ENTPE/INRETS, Vaulx-en-Velin, France
Replications in stochastic traffic flow models: incremental method to determine sufficient number of runs
Nicolas CHIABAUT, Christine BUISSON
This paper tackles the issues of the minimal and sufficient number of replication needed to evaluate correctly the mean value of a stochastic simulation results but also the shape of the results' distribution. Indeed, stochasticity is more and more widespread in traffic flow models.
On the one hand, microscopic models try to reproduce inter-vehicular deviation through stochastic algorithm. Distributions are sources of randomness. Even if many articles discuss the need for a certain number of simulations to obtain reliable results, they seldom if ever suggest a way to determine this number. Different simulations runs can produce various results due to a randomly assignment of desired speed of each car for example.
On the other hand, most of macroscopic models are totally deterministic such as the Lighthill-Whitham-Richards (LWR) model. This lack prevents them from representing some traffic phenomena as roundabout insertion; lane-changing; various desired speed... Stochasticity can overcome those weaknesses. Recently [1] presents a microscopic Lagrangian solution of LWR model allowing individual fundamental diagram.
Thereby, numerous runs have to be computed to estimate the mean value of a measurement of effectiveness (MOE) but also to test if the results come from a particular stochastic process. The knowledge of the whole distribution allows us to determine every percentile needed (for example the 5% worst situations). The aim of this paper is to propose a way to identify such distributions and to estimate the minimal number of replications that one should make to obtain a given confidence level. We will focus on car-following component of models while lane changing, insertion (ramps, roundabout), and assignment will not be considered. However the proposed methodology can be applied to any components of traffic models.
Section 1 will first present the role of distributions in traffic simulation and will focus on what the experimental processes of measurable stochastic variables are and how various microscopic models take those distributions into account. Section 2 will briefly present the Lagrangian resolution of the LWR model based on a GODUNOV scheme. Then, section 3 will explain the incremental method chosen based on three statistical tests: the STUDENT test, the JARQUE-BERA test and the KOLMOGOROV-SMIRNOV test [2,3]. Section 4 will illustrate the proposed framework for the particular case of a stochastic simulation model based on Lagrangian resolution of the LWR model and analyze the results obtained.
[1] LECLERCQ, L. and LAVAL, J. and CHEVALLIER, E. 2007. The lagrangien coordinates and what it means for first order traffic flow models, from 17th International Symposium on Transportation and Traffic Theory. accepted for presentation, London, U.K..
[2] BURGHOUT, W. 2004. A note on the number of replication runs in stochastic traffic simulation models. Technical report. Center for Traffic Research.
[3] LAW, A. M., and KELTON, W. D. 2000. Simulation, Modeling and Analysis. McGraw-Hill. Boston.
[4] BRACKSTONE, M. and McDONALD, M. 1999. Car-following: a historical review. Transportation Research Part F: Psychology and Behaviour(2), pages 181-196.
[5] CHIABAUT, N. and BUISSON, Ch. 2006. Towards a determination of the minimal number of replications for stochastic traffic models, from International Symposium of Transport Simulation 2006. Lausanne, Ch..
[6] SAPORTA, G. 2006. ProbabilitÚs, analyse des donnÚes et Statistique. Editions TECHNIP. Paris.

Delft University of Technology
"Faster is slower" effects and capacity drop of revolving doors
Winnie Daamen, Serge Hoogendoorn, and Ramon Landman
Revolving doors are mainly used to control the climate inside a building, while maintaining the building's accessibility. The performance of revolving doors depends on the interaction between their characteristics, the pedestrian flow and individual pedestrian walking behaviour. In order to increase the insights into the performance of revolving doors controlled walking experiments have been performed. In nine experiments, door rotation speed, pedestrian flow composition, and pedestrian load have been varied. Using cameras, detailed pedestrian trajectories have been gathered, coupled to individual pedestrian characteristics.
One of the often observed characteristics of pedestrian flows is the "faster is slower" effect: when individual pedestrians aim to go faster or are more in a hurry, the total system performance drops. Analyses of the trajectory data from the revolving door experiments show that this "faster is slower" effect also occurs upstream of revolving doors. Pedestrians with a high walking speed overtake slower pedestrians and form passing lanes. Just upstream of the bottleneck, these fast pedestrians enter the flow, causing severe congestion.
Another aspect of the "faster is slower" effect is related to the rotation speed of the revolving door. Because of safety reasons, the revolving door stops when someone enters one of its safety sensors. Faster pedestrians are more likely to move into the safety sensor area (less cautious behaviour), thus causing more stops. The same effects are observed in higher densities, leading to lower average rotation speeds. This paper presents a relation between average rotation speeds of revolving doors and the number of pedestrians in the approach area.
A third aspect discussed in this paper is the capacity drop of a revolving door. The throughput of a revolving door is smaller when congestion occurs than in free flow conditions. In car traffic, similar effects are observed. However, the cause of the capacity drop is different. While in car traffic capacity drops are caused by changes in driver behaviour (smaller headways and better anticipation), capacity drops for revolving doors are caused by an increased number of revolving door stops.
A strong correspondence can be found to car traffic: when the speed variance increases, both the probability of traffic breakdowns and the probability of capacity drops increase.

