Andreas Schadschneider

Institute for Theoretical Physics

University of Cologne

Germany

www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

Modelling of Traffic Flow and

Related Transport Problems

Overview

• Highway traffic• Traffic on ant trails• Pedestrian dynamics• Intracellular transport

•basic phenomena

•modelling approaches

•theoretical analysis

•physics

Topics:

Aspects:

General topic: Application of nonequilibrium physics to various transport processes/phenomena

Introduction

Traffic = macroscopic system of interacting particles

Nonequilibrium physics:Driven systems far from equilibrium

Various approaches:• hydrodynamic• gas-kinetic• car-following• cellular automata

Cellular Automata

Cellular automata (CA) are discrete in• space• time• state variable (e.g. occupancy, velocity)

Advantage: very efficient implementation for large-scale computer simulations

often: stochastic dynamics

Asymmetric Simple

Exclusion Process

Asymmetric Simple Exclusion Process

Asymmetric Simple Exclusion Process (ASEP):

1. directed motion2. exclusion (1 particle per site)

3. stochastic dynamics

qq

Caricature of traffic:

“Mother of all traffic models”

For applications: different modifications necessary

Update scheme

• random-sequential: site or particles are picked randomly at each step (= standard update for ASEP; continuous time dynamics)

• parallel (synchronous): all particles or sites are updated at the same time

• ordered-sequential: update in a fixed order (e.g. from left to right)

• shuffled: at each timestep all particles are updated in random order

In which order are the sites or particles updated ?

ASEP

• simple• exactly solvable• many applications

Applications:

• Protein synthesis

• Surface growth

• Traffic

• Boundary induced phase transitions

ASEP = “Ising” model of nonequilibrium physics

Periodic boundary conditions

no or short-range correlations

fundamental diagram

Influence of Boundary Conditions

open boundaries: density not conserved!

exactly solvable for all parameter values!

Derrida, Evans, Hakim, Pasquier 1993

Schütz, Domany 1993

Phase Diagram

Low-density phase

J=J(p,)

High-density phase

J=J(p,)

Maximal current phase

J=J(p)

1.order

transition

2.order

transitions

Highway

Traffic

Spontaneous Jam Formation

Phantom jams, start-stop-waves

interesting collective phenomena

space

time

jam velocity:

-15 km/h

(universal!)

Experiment

Relation: current (flow) $ density

Fundamental Diagram

free flow

congested flow (jams)

more detailed features?

Cellular Automata Models

Discrete in • Space • Time• State variables (velocity)

velocity ),...,1,0( maxvv

dynamics: Nagel – Schreckenberg (1992)

Update Rules

Rules (Nagel, Schreckenberg 1992)

1) Acceleration: vj ! min (vj + 1, vmax)

2) Braking: vj ! min ( vj , dj)

3) Randomization: vj ! vj – 1 (with probability p)

4) Motion: xj ! xj + vj

(dj = # empty cells in front of car j)

Example

Configuration at time t:

Acceleration (vmax = 2):

Braking:

Randomization (p = 1/3):

Motion (state at time t+1):

Interpretation of the Rules

1) Acceleration: Drivers want to move as fast as possible (or allowed)

2) Braking: no accidents

3) Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions

4) Driving: Motion of cars

Realistic Parameter Values

Standard choice: vmax=5, p=0.5

Free velocity: 120 km/h 4.5 cells/timestep

Space discretization: 1 cell 7.5 m

1 timestep 1 sec

Reasonable: order of reaction time (smallest relevant timescale)

Discrete vs. Continuum Models

Simulation of continuum models:

Discretisation (x, t) of space and time necessary

Accurate results: x, t ! 0

Cellular automata: discreteness already taken into account in definition of model

Simulation of NaSch Model

• Reproduces structure of traffic on highways

- Fundamental diagram

- Spontaneous jam formation

• Minimal model: all 4 rules are needed

• Order of rules important

• Simple as traffic model, but rather complex as stochastic model

Simulation

Analytical Methods

Mean-field: P(1,…,L)¼ P(1) P(L)

Cluster approximation:

P(1,…,L)¼ P(1,2) P(2,3) P(L)

Car-oriented mean-field (COMF):

P(d1,…,dL)¼ P(d1) P(dL) with dj = headway of car j (gap to car ahead)

1 2 3 4

1 2 3 4

1 2 3 4

d1=1 d2=0 d3=2

Particle-hole symmetry

Mean-field theory underestimates flow: particle-hole attraction

Fundamental Diagram (vmax=1)

vmax=1: NaSch = ASEP with parallel dynamics

ASEP with random-sequential update: no correlations (mean-field exact!)

