Headway distribution of the asymmetric simple exclusion model
Milan Krbalek$^{a,b}$ %and V. B. Priezzhev$^{c}$
Czech Academy of Science
Cukrovarnick\'a 10
Prague
Czech Republic
$^{b}$ Faculty of Nuclear Sciences and Physical Engineering
Trojanova 13
We present an exact solution of headway distribution of the asymmetric simple exclusion model with open boundary conditions and compare it to the headway distributions of the highway traffic.
Article.tex
\documentclass [11pt]{article}
\usepackage[T1]{fontenc}
\usepackage[final]{epsfig}
 \leftskip -2 cm
 \rightskip -1 cm
 \topmargin -0.5 cm
 \textheight 22 cm

\begin{document}

\include{incpic}

\emptyplace{3.3in \special{psfile="Figure1.eps" hscale=50
vscale=50 hoffset=-30 voffset=-300}} {3.1in} {\footnotesize
\noindent FIGURE 1. Headway distribution of the ASEP . The
curve is (\ref{Poisson}). Bars and plus signs show the result
(\ref{Finalk}) after the re-scaling to mean headway equal to one
for $\alpha=0.15,\beta=0.6$ and $\alpha=0.3,\beta=0.5,$
respectively.}


\title{Headway distribution of the asymmetric simple exclusion model}
\author{Milan Krbalek$^{a,b}$
%and V. B. Priezzhev$^{c}$
\\
    $^{a}$ Institute of Physics, Czech Academy of Science,\\
    Cukrovarnick\'a 10, Prague, Czech Republic\\
    $^{b}$ Faculty of Nuclear Sciences and Physical Engineering,\\
      Trojanova 13, Prague, Czech Republic\\
    e-mail: milan.krbalek@uhk.cz\\}
    %priezzvb@thsun1.jinr.ru,\,milan.krbalek@uhk.cz\\}
 %   $^{c}$ Joint Institute for Nuclear Research,\\
 %  Dubna 141980, Russia\\

\maketitle
\begin{abstract}
We present an exact solution of headway distribution of the
asymmetric simple exclusion model with open boundary conditions
and compare it to the headway distributions of the highway
traffic.
\end{abstract}

\section{Introduction}
Recently, a new tendency has appeared in the physics of traffic
systems. Besides macroscopic characteristics of traffic (flow,
density, etc.), researchers are focused on microscopic properties
as well (\cite{Book}). One of these properties is the distribution
of the distances (or time intervals) between neighboring cars, so
called headway distribution. This function was investigated for
existing traffic models and compared to the various regimes of the
real traffic. It was observed that the real traffic data as well
as many traffic models lead to a headway distribution which
resembles the spacing distribution of the random-matrices
eigenvalues (\cite{Krbalek1},\cite{Krbalek2},\cite{Seba}). To
verify this observation, we investigate the asymmetric simple
exclusion process (ASEP) which is the well known exactly solvable
model of traffic (\cite{Derrida1},\cite{Derrida2},\cite{Santen}).
We find the exact formula for headway distribution of the
asymmetric simple exclusion model and compare it with the
eigenvalues distributions obtained from random matrix theory.

%\section{Headway distribution of the real-roads traffic}


In the traffic theory, the spacing distribution which was often
called \textit{headway distribution} $P(s)$ is defined as
probability distribution of the distances $s$ between neighboring
cars (particles of the model). For the sake of simplicity we
assume that $\langle s \rangle \equiv \int_0^\infty s\,P(s)\,ds
=1.$ Among many others, we discern two special traffic systems in
real roads. The first is the \textit{bus transport} where no fixed
schedule exists (see \cite{Krbalek1},\cite{Seba}) and where the
interaction between bus drivers exists owing to the check points
in which the information about the time interval to the previous
bus is transmitted to a driver. It was found that the headway
distribution of that bus transport behaves as
\begin{equation}
P(s)=A\,s^2 \, e^{-B s^2}, \label{RMT-LS}
\end{equation}
where  $A,B$ are constants determined by conditions $\int_0^\infty
P(s) \, ds=1$ and $\int_0^\infty s\,P(s) \, ds=1.$ The equation
(\ref{RMT-LS}) originates from random matrix theory where it
describes the distribution of distances between eigenvalues of the
unitary random matrices.\\ The second system described here is
\textit{highway traffic} for which we distinguish three basic
regimes (\cite{Krbalek1}):\begin{itemize}
\item a) \textit{free-flow regime}, where the density of cars (number of
cars per unit of length) is low and the cars are therefore
moving as free. Headway of this regime is reduced to the
Poissonian distribution

\begin{equation}
P_0(s)=e^{-s}
\label{Poisson}
\end{equation}

\item b) \textit{synchronized regime}, where the mean velocity is
considerably reduced compared to the free-flow states due to a
higher density of cars, but cars are still moving (no traffic
jam). Headway distribution of these states is well described by
the formula

\begin{equation}
P_\nu(s)  =  \frac{(\nu+1)^{\nu+1}}{\Gamma(\nu +1)}s^{\nu}
e^{-(\nu +1) \, s}, \label{Bogo}
\end{equation}

where $\nu$ is a free parameter and $\Gamma$ is Gamma function.