INRETS (Arcueil, France)
Risk index model for motorway traffic
Habib HAJ SALEM, Jean-Patrick LEBACQUE
1. Introduction
In the field of safety analysis, the classical traffic evaluation approaches consist in collecting incident/accidents traffic data during the experimented scenarios (traffic control strategies, modification of the infrastructure etc.), and in proceeding to traffic impact and statistical safety analysis of the number of accidents before and after the implementation of these scenarios. Generally, the collection of the accident numbers must get a statistical significance before undertaking an evaluation process. This remark imposes a long time of field data collection (3-5 years), which is the "price to pay" for having a correct safety evaluation.
This paper aims at developing a risk index based on real-data measurements, which can be used either off-line as an evaluation index during the evaluation process leading to the dramatically reduction of the field test periods, or in real-time like: a safety monitoring tool (e.g. safety user warning system), a multi-criterion function to be optimized in real time (safety index combined with a traffic index) within several control strategies such as coordinated ramp metering, speed limit control, route guidance etc.
The developed index is based on the collection of measured traffic data synchronized with incident/accidents data on two sites in France: the ring way of Paris and the A6W motorway toward Paris direction.
2. Data base characteristics
For the two sites, traffic dataset and accident characteristics are collected from historical database stored in the SISER/SIRIUS and the Ville de Paris operating system. The considered sites are fully equipped with real traffic measuring sensors located at around every 500 meters apart. The incidents/accidents data characteristics include: time of day, location of the accident, weather conditions, severity. The collected traffic data covers 2 hours (one before and one after the crash) at two upstream and two downstream measurement stations. The time intervals of the traffic measurements are equal to six minutes and one minute, for the SIRIUS and the Ville de Paris systems respectively.
The final constituted database includes the overall accidents occurred and traffic data during 4 years (2002-2004) and 6 month for A6W. The total number of accidents collected is around 900 on the ring way of Paris, whereas on A6W this number is around 20.
3. Methodology
The applied methodologies are mainly based on first order traffic flow modelling and statistical analysis of the traffic conditions before the accident. A series of multivariate statistical methods are used, with the aim of finding the relationship between the accident and the traffic conditions. Two well-known statistical methods are applied: cluster analysis and the most common form of factors analysis [3]. In particular, the principal components analysis is applied to find the non-correlated variables to be used for building the risk model. In our case, the total number of variables characterizing the dataset is equal to 4(stations)*2(volume, occupancy rate)*4 (number of lanes) = 32 variables. For the clustering analysis, several possibilities are investigated:
 Clustering by upstream occupancy rates/lane
 Clustering by downstream occupancy rates/lane
 Clustering by all occupancy rates/lane
The same clustering method is applied for the measurement stations including the four lanes. Lastly, based on the clustering output results, linear regression and non linear logistic modelling approaches are applied for computing the risk index.
The hierarchical ascending clustering via SAS is performed, using a Ward┬s criterion [2], in order to exhibit the particular class of traffic conditions which prevail at the time just before the accident.
Screening the time evolution of the clusters (one hour before the crash), the results obtained demonstrate that among all found clusters a critical cluster leads (48% in total) to the occurrence of the accident. Consequently, the risk modelling is based on the traffic state of this cluster.
3.1.1. Logistic regression
The database is split into two parts. The first half is dedicated to the calibration of the linear regression using SAS tool. The second half is used for the validation of the found risk model. During the calibration process, the results given by the clustering are used. The logistic regression is performed by considering that cluster 3 presents the highest level of crash. In this case, the risk model value is set to 1, otherwise to 0.
In order to simplify the model and its interpretation, we aggregate our variables by averaging the occupancy rates on the lanes of a same station and by summing the flows at each station. Our variables then reduce to one flow and one occupancy rate at each station.
The dataset rest (20%) of the observations are used for the model validation. The same obtained model is applied on 1000 observations which are not used for the calibration. The output results of this model indicate that the risk model is maximum when the accident is occurred.
The obtained results are very promising. The number of parameters was limited to 5 which can minimise the effort for the calibration process. However, more investigations are needed in order to take into account other parameter conditions such as the weather, luminosities (night) and the modification of the geometric topology (different lane number of the upstream and downstream measurement stations).
[1] B. Everitt, Cluster Analysis Heinemann Educ. Books, London 1974;
[2] EURAMP Project, Deliverable D3.5, Safety Critical Issue For Ramp metering, European commission, Brussels 2006.
[3] F. Thomas Glob, Wilfred W. Recker, ┬ A method for relating type of crash to traffic characteristics on urban freeway┬, Transportation Research Part A 38, p. 55-80,

University of Maryland, College Park
Colliding Particles: Beyond Accident-Free Car Following Model
Samer Hani Hamdar and Hani S. Mahmassani
This paper explores specifications of microscopic traffic models that could capture congestion dynamics and model accident-prone behaviors on a highway section in greater realism than existing models currently used in practice (commercial software). A comparative assessment of several major acceleration models is conducted, especially in regards to congestion formation and incident modeling. Based on this assessment, alternative specifications for a car-following/lane changing model are developed and implemented in a microscopic simulation framework. The models are calibrated and compared in terms of resulting vehicle trajectories and macroscopic flow-density relationships. Experiments are conducted with the models under different degrees of relaxation of the safety constraints typically applied in conjunction with simulation codes used in practice. The ability of the proposed specifications to capture traffic behavior in extreme situations is examined. The results suggest that these specifications offer an improved basis for microscopic traffic simulation for situations that do not require an accident free environment. As such, tthe same basic behavior model structure could accommodate both extreme situations (evacuation scenarios, over-saturated networks) as well as "normal" daily traffic conditions.

