaton simulating the behavior of public
bus transport in several Mexican cities. The headway statistics
obtained from the model is compared to the measured time intervals
between subsequent bus arrivals to a given bus stop and to a
spacing distribution resulting from a random matrix theory.

\vspace*{3mm}
 \noindent KEYWORDS: transport, cellular automata, random matrix theory\\
%\pacs{PACS Nos: *************************}
\end{abstract}

%------------------------------------------------------------------------

The public transport in Mexico is organized differently to that
known in Europe. First, no leading companies are responsible for
the city transport. Thus, there are no timetables for the city
buses and sometimes even not well defined bus stops. Moreover, the
driver is usually the owner of the bus and so his aim is to
maximize the income. Since every passenger entering the bus has to
pay, the driver tries to collect the largest possible number of
passengers. When not regulated the time interval under which two
subsequent buses pass a given point will display a Poissonian
distribution. This is a consequence of the absence of
correlations between the motion of different buses. Such situation
is, however, not welcomed by the drivers since then the
probability density that two buses arrive to a bus stop within
short time interval is large. In this case the first bus collects
all waiting passengers  and the second one that arrives bit later
will find the stop practically empty. This simple reasoning makes
clear that  existence of certain correlations between buses, that
will change the Poisson process, will be of favor. Indeed, in
Mexico various strategies have been developed to create such
correlations. Here we discuss the situation in three cities:
Cuernavaca, Puebla and Mexico City.\\

In Puebla there is not an additive mechanism that helps to
increase the bus correlation. Hence the headway statistics of
the buses in this city should be close to Poissonian. In Cuernavaca in
turn, the information about the times, when the buses pass certain
points, are notified and then sold to the bus drivers (there is
a real market with this information). The driver can in such a way
change the velocity of the bus in dependence on the position of
the bus in front of him and a bus-bus correlation appears. \\

To describe the correlations we use a modified version of the
celebrated Nagel-Schreckenberg cellular automaton (see
\cite{Helbing} and \cite{Chowdhury}). Consider $N$ equal cells on
the line (for our purpose we define the length of  one cell as 30
meters) and $n$ particles (buses) moving along it. Thus,
$c=\frac{n}{N}$ is the bus density and $\bar{d}=c^{-1}$ is the
mean distance between two neighboring buses. Furthermore, define
the maximal velocity $v_{max}$ of the bus and its probability $p$
to slow down $p \in [0,1].$ The Nagel-Schreckenberg model
describes the dynamics of the system with the help of the
following update rules. In time $t=0$ the positions $x_i$ of the
particle $i=1\ldots n$ are integer numbers randomly chosen from
the set $1\ldots N$ satisfying the condition $x_{i-1}>x_i$ for
every $i=2\ldots n.$ Furthermore, in time $t=0$ the velocity $v_i$
of the $i$-th particle is set to zero for  all $i:$ $v_i(0)=0$.
The buses start to move with velocity $v$ according to the update
rules that has to be applied simultaneously  to all particles.

\begin{itemize}
\item \underline{Step one:} When velocity of the bus  is smaller than a maximal velocity $v_{max},$
it  increases its velocity by one

\begin{equation}
v_i(t+1)=\min \left \{v_i(t)+1;v_{max} \right\} \label{rule1}
\end{equation}

\item \underline{Step two:} Particles with positive velocities are
randomly slowed down

\begin{equation}
v_i(t+1)=v_i(t+1)-1 \label{rule2}
\end{equation}

 with probability $p$

\item \underline{Step three:} Particles update their positions
according to

\begin{equation}
x_i(t+1)=\min \left \{x_i(t)+v_i(t+1);x_{i-1}(t+1)-1 \right \} \label{rule3},
\end{equation}

i.e., the particles move according to the rule $
x_i(t+1)=x_i(t)+v_i(t+1)$ with the restriction that they cannot
occupy the same cell or overtake each other. In that case the
particle hops to the cell behind the occupied one.

\end{itemize}

To create  a modification of the model and to adapt it to the
situation in Mexico we change the update rules on a subset $M$ of possible bus positions
where the information is passed to the driver. (The density of those points will be denoted
by $a$; $a=M/N$.)