\item c) \textit{stop-and-go regime}, where cars are moving in traffic
jams
\end{itemize}

Note that it is possible to obtain the formula (\ref{Poisson})
taking the limit $\nu \rightarrow 0^+$ in formula (\ref{Bogo}).
Distribution $P_1(s)$ used to be called as semi-Poissonian
statistics.\\

 Roughly,  particles in the free-flow regime
do not interact between each other, contrary to the synchronized
states when the interaction leads to the particle repulsion and to
the fact that $\lim_{s \rightarrow 0^+} P_\nu(s)=0$ for positive
$\nu.$ One of aims of this article is to answer whether it is
possible to get a synchronized state for the asymmetric simple
exclusion model by an appropriate choice of input parameters, or
the model is in the free regime for all parameters.\\

\section{ASEP with open boundaries}

Consider sites located on a chain of length $N$. Each site $i$
($1\leq i\leq N$) is either occupied by a particle, or
empty. During the time interval $dt$, each particle jumps to the
next site (in the defined direction) with probability $dt$ if the
target site is empty and does not hop otherwise. Moreover, during
the time interval $dt$, a particle can enter the first chain
position $(i=1)$ with probability $\alpha\,dt$ $(\alpha \in
\langle 0,1\rangle)$ if this position is empty and a particle at
the site $i=N$ can leave the chain with probability $\beta\,dt$
$(\beta \in \langle 0,1\rangle)$ if this site is occupied. These
rules define a simple traffic model with a hard-core repulsion
which  decreases the probability for
occurrence of the two particles close each other. This was
\textit{de facto} the reason why we have chosen ASEP for our
investigation.\\
Two alternative methods of exact solution of the model are outlined in
\cite{Derrida1}. We will use the \textit{matrix method} in our
computations. Consider the matrices $D,E$ and vectors $\langle w
|$ and $| v \rangle$ satisfying the following algebraic rules
\begin{equation}DE=D+E \label{Rule1}\end{equation}
\begin{equation} \langle w |E=\frac{1}{\alpha}\langle w
|\label{Rule2}\end{equation}
\begin{equation}D| v \rangle=\frac{1}{\beta}| v
\rangle\label{Rule3}\end{equation} Then we can determine the
probability of the arbitrary configuration
$\mathcal{C}=(n_1,m_1,n_2,m_2,\ldots)$ of the model through the
formula \begin{equation}P(\mathcal{C})=\frac{1}{Z_N} \,
\frac{\langle w |D^{n_1}E^{m_1}D^{n_2}E^{m_2}\ldots | v
\rangle}{\langle w | v \rangle}\label{Config},\end{equation} where
$n_1$ is the number of the empty sites, starting with the entering
site of the chain (and $(n_1+1)$-th site is occupied), $m_1$ is
the number of occupied sites starting with the site number $n_1+1$
(and $(n_1+m_1+1)$-th site is empty), and so on. $Z_N$ is a normalization
constant. Hence $\sum_i n_i + \sum_j m_j =N.$ Owing to the fact

\begin{equation}(D+E)^n=\sum_{m=1}^{n}
\frac{m\,(2n-m-1)!}{n!(n-m)!} \sum_{i=0}^{m} E^i D^{m-i}
\label{D+E}\end{equation}

 we can rewrite the normalization factor $Z_N$ to the form

 \begin{equation}Z_N\equiv\frac{\langle w|
(D+E)^N| v \rangle}{\langle w| v \rangle} = \sum_{m=1}^N
\frac{m\,(2N-m-1)!}{N!(N-m)!} \sum_{i=0}^m
\left(\frac{1}{\alpha}\right)^i \,
\left(\frac{1}{\beta}\right)^{m-i} \label{ZN}\end{equation}

Hence,

\begin{equation}Z_N=\sum_{m=1}^N \frac{m\,(2N-m-1)!}{N!(N-m)!}
\frac{\beta^{-m-1} - \alpha^{-n-1}}{\beta^{-1} - \alpha^{-1}}
\label{ZN2}\end{equation}

These are the known results  \cite{Derrida1} which will be used
below.

\section{Headway distribution in the ASEP (a special case)}

Our aim is an analytical expression for
the spacing distribution, which means
the probability of the gap of the length $k$ behind the particle
being on the $i$-th site. One can determine this probability
$P^{(i)}(k)$ from the equation (\ref{Config}) as follows
\begin{equation}
P^{(i)}(k)=\frac{\mathcal{A}}{Z_N\,\langle w| v \rangle} \,
\langle w| (D+E)^{i-1} D\,E^{k-1} D\,(D+E)^{N-i-k}\,|v\rangle,
\label{gap}
\end{equation}
since the first $i-1$ sites could be arbitrary occupied or not,
$i$-th site is occupied, the following $k-1$ sites are empty, the
site $i+k$ is occupied and the rest sites could be
arbitrary occupied or not. If these rules are satisfied, one can
see the gap of the length $k$ behind the $i$-th position.
The constant $\mathcal{A}$ can be calculated from the condition
$$\sum_{k=1}^{N-i} P^{(i)}(k)=1 %\label{normalisation}
$$

Generally, the matrices $D,E$ are infinite-dimensional, but
provided the condition \begin{equation} \alpha + \beta =1
\label{special-case} \end{equation}  one can find
$D,E$ to be real numbers. The equations (\ref{Rule2}) and (\ref{Rule3})
imply that $D=1/\beta,E=1/\alpha$ and the rule (\ref{Rule1})
imposes that ${}^1/{}_\alpha + {}^1/{}_\beta = {}^1/{}_{\alpha
\beta}$, which leads to the equation (\ref{special-case}).
Instituting (\ref{special-case}) and (\ref{ZN}) into the
(\ref{gap}) we obtain
\begin{equation}
P^{(i)}(k)=\mathcal{A}\,\alpha^2\,\beta^{k-1} \, (k \geq 1)
\label{headway-special} \end{equation}