HAN Xianglin
Shanghai Institute of applied mathematics and mechanics, China
Control of Traffic Congestion in the Modified Coupled Map Car-Following Model Based on Intelligent Transportation system Application
Han Xiang-Lin, Ge Hong-Xia, Jiang Chang-Yuan, Dai Shi-Qiang
With the consideration of the application of ITS information, a modified coupled-map car-following model for traffic flows is proposed to describe the dynamical behavior of vehicles moving along a single-lane road. It is assumed that there exists the vehicle-bound navigation systems providing information about the vehicles both preceding and following the vehicle studied. According to the previous work and the real traffic, only the information of three vehicles ahead and one following vehicle is investigated in this paper. A new kind of the optimal velocity function is introduced to include both the forward- and backward-looking effects on a vehicle in the optimal velocity model used by Bando et al. The parameter P in the governing equation stands for the relative roles of the forward- and backward-looking effects, which is determined by the numerical simulation. Then by utilizing a dynamical version of the decentralized delayed-feedback control scheme, a strategy for controlling the traffic congestion is worked out through adjusting the control parameters in the presented model. Compared with the pioneering work by Konishi et al. in 1999, a simpler form of the control signal is adopted. For the present model, only one feedback gain needs to be determined instead of four feedback gains in the model of Konishi et al, and the feedback gain in our method can be set in a range instead of fixed values in the model of Konishi et al. More important is that there are strong interactions among the studied vehicle and its preceding and following ones. Based on the new principle of feedback control, the stability criteria are given as the speed of the preceding vehicle changes. The theoretical analysis shows that taking into account more information on preceding and following vehicles could lead to the stabilization for traffic flows, that is, the stability condition could be considerably weakened. The corresponding numerical simulations confirm the correctness of the theoretical analysis. Compared with the work of Konishi et al, the control strategy in this paper is more effective in suppressing the formation of traffic congestion.

Delft University of Technology
Dynamic first-order modeling of phase-transition probabilities
S. P. Hoogendoorn, H. van Lint, V.L. Knoop
Modeling breakdown probabilities or phase transition probabilities is an important issue when assessing and predicting the reliability of traffic flow operations. Looking at empirical spatio-temporal patterns, these probabilities clearly are not only a function of the local prevailing traffic conditions (density, speed), but also of time and space.
The dynamics of the breakdown probabilities are the topic of this paper. We propose a simple partial differential equation that can be used to model the dynamics of breakdown probabilities, in conjunction with a first-order model. The main assumption is that the breakdown probability dynamics satisfy the way information propagates in a traffic flow, i.e. they move along with the characteristics.
The main result is that we can reproduce the main characteristics of the breakdown probabilities, such as observed by Kerner. This is illustrated by means of two examples: free flow to synchronized flow (F-S transition) and synchronized to jam (S-J transition). We show that the probability of an F-S transition increases away from the on-ramp in the direction of the flow; the probability of an S-J transition increases as we move upstream in the synchronized flow area. Note that all the examples shown in the paper are deterministic.

HUANG Ding-Wei
Department of Physics, Chung Yuan Christian University
Traffic dynamics on a rotary
Ding-wei Huang
We present a simple cellular automaton model to study the traffic dynamics on a 4-ramp rotary (as an alternative to signaled crossroads). The rotary is an open system. Vehicles can move in and out of the rotary through on-ramp and off-ramp, respectively. On the rotary, vehicles move deterministically; while the ramps are operated stochastically. We show that, both numerically and analytically, the traffic states on the rotary are completely determined by the ramps (which can be taken as stochastic boundaries to trigger the phase transitions in bulk). The ramps provide a means to stabilize the density difference on the rotary and to support the maximum flow as a distinct phase. We are able to obtain exact solutions in the full parameter space. The complete phase diagram can be derived. When the north-bound vehicles can be distinguished from the east-bound vehicles, there are 5 different phases. When the vehicles are indistinguishable, there are only 4 different phases. We also discuss the results in a closed system, where a kind of periodic boundary condition is further imposed.

Technische Universitńt Dresden, Germany
Calibration of car-following models using floating car data
Arne Kesting, Martin Treiber and Dirk Helbing
As microscopic traffic flow models are mainly used to describe collective phenomena such as traffic breakdowns, traffic instabilities, and the propagation of stop-and-go waves, these models are traditionally calibrated with respect to macroscopic traffic data, e.g., one-minute flow and velocity data collected by double-loop detectors. Nowadays, as microscopic traffic data have become more and more available, the problem of analyzing and comparing microscopic traffic flow models with real microscopic data has recently raised some interest.
We study the car-following behavior of individual drivers in real city traffic on the basis of (publicly available) floating car data sets recorded by a vehicle equipped with an ACC sensor. By means of a nonlinear optimization procedure based on a genetic algorithm, we calibrate common car-following models by minimizing the deviations between the observed driving dynamics and the simulated trajectory when following the same leading vehicle. Furthermore, the calibrated parameter sets are applied to the other trajectories allowing for model validation. The results indicate that `intra-driver variability' rather than `inter-driver variability' accounts for a large part of the fit errors.
Furthermore, we compare the performance of different models. The reliability and robustness of the nonlinear fits can be assessed by applying different optimization criteria, i.e., different measures for the deviations between two trajectories. We also investigate the sensitivity of single model parameters and the correlation of pairs of parameters. The results will be used to suggest some criteria towards a benchmarking of car-following models.

INRIA, Le Chesnay, France
A queueing theory approach for a multi-speed exclusion process.
Furtlehner, Cyril and Lasgouttes, Jean-Marc
We consider a one-dimensional stochastic reaction-diffusion model, generalizing the totally asymmetric simple exclusion process and aiming at describing single lane roads with vehicles that can change speed between fast and slow. The site state is described by three letters corresponding to ``empty'', ``fast vehicle'' and ``slow vehicle'', and the particular dynamics that we choose (based on 3-sites patterns) ensures that clusters of occupied sites are of uniform type. To each type of cluster is associated a different escape rate, which encodes the speed of the vehicles. This setting is different from usual exclusion processes with multi-type particles, each having its own jump rate ; it is more in line with the Nagel-Schreckenberg model, with the difference that only local jumps are allowed and speed is replaced by jump rate.
When this model is set on a circle or an infinite line, classical arguments allow to map it to a linear network of queues (an urn model in theoretical physics parlance) with exponential service times, but with a twist: the service rate remains constant during a busy period, but can change at renewal events. By translation invariance, the stochastic point process that represents the input to any queue is identical in distribution to the output of the queue. Starting from this fact, we use the tools of queueing theory to evaluate the size, type and lifetime of clusters and investigate the dynamics of the system through simulation. This yields results on the global throughput, the proportion of vehicles that are in slow clusters, and the number of clusters.

