Brno University of Technology, Czech republic
Pattern formation in traffic microscopic model
Apeltauer Tomas, Holcner Petr, Macur Jiri
Long-range patterns can be often observed in dynamic systems where only the short-range interactions between components exist. The progress in nonlinear dynamics theory within the past twenty years has helped us to understand the emergence of order in such complex systems. Nobel Prize winner Ilya Prigogine has coined the term ~Sdissipative structure~T for the ordered spatial patterns that emerge in dynamic systems. These patterns are independent on the geometry of system boundaries and often become time-dependent even when all the system parameters are time independent.
There are three necessary conditions for such pattern-producing systems: Energy flow between the parts (the system is driven sufficiently far from equilibrium), energy dissipation which allows the system to settle into definite structures, presence of nonlinearity in interactions between the parts of the system. All properties mentioned above can be found in single-lane microscopic models of traffic as well even in case of vehicles with identical properties.
In our paper we investigated behavior of such model from the point of view of nonlinear dynamic theory. Intelligent Driver Model (IDM) has been studied with respect to finite reaction interval (retarded interaction between vehicles). We used mean density of the vehicles as the main external parameter of the system.
During the extensive set of simulation experiments we detected following general properties of the system stationary behavior:
1. There is a certain critical mean density for which the spontaneous pattern (congestion) appears. As the mean traffic density further increases the new congestions are formed and the sequence of critical densities can be obtained (congestions are considered to be in steady state after some transient phase). The geometric shapes of these local density fluctuations are in case of simultaneous congestions very similar.
2. During the transient state the propagation rate of different congestions varied (group velocity) and congestions can interact. In steady state all congestions move upstream synchronized with the group velocity depending on the external parameter and the number of congestions.
3. In case of post-critical mean density a number of stable states can exist simultaneously (e.g. homogenous traffic flow, one or more stable congestions in the system). Only initial conditions arbitrate which state will be reached after transient phase.
4. The step changes of the system steady state depending on the mean density formed classical bifurcation diagram used in nonlinear dynamics (e.g. catastrophe theory).
5. There are some hysteresis phenomena during an increasing/decreasing external parameter ~V mean traffic density. If the system passes into a new qualitative state at some critical value of the growing mean density, the return to the original state proceeds generally at the lower value of the density ~V another typical property of the nonlinear system. All these features observed in simulation experiments retain generic character ~V they are resisting the perturbations of the original system. Even in case of non-identical vehicles with the various internal parameters (maximal velocity, acceleration, reaction time etc.) very similar results can be obtained.

Lab. Theoretical Physics, Univ. Paris-Sud XI, France
Velocity / time headway correlations and platoons: data analysis on an expressway
in collaboration with the LCPC (Laboratoire Central des Ponts et Chaussées), Paris, France
This study is based on single vehicle data collected by magnetic loops on an expressway located in the suburb of Paris.
Using time windows with a homogeneous traffic state, velocity correlations have been calculated, as a fonction of the time headway, for vehicles which are not necessarily nearest neighbors. We show that strong velocity correlations can be observed even for vehicles which are separated by six other vehicles.
The environment of vehicles with a short time-headway on the right lane has been studied. It turns out that those short time-headways cannot be interpreted only as a preparation for overtaking.

Lab. Theoretical Physics, Univ. Paris-Sud XI, France
Multi-lane cellular automaton : influence of aggressive driving
Xavier Faure-miller, Olivier Ratier, Philippe Ribagnac, Alain Schmutz, Olivier Sélébran, Jérome Senot, Qiang Wang (Ecole Polytechnique),
Cécile Appert-Rolland

A multi-lane cellular automaton inspired from [1] was developped to model highway traffic. A concept of "ghost car" was introduced in order to implement the lane changing manoeuvers. Aggressivity / Courtesy characteristics have been associated to the vehicles : an aggressive driver is supposed to change lane more often, to have a shorter reaction time, and to ignore the other drivers intentions. Courteous drivers adapt their behavior so as to facilitate the lane changes of other drivers.
The individual benefit of agressive driving in terms of travel time has been studied as a function of the total fraction of aggressive drivers, and of traffic states.
[1] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, ``Towards a realistic microscopic description of highway traffic'', J. of Physics A: Math. and Gen., {\bf 33}, L477 (2000).

BUSLAEV Alexander
Moscow State Automobile and Road Technical University, Russia
On stability of flow on a ring with three links
Buslaev A.P., Tatashev A.G., Yashina M.V.
The qualitative properties of systems on nonlinear differentional equations solutions that describing of traffic flow on a ring with three links are developed.
The necessary and sufficient conditions of system outcome to critical mode are searched out.