Strictly speaking, the derivation of the formula
(\ref{headway-special}) from the equation (\ref{gap}) is possible
only for $k\geq3$ since for $k=1,2$ the relation (\ref{gap}) has a
different form. But it can easily be verified that
(\ref{headway-special}) holds true also for $k=1,2$. Note now that
the result (\ref{headway-special}) is independent of the value
$i$. For large $N$, after the re-scaling to $\langle s \rangle=1,$
equation (\ref{headway-special}) changes to the form $P(s):=
P(k-1)=e^{-s}$, which is the Poissonian distribution. This clearly
shows that under the condition (\ref{special-case}), the
interaction among the particles vanishes in the continuous limit
 and hence the movement of
the particles is asymptotically free (in the sense mentioned in the section
$2$).\\ Also the real headway distribution (without the
re-scaling) can be determined in this section. If
(\ref{special-case}) holds true then
\begin{equation} P^{(i)}(k)=\mathcal{N}
\left(\frac{1}{\alpha}
\right)^{N-2}\,\left(\frac{1}{\beta}\right)^{N+1-k},
\label{real-gap}
\end{equation} when the normalization pre-factor $\mathcal{N}$ can be
computed either from the condition $\sum_{k=1}^{N-i} P^{(i)}(k)
=1$ (through all the lengths of the gap) or from the equation
$$\frac{1}{\mathcal{N}}=\frac{\langle
w|(D+E)^{N-i}D\,(D+E)^{N-i}|v\rangle\,-\,\langle
w|(D+E)^{i-1}D\,E^{N-i}|v\rangle}{\langle w|v \rangle},$$ which
takes into account all the possible combinations with occupied
site number $i$ and rejects the combinations when all the sites
(behind the site number $i$) are empty. This provides


$$\frac{1}{\mathcal{N}}=\left(\frac{1}{\alpha}\right)^{N-i}
\left(\frac{1}{\beta}\right)^{N}(1-\beta^{N-i}),$$

which together with (\ref{real-gap}) yields $$
P^{(i)}(k)=\frac{\alpha}{1-\beta^{N-i}}\,\beta^{k-1}
%\label{real-gap2}
$$

In the large $N$ limit, one can calculate from
(\ref{special-case}) that $$ P^{(i)}(k)=\alpha\,\beta^{k-1}
%\label{real-gap3}
$$ and hence $$ \langle k \rangle \equiv \sum_{k=1}^\infty k\,
P^{(i)}(k)=\alpha\,\sum_{k=1}^\infty
k\,\beta^{k-1}=\frac{1}{\alpha} $$\label{mean-gap}
%\end{equation}

for the mean headway.


\section{Headway distribution of the ASEP (general case)}

In this section we are solving the ASEP (equation (\ref{gap}),
strictly speaking) for arbitrary values of the parameters
$\alpha,\, \beta.$ Introduce now the following pre-factors

$$\omega=\omega(m,k,i,N,p)=\frac{m\,p\,(2N-2i-2k-m-1)!(2i-p-3)!}{(N-i-k)!(N-i-k-m)!(i-1)!(i-p-1)!}$$


$$ A_1=A_1(p,q,a,w)=\left( \begin{array}{c} p-q-a+w-1
\\ p-q-1 \end{array} \right)
$$

$$ A_2=A_2(p,q,b,w)=\left( \begin{array}{c} p-q-b+w-1
\\ w-1 \end{array} \right)
$$

With the help of them and the computations presented in an
Appendix we obtain the desired result

\inspicture

\begin{equation}
P^{(i)}(k)=\frac{\mathcal{A}}{Z_N} \sum_{p=1}^{i-1}
\sum_{m=1}^{N-i-k} \sum_{l=0}^m \,\omega\, (S_1+S_2+S_3+S_4+S_5)
\label{Finalk}
\end{equation}

where

$$S_1=\left(\frac{1}{\beta}+l \right)\sum_{q=0}^{p-1} \left(
\frac{1}{\alpha} \right)^q\left( \frac{1}{\beta}
\right)^{p-q+m-l+1}$$

$$S_2=\sum_{q=0}^{p-1} \sum_{w=1}^{k-1}\left(\, \sum_{a=1}^w
A_1\left( \frac{1}{\alpha} \right)^{q+a}\left( \frac{1}{\beta}
\right)^{m-l+1} + \sum_{b=1}^{p-q} A_2\left( \frac{1}{\alpha}
\right)^q\left( \frac{1}{\beta} \right)^{m+b-l+1} \right)$$

$$S_3=l\,\sum_{q=0}^{p-1} \sum_{w=1}^{z+k-1}\left(\, \sum_{a=1}^w
A_1\left( \frac{1}{\alpha} \right)^{q+a}\left( \frac{1}{\beta}
\right)^{m-l} + \sum_{b=1}^{p-q} A_2\left( \frac{1}{\alpha}
\right)^q\left( \frac{1}{\beta} \right)^{m+b-l} \right)$$

$$S_4=\left( \frac{1}{\alpha} \right)^p\left( \frac{1}{\beta}
\right)^{m-l+2}+\sum_{w=1}^{k-1} \left( \frac{1}{\alpha}
\right)^{p+w}\left( \frac{1}{\beta} \right)^{m-l+1}$$

$$S_5=l\left( \frac{1}{\alpha} \right)^p\left( \frac{1}{\beta}
\right)^{m-l+1}+l\sum_{w=1}^{z+k-1} \left( \frac{1}{\alpha}
\right)^{p+w}\left( \frac{1}{\beta} \right)^{m-l}$$


The equation (\ref{Finalk}) presents the final result of this
article. In the special case (\ref{special-case}) the equations
(\ref{Finalk}) and (\ref{real-gap}) provide the same values, which
supports the correctness of our result.  Now, one can show that
the headway distribution of the ASEP (after re-scaling) is of a
Poissonian type for any parameters $\alpha,\beta$ (see Figure 1).