LEBACQUE Jean-Patrick
INRETS (Arcueil, France)
A stochastic but diffusionless macroscopic traffic flow model
M.M. Khoshyaran, J.P. Lebacque
Most macroscopic trafic flow models are purely deterministic, described by single or systems of conservation laws with the possible addition of some relaxation term (the so-called first- respective second-order models). One notable exception are the kinetic models family, in which the speed / desired speed distribution can be given a stochastic interpretation. Nevertheless kinetic models are usually closed in such a way as to yield deterministic systems of conservations laws. Another exception is constituted by some attempts, limited in scope, have been made to add some stochastic noise terms to deterministic models described by partial differential equations, such as Wet's model. One drawback of such an approach is that the stochastic term affects the speed/acceleration terms and induces a diffusion-like effect, in contravention with normal trafic behaviour.
The work introduced in the presentation attempts to build a macroscopic stochastic model devoid of any diffusion-like effects. The purpose of the stochastic part of the model is to emulate the randomness of interactions between drivers . The model is based on the EARZ concept, introduced by the first author with some co-authors in the recent paper [2]. The EARZ model is essentially a first order model in which the fundamental diagram is not unique but is attached to, and dependent on, the driver. In the stochastic extension of the EARZ model, the parameter controlling the driver fundamental diagram is a random variable. Thus the model expresses how the driver reacts to the randomness of trafic conditions and interactions by a change in behaviour.
The model results, in eulerian coordinates, in a system of conservation laws with a stochastic corrective term. It is better expressed in lagrangian coordinates [1,2], as the stochastic corrective term is attached to vehicles. The latter approach is the key to the discretization of the model. The usual concepts of shifted supply and demand apply with some modifications, and provide both boundary conditions and intersection models. Some first numerical results and comparisons with field data will be presented.
[1] Aw, A., Klar, A., Materne, T. and Rascle M. (2002). Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM Journal of applied Mathematics, 63, 259-278.
[2] Lebacque J.P., Mammar, S., Haj-Salem, H. Generic second order trafic flow modelling. Accepted for publication in the Prcoceedings of the 17th ISTTT.

A multiclass car-following rule based on the LWR model
Ludovic Leclercq, Laboratoire IngÚnierie circulation Transport (ENTPE / INRETS), Vaulx-en-Velin, France
Jorge A. Laval, Georgia Institute of Technology, Atlanta, USA

Recent developments in traffic flow theory have led to efficient numerical schemes for solving macroscopic models. They are obtained by solving the models in Lagrangian rather than the traditional Eulerian coordinates; see for example Newell (2002), Daganzo (2006), Leclercq (2007) or Leclercq et al (2007) for first order models and Aw et al (2002) for higher order models. Additionally, variational theory (Daganzo, 2005a, 2005b, and Daganzo and Menendez, 2005) and its extension in the Lagrangian framework (Leclercq et al, 2007) make it possible to prove that these schemes are exact for first order models when the fundamental diagram is triangular. This is an important leap forward since current methods introduce numerical errors that can be devastating in practice.
The aim of this paper is to extend the framework in (Leclercq et al, 2007) in order to incorporate multiple vehicle types, each one with a different car-following rule. In this way, the free-flow speed, the jam density and the wave-speed can be defined for each individual vehicle class. Note that the one-class car-following rule has already been coupled with a lane-changing model (Laval and Leclercq, 2007) and thus the proposed extension is fully compatible with the latter.
The sketch of the paper is as follows: section 1 will recall the Lagrangian formulation of first order macroscopic models and its numerical resolution using (i) the Godunov scheme and (ii) the variational theory. Section 2 will introduce the proposed extension, the resulting numerical schemes using (i) and (ii) above, and the conditions for the exactness of the numerical schemes. Finally, section 3 will present some numerical results focusing on the representation of a bimodal flow (trucks and cars).
Aw, A., Klar, A., Materne, T., Rascle, M. (2002). Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM Applied Mathematics 63(1), 259-278.
Daganzo, C.F. (2006). In traffic flow, cellular automata = kinematic waves. Transportation Research B, 40(5), 396-403.
Daganzo, C.F. (2005a). A variational formulation of kinematic waves: basic theory and complex boundary conditions. Transportation Research B, 39(2), 187-196.
Daganzo, C.F. (2005b). A variational formulation of kinematic waves: Solution methods. Transportation Research B, 39(10), 934-950.
Daganzo, C.F. and Menendez, M. (2005). A variational formulation of kinematic waves: bottlenecks properties and examples. In: Mahmassani H.S. (Ed.), 16th ISTTT, Pergamon, London, 345-364.
Laval, J.A., Leclercq, L. (2007░. A microscopic theory of lane-changing. Submitted for publication. Avalaible as Research Report LICIT 06-05, INRETS, Bron, France.
Leclercq, L (2007). Hybrid approaches for the solutions of the Lighthill-Whitham-Richards model. Transportation Research part B, In press.
Leclercq, L., Laval, J. and Chevallier, E (2007). The Lagrangian coordinates and what it means for first order traffic flow models. Journal of the 17th International Symposium on Transportation and Traffic Theory, accepted for publication.