Faculty of Civil Engineering and Geosciences, Delft University of Technology
Improving pedestrian micro-simulations using an event-driven time step
M.C. Campanella, S.P. Hoogendoorn, and W. Daamen
This paper focuses on improving the quality of micro-simulation models in situations where pedestrian flow is very dense, implying that pedestrians may collide or already experience physical contact. We propose a hybrid approach to the discrete time step that combines traditional constant-time step with a variable event-driven time step. These different time steps allow different pedestrian behaviours to be employed selectively according to the density improving the overall quality of the simulation results.
Simulations using models in which the walker dynamics are described in terms of systems of differential equations (such as the social forces model and the NOMAD model) involve numerical approximations of these equations. This is usually done using explicit approximation schemes, which generally implies that the (constant) discrete time must be set very small in the order of fractions of the second to permit good results. This is even more aggravated when the density of pedestrians is very high or the simulated area is populated with small obstacles or present sharp corners. However, due to computation performance, practical applications must put a limit to the length of the time step. The consequence of the fixed time step is that in dense situations the collisions between pedestrians and between pedestrians and obstacles that may occur during the simulation step must be approximated. Another problem that arises with these models is due to the exponential formulation for the interaction between pedestrians and obstacles that these models also share. In very short distances the interactions grow tremendously generating unrealistic values for the acceleration of pedestrians in a time step.
To overcome these problems we propose a hybrid approach to the management of pedestrians (the algorithm responsible to determine how pedestrians perform their walking behaviour). The hybrid management will split the constant time step for those pedestrians near or in-collision into smaller collision events steps and apply a more adequate walking behaviour for this situation. In this research we propose four behavioural hypotheses that support the modelling of pedestrians in the event step as colliding particles in a similar manner as discrete granular flow models.
The implication of this proposal is a more refined assignment of pedestrian behaviours that are justified by behavioural hypothesis and do not require the introduction of maximum values of accelerations to prevent overshooting or any other parameterized approximation for short range interactions between pedestrians and obstacles. To validate the results of our hypothesis we compare a normative model with the hybrid pedestrian management and a normative model with constant time step with pedestrian trajectory data extracted from video recordings.
Keywords: pedestrian micro-simulation, social-force models, pedestrian trajectories video data.

INRIA, France
Road Traffic Models using Petri Nets and Minplus Algebra
Nadir Farhi, Maurice Goursat, and Jean-Pierre Quadrat
In this paper, we present a microscopic model based on Petri nets whose dynamic is written using minplus algebra [1]. Our approach is not far from the cellular automata approach. As in statistical physics, we study the individual movement of vehicles to derive macroscopic laws. The effectiveness of this type of model in determining fundamental traffic diagrams has been shown on a system without intersection [3].
First, we show that crossings with traffic lights can be modeled using time invariant Petri nets with negative multiplicities on some arcs. Indeed with negative multiplicities we can model tokens staying only a finite time in a place. This property is necessary to be able to represent the phases of the traffic light by tokens. An alternative way of modeling is to consider that the Petri net change with the phases but in this case the Petri net become time variant.
Second, we use a system point of view to build a large Petri net representing a regular city. A regular city is an american like, one-way parallel streets regularly spaced, one in a direction the next one in the other direction. To build the dynamics of a regular city we introduced a new system theory. Systems are seen as operators corresponding to Petri nets with named inputs and outputs. The dynamics of such systems can be defined by three standard matrices and three maxplus algebra matrices. On such systems some operations can be defined: parallel and series composition and different sort of feedbacks. Then a regular city dynamic is obtained by combining three elementary systems with these combinators. The dynamics of free choice Petri nets with negative multiplicites is also introduced, it is useful to model crossings with control lights.
Third, by simulation of the built system, the fundamental diagrams of regular cities can be computed using the maxplus algebra toolbox of Scilab. Some diagram examples are given. On these diagram appears clearly three phases which have been previously commented on simple systems see [2]. Lastly, the improvement of setting the traffic lights in feedback on the state of the traffic is shown.It is to avoid unbalanced stationary regimes which can be reached using traffic lights with an open loop.
[1] F. Baccelli, G. Cohen, G. J. Olsder, and J.P. Quadrat : ``Synchronization and Linearity'', Wiley, 1992 available
[2] N. Farhi, M. Goursat, and J.P. Quadrat : ``Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling'', In Proceedings of the 44th IEEE - CDC-ECC Seville December 2005.
[3] P. Lotito, E. Mancinelli, J.P. Quadrat : ``A Min-Plus Derivation of the Fundamental Car-Traffic Law'', IEEE-AC V.50, N.5, pp.699-705, 2005.

Zanjan University, Iran
Vehicular traffic flow at a non-signalised intersection
M. Ebrahim Foulaadvand and Sommayeh Belbaasi
We have developed a modified Nagel-Schreckenberg cellular automata model for describing a conflicting vehicular traffic flow at the intersection of two streets. No traffic lights control the traffic flow. The approaching cars to the intersection yield to each other to avoid collision. Closed boundary condition is applied to the streets. Extensive Monte Carlo simulations is taken into account to find the model characteristics. In particular, we obtain the fundamental diagrams and show that the effect of interaction of two streets can be regarded as a dynamic impurity located at the intersection point. Our results suggest that yielding mechanism gives rise to a high total flow throughout the intersection especially in the low density regime. In some ranges of densities, yielding mechanism even improves and regulates the flow in comparison to the absence of perpendicular flow.