ASEP with parallel update: correlations, mean-field not exact, but 2-cluster approximation and COMF

Origin of correlations?

Paradisical States

Garden of Eden state (GoE)

in reduced configuration space without GoE states: Mean-field exact!

=> correlations in parallel update due to GoE states

not true for vmax>1 !!!

(AS/Schreckenberg 1998)

(can not be reached by dynamics!)

Fundamental Diagram (vmax>1)

No particle-hole symmetry

Phase Transition?

Are free-flow and jammed branch in the NaSch model separated by a phase transition?

No! Only crossover!!

Exception: deterministic limit (p=0)

2nd order transition at 1

1

max vc

Andreas Schadschneider

Institute for Theoretical Physics

University of Cologne

Germany

www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

Modelling of Traffic Flow and

Related Transport Problems

Lecture II

Nagel-Schreckenberg Model

velocity ),...,1,0( maxvv

1. Acceleration

2. Braking

3. Randomization

4. Motion

vmax=1: NaSch = ASEP with parallel dynamics

vmax>1: realistic behaviour (spontaneous jams, fundamental diagram)

Fundamental Diagram II

free flow

congested flow (jams)

more detailed features?

high-flow states

Metastable States

Empirical results: Existence of

• metastable high-flow states

• hysteresis

VDR Model

Modified NaSch model: VDR model (velocity-dependent randomization)

Step 0: determine randomization p=p(v(t))

p0 if v = 0

p(v) = with p0 > p p if v > 0

Slow-to-start ruleSimulation

NaSch model

VDR-model: phase separation

Jam stabilized by Jout < Jmax

VDR model

Jam Structure

Fundamental Diagram III

Even more detailed features?

non-unique flow-density relation

Synchronized Flow

New phase of traffic flow (Kerner – Rehborn 1996)

States of• high density and relatively large flow• velocity smaller than in free flow• small variance of velocity (bunching)• similar velocities on different lanes (synchronization)• time series of flow looks „irregular“• no functional relation between flow and density• typically observed close to ramps

3-Phase Theory

free flow

(wide) jams

synchronized traffic

3 phases

Cross-Correlations

free flow

jam

synchro

free flow, jam:

synchronized traffic:

1)(, Jcc

0)(, Jcc

Cross-correlation function:

cc J() / h (t) J(t+) i - h (t) i h J(t+)i

Objective criterion for classification of traffic

phases

Time Headway

free flow synchronized traffic

many short headways!!! density-dependent

Brake-light model

Nagel-Schreckenberg model

1. acceleration (up to maximal velocity)

2. braking (avoidance of accidents)

3. randomization (“dawdle”)

4. motion

plus:

slow-to-start rule

velocity anticipation

brake lights

interaction horizon

smaller cells

…

Brake-light model

(Knospe-Santen-Schadschneider-Schreckenberg 2000)

good agreement with single-vehicle data

Fundamental Diagram IV

a) Empirical results

b) Monte Carlo simulations

Test: „Tunneling of Jams“

Highway Networks

Autobahn networkof North-Rhine-Westfalia

(18 million inhabitants)

length: 2500 km67 intersections (“nodes”)830 on-/off-ramps (“sources/sinks”)

Data Collection

online-data from

3500 inductive loops

only main highways are densely equipped with detectors

almost no data directly

from on-/off-ramps

Online Simulation

State of full network through simulation based on available data “interpolation” based on online data: online simulation

classification into 4 states (available at www.autobahn.nrw.de)

Traffic Forecasting

state at 13:51forecast for 14:56actual state at 14:54

2-Lane Traffic

Rules for lane changes (symmetrical or asymmetrical)• Incentive Criterion: Situation on other lane is better• Safety Criterion: Avoid accidents due to lane changes

Defects

Locally increased randomization: pdef > p

Ramps have similar effect!

shock

Defect position

City Traffic

BML model: only crossings

Even timesteps: " moveOdd timesteps: ! move

Motion deterministic !

2 phases:

Low densities: hvi > 0

High densities: hvi = 0

Phase transition due to gridlocks

More realistic model

Combination of BML and NaSch models

Influence of signal periods,

Signal strategy (red wave etc), …

Chowdhury, Schadschneider 1999

Summary

Cellular automata are able to reproduce many aspects of

highway traffic