MA Tai-Yu
INRETS (Arcueil, France)
A cross entropy based multi-agent approach to predictive dynamic traffic assignment based on activity
T.Y. Ma, J.P. Lebacque
Modeling dynamic traveler choice behavior under uncertainty is an important issue in transportation research. As the road choice behavior of travelers is closely related to their activity planning and location choice, this has led to the development of multi-agent and activity-oriented modeling to implement individual learning process and interaction with its environment. In this work, we focus on travelers┬ departure time/route/activity choice behavior in a queuing network. In this respect, we propose a predictive dynamic traffic assignment model based on two interaction levels: travelers┬ adaptive reaction level and network propagation level. For the first level, each individual driver is modeled as a simple autonomous agent. With the assumption of rational decision-making, agents make their departure time/route/activity choice based on maximizing their received activity value minus travel cost and early/late arrival penalty. The departure time/route/activity choice is made jointly so that, for each OD pair, agents consider a set of reasonable paths and discretized departure intervals. In order to reflect traveler behavior uncertainty, the departure time instant is randomly decided within a chosen departure interval. Traffic condition changes according to travelers' departure time/route/activity choice behavior, dynamic traffic information and network supply constraints. For the second level, traffic congestion is modeled by extension of point queue dynamics concept. An approach of similar computational complexity by greater numerical precision is the lagrangian discretization of the EARZ model, currently under study. The dynamic user equilibrium is formulated as a variational inequality problem. As a solution method, we propose an algorithm based on Cross Entropy (CE) Method (Rubinstein, 1999). The CE method is a stochastic optimization technique for finding the optimal solution of difficult combinatorial nonconvex optimization problems. The algorithm uses a family of probability distributions to generate agents' departure time/route/activity choice. These probability distributions over a choice set depend on a choices┬ average performance experienced by agents on previous day. Based on the minimisation cross entropy concept, optimal probability distributions are derived iteratively such that high quality paths are more attractive to agents. The advantage of the CE method is that it is based on a mathematical framework and sampling theory, in order to derive the optimal probability distribution guiding agents to the dynamic system equilibrium. The proposed algorithm provides a fast approach for solving dynamic user equilibrium problems under more realistic traffic flow models or more complicate multimodal transport systems. We develop a discrete-event traffic simulator to simulate the travelers┬ behavior of adaptation to network traffic characteristics while satisfying their activity requirements and to approximate the network dynamics equilibrium. A numerical example is given to illustrate the performance of the proposed method.

MAURY Bertrand
Labo. Mathematiques de l'Univ. Paris-Sud 11
Numerical modelling of crowd motion in panic situation
B. Maury, J. Venel
We propose a microscopic model for crowd motion. We are especially interested in describing panic situations : people want to leave a room, building, railway station or a plane, that may contain obstacles. Our model rests on two principles. On the one hand, we define a spontaneous velocity, which corresponds to the velocity each individual would like to have in the absence of other people. On the other hand, individuals (which are identified to rigid discs) must obey a non-overlapping constraint. Those two principles lead us to define the actual velocity field as the projection of the spontaneous velocity over the set of admissible velocities (regarding to the non-overlapping constraints). The model takes the form of a differential inclusion, for which well-posedness can be established by means of recent abstract results in convex analysis. The spontaneous velocity of a person depends on his own position and on the room's geometry. People want to reach the exit as quickly as possible by avoiding the obstacles. Moreover people can have individual strategies. In case of congestion, they may try to circumvent the jam, instead of keeping pushing inefficiently. In this case, their spontaneous velocity is made dependent upon the position of people that they can see in front of them.
Contacts are handled by means of Lagrange multipliers that can be interpreted as pressure forces applied on each individual by its neighbors. We propose a time discretization scheme based on granular flow principles, which makes it possible to simulate the evacuation of thousands of highly-packed individuals. Simulations of some typical situations will be presented. In particular, those simulations make it possible to investigate the influence of individual optimization strategies upon the global efficiency of the evacuation.

MO Ye-Liu
School of Physical science and engeering, Guangxi University,China
Optimal velocity model via considering multi-velocity difference ahead
Ye-liu Mo,Yu Xue, Wei Shi ,Zhang Liu
Many traffic models attempt to simulate driver behavior in enough detail to reproduce the observed features of traffic flow. The optimal velocity model proposed by Bando et al in 1995 is an important car-following model. Although optimal velocity model is able to reproduce the stop-and-go traffic phenomenon, it can not avoid crash at values of the sensitivity smaller than about one, a≤1/s for it shows unrealistic values of the acceleration for various optimal functions. Wilson et al. & Lenz et al. proposed two multiple look-ahead car-following models considering many-neighbor interactions and non-locality of traffic flow, which can reduced the unrealistic magnitudes of acceleration and is beneficial to enhanced stability. However, all of multiple look-ahead models do not concern with the effects of the velocity difference of vehicles in front, i.e. relative velocity effects. Helbing et al have ever pointed out that the relative velocity has to been taken into account when multiple vehicles in front approach each other.
Based on the follow-the~Vleader model proposed by Gazis et al and symmetrical consideration, we present the following optimal velocity model with multi-velocity difference ahead, where the coefficient term of each relative velocity has nonlinear property with the function of velocity and headway. The stability condition is obtained by the linear analysis. The result shows that the effect of multi-velocity difference enhances stability. We study the stable properties of the model from consumption of energy on the other hand.
We make use of the following formula presented by Nakayama et al. to estimate the additional energy consumption of flow in the stable region. Finally, the distribution of change of kinetic energy of each vehicle in the congestion region can be investigated in details. The numerical simulation results show that we just focus on the information of smaller cars ahead in the model to reach best traffic and minimizing energy consumption of the car flow. Additionally, the profile curve in each model with a asymmetric hysteresis loop indicates that consumed energy in acceleration process is smaller than one in deceleration process.