Moscow Technical University for Radioengineering
Complex fundamental diagram and phase portrait of traffic flow in deep Lefortovo's tunnel (Moscow)
Ihor Lubashevsky, Cyril Garnisov, Boris Lifshits, Mikhail Pechersky
The basic properties exhibited by tunnel traffic are analyzed using the empirical data collected during the last two years in the deep long branch of Lefortovo's tunnel on the 3-rd circular highway of Moscow. This tunnel of length about 3 km is equipped with a dense system of stationary radiodetetors distributed uniformly along it chequerwise at spacing of 60 m. Because of the detector technical features traffic flow on the left and right lanes is measured at spacing of 120 m whereas on the middle lane the spacial resolution gets 60 m. The data were averaged over 30 s. Each detector measures three characteristics of the vehicle ensembles; the flow rate, the car velocity, and the occupancy for three lanes individually. The occupancy is an analogy to the vehicle density and is defined as the total relative time during with vehicles were visible in the view region of a given detector within the averaging interval.
The present work continues the investigations of tunnel traffic reported previously. First, the detectors themselves and their records were analyzed in a spacial way to justify the reliability of the collected data. Then based on the collected data the existence of cooperative vehicle motion is demonstrated again using now a more sophisticated analysis. In particular, the phase portrait of the vehicle ensemble dynamics on the occupancy-velocity plane is studied for individual detectors and lanes as well as for the tunnel as a whole. The observed drift fields in this plane demonstrate the presence of dynamical traps.
New features are found for the fundamental diagram constructed for the detectors and lanes individually as well as for the tunnel as a whole. Complex structures of various nature are obtained for the free flow state and jam. The phase of cooperative vehicle motion turns our to be rather homogeneous and separated from the two phases by sharp boundaries. We assume that these structures are due to the heterogeneity of city traffic for free flow and the depression of lane change maneuvers in jammed traffic.

GE Hongxia
NingBo University, China
Traffic anticipation effect in the lattice hydrodynamic model
H.X. Ge
The seminal work of Lighthill, Whitham and Richards (short for, LWR) on kinematic waves of vehicular traffic flow does not allow any fluctuation of speed around the equilibrium values, and not have the ability to explain the amplification of small disturbances on heavy traffic. In 1971, Payne introduced a high-order continuum traffic flow model including a dynamic equation to overcome the shortcomings in the LWR model. In order to analyze continuum model conveniently, Nagatani first proposed the lattice version of the Payne model. In the meantime the author incorporated the idea of the optimal velocity model proposed by Bando, that is, the tendency of traffic flow at a given density relaxes to some natural average flow. So the lattice hydrodynamic (short for, LH) model is an effective way of building bridge between macroscopic and microscopic model. Unfortunately the anticipation factor has been omitted in the lattice model. As we know the anticipation term reflects a driver's reaction to the vehicles in front, it is an important factor in traffic flow models and can not be omitted. We presented the anticipation lattice hydrodynamic (short for, ALH) model of traffic flow for the purpose of constructing a cooperative driving

Universidad de los Andes, Bogotá, Colombia
Statistical Behavior Of Domain Systems
Diego Luis Gonzalez, Gabriel Tellez
In this paper we study the statistical behavior of two out of equilibrium systems. The first one is a quasi one dimensional gas with two species of particles under the action of an external field, the second is a one dimensional spin system with nearest neighbor interaction also under the influence of an external driving force. The statistical behavior of these systems is compared with their counterpart in the coalescing random walk and the interacting random walk. In the gas and spin systems the length of domains satisfies the Wigner distribution which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices. At the end, we find that the statistical behavior of both gas and spin systems is not well described by the coalescing random walk, interacting random walk or circular orthogonal ensemble. Nevertheless a simple model of independent domains describe more closely the statistical behavior of these systems.

Institut fuer Theoretische Physik Koeln
driven diffusive systems with sitewise disorder and open boundary conditions
Philip Greulich, Andreas Schadschneider
We investigate driven diffusive systems with open boundary conditions in presence of defect sites with reduced hopping rate. These systems are characterized by a decrease of the maximum current in comparison to pure systems. We are especially interested in randomly distributed defects (disorder) and their influence on the phase diagram and the phase transitions. The models are biologically motivated by intracellular transport processes of motor proteins attached to microtubuli. Defects on the microtubuli, e.g. due to other attached molecules, can impede forward movement of the motors. Those molecules can be protein agglomerations caused by diseases (e.g. Alzheimer disease) or drug molecules. Knowledge of the relation of the maximum current (transport capacity) and the concentration of defect molecules in intracellular transport systems can be of greater relevance in medicine and pharmacology.

Delft University of Technology, the Netherlands
Generic Driving Behavior Modeling by Differential Game Theory
Serge P. Hoogendoorn and Saskia Ossen
In the last few decades, several driving models have been put forward based on the analogy with self-driven many particle systems. Examples of such models are the social-forces type models for car-following behavior, the Ideal Driver Model, and the MOBIL model describing lane-changing behavior.
Although these models can describe many phenomena in motorway traffic flow operations, a clear behavioral foundation is lacking. This paper puts forward a new generic theory of driving behavior, based on the principle of least effort. In this theory, drivers are assumed to minimize the predicted effort of their control actions, including acceleration towards the free speed, car-following and lane changing. In this game-theoretic approach, drivers may or may not anticipate the reactions of the other drivers on their control decisions. Furthermore, both non-cooperative and cooperative driving rules will be discussed.
We present the main behavioral assumptions, the model derivation, and the resulting car-following and lane changing models. We also show the analogy and differences between the proposed model and other driving models, and discuss under which behavioral assumptions the proposed model reduces to well known driving models based on principles from physics (e.g. social forces). Finally, other model properties (such as the fundamental diagram and emergent lane distributions) will be presented.