To change the model we add to the steps
(\ref{rule1}),(\ref{rule2}),(\ref{rule3})
 an additive step (\ref{rule4}) that takes into account the processing of the information :



At the points $j \in M$ the information about the time interval
$\Delta t$ to a preceding bus, which passed that point, is
available to the driver. Using it the driver can modify the bus
velocity - to speed up if  $\Delta t>\bar{t}$ or to slows down for
$\Delta t <\bar{t}$. (Recall that $\bar{t}$ is the mean time
interval between subsequent buses.) Hence we change the model by
adding

\begin{itemize}
\item{\underline{Step four:}}

$$ v_j(t)=v_j(t) +1 $$

if $ t_i(j)- t_{i-1}(j)< \bar{t}$
\begin{equation}
v_j(t)=v_j(t) -1 \label{rule4}
\end{equation}
if $ t_i(j)- t_{i-1}(j)> \bar{t}$ where $ t_i(j)$ denotes the time when a bus
$i$ passed through the point $j$: $x_i(t_i(j))=j$.

\end{itemize}

Using the modified  model we focus on the $headway$ $statistics,$
i.e. on the probability distribution of the time intervals $\Delta
t$  between two subsequent buses passing a given point and compare
it with the results obtained in the cities. For the simulation of
the transport in Puebla we choose $c=1/40,\,v_{max}=2,p=0.5,$ and
$a=0$ taking into account the fact that there are no checking
points. It is not surprising that the headway statistics is in
this case  close to the Poisson distribution (see Figure 1)

\begin{equation}
P(t)=e^{-t} \label{Poisson},
\end{equation} when $t$ is the spacing re-scaled to the mean distance equal
1. Besides the headway statistics, we compare also a $number$
$variance$ $\Sigma^2(t)$ that is defined as

\begin{equation}
\Sigma^2(t)=\langle \left( n(t)-t \right)^2 \rangle \label{NV-definition},
\end{equation}

where $n(t)$ is the number of bus arrivals to a given point during
the time period of the length $t.$ Note that $\langle n(t)
\rangle=t$ due to the fact that $\langle t \rangle = 1.$ It can
easily be checked that for a Poissonian process

\begin{equation}
\Sigma^2(t)=t \label{NV_Poisson}.
\end{equation}

\inspicture

The number variance obtained from the data and from the simulation fits quite
well with this prediction (see Figure 3). The data show, however, a small
deviations from  (\ref{NV_Poisson}). This is a manifestation of a weak
interaction between the buses which probably originates from the fact that the buses
interact through the number of passengers waiting on the stops.
Namely when the distance between the buses is
large, more passengers are waiting in the stop and the delay of bus in the
stopping-place is longer.\\

To simulate the situation in Cuernavaca and in Mexico City we use parameters
$c=1/40,\,v_{max}=2,p=0.5,\,a=1/36,$ which represents one bus per 1.2
kilometers and one check point per one kilometer approximately. The modified
cellular automaton leads in this case to significant changes in the time
interval distribution (see Figure 2).  The distribution obtained from
the automaton fits well the observed time interval distribution. Moreover both distributions
conform well with the distribution (see Ref. \cite{Seba})

\begin{equation}
P(t)=\frac{32}{\pi^2}t^2 e^{\frac{4}{\pi}t^2} \label{Wigner},
\end{equation}

that describes  the spacing distribution of a Gaussian unitary ensemble of
random matrices (GUE). (It is known that this function describes also  the
distance distribution of certain one-dimensional interacting gases (see Ref.
\cite{Scharf}).

Similar agreement is observed also for the number variance (see Figure 3) where
GUE leads to


\begin{equation}
\Sigma^2(t)=\frac{1}{\pi^2}(\ln 2\pi t + \gamma +1) \label{NV_GUE}
\end{equation}

with $\gamma\approx0.57721566$ (see \cite{Mehta}).

\inspicture

However, as evident on the Figure 3, the interaction between the
buses in Cuernavaca and in Mexico City is stronger than that
resulting from  the cellular model and leads to the stronger
correlations. Whereas in the automaton model the correlations
exist between the nearest neighbors only, in Cuernavaca and Mexico
City one can recognize that interaction exists also between the
first, second and third neighbor. \\

We conclude that the modified Nagel-Schreckenberg cellular
automaton successfully describes microscopic properties of the bus
transport in some Mexican cities.  The velocity of the buses is
influenced by the information about their mutual positions so that
the drivers can optimize their rank in competing on the passengers
to be transported. This finally increase the coordination of the
bus motion and changes the time headway statistics.\\

Acknowledgement: This work has been supported by the grant
A1048101 and GACR 202/02/0088 of the Czech grant agency. The data
in the Mexican cities were collected with the friendly help of
Dr.Markus Mueller from the University in Cuernavaca and Dr.
Antonio Mendes-Bermudes from the University of Puebla. Authors are
also very grateful to Patrik Kraus from the University of Hradec
Kr\'alov\'e for entering the collected data into the computer.