\section{Conclusion}

To conclude, the ASEP shows the Poissonian behavior for all
possible values of the parameters $\alpha$ and $\beta$. Thus, this
model can simulate only the traffic in free-flow regime, which
means that
\begin{equation} \lim_{s \rightarrow 0^+} P(s) \gg 0
\label{ConditionP} \end{equation} To be precise, we have to
express that as the parameter $\beta$ goes to zero ($\beta
\rightarrow 0^+$) the Poissonian distribution changes since one
can easily to follow that $P(s)=\delta(s)=\delta(k-1)$ for
$\beta=0$ and $\alpha\neq0$. This holds true due to the shift of
the probability $P(s):=P(k-1)$. The above mentioned shift is done
because $P(k)=0$ for every $k<1$ in the original model. However,
for every $\beta$ we get the headway distribution satisfying the
condition (\ref{ConditionP}).\\

On contrary, in \cite{Gier} a different variant of the ASE-model
(high speed traffic model with open boundaries) is presented. The
particles of this model also enter the system at the left and
leave at the right of the chain. Moreover, all particles will move
with their maximal possible speed, which is given by the speed
limit $v_{max}$ (a maximal length of the jump). For $v_{max}=1$
one can obtain the model investigated in this article. Numerical
simulations of the high speed model with $v_{max}\geq2$ clearly
show that the gap distribution can start from the origin (i.e.
$\lim_{s \rightarrow 0^+} P(s) = 0$) for some values
$\alpha,\beta.$\\ Hence, the high speed model has the synchronized
states as well, on contrary to the asymmetric simple exclusion
process and it can be used to the full simulation of the real-road
traffic.

\section{Acknowledgement}
Author is very grateful to V.B. Priezzhev and Petr \v Seba for
valuable advice and for the help with writing of this paper.

\begin{thebibliography}{999}

\bibitem{Book}
B.~Tilch and D.~Helbing: Evaluation of single vehicle data in
dependence of the vehicle-type, lane, and site, Traffic and
Granular Flow '99, Springer-Verlag Berlin Heidelberg 2000, 333 --
338
\bibitem{Krbalek1}
P.~Seba and M.~Krbalek: The statistical properties of the city
transport in Cuernavaca (Mexico) and random matrix ensembles, J.
Phys. A {\bf 33} (2000), L229 -- L234
\bibitem{Krbalek2}
M.~Krbalek, P.~Seba, and P.~Wagner: Headways in traffic flow -
remarks from a physical perspective, Phys. Rev. E. {\bf 64}
(2001), 066119 -- 066125
\bibitem{Derrida1}
B.~Derrida: An exactly soluble non-equilibrium system: The
asymmetric simple exclusion process, Physics Reports {\bf 301}
(1998), 963 -- 977
\bibitem{Derrida2}
B.~Derrida, J.~L.~Lebowitz: Exact large deviation function in the
asymmetric exclusion process, Phys. Rev. Lett. {\bf 80} (1998),
209 -- 213
\bibitem{Santen}
N.~Rajewski, L.~Santen, A.~Schadschneider, and M.~Schreckenberg:
The asymmetric exclusion process: Comparison of update procedures,
Journal of Statistical Physics, {\bf 92} (1998), 151 -- 194.
\bibitem{Gier}
J.~de Gier: Exact stationary state for a deterministic high speed
traffic model with open boundaries, J. Phys. A {\bf 34} (2001),
3707 -- 3720
\bibitem{Seba}
M.~Krbalek and P.~Seba: Headway statistics of public transport in
Mexican cities, to appear

\end{thebibliography}

\section{Appendix}


At the beginning, we cite some useful lemmas.

\vspace{0.5 cm} \underline{Lemma 1}

 \vspace{0.5 cm} Let $m,n,a$ are the natural numbers and $a \leq m.$ Then
\begin{equation} \sum_{i=a}^m \left( \begin{array}{c} n-i+m-1 \\
n-1 \end{array} \right) = \left( \begin{array}{c} n+m-a \\ n
\end{array} \right) \label{lemma1} \end{equation}

\textit{Proof:} by means of the mathematical induction for the
numbers $m$ and $n$ \\

\vspace{0.5 cm} \underline{Lemma 2}

 \vspace{0.5 cm} Assume that $D,E$ are arbitrary matrices
 fulfilling the equation $DE=D+E.$ Let $m,n$ are the natural
 numbers. Then

\begin{equation} D^nE^m = \sum_{i=1}^m \left( \begin{array}{c} n-i+m-1 \\
n-1 \end{array} \right) \, E^i + \sum_{j=1}^n  \left(
\begin{array}{c} n-j+m-1 \\ m-1
\end{array} \right) \, D^j \label{lemma2} \end{equation}

\textit{Proof:} by means of the mathematical induction for the
numbers $m$ and $n$ and (\ref{lemma1})

\vspace{0.5 cm}

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

Now we can rewrite the equation (\ref{gap}) in order to use the
rules (\ref{Rule2}),(\ref{Rule3}). Thus, it is necessary to
replace the matrices $D$ on the right-hand side and matrices $E$
to the left-hand side of the equation (\ref{gap}). The expression
(\ref{D+E}) yields

$$(D+E)^{N-i-k} = \sum_{m=1}^{N-i-k}
\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!} \sum_{l=0}^m
E^lD^{m-l} %\label{101}
$$

Thus,

 \begin{eqnarray} D\,(D+E)^{N-i-k} = \sum_{m=1}^{N-i-k}
\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!} \sum_{l=0}^m
D\,E^lD^{m-l}=  \nonumber \\ = \left| \begin{array}{c} D\,E^l = D
+ \sum_{z=1}^l E^z
\\ D\,E^lD^{m-l} = D^{m-l+1} + \sum_{z=1}^l E^zD^{m-l} \end{array}
\right| =  \nonumber \\ = \sum_{m=1}^{N-i-k}
\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!} \sum_{l=0}^m \left(
D^{m-l+1} + \sum_{z=1}^l E^zD^{m-l} \right)  \nonumber %\label{102}
\end{eqnarray}