HUANG Wei-Neng
Department of Physics, Chung Yuan Christian University, TAIWAN
A model for city traffic in rush hour
Wei-neng Huang
We propose a cellular automaton to simulate the city traffic in rush hours. Vehicles on road are treated as particles hopping on lattice. In the morning rush hours, downtown parking lots are taken as the sinks for particles moving in from suburban; in the afternoon rush hours, the parking lots become the sources for particles moving out of town. The traffic is regulated by the operation of a series of traffic lights. Both the traffic flow (from global perspectives) and the travel time (from personal perspectives) are studied. As the traffic patterns response to the traffic lights, the phase transition is defined by the pattern shift. The influence of various types of noise is also analyzed.

Jiaotong University, Beijing, P.R.China
Traffic behaviors of mixed bicycle system in the multi-value cellular automata model
Bin Jia, Xin-Gang Li, Rui Jiang and Zi-You Gao
Traffic problems have attracted much attention in the past few decades. Traffic flow is a kind of many-body system of strongly interacting vehicles, and it can perform very complex behaviors. Many theoretical models have been proposed to explore the mechanisms in traffic flow. Among themwe will focus on cellular automata (CA) models. CA has become an efficient tool for simulating traffic flow, for it is conceptually simpler and can be easily implemented on computers for numerical investigations.
Recently, Nishinari and Takahashi have proposed a family of multi-value CA models. Different from previous ones, in these models, each site is assumed to hold $L$ vehicles at most. The basic of the family is obtained from an ultradiscretization of the Burgers equation, so it is called Burger cellular automata (BCA). If it is assumed that the road is an $L$-lane freeway, then the model can describe the multi-value traffic without explicitly considering the lane-changing rule.
The maximum velocity of vehicles in BCA is 1, and Nishinari and Takahashi have extended BCA for the case of maximum velocity 2 and presented extended BCA (EBCA) models. In our previous work, the EBCA models(EBCA1 and EBCA2) are used to model the bicycle flow, and the stochastic randomization is introduced in the models.
In real traffic, bicycles are not with the same maximum velocity due to differences in the personalities of riders. For example, young riders always tend to ride with high speed, while the older usually ride with low speed. To this end, the mixed bicycle flow is investigated using the EBCA1 model. Both the deterministic case and the stochastic case are studied.
The simulation results show that in deterministic case, the multiple states effect exists both in free flow region and congested region. The site which contains the largest number of slow bicycles forms a moving bottleneck to the system and the flux is constrained by this bottleneck. The simulation results are consistent with the analytical ones. To analize the multiple states effect exists in free flow region, the spacetime plots are also investigated
In stochastic case, the multiple states effect disappears in free flow region when $p_s$ greater than $0.0$, and it disappears in congested region when $p_f$ greater than $0.0$. The multiple states effect disappears when both slow bicycles and fast bicycles are randomized.
In out future work, we will extend the EBCA2 model to study the mixed bicycle flow and compare the results of the two models. The calibration and verification of the models using real bicycle flow data will also be performed.

KAUSHIK Inderjeet
Institute of Technology, Banaras Hindu University, Varansi(UP), India
Modeling Traffic Arrival Patterns on a Cross Section of Highway
Inderjeet Kaushik, Anurag Ohri
The dependence of vehicle arrival patterns at a section on a multilane highway necessitates the use of models which describe vehicle arrivals patterns on highway. The assumptions of arrival rate of pattern may lead to inaccuracy in a number of design applications like queue models for predicting the speed-flow relations. In this paper an attempt was made to model the arrival pattern of vehicles in terms of probability distribution which depends upon the traffic volume.

KIRIK Ekaterina
Institute of Computational Modelling of Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia
Influence of model parameters to people behavior simulated by intelligent flour field CA model
Ekaterina Kirik and Tat'yana Yurgel'yan
A stochastic cellular automation (CA) model to simulate pedestrian dynamics is presented. The model obtained takes its inspiration from floor field (FF) model~\cite{ExtFFCAMod}. It uses idea of providing pedestrian with weighed map that shows a distance to a target point (exit, for instance) and realizes pedestrian's environment analysis~\cite{MalStep} as well. Regular situations imply that pedestrians analyze environment and choose their route more carefully. Pedestrians minimize efforts to reach their destinations: feel strong aversion to taking detours or moving opposite to their desired direction, however, normally choose the fastest rout (but not the shortest). Pedestrians keep a certain distance from other people and obstacles. As well as it's necessary to take into account that some effects are reside for certain regions in a space, e.g., clogging situations are more pronounced in the nearest to an exit regions. It implies that model parameters have to be at least spatially adapted. The aspects mentioned make simulation of pedestrians behavior realistic. And model obtained is made to reproduce such aspects and named ``intelligent flour field CA model''.
One part of model investigation is devoted to the following. We studied influence of model parameters to a type of people behavior that simulated by the model. From one side this study shows the model intuitive validity. From the other hand we saw the connection of parameters values and styles of particles moving.