\inspicture

\begin{thebibliography}{100}

\bibitem{Helbing}
D.~Helbing, Traffic and related self-driven many-particle systems,
Rev. Mod. Phys. {\bf 73} (2001), 1067 -- 1141

\bibitem{Chowdhury}
K.~Ghost,A.~Majumdar, and D.~Chowdhury, Distribution of time-headways in a
particle-hopping model of vehicular traffic, Phys. Rev. E {\bf 58} (1998), 4012
-- 4015

\bibitem{Seba}
P.~Seba and M.~Krbalek: Statistical properties of the city transport in
Cuernavaca (Mexico) and random matrix ensembles, J. Phys. A {\bf 33} (2000),
L229 -- L234.

%\bibitem{Yokoya}
%J.~Yokoya: Independence between route decision and traffic flow
%through the medium position of dynamics information, najdu pak

\bibitem{Mehta}
K.~Mehta: Random Matrices. Academic Press, New York 1991

\bibitem{Scharf}
R.~Scharf, F.M.~Izrailev: Dyson's Coulomb gas on a circle and intermediate
eigenvalue statistics, J.Phys.A {\bf 23} (1990) 963-977

\bibitem{Wagner}
M.~Krbalek, P.~Seba, and P.~Wagner: Headways in traffic flow -
remarks from a physical perspective, Phys. Rev. E. {\bf 64}
(2001), 066119 -- 066125