It can be verified that \begin{eqnarray} E^{k-1}D\,(D+E)^{N-i-k}=
\nonumber \\ =\,\sum_{m=1}^{N-i-k}
\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!} \sum_{l=0}^m
\left(E^{k-1}D^{m-l+1} + \sum_{z=1}^l E^{z+k-1}D^{m-l}\right)
\nonumber %\label{111}
\end{eqnarray}

Furthermore,

 \begin{eqnarray} D\,E^{k-1}D\,(D+E)^{N-i-k}=
 \left|\begin{array}{c} D\,E^{k-1} = D + \sum_{w=1}^{k-1} E^w
\\ D\,  E^{z+k-1} = D + \sum_{w=1}^{z+k-1} E^w \end{array}
\right| =  \nonumber \\ = \sum_{m=1}^{N-i-k}
\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!} \nonumber \\
\sum_{l=0}^m \left( D^{m-l+2} + \sum_{w=1}^{k-1}  E^wD^{m-l+1} +
l\,D^{m-l+1} + \,l\, \sum_{w=1}^{z+k-1} E^wD^{m-l}\right)
\label{vlozeno}
\end{eqnarray}

Another use of the expression (\ref{D+E}) gives \begin{equation}
(D+E)^{i-1}= \sum_{p=1}^{i-1} \frac{p\,(2i-p-3)!}{(i-1)!(i-p-1)!}
\sum_{q=0}^p E^qD^{p-q} \label{112} \end{equation}

Let now \begin{equation}
\gamma=\gamma(m,k,i,N)=\frac{m\,(2N-2i-2k-m-1)!}{(N-i-k)!(N-i-k-m)!}
\label{113a} \end{equation}

\begin{equation}
\delta=\delta(i,p)=\frac{p\,(2i-p-3)!}{(i-1)!(i-p-1)!}
\label{113b}
\end{equation}

Applying (\ref{113a}),(\ref{113b}), you can transform the
equations (\ref{vlozeno}) and (\ref{112}) to

\begin{eqnarray} D\,E^{k-1}D\,(D+E)^{N-i-k}=\nonumber \\ =\sum_{m=1}^{N-i-k}
 \sum_{l=0}^m \gamma \left( D^{m-l+2} + \sum_{w=1}^{k-1}  E^wD^{m-l+1} +
 l\,
D^{m-l+1} + \,l\, \sum_{w=1}^{z+k-1} E^wD^{m-l}\right)
\nonumber %\label{114a}
\end{eqnarray}

$$ (D+E)^{i-1}=\sum_{p=1}^{i-1} \sum_{q=0}^p \delta \,E^qD^{p-q}
%\label{114b}
$$

Hence, \begin{eqnarray} (D+E)^{i-1}D\,E^{k-1}D\,(D+E)^{N-i-k} =
\sum_{p=1}^{i-1} \sum_{q=0}^p \sum_{m=1}^{N-i-k} \sum_{l=0}^m
\delta \, \gamma \nonumber \\ \left( E^qD^{p-q+m-l+2} +
\sum_{w=1}^{k-1} E^q \overbrace{D^{p-q}E^w} D^{m-l+1} \, +
\nonumber \right. \\ \left. l\, E^qD^{p-q+m-l+1} \, + \,l\,
\sum_{w=1}^{z+k-1} E^q \overbrace{D^{p-q}E^w} D^{m-l} \right)
\label{115}
\end{eqnarray}

In the expression (\ref{115}) we can change the products indicated
by the horizontal braces. Using (\ref{lemma2}) one can obtain

\begin{equation} D^{p-q}E^w =\sum_{a=1}^w \left(
\begin{array}{c} p-q-a+w-1 \\ p-q-1 \end{array} \right) E^a +
\sum_{b=1}^{p-q} \left(
\begin{array}{c} p-q-b+w-1 \\ w-1 \end{array} \right) D^b
\label{116} \end{equation}

Note that (\ref{116}) is valid only for $p>q.$ The remaining
arrangement is the substitution of the (\ref{116}) into the
equation (\ref{115}). With the help of the pre-factors

$$ A_1=\left( \begin{array}{c} p-q-a+w-1
\\ p-q-1 \end{array} \right) %\label{121a}
$$

$$ A_2=\left( \begin{array}{c} p-q-b+w-1
\\ w-1 \end{array} \right) %\label{121b}
$$

the expression (\ref{115}) turns into the form

\begin{eqnarray}
(D+E)^{i-1}D\,E^{k-1}D\,(D+E)^{N-i-k} = \sum_{p=1}^{i-1}
\sum_{q=0}^{p-1} \sum_{m=1}^{N-i-k} \sum_{l=0}^m \delta\,\gamma
\nonumber \\ \left(  E^qD^{p-q+m-l+2} + \,l\, E^qD^{p-q+m-l+1} +
\sum_{w=1}^{k-1} \sum_{a=1}^w A_1 E^{q+a} D^{m-l+1}\right. \, +
\nonumber
\\ + \left. \sum_{w=1}^{k-1} \sum_{b=1}^{p-q} A_2 E^q D^{m+b-l+1} \,+\,
l\, \sum_{w=1}^{z+k-1} \sum_{a=1}^w A_1 E^{q+a}D^{m-l} \,+\,l\,
\sum_{w=1}^{z+k-1} \sum_{b=1}^{p-q} A_2 E^qD^{b+m-l} \right) +
\nonumber \\ + \sum_{p=1}^{i-1} \sum_{m=1}^{N-i-k} \sum_{l=0}^m
\delta\,\gamma \left( E^pD^{m-l+2}+\sum_{w=1}^{k-1}
E^{p+w}D^{m-l+1} + l\,E^pD^{m-l+1} + \,l\, \sum_{w=1}^{z+k+1}
E^{p+w}D^{m-l} \right) \nonumber %\label{123}
\end{eqnarray}