KNOOP Victor
Delft University of Technology, the Netherlands
Impact of data averaging methods on macroscopic traffic flow characteristics
Victor Knoop, Serge Hoogendoorn, Henk van Zuylen
In gaining insight into the characteristics of traffic flow, local measurement techniques (e.g., inductive loops) are the most common source of information. To determine macroscopic flow characteristics, aggregation of the microscopic information (i.e. speeds) over a specific period of time (e.g., 1 minute or 5 minutes) is required.
In most cases, time-mean averaged information is determined and used for analyzing the macroscopic flow properties. However, it is well known that averaging over a time period leads to overestimation of the influence of faster vehicles, and consequently to an overestimation of the mean speed. As a direct result, densities, computed directly from flow and speed, are underestimated substantially.
This paper discusses the impact of not correctly averaging the local individual vehicle measurements. We compare the time mean speed and space mean speed, using data of individual car passages on a motorway road stretch. We show that the differences between time-mean and space mean averages are substantial, up to a factor 4. In particular in the lower speed regions the error is big. We also show that this has considerable consequences for the jam density and shock wave speed, and that when correctly averaging the microscopic data, one can get a much better fit of the fundamental diagram.

LI Xing-Li
Shanghai Institute of Applied Mathematics and Mechanics, P.R. China
Jamming transition in extended cooperative driving lattice hydrodynamic models considering effect of looking backward on traffic flow
Phase transition and critical phenomenon are investigated in extended cooperative driving lattice hydrodynamic models analytically and numerically. The two extended cooperative driving lattice hydrodynamic models of traffic are proposed by incorporating intelligent transportation system and the backward looking effect under certain condition in traffic flow. They are the lattice versions of the hydrodynamic model of traffic: the one (model A) is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other (model B) is the difference equation in which both time and space variables are discrete. Based on the real traffic situation, the appropriate forward and backward optimal velocities are selected, respectively. Then the stability criterions are investigated by the linear stability analysis with finding that new consideration theoretically lead to the improvement of the stability of traffic flow, and the validity of our theoretical analysis is confirmed by direct simulations. The simulation results show that the traffic jam could be suppressed efficiently via taking into account the information about the motion of two preceding vehicles and one following vehicle. In addition, the jamming transitions among the freely moving phase, the coexisting phase, and the uniform congested phase are studied by using the nonlinear perturbation methods. The traffic jam is described by the kink solution of the modified Korteweg~Vde Vries (mKdV) equation. The theoretical coexisting curve is in good agreement with the simulation result.

A.M.Prokhorov General Physics Institute, Moscow, Russia
Phase transitions in traffic flow caused by dynamical traps
Ihor Lubashevsky, Julia Hinkel, Namik Gusein-zade, Reinhard Mahnke
Phase transitions in traffic flow induced by dynamical traps are under consideration. The notion of dynamical traps is introduced to describe difference in the driver control over the relative velocity and headway. This effect manifests itself in the substantial increase of the driver delay time when the relative velocity becomes less than the driver threshold for recognizing small relative speeds. Its value is estimated as 1-2 m/s using the available empirical data on driver action points.
Drivers can control the relative speed within high accuracy in comparison with that of headway distance. So when the relative velocity becomes rather small it takes for drivers much longer time to recognize the necessity of correcting the headway. As a result the human activity in driving is relatively depressed for small relative speeds, which is the implementation of the driver bounded rationality.
By way of example, two problems are analyzed. The former one is the car following problem with dynamical traps. The leading car is assumed to move with a fixed speed, motion of the following car is described within the Bando model, where the delay time exhibits essential increase as the relative speed tends to zero. The acceleration of the following car is also affected by additive white noise. It is demonstrated numerically that in this case the car motion can become unstable and exhibit anomalous properties such as a bimodal distribution of the headway, the Laplace distribution of the relative velocity, long time correlations, etc.
The second problem is the dynamics of car ensemble on a circular one-lane highway that is affected by dynamical traps. The car motion is again simulated numerically within the Bando model with the delay time depending on the relative velocity in the same way. It is found that in addition to the previous effects dynamical traps can induce self-organized criticality in the car motion.

MAHNKE Reinhard
Rostock University, Germany
Understanding Traffic Breakdown: A Stochastic Approach
Reinhart Kuehne and Reinhard Mahnke
Observations from different highways indicate that traffic breakdowns do not necessary occur at maximum flow and can occur at flows lower or higher traditionally accepted as capacity value. The notion breakdown means a stochastic transition from an uncongested traffic state to a congested state which is compariable wih a phase transition in nature between different states of matter.
With the balance master equation it is feasible to calculate the dynamics of traffic pattern formation especially the first passage time distribution.