Central Institute for Applied Mathematics
Research Centre Juelich
New insights into pedestrian flow through bottlenecks
Armin Seyfried (1), Tobias Rupprecht (2), Oliver Passon (1), Bernhard Steffen (1), Wolfram Klingsch (2) and Maik Boltes (1)
(1) Central Institute for Applied Mathematics, Research Centre Julich, 52425 Julich, Germany
(2) Institute for Building Material Technology and Fire Safety Science, University of Wuppertal Pauluskirchstrasse 7, 42285 Wuppertal, Germany

One central problem for modeling of pedestrian traffic is the lack of experimental data.
In our contribution results from two experiments under laboratory conditions are presented. One considers the movement of pedestrians along a line while the second inspects the pedestrian flow through a bottleneck. The time development of quantities like individual velocities, densities and individual time gaps in bottlenecks of different width are analyzed. The comparison of our data with experimental measurements from other authors gives new insights into the pedestrian flow through bottlenecks. In accordance with a previous experimental studies of Hoogendoorn and Daamen \cite{Hoogendoorn04,Hoogendoorn05a} we found a formation of lanes inside the bottleneck but contrary to their results a linear growth of the flow with the width.
The main result concerns the common assumption, that a jam occurs when the incoming flow exceeds the capacity of the bottleneck defined by the maximum of the flow according to the fundamental diagram. At first glance this seems to be a reasonable assumption which can be justified by reference to the continuity equation. However our measurements of densities and velocities inside the bottleneck under jamming conditions show a stationary flow of $J_s \approx 1.7\;(ms)^{-1}$ tuning in around densities of $\rho \approx 1.8\; m^{-2}$, while other authors measured significant higher flow values by increasing the initial density in front of the bottleneck. Thus the capacity defined by the maximum of the fundamental diagram is expected for densities substantial higherthan $\rho=1.8\;m^{-2}$ and a jam occurs even if the incoming flow does not overrun the capacity.

\bibitem{Hoogendoorn04} S.~P. Hoogendoorn and W.~Daamen. \newblock Self-organization in walker experiments. \newblock Traffic and Granular Flow 2004. \newblock \texttt{\small http://www.pedestrians.tudelft.nl/publications/TGF04.pdf}.
\bibitem{Hoogendoorn05a} S.~P. Hoogendoorn and W.~Daamen. \newblock Pedestrian behavior at bottlenecks. \newblock {\em Transportation Science}, 39 2:0147--0159, 2005.

SIEBEL Florian
University of Munich
Modelling synchronized flow at highway bottlenecks
Florian Siebel, Wolfram Mauser, Salissou Moutari and Michel Rascle
Many experimental studies have shown the appearance of synchronized flow at highway bottlenecks. We study highway bottlenecks within the macroscopic BVT model [1,2]. The BVT model describes traffic flow as a hyperbolic system of balance laws. It generalizes the traffic model of Aw and Rascle [3] and Greenberg [4] by introducing in the momentum equation a new source term, which can become negative due to the finite reaction and relaxation times of drivers. The model is capable of reproducing multivalued fundamental diagrams, the metastability of free traffic flow at the onset of instabilities and wide moving jams. Based on previous work [5,6] we describe the coupling conditions for the Riemann problem of the system and apply them to highway bottlenecks. We focus our study on the situation where the bottlenecks are either caused by the reduction of of the number of lanes or by on-ramps or off-ramps. Our numerical simulations reproduce the appearance of synchronized flow at these highway bottlenecks [7]. The analysis of the lane reduction setup shows that the outflow from the synchronized flow region in front of the bottleneck is constant and below the maximum free flow. This observation can be understood from the study of the static solutions within the model. As a consequence of the coupling conditions static solutions have to cross the jam line, one of the additional equilibrium solutions within the BVT model, and this crossing determines the flow value of the static solution.
[1] F. Siebel and W. Mauser, SIAM Journal on Applied Mathematics 66 1150-1162 (2006)
[2] F. Siebel and W. Mauser, Physical Review E 73 066108 (2006)
[3] A. Aw and M. Rascle, SIAM Journal on Applied Mathematics 60 916-938 (2000)
[4] J. Greenberg, SIAM Journal on Applied Mathematics 62 729-745 (2001)
[5] B. Haut, G. Bastin, Proceedings of the 8th International IEEE Conference on Intelligent Transportation Systems 178-184 (2005)
[6] M. Herty, S. Moutari, M. Rascle, Networks and Heterogenous Media 1 275-294 (2006)
[7] F. Siebel, W. Mauser, S. Moutari and M. Rascle, submitted to Physica A,

Institute for Transport & Economics, TU Dresden
MOBIL: General Lane-Changing Model for Car-Following Models
Martin Treiber, Arne Kesting, and Dirk Helbing
We propose a general model to derive lane-changing rules for discretionary and mandatory lane changes for a wide class of car-following models. Both the utility of a given lane and the risk associated with lane changes is determined in terms of longitudinal accelerations calculated with the microscopic traffic models. This allows for the formulation of compact and general safety and incentive criteria both for symmetric and asymmetric passing rules. Moreover, anticipative elements and the crucial influence of velocity differences of these car-following models are automatically transferred to the lane-changing rules.
While the safety criterion prevents critical lane changes and collisions, the incentive criterion also takes into account the (dis-)advantages of other drivers associated with a lane change via the 'politeness factor'. The parameter allows to vary the motivation for lane-changing from purely egoistic to more cooperative driving behavior. This novel feature allows first to prevent change lanes for a marginal advantage if this obstructs other drivers, and, second, to let a 'pushy' driver induce a lane change of a slower driver ahead in order to be no longer obstructed. This is a common phenomenon for asymmetric passing rules with a dedicated lane for passing.
We apply the model to traffic simulations of cars and trucks with the Intelligent Driver Model (IDM) as underlying car-following model. We study an open system with an on-ramp and investigate the resulting lane-changing rate as a function of the spatial coordinate as well as a function of traffic density.
In a more general context, we show that applying the MOBIL concept with zero politeness to simple car-following models and cellular automata results in lane changing models already known in the literature.