which together with the rules (\ref{Rule2}),(\ref{Rule3}) yields

\begin{center}
\begin{eqnarray}
P^{(i)}(k) =\frac{\mathcal{A}}{Z_N} \sum_{p=1}^{i-1}
\sum_{q=0}^{p-1} \sum_{m=1}^{N-i-k} \sum_{l=0}^m  \delta \, \gamma
\left\{ \left( \frac{1}{\alpha} \right)^q\left( \frac{1}{\beta}
\right)^{p-q+m-l+2} + \,l\, \left( \frac{1}{\alpha}
\right)^q\left( \frac{1}{\beta} \right)^{p-q+m-l+1} \right. \,+
\nonumber
\\ + \sum_{w=1}^{k-1} \sum_{a=1}^w A_1 \left( \frac{1}{\alpha}
\right)^{q+a} \left( \frac{1}{\beta} \right)^{m-l+1} \,  +
\sum_{w=1}^{k-1} \sum_{b=1}^{p-q} A_2 \left( \frac{1}{\alpha}
 \right)^q \left( \frac{1}{\beta} \right)^{m+b-l+1} \,+ \nonumber
 \\ + \left.
\,l\, \sum_{w=1}^{z+k-1} \sum_{a=1}^w A_1\left( \frac{1}{\alpha}
\right)^{q+a}\left( \frac{1}{\beta} \right)^{m-l} \,+\,l\,
\sum_{w=1}^{z+k-1} \sum_{b=1}^{p-q} A_2\left( \frac{1}{\alpha}
\right)^q\left( \frac{1}{\beta} \right)^{b+m-l} \right\} +
\nonumber \\ + \frac{\mathcal{A}}{Z_N} \sum_{p=1}^{i-1}
\sum_{m=1}^{N-i-k} \sum_{l=0}^m \delta\,\gamma \left\{ \left(
\frac{1}{\alpha} \right)^p\left( \frac{1}{\beta}
\right)^{m-l+2}+\sum_{w=1}^{k-1} \left( \frac{1}{\alpha}
\right)^{p+w}\left( \frac{1}{\beta} \right)^{m-l+1} \,\right.+
\nonumber
\\ \left.+ \,l\,\left( \frac{1}{\alpha} \right)^p\left( \frac{1}{\beta}
\right)^{m-l+1} + \,l\, \sum_{w=1}^{z+k+1} \left( \frac{1}{\alpha}
\right)^{p+w}\left( \frac{1}{\beta} \right)^{m-l} \right\}
\end{eqnarray}
\end{center}


\end{document}

incpic.tex
% Macro for more comfortable treating pictures (pictures placed
% on left or right side of page are imbedded into paragraphs, pictures
% that are centered can be imbedded between lines of paragraph).
% For more details see the end of this file.
%
% Oldrich Ulrych                                     July 30, 1992, Prague
%
%
\edef\catcodeat{\the\catcode`\@ }     \catcode`\@=11
%
\newbox\p@ctbox                       % box for the current picture to be placed in
\newbox\t@mpbox                       % box for temporary storing
\newbox\@uxbox                        % auxiliary box
\newbox\s@vebox                       % stack of scanned pictures
%
\newtoks\desct@ks \desct@ks={}        % stack for of descriptions under pictures
\newtoks\@ppenddesc                   % toks for appended local description
\newtoks\sh@petoks                    % token for description of picture
%
\newif\ifallfr@med  \allfr@medfalse   % true if all picture are being fr@med
\newif\if@ddedrows                    % true if round brackets appeared
\newif\iffirstp@ss  \firstp@ssfalse   % flag for paragraph test
\newif\if@mbeeded                     % flag to indicate just to copy picture
\newif\ifpr@cisebox                   % flag to copy picture in natural width
\newif\ifvt@p                         % inserted picture is \vtop
\newif\ifvb@t                         % inserted picture is \vbot
\newif\iff@nished    \f@nishedtrue    % text is around the picture
\newif\iffr@med                       % picture will be framed
\newif\ifj@stbox     \j@stboxfalse    % only framed box is shown
%
\newcount\helpc@unt                   % auxiliary counter
\newcount\p@ctpos                     % position of the picture (0-l,1-c,2-r)
%
\newdimen\r@leth      \r@leth=0.4pt   % thickness of rules
\newdimen\x@nit                       % x-size of user's horizontal units
\newdimen\y@nit                       % y-size of user's vertical units
\newdimen\xsh@ft                      % x-shift of user's coordinates
\newdimen\ysh@ft                      % y-shift of user's coordinates
\newdimen\@uxdimen                    % auxiliary dimension
\newdimen\t@mpdimen                   % auxiliary dimension
\newdimen\t@mpdimeni                  % auxiliary dimension
\newdimen\b@tweentandp                % marginal space between text and picture
\newdimen\b@ttomedge                  % bottom edge of the picture on page
\newdimen\@pperedge                   % upper edge of the picture on page
\newdimen\@therside                   % space on the outer side of picture
\newdimen\d@scmargin                  % space around marks inside picture
\newdimen\p@ctht                      % height+depth of picture
\newdimen\l@stdepth
\newdimen\in@tdimen
\newdimen\l@nelength
\newdimen\re@lpictwidth
%
\def\justframes{\global\j@stboxtrue}  % only framed empty place is shown
\def\picturemargins#1#2{\b@tweentandp=#1\@therside=#2\relax}
\def\allframed{\global\allfr@medtrue} % all picturew will be fr@med
\def\emptyplace#1#2{\pl@cedefs        % the empty vbox will be created
    \setbox\@uxbox=\vbox to#2{\n@llpar% all paragraph values are reset
        \hsize=#1\vfil \vrule height0pt width\hsize}% vbox to prescribed dimensions
    \e@tmarks}
\def\boxplace{\pl@cedefs\afterassignment\re@dvbox\let\n@xt= }
\def\re@dvbox{\setbox\@uxbox=\vbox\bgroup% all is stored in vbox
         \n@llpar\aftergroup\e@tmarks}
\def\fontcharplace#1#2{\pl@cedefs     % picture is in font
    \setbox\@uxbox=\hbox{#1\char#2\/}%
    \xsh@ft=-\wd\@uxbox               % width of character and shift of origin
    \setbox\@uxbox=\hbox{#1\char#2}%
    \advance\xsh@ft by \wd\@uxbox     % horizontal shift of the origin
    \helpc@unt=#2
    \advance\helpc@unt by -63         % fontdimen code-63: size of user's x-unit
    \x@nit=\fontdimen\helpc@unt#1%
    \advance\helpc@unt by  20         % fontdimen code-43: size of user's y-unit
    \y@nit=\fontdimen\helpc@unt#1%
    \advance\helpc@unt by  20         % fontdimen code-23: y-shift of origin
    \ifnum\helpc@unt<51
      \ysh@ft=-\fontdimen\helpc@unt#1%
    \fi
    \e@tmarks}
\def\n@llpar{\parskip0pt \parindent0pt% paragraph values are reset
    \leftskip=0pt \rightskip=0pt
    \everypar={}}
\def\pl@cedefs{\xsh@ft=0pt\ysh@ft=0pt}% by default user's origin is not shifted
\def\e@tmarks#1{\setbox\@uxbox=\vbox{ % reset the box with picture
      \n@llpar
      \hsize=\wd\@uxbox               % to preserve its width
      \noindent\copy\@uxbox           % the box itself
      \kern-\wd\@uxbox                % jump to the left margin
      #1\par}%                        % mark into the picture
    \st@redescription}
\def\t@stprevpict#1{\ifvoid#1\else    % if previous picture is not finished
   \errmessage{Previous picture is not finished yet.}\fi} % then error message