MENG Jianping
Shanghai Institute of Applied Mathematics and Mechanics, China
Lattice Boltzmann model for traffic flow
Meng Jianping, Qian Yuehong, Li Xingli, and Dai Shiqiang
Mesoscopic models for traffic flows are usually difficult to be employed because of the appearance of integro-differential type Boltzmann equations in the models. In this work, a lattice Boltzmann model for traffic flow is established on the basis of the existing kinetics models by introducing the BGK-like interaction term in the Boltzmann equation and discretizing it in time and phase-space. Because of its discrete nature, the proposed model could be easily used to simulate traffic flow numerically. In order to validate the model, some concrete numerical simulation are conducted under the periodic boundary condition. It is found that the obtained fundamental diagram agrees quite well, even quantitatively, with the realistic one. Meanwhile, the most basic and important physical phenomena could be explained, such as the transition phenomena from the free flow to the congested flow, as is observed in the real traffic and in the Nagel-Schreckenberg cellular automaton(NaSch) model. Furthermore, compared with the NaSch model, the present model has much higher computational efficiency and that the parameters of the model are physically meaningful.

Meijo University, Japan
Detailed data of traffic jam experiment
Minoru Fukui, Katsuya Hasebe, Macoto Kikuchi, Akihiro Nakayama, Katsuhiro Nishinari, Yuki Sugiyama, Shin-ichi Tadaki, Satoshi Yukawa
We show detailed data of two traffic jam experiments on a circuit. In the experiments, a traffic jam emerges spontaneously without any artificial bottlenecks and any instructions to drivers. Positions of cars are obtained from video records every 1/3 second, and other quantities are calculated from these data. We found the power law nature in time series of average velocity, and also found the homogeneous flow with large velocity is temporarily made before a jam cluster is formed.

Université de Cergy-Pontoise, France
An integrated transportation, land-use and demographic model: Application in Paris region
André DE PALMA, Kiarash MOTAMEDI, Hakim OUARAS, Nathalie PICARD
Our project aims to develop a model of interactions between transports, land-use and demography. We use three models: UrbanSim which is a land-use model, METROPOLIS which is a dynamic traffic model and a demographic model. The system will run on disaggregated data, the economic agents are households and jobs. The Dynamic simulation uses a one year stage for UrbanSim and the demographic model and three years for the traffic model. The period of calibration goes from 1990 to 1999, the period of the simulation from 1999 to 2026. The system is led at two levels, global and local. Global, because it takes into account the environmental and political constraints and accessibility in transport. Local, because it takes into account the urban features, distribution of the population and the price of the real estate in each cell. The location choices of the households and jobs are estimated by the discrete choice models (logit). It is the same for the probability of transition between the various urban types. While the model of the real estate price, it is modeled like a hedonic regression. To make evolve the population (demographic model); we use the micro-simulation approach on individuals and households levels. This project will be used to forecast the 2026 land use of Paris Region as well as the distribution of households and jobs among cells.

ROGSCH Christian
University of Wuppertal, Germany
Prediction accuracy of evacuation times for high-rise buildings and simple geometries by using different software tools
C. Rogsch, W. Klingsch, A. Seyfried
Evaluation and optimization of emergency systems can be done by means of several engineering methods, which are entirely different: macroscopic hydraulic models, which can be calculated by hand, and microscopic computer simulation methods. Both allow forecasting of evacuation-times for various settings. The authors compare results of four commercial software-tools (ASERI, buildingEXODUS, PedGo, Simulex) and some macroscopic hydraulic models with real evacuation trials in high-rise buildings. Furthermore evacuation times for simple layouts of rooms, floors and stairs are calculated and the results are compared against each other.
The analysis shows that chosen software tools are able to predict total evacuation times for such kind of buildings. However by taking a closer look at evacuation times for selected floors, calculated results differ from the results of the trials. To get a deeper insight we decided to proof the software tools by means of simple layouts. Thus compensation effects are excluded and it is possible to recognize weak points of the software and the implemented algorithm. The comparison of the evacuation times reveal differences up to a factor of four among the tools, a problem which is probably based upon the update algorithm used for description of individual movement of pedestrians. Moreover we found problems in the definition of boundary conditions. Due to the simulation of a simple room geometry with different boundary conditions we achieved results differing up to 20%. Thus it is possible, that only compensation effects provide the correspondence between calculated evacuation times for high-rise buildings and times obtained by the trials. In particular it is obvious that simulated pedestrian flow inside the building is affected by non-negligible uncertainties and simple macroscopic hydraulic models for the prediction of total evacuation times are as good as more detailed models.
For software-users it is crucial to know this weak points if they are planning evacuation-routes in buildings or analyzing evacuation quality for large numbers of people. Our investigations will help to assess simulation results in a better way by knowledge of strengths and weaknesses of commercial software tools for evacuation calculation.