Universidad Autˇnoma Metropolitana (Mexico)
Kinetic derivation for a traffic flow model
A. R. MÚndez, R. M. Velasco
Macroscopic traffic flow equations can be derived from some phenomenological considerations, however they can also be supported by a kinetic equation. Firstly, in this work we solve the Paveri-Fontana equation [1] in the homogeneous steady state for a specific model of the average desired velocity of drivers. By means of a maximization procedure in the informational entropy [2], a local distribution function is also obtained. Such a distribution function is taken as a basis to develop the Grad's method [3] which allows us to expand any distribution function in terms of a complete set of orthonormal polynomials. The distribution function obtained through this method contains the density and the average velocity of vehicles as macroscopic variables to describe the time evolution in the system. In order to consider the spatial inhomogeneities we assume a relaxation behavior in the perturbed distribution function [4], as a result we obtained the contribution of the gradients in the density and velocity. In the following step we used the distribution function to calculate the traffic pressure and it gave us a closure relation to be introduced in the macroscopic equations coming from an integration of the Paveri-Fontana equation. The closed macroscopic model is simulated with periodic boundary conditions and two different initial situations. The simulation results show a general agreement [5] with the characteristics of traffic flow and the corresponding behavior is shown.

[1] S. L. Paveri-Fontana; Transp. Res. 9 (1975) 225-235.
[2] R. M. Velasco, A. R. M\'endez; in Statistical Physics and Beyond, (2005), AIP Conference Proceedings, 200-206.
[3] H. Grad; Comun. Pure \& Appl. Math. 2 (1949) 331-407.
[4] R. M. Velasco, W. Marques Jr., Phys. rev. E72 (2005) 046102.
[5] D. Helbing; Revs. Mod. Phys. 73 (2001) 1067-1141.

WU Qing-Song
School of Engineering Science, University of Science and technology of China, Hefei
Effect of adaptive cruise control vehicles on phase transition in a mixture with manual vehicles
Rui Jiang, Qing-Song Wu, Bin Jia, Ruili Wang, and Mao-Bin Hu
Recently, the research on vehicles equipped with adaptive cruise control (ACC) system has attracted the interest from both physicists and engineers. ACC is a driver assistance system designed to provide improved convenience and comfort. An ACC-equipped vehicle detects the presence of a preceding vehicle and measures the distance (range) as well as the relative speed (range rate) by using a forward-looking sensor. It automatically adjusts the vehicle speed to keep a proper range when a preceding vehicle is detected. Obviously, ACC vehicles will impact traffic characteristics, including highway safety, efficiency, and capacity because of their different behavior compared with human drivers. Therefore, before ACC vehicles are deployed on a large scale, their effects on traffic flow characteristics need to be carefully investigated.
Previous studies on ACC vehicles focus on the range policy, effect of ACC vehicles on traffic flow stability, road capacity, traffic safety, and environmental benefits. However, to our knowledge, the effect of ACC vehicles on the phase transition behavior in a mixture of ACC vehicles and manual vehicles has not been studied.
In this paper, we investigate the mixture of ACC vehicles with manual vehicles described by a CA model, which can describe first order phase transition from free flow to synchronized flow.
Our simulations show when the ratio of ACC vehicles $R$ is small, the introduction of ACC vehicles will reduce the phase transition threshold $\rho_{c2}$, and accordingly reduce the maximum flow rate $q_{\max}^f$. When the ratio is large, $q_{\max}^f$ will increase only when the time headway $T$ used in the ACC system is small. Nevertheless, the introduction of ACC vehicles will enhance the flow rate in synchronized flow for not so large $T$. Different from the phase transition from free flow to synchronized flow, the introduction of ACC vehicles will generally enlarge the threshold from synchronized flow to jams (except at small $T$ and $R$, under which the transition curve is essentially the same).
Based on these results, it may be suggested that in free flow of high flow rate, it is better to switch off the ACC system and drive manually. But after the breakdown and the traffic transits into synchronized flow, it is better to use ACC system (if $T$ is not so large). When jams appear, it is also suggested to use the ACC system if $T$ is not so large, because the propagation velocity of jams increases with the introduction of ACC vehicles.
Furthermore, the spatiotemporal patterns of mixture of ACC vehicles and manual vehicles and the velocity distribution are also studied. It is interesting to report that when $T$ is small and the traffic is in synchronized flow, a mixture of free flow stripes and synchronized flow stripes will appear, even though $R=0.99$. In other words, only several manual vehicles will seriously destroy the homogeneity of traffic flow. This is undesired and we need to find a way to suppress this phenomenon.