\def\st@redescription#1\par{%         % puts picture's description on the stack
    \global\setbox\s@vebox=\vbox{\box\@uxbox\unvbox\s@vebox}%
    \desct@ks=\expandafter{\the\desct@ks#1\@ndtoks}}%remembers the description
%\def\@ndtoks{}
\def\def@ultdefs{\p@ctpos=1         % picture is centered by default
      \def\lines@bove{0}%           % no full lines above picture by default
      \@ddedrowsfalse               % no extra full lines read by default
      \@mbeededfalse                % paragraph flows around picture by default
      \pr@ciseboxfalse
      \vt@pfalse                    % output box will be vbox
      \vb@tfalse                    % output box will be vbox
      \@ppenddesc={}%               % no append to description
      \ifallfr@med\fr@medtrue\else\fr@medfalse\fi
      }

\def\descriptionmargins#1{\global\d@scmargin=#1\relax}
\def\@dddimen#1#2{\t@mpdimen=#1\advance\t@mpdimen by#2#1=\t@mpdimen}
\def\placemark#1#2 #3 #4 #5 {\unskip    % ignore previous spaces
      \setbox1=\hbox{\kern\d@scmargin#5\kern\d@scmargin}% form the mark
      \@dddimen{\ht1}\d@scmargin        % add some space above mark
      \@dddimen{\dp1}\d@scmargin        % add some space below mark
      \ifx#1l\dimen3=0pt\else           % the correction
        \ifx#1c\dimen3=-0.5\wd1\else
          \ifx#1r\dimen3=-\wd1
     \fi\fi\fi
     \ifx#2u\dimen4=-\ht1\else          % and @dd the correction
       \ifx#2c\dimen4=-0.5\ht1\advance\dimen4 by 0.5\dp1\else
         \ifx#2b\dimen4=0pt\else
           \ifx#2l\dimen4=\dp1
     \fi\fi\fi\fi
     \advance\dimen3 by #3%             % x position
     \advance\dimen4 by #4%             % y position
     \advance\dimen4 by-\dp1
     \advance\dimen3 by \xsh@ft         % influence of shifting of the origin
     \advance\dimen4 by \ysh@ft         % influence of shifting of the origin
     \kern\dimen3\vbox to 0pt{\vss\copy1\kern\dimen4}% put the mark
     \kern-\wd1                        % jump back of the width of mark
     \kern-\dimen3                     % jump back of the x position
     \ignorespaces}                    % ignore following spaces
\def\fontmark #1#2 #3 #4 #5 {\placemark #1#2 #3\x@nit{} #4\y@nit{} {#5} }
\def\fr@msavetopict{\global\setbox\s@vebox=\vbox{\unvbox\s@vebox
      \global\setbox\p@ctbox=\lastbox}%
    \expandafter\firstt@ks\the\desct@ks\st@ptoks}
\def\firstt@ks#1\@ndtoks#2\st@ptoks{%
    \global\desct@ks={#2}%
    \def\t@mpdef{#1}%
    \@ppenddesc=\expandafter\expandafter\expandafter
                        {\expandafter\t@mpdef\the\@ppenddesc}}
\def\testf@nished{{\let\s@tparshape=\relax
    \s@thangindent}}
\def\inspicture{\t@stprevpict\p@ctbox
    \def@ultdefs                  % set default values
    \fr@msavetopict
    \iff@nished\else\testf@nished\fi
    \iff@nished\else
      \immediate\write16{Previes picture is not finished yet}%
    \fi
    \futurelet\N@xt\t@stoptions}  % look ahead
\def\t@stoptions{\let\n@xt\@neletter% scans parameters
  \ifx\N@xt l\p@ctpos=0\else                % picture on left inside par
   \ifx\N@xt c\p@ctpos=1\else               % picture copied centered
    \ifx\N@xt r\p@ctpos=2\else              % picture on right inside par
     \ifx\N@xt(\let\n@xt\e@tline\else        % number of full row above
      \ifx\N@xt!\@mbeededtrue\else           % just copy the picture
       \ifx\N@xt|\fr@medtrue\else            % picture will be framed
        \ifx\N@xt^\vt@ptrue\vb@tfalse\else  % picture as vtop
         \ifx\N@xt_\vb@ttrue\vt@pfalse\else % picture as vbot
          \ifx\N@xt\bgroup\let\n@xt\@ddgrouptodesc\else
           \let\n@xt\@dddescription % all parameters are read
  \fi\fi\fi\fi\fi\fi\fi\fi\fi\n@xt}
\def\e@tline(#1){\def\lines@bove{#1}% define number of full rows above picture
    \@ddedrowstrue
    \futurelet\N@xt\t@stoptions}
\def\@neletter#1{\futurelet\N@xt\t@stoptions} % eats one parameter
\def\@ddgrouptodesc#1{\@ppenddesc={#1}\futurelet\N@xt\t@stoptions}
\def\fr@medpict{\setbox\p@ctbox=