Similarity between temporal and spatial structures in pattern formation in dissipative non-equilibrium system - Analytical explanation by an exactly solvable model for traffic flow -
Rintarou Akiyama, Hiroshi Watanabe, Yuki Sugiyama
Similarity between the temporal structure of emerging patterns and the spatial structure of stable patterns is observed in nature, such as biological development, growth of trees, forming livers, etc. These phenomena can be originated in a general property in dissipative non-equilibrium systems. In traffic flow, the same aspect is observed in the relation between forming a jam in time development on circle road and the stable sequence of patterns in jam flow upstream of a bottleneck. The Optimal Velocity (OV) model succeeds in forming a jam as a traveling cluster solution in dissipative non- equilibrium system. The delayed-OV model, which is a first-derivative equation with time delay, shows the same behavior in forming a jam. Fortunately, the model is exactly solvable and traveling cluster solutions are described by elliptic functions. Cluster solutions are characterized by modulus parameter in elliptic function. We can explain the similarity in pattern between temporal and spatial structures by the fact that the instabilities of cluster solutions are dependent on modulus parameter.

Institute of Physics, Bratislava
Microcanonical and canonical approach to trafic flow
Anton Surda
Traffic flow of a system of identical cars on a single-lane road is treated as a microcanonical ensemble. Behaviour of the cars is described by the probability of car velocity at given headway and velocity of the car ahead. The calculations are performed on a discrete 1D lattice and the car velocities are taken also discrete.
Number of states (entropy) at fixed size of the system and total velocity is calculated. Dividing a large system into subsystem and reservoir and allowing the length of the group to vary, a canonical approach is developed when the reservoir of cars exerts a "pressure" on the subsystem. Thermodynamic properties of the subsystem of cars are calculated and a phase transition between free flow and jammed phase is found. The jammed phase appears when the density of cars is high, ability of cars to decelarate low, or fluctuations of car velocities is large.
Our description of a system of cars differs from the standard approach of statistical mechanics, so that conserved energy is replaced by non-conserved car velocity, and the states of the system are not equally probable as is the case in statistical physics.

Dept.of Aeronautics and Astronautics, Tokyo, Japan
A public conveyance model and analysis on clustering of vehicles
Akiyasu Tomoeda, Katsuhiro Nishinari, Debashish Chowdhury and Andreas Schadschneider
A new public conveyance model (PCM) applicable to buses and trains is proposed in this study by using stochastic cellular automaton. We apply our PCM to the bus system and measure the efficiency of the system. We have found the optimal density of vehicles, at which the average velocity becomes maximum, significantly depends on the number of stops and passengers behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed by comparing the different behavior of passengers. By using mean field analysis, we estimate the average velocity and the waiting passengers in the low density limit. We have obtained the theoretical results which are in good agreement with numerical simulations. It is also observed that clustering of vehicles which is caused by the time delay effect of passengers when they get on a vehicle. We have found that the big cluster of vehicles is divided into small clusters, by incorporating information of the number of vehicles between successive stops. This result shows that the bus system is more effective by introducing information.

Institute for Transport and Economics, Dresden, Germany
Modelling and simulating several time-delay mechanisms in human and automated driving
Martin Treiber and Arne Kesting
In vehicular traffic, reactions to new traffic situations are subject to several mechanisms of time delay. Besides the reaction time of the drivers (or sensors), the finite acceleration capabilities lead to a nonzero ``velocity adaptation time'' to perform the action itself (e.g. changing the velocity). The commonly used explicit integration schemes for simulating the models introduce the update time as a third delay parameter. By means of numerical simulations with a time-continuous car-following model, we investigate how these times interplay with each other and influence the instability mechanisms effective for a platoon of vehicles. We show that the stability properties essentially depend on these times normalized to the time gap to the leading vehicle. Moreover, the long-wavelength string instability is mainly driven by the velocity adaptation time while short-wavelength local instabilities arrise for sufficiently high reaction and update times. Remarkably, the numerical update time is dynamically equivalent to about half the reaction time. We show that, with respect to stability, there is an `optimal' adaptation time as a function of the reaction time and draw implications for human \textit{vs.} semi-automated driving.

Delft University of Technology, The Netherlands
A new multi-class first order traffic flow model
Hans van Lint, Serge Hoogendoorn
The heterogeneity of traffic, e.g. due to different vehicle types and drive styles, constitutes a significant if not dominant factor in accurately modeling freeway traffic flow operations. For example, high truck percentages may induce congestion at much lower volumes and hence result in very different network traffic conditions than low track percentages. This implies that traffic models for real time decision support systems in traffic management centers should provide the means to account for traffic heterogeneity.
This paper presents a new multi-class first order traffic model (FASTLANE~Y), which provides these means and is suitable for simulating heterogeneous traffic on (large) freeway networks. It differs from earlier multiclass first order macroscopic traffic models in that it calculates the dynamics in terms of person car equivalents. This allows FASTLANE~Y to be numerically solved by a new efficient and stable Godunov-based solver, while maintaining a dynamic and realistic representation of class-specific flows and densities throughout the network. In a real testcase the paper demonstrates the workings of FASTLANE~Y under different truck percentages and different conditions (regular congestion and a large incident).

WEBER Daniel
University Duisburg-Essen, Germany
Microsimulation of Urban Traffic
Daniel Weber, Stephan Nies, Michael Schreckenberg
We present a microsimulation of urban traffic derived from the Nagel-Schreckenberg cellular automata model. The simulator was developed to provide information about the traffic state and link travel times in the cities of the Ruhr area. The simulation integrates different kinds of data, like current traffic measurements from loop and infrared detectors, historical traffic data, programs of traffic signals, and information about road blockings. We describe the model dynamics, the technical layout of the whole system, and show results of the validation of the model by floating car data.