\end{thebibliography}


\end{document}

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      }%
      \ifvt@p
       \setbox\p@ctbox=\vtop{\unvbox\p@ctbox}%
      \else
        \ifvb@t\else
          \@uxdimen=\ht\p@ctbox
          \advance\@uxdimen by -\p@ctht
          {\vfuzz=\maxdimen
           \global\setbox\p@ctbox=\vbox to\p@ctht{\unvbox\p@ctbox}%
          }%
          \dp\p@ctbox=\@uxdimen
        \fi
      \fi
      }
\def\picname#1{\unskip\setbox\@uxbox=\hbox{\bf\ignorespaces#1\unskip\ }%
      \hangindent\wd\@uxbox\hangafter1\noindent\box\@uxbox\ignorespaces}
\def\justc@py{\ifinner\box\p@ctbox\else\kern\parskip\unvbox\p@ctbox\fi
  \global\setbox\p@ctbox=\box\voidb@x}
\def\vjustc@py{\vadjust{\kern0.5\baselineskip\unvbox\p@ctbox}%
      \global\setbox\p@ctbox=\box\voidb@x}
\def\tr@toplacepicture{%             % tries to find the good place
      \ifvmode\l@stdepth=\prevdepth  % remember the depth of last box
      \else   \l@stdepth=0pt         % otherwise the depth of last box is zero
      \fi
      \vrule height.85em width0pt\par% to define exact totalpage
      \r@memberdims                  % remembers the significant dimensions
      \global\t@mpdimen=\pagetotal
      \t@stheightofpage              % tests if the picture fits here
      \ifdim\b@ttomedge<\pagegoal    % if the picture ends on the current page
         \let\N@xt\f@gurehere        % the picture can be placed here
         \global\everypar{}%         % clear up the everypar token
      \else
         \let\N@xt\relax             % we must wait till the new page
         \penalty10000
         \vskip-\baselineskip        % but jump back to the reference point
         \vskip-\parskip             % of the last line of the previous par
         \immediate\write16{Picture will be shifted down.}%
         \global\everypar{\sw@tchingpass}%  % check the begin of every paragraph
      \fi
      \penalty10000
      \N@xt}
\def\sw@tchingpass{%                 % on every odd pass t@sts the possibility
    \iffirstp@ss                     %   to place picture here
      \let\n@xt\relax
      \firstp@ssfalse                % on every even pass does nothing
    \else
      \let\n@xt\tr@toplacepicture
      \firstp@sstrue
    \fi  \n@xt}
\def\r@memberdims{\global\in@tdimen=0pt
    \ifnum\p@ctpos=0
        \global\in@tdimen=\re@lpictwidth
      \fi
      \global\l@nelength=\hsize
      \global\advance\l@nelength by-\re@lpictwidth
      }
\def\t@stheightofpage{%
     \global\@pperedge=\t@mpdimen
     \advance\t@mpdimen by-0.7\baselineskip % jump back to the begin of par
     \advance\t@mpdimen by \lines@bove\baselineskip % and down by full lines
     \advance\t@mpdimen by \ht\p@ctbox      % and down by the height of new box
     \advance\t@mpdimen by \dp\p@ctbox      % and down by the depth of new box
     \advance\t@mpdimen by-0.3\baselineskip % and down by the depth of last box
     \global\b@ttomedge=\t@mpdimen          % and remember bottom edge
     }
\def\f@gurehere{\global\f@nishedfalse
      \t@mpdimen=\lines@bove\baselineskip   % and remember the amount of used
      \advance\t@mpdimen-0.7\baselineskip   %   space
      \kern\t@mpdimen
      \advance\t@mpdimen by\ht\p@ctbox
      \advance\t@mpdimen by\dp\p@ctbox
      {\t@mpdimeni=\baselineskip
       \offinterlineskip
       \unvbox\p@ctbox
       \global\setbox\p@ctbox=\box\voidb@x
       \penalty10000   \kern-\t@mpdimen     % and jump back to the reference
       \penalty10000   \vskip-\parskip      % point of the last line of previous
       \kern-\t@mpdimeni                    % paragraph
%       \hbox{\vrule height\l@stdepth width0pt}%
      }%
      \penalty10000                         % prevent page breaking
      \global\everypar{\s@thangindent}%     % and mark that paragraph is special
      }
\def\s@thangindent{%
    \ifdim\pagetotal>\b@ttomedge\global\everypar{}%
      \global\f@nishedtrue             % if we continue under the picture
      \else
        \advance\@pperedge by -1.2\baselineskip
        \ifdim\@pperedge>\pagetotal\global\everypar{}%
          \global\f@nishedtrue
        \else
          \s@tparshape                 % if we continue around the picture
        \fi
        \advance\@pperedge by 1.2\baselineskip
      \fi}
\def\s@tparshape{\t@mpdimen=-\pagetotal% minus upperedge of the picture
   \advance\t@mpdimen by\b@ttomedge    % plus bottomedge of the pictured
   \divide\t@mpdimen by\baselineskip   % divided by baselineskip gives the
   \helpc@unt=\t@mpdimen               %  number of shorter lines
   \advance \helpc@unt by 2            % rounding correction and one line more
   \sh@petoks=\expandafter{\the\helpc@unt\space}% \sh@petoks is for \parshape
   \t@mpdimeni=\lines@bove\baselineskip
   \t@mpdimen=\pagetotal
   \gdef\lines@bove{0}%              % in next paragraphs no lines are full
   \loop \ifdim\t@mpdimeni>0.999\baselineskip % counts the full lines
     \advance\t@mpdimen  by \baselineskip
     \advance\t@mpdimeni by-\baselineskip
     \sh@petoks=\expandafter{\the\sh@petoks 0pt \the\hsize}%
   \repeat
   \loop \ifdim\b@ttomedge>\t@mpdimen         % counts the shorter lines
     \advance\t@mpdimen by \baselineskip
     \sh@petoks=\expandafter{\the\sh@petoks \in@tdimen \l@nelength }%
   \repeat
   \sh@petoks=\expandafter
      {\the\sh@petoks 0pt \the\hsize}%        % and next lines are full again
   \expandafter\parshape\the\sh@petoks
   }

\descriptionmargins{2pt}
\picturemargins{15pt}{0pt}

\catcode`\@=\catcodeat        \let\catcodeat=\undefined

\endinput
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Short syntax of control sequences:
%
% \allframed  all following pictures will be framed by thin line
%
% \picturemargins{<dimen1>}{<dimen2>}  sets the space between text
%      of paragraph and picture to the value <dimen1> and space on the
%      other side of picture to the value <dimen2>. Default values are
%      15pt and 0pt. These values are taken into account only when
%      pictures are placed inside paragraphs.
%
% \descriptionmargins{<dimen>}  sets the space around marks to the value
%      <dimen> (so the mark seems a little bigger and doesn't stick to the
%      point where placed (if placed by point on its border). Default
%      value is 1pt.
%
% \justframes means that all following pictures will be shown only by
%      the outlines.
%
% Control sequences that create object and put it onto the stack of pictures
%
% \emptyplace{<width><whatever>}{<height>}{<marks>}<description>\par
%
% \boxplace{<vbox>}{<marks>}<description>\par
%
% \fontcharplace{<font>}{<char>}{<marks>}<description>\par
%      <width>   is the width of free place
%      <whatever> material placed into created vbox
%      <height>  is the height of free place
%      <vbox>    the vbox with picture
%      <font>    is the cs of font with picture
%      <char>    number of char with picture (65,66,...
%      <marks>   are either \placemark's with their parameters
%                or \fontmark's with their parameters (in the case of
%                \fontcharplace)
%      <description> is text that will be placed under the picture
%                as the description of the picture
%
% \placemark <ph><pv> <xdim> <ydim> <mark>
%
% \fontmark  <ph><pv> <xrel> <yrel> <mark>
%      <ph>     one of the letter  l, c, r  indicates that the left
%               margin, center or right margin (after adding value
%               determined by \descriptionmargins) will be placed into
%               the prescribed position.
%      <pv>     one of the letter  u, c, b, l  indicates that the upper
%               edge, center, bottom line or lower edge (after adding value
%               determined by \descriptionmargins) will be placed into
%               the prescribed position.
%       <xdim> or <ydim> are dimensions describing the position of
%               mark with respect to the lower left corner of
%               picture (this corner can be seen when you use
%               \allframed at the begin of your file)
%       <xrel> or <yrel> are relative dimensions describing the
%               position of mark with respect to the origin of user's
%               coordinate system (as specified in METAFONT macro
%               INCPIC.MF). It works only in \fontcharplace and for
%               pictures that are created by METAFONT using macro INCPIC.MF.
%       <mark>  (terminated by space) is the text of mark.
%
% \picname{<title>}  the <title> is set in bold face font in front of
%       the <description> from the left margin of the picture and the
%       <description> is indented by the width of the title. (This macro
%       could be used in the description of the picture.)
%
% Control sequences take the picture from the stack and put it into text.
%
% \inspicture lcr(<n>)!|^_{<description>} takes the first picture from
%      the stack and puts it into the text. All parameters are optional
%      and their meaning is:
%      l           the picture is placed on the left side of the pages
%      c           the picture is centered (this is default)
%      r           the picture is placed on the right side of the pages
%      <n>         number of full rows (default is 0)
%                  above the picture when the picture is placed on the
%                  left or right side of page.
%       !          forces to copy the picture here. In this case the
%                  options lrc(<n>) are ignored.
%       |          picture will be framed by thin line
%       ^          the picture forms \vtop (influnces the aligning with
%                  surrounding text.
%       _          the picture forms \vbox with zero depth
%                  (influnces the aligning with surrounding text.
%       When \inspicture is used inside a paragraph, it is
%       recomended to put empty group after its last parameter.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

nlin0206043.extracted_bib
\begin{thebibliography}{100}

\bibitem{Helbing}
D.~Helbing, Traffic and related self-driven many-particle systems,
Rev. Mod. Phys. {\bf 73} (2001), 1067 -- 1141

\bibitem{Chowdhury}
K.~Ghost,A.~Majumdar, and D.~Chowdhury, Distribution of time-headways in a
particle-hopping model of vehicular traffic, Phys. Rev. E {\bf 58} (1998), 4012
-- 4015

\bibitem{Seba}
P.~Seba and M.~Krbalek: Statistical properties of the city transport in
Cuernavaca (Mexico) and random matrix ensembles, J. Phys. A {\bf 33} (2000),
L229 -- L234.

%\bibitem{Yokoya}
%J.~Yokoya: Independence between route decision and traffic flow
%through the medium position of dynamics information, najdu pak

\bibitem{Mehta}
K.~Mehta: Random Matrices. Academic Press, New York 1991

\bibitem{Scharf}
R.~Scharf, F.M.~Izrailev: Dyson's Coulomb gas on a circle and intermediate
eigenvalue statistics, J.Phys.A {\bf 23} (1990) 963-977

\bibitem{Wagner}
M.~Krbalek, P.~Seba, and P.~Wagner: Headways in traffic flow -
remarks from a physical perspective, Phys. Rev. E. {\bf 64}
(2001), 066119 -- 066125

\end{thebibliography}