    \vbox{\n@llpar\hsize=\wd\p@ctbox
       \iffr@med\else\r@leth=0pt\fi
       \ifj@stbox\r@leth=0.4pt\fi
       \hrule height\r@leth \kern-\r@leth
       \vrule height\ht\p@ctbox depth\dp\p@ctbox width\r@leth \kern-\r@leth
       \ifj@stbox\hfill\else\copy\p@ctbox\fi
       \kern-\r@leth\vrule width\r@leth\par
       \kern-\r@leth \hrule height\r@leth}}
\def\@dddescription{\fr@medpict     %  adds description to the picture
    \re@lpictwidth=\the\wd\p@ctbox
    \advance\re@lpictwidth by\@therside
    \advance\re@lpictwidth by\b@tweentandp
    \ifhmode\ifinner\pr@ciseboxtrue\fi\fi
    \createp@ctbox
    \let\N@xt\tr@toplacepicture
    \ifhmode                         % if inside paragraph then use \vadjust
      \ifinner\let\N@xt\justc@py
      \else\let\N@xt\vjustc@py
      \fi
    \else
      \ifnum\p@ctpos=1               % if centered then just copy the box
        \let\N@xt\justc@py
      \fi
    \fi
    \if@mbeeded\let\N@xt\justc@py\fi % if forced to copy, just copy the box
    \firstp@sstrue
    \N@xt}
\def\createp@ctbox{\global\p@ctht=\ht\p@ctbox
    \advance\p@ctht by\dp\p@ctbox
    \advance\p@ctht by 6pt
    \setbox\p@ctbox=\vbox{%        % make new vbox
      \n@llpar                     % reset par's parameters
      \t@mpdimen=\@therside          % and set the leftskip and rightskip
      \t@mpdimeni=\hsize             %  according to the position of picture
      \advance\t@mpdimeni by -\@therside
      \advance\t@mpdimeni by -\wd\p@ctbox
      \ifpr@cisebox
        \hsize=\wd\p@ctbox
      \else
        \ifcase\p@ctpos
               \leftskip=\t@mpdimen    \rightskip=\t@mpdimeni
        \or    \advance\t@mpdimeni by \@therside
               \leftskip=0.5\t@mpdimeni \rightskip=\leftskip
        \or    \leftskip=\t@mpdimeni   \rightskip=\t@mpdimen
        \fi
      \fi
      \hrule height0pt             % invisible rule and nobreak penalty
      \kern6pt                     % small space above
      \penalty10000
      \noindent\copy\p@ctbox\par     % put the picture with marks
      \kern3pt                       % small space between pict and description
      \hrule height0pt
      \hbox{}%
      \penalty10000
      \interlinepenalty=10000
      \the\@ppenddesc\par            % follows text of description
      \penalty10000                  % prohibite page break
      \kern3pt                       % small space below
      }%
      \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.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

nlin0206042.extracted_bib
\begin{thebibliography}{999}

\bibitem{Book}
B.~Tilch and D.~Helbing: Evaluation of single vehicle data in
dependence of the vehicle-type, lane, and site, Traffic and
Granular Flow '99, Springer-Verlag Berlin Heidelberg 2000, 333 --
338
\bibitem{Krbalek1}
P.~Seba and M.~Krbalek: The statistical properties of the city
transport in Cuernavaca (Mexico) and random matrix ensembles, J.
Phys. A {\bf 33} (2000), L229 -- L234
\bibitem{Krbalek2}
M.~Krbalek, P.~Seba, and P.~Wagner: Headways in traffic flow -
remarks from a physical perspective, Phys. Rev. E. {\bf 64}
(2001), 066119 -- 066125
\bibitem{Derrida1}
B.~Derrida: An exactly soluble non-equilibrium system: The
asymmetric simple exclusion process, Physics Reports {\bf 301}
(1998), 963 -- 977
\bibitem{Derrida2}
B.~Derrida, J.~L.~Lebowitz: Exact large deviation function in the
asymmetric exclusion process, Phys. Rev. Lett. {\bf 80} (1998),
209 -- 213
\bibitem{Santen}
N.~Rajewski, L.~Santen, A.~Schadschneider, and M.~Schreckenberg:
The asymmetric exclusion process: Comparison of update procedures,
Journal of Statistical Physics, {\bf 92} (1998), 151 -- 194.
\bibitem{Gier}
J.~de Gier: Exact stationary state for a deterministic high speed
traffic model with open boundaries, J. Phys. A {\bf 34} (2001),
3707 -- 3720
\bibitem{Seba}
M.~Krbalek and P.~Seba: Headway statistics of public transport in
Mexican cities, to appear

\end{thebibliography}