School of Physical science and engeering, Guangxi University
Phase transition induced by initial uniform distribution of cars
Yu Xue,Yanwang Wei,Wei Shi
A very simple microscopic model for single-lane traffic is the cellular automaton model presented by Nagel and Schreckenberg in 1992, called the NaSch model. Recent experiments have shown that the flow-density in the fundamental diagram is more complicated. Analysis for measurements in highway traffic indicates that the flow is not a unique function of the density in some situations. The different scenario for the jam formation was proposed in 1994 by Kerner and Konhauser at DaimlerChrysler. Moreover, hysteresis effects encountered in empirical observations are related to the existence of metastable states in certain density regimes, which is not observed in the NaSch model. For a better understanding of such complex traffic phenomena, a variety of modifications to the NaSch model have been proposed by introducing the slow-to-start rules, among which are the VDR model, the TT model, the BJH model and so on. They are able to reproduce meta-stable states and exhibit the clear separation of the congestion and free-flow regions in a space-time plot. However, the nature of phase transition to induce jamming formation has always attracted considerable arguments. Roters et al. considered the stochastic NaSch model displays criticality via investigating the dynamical structure factor of the nondeterministic NaSch model. Conversely, Chowdhury et al. showed the existence of a crossover instead of a critical point. Recently, Levine et al. also discussed the same question by studying existence of a jamming phase transition in the asymmetric chipping model and found that the system exhibits a smooth crossover between free flow and jammed states, as the car density is increased. Cheybani et al have study the transition from freely (jammed) to jammed and super-jammed traffic in the stochastic NaSch model using the spatiotemporal correlation function. They have turned out that both the transition from freely moving to jammed traffic and from jammed to super-jammed traffic in the stochastic NaSch model is not sharp but rather like a crossover. In theory of car-following model, however, traffic jamming will occur when traffic flow loses its stability starting from initial uniform distribution of cars. Thus, we can infer that complicated phase transition to induce jamming formation is not only related to the rule of model but also to the initial distribution and the maximum velocity of cars. In this paper, we systematically study the phase transition induced by initial condition to the typical cellular automaton traffic model, such as the cruise-control limit of the NaSch ,VDR,BJH and TT model. Under the homogeneous initial condition of identical vehicles and without stochastic brake, it is found that the initial condition has a great effect on the fundamental diagram and phase transition of traffic flow. There exist three different traffic phases: free flow, lowering-velocity flow and traffic jamming. The analytical solutions corresponding to these models are obtained respectively. Moreover, we have found the phase transition from free flow to lowering-velocity takes on criticality and traffic jamming via the local first phase transition from lowering ~Vvelocity to jamming occurs starting at the density k=0.5. And then, we make use of the several the definition of order parameter to analyze the characteristics of phase transition of the several models above mentioned under the homogeneous initial condition, it is found that the definition of order parameter in being is not better to describe the characteristics of phase transition of traffic. It brings forward an open question how to define order parameter to correctly depict the characteristics of phase transition of traffic.

YOKOYA Yasushi
Japan Automobile Research Institute
Qualitative change of car-following behavior observed in real traffic
Yasushi Yokoya, Youichi Asano, Nobuyuki Uchida
The phase transition of vehicular traffic is a universal feature of traffic flow [1] - [5]. The empirical facts characterizing the dynamical process in the phase transition of vehicular traffic are given by the hysteresis effects or the metastable branch given by the flux-density relation (i.e., the fundamental diagram), where spontaneous formation of high-density flow and its decay are observed in measurement of variations of flux with density. In this relation, flux increases in proportion to density under the critical point. However, a discontinuous reduction of flux occurs beyond the critical point, and jams eventually emerge through a mixed state of freely flowing traffic and jams (e.g., stop-and-go traffic).
It is important to interpret the flux-density relation, given by empirical data of the time series of the velocity and the headway of individual vehicles. However, difficulties with measurement hinder the direct comparison of empirical data on real traffic and the results of the numerical simulations or the analytical calculations. Traffic measurements are typically local-point measurements using roadside detectors or counting loops, whereas velocity and headway are not local quantities. In these measurements, the assumption of constant velocity and the following distance of individual vehicles on a stretch of road, which is divided by sensors, restricts the resolving power of data collection [6] - [8]. Consequently, it is necessary to aggregate and average experimental data for certain periods of time. Unfortunately, this assumption becomes unsuitable in the critical density region because the metastable states, especially the high-flow part of the branch, are unstable, with strong fluctuations of velocity [9].
We measured time series of the single-vehicle data in urban traffic with an onboard apparatus and determined the nature of fluctuations around the nonequilibrium phase transition of local vehicular traffic by analyzing a time series of successive variations of velocity. At low vehicle density, small fluctuations due to the uncertainty of the driver's reaction or various external conditions on the roads add statistical noise to the time series of the states of local vehicular traffic. An increased density leads to amplification of the fluctuations as a result of interaction between vehicles, and instability of traffic flow is induced at the critical density.
Regarding time evolution of local flux-density relation, the single-vehicle data indicates that the metastable branch distributes around the intermediate regime of vehicle density, which lies between the uniform flow phase and the congested flow phase. The dynamical process across the branch point lying between the uniform flow phase and the coexisting phase is reversible, and a dynamical property of vehicular traffic, e.g., relation between velocity and the following distance, is qualitatively unchanged. In contrast, the dynamical process across the decay point lying between the coexisting phase and the congested flow phase is irreversible, corresponding to the spinodal decomposition. In this dynamical process, the metastable branch decays spontaneously as a result of the noise caused by a driver's reaction to various external conditions on the road.
In the decay process of the metastable branch, dynamical properties of vehicular traffic change qualitatively. We found that the probability density function calculated from the time series of variation of velocity is transformed around the decay point, where a Gaussian distribution changes into a L├ęvy stable symmetrical distribution. The power-law tail in the L├ęvy distribution indicates that the time series of variation of velocity exhibits the nature of the critical fluctuations, as generally observed in phase transitions driven far from equilibrium. Our experimental data suggests that the vehicular traffic system spontaneously approaches a delicate balance between the metastable states and the congested flow states in the critical region.
We attempted to analyze microscopic dynamical processes of traffic flow in the critical region, in connection with a discrete stochastic process involving random amplification with additive external noise, which exhibits a power-law probability density distribution of the variation of velocity. As a result, we found the relation between the nature of the driver's response to the stimulus and a power-law probability density distribution. This relation is expected to be a clue to the microscopic mechanisms behind the spontaneous decay of the metastable branch at the phase transition driven far from equilibrium.
Furthermore, we carried out car following experiments by a driving simulator, in order to study the nature of the driver's response to the deceleration of the leader vehicle. We found characteristic behavior of the acceleration as function of distance and speed difference. Subjects for the experiment did not respond to the change of distance and speed difference immediately, and sometimes ignored the change of them. These results suggest that the desire for smooth and comfortable driving is responsible for the occurrence of complex spatiotemporal structures observed in traffic flow [10].
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