Lulea University of Technology, Sweden
Thermodynamics of Traffic Flow
Hans Weber, Reinhard Mahnke, Jevgenis Kaupuzs
Application of thermodynamics to traffic flow is discussed. On a microscopic level, traffic flow is described by Bando's optimal velocity model in terms of accelerating and decelerating forces. It allows us to introduce a kinetic, potential, as well as a total energy, which is the internal energy of the car system in view of thermodynamics. The total energy is however not conserved, although it has a certain value in any of the two possible stationary states corresponding either to a fixed point or to a limit cycle solution in the space of headways and velocities.
On a mesoscopic level of description, the size $n$ of a car cluster is considered as a stochastic variable in the master equation for the system. Here $n=0$ corresponds to the fixed--point solution with no cluster of the microscopic model, whereas the limit cycle is represented by the coexistence of a car cluster with $n$ strictly positive and a free flowing phase.
The stationary solutions obey a detailed balance condition, which allows us to describe some properties of the model by equilibrium thermodynamics in analogy to the liquid--vapour system. We define the free energy and the chemical potential of the car system. In this sense the behaviour of traffic flow can be described by equilibrium thermodynamics in spite of the fact that it is a driven system. We find, however, that the chemical potential of the cluster phase of traffic flow depends on an outer parameter -- the density of cars in the free--flow phase. It allows to distinguish between the traffic flow as a driven system and purely equilibrium systems.

Unversitaet Duisburg-Essen, Germany
Exact solutions for simple models of traffic flow
Marko Woelki, Michael Schreckenberg, and Andreas Schadschneider
We investigate simple models for one-dimensional vehicular traffic that can be regarded as generalizations of the asymmetric exclusion process (ASEP) to increased maximum speed. In contrast to the well-known Nagel-Schreckenberg model for highway traffic, we consider models without temporal memory, i.e. the rules for cars to move at a given time-step only depend on the actual configuration of cars (and not on their velocities for instance). We present exact expressions for the probabilities of configurations as well as correlation functions in the steady state using the so called matrix-product ansatz.

Department of Aeronautics and Astronautics, Tokyo, Japan
Simulation and Mean Field Analysis on Pedestrian Dynamics around Exit
Daichi Yanagisawa, Katsuhiro Nishinari
Pedestrian dynamics has received growing interest over the last decades. Many intersting models are introduced and studied. Floor field model is one of such a model, which is a cellular automata model and reproduce characteristics of pedestrian dynamics such as arch phenomenon and lane formation. When we simulate an egress behavior from a room, pedestrians move similarly whether they are near the exit or not in floor field model. In reality, pedestrian move fast if they are far from the exit, but move slowly around the exit since they avoid collisions with other pedestrians. The new parameter which represents this phenomenon is introduced to floor field model. We studied egress processes by simulation and 4 clusters mean field analysis and obtained an expression of an average flow of pedestrians going out from the exit when they move to the exit directly. Three interesting phenomena are discovered by our expression. First, we find the most suitable pedestrian behaviour according to the width of the exit. Secondly, there are different optimal positions of the exit depending on the pedestrian behaviours. Thirdly, there is a different optimal gap between two exits depending on the pedestrian behaviours.

ZHU Hui-bing
Shanghai Institute of Applied Mathematics and Mechanics, China
A density-dependent NaSch model for traffic flow controlled by traffic lights
H.B.Zhu, S.Q.Dai
Traffic flow controlled by the traffic lights on a single-lane roadway is studied by using a modified NaSch model which has been proposed by the authors. Simulation results has shown that this modified NaSch model with the density-dependent randomization (abbreviated as the DDR model) has the ability to capture the essential features of traffic flow, e.g., synchronized flow, metastable state, hysteresis and phase separation at higher densities. In this paper a signal model based on the DDR model is proposed. A single light situation is selected on account of the fact that investigating the simpler problem of a single road with one traffic light is sufficient to give appropriate results and to adjust the optimal traffic light periods. Simulation results show that this signal model is able to reproduce the characteristic features of traffic flow with a traffic light. There are three different traffic density regions in the flow-density diagrams when cycle time is large and four traffic density regions when cycle time is small. This phenomenon is different from that using the NaSch model and the underlying mechanism has been analyzed. Further it is found that mean traffic flow strongly depends on the cycle time of traffic lights, i.e., strong oscillations are observed in the flow both for lower and higher densities. Similarly the mean velocity also depends on the cycle time and reaches its maxima and minima at regular distances. On the basis of above investigations an application of the signal model on optimizing traffic lights in order to enhance the global throughput is studied. All the results indicate that the presented model is reasonable and has realistic meaning. Although the model is simulated in a single-lane road with one traffic light, it is possible to apply it to complex highway topologies and a series of traffic lights. Moreover it is possible to apply it to optimize traffic lights in a more realistic fashion and to provide some theoretical reference for traffic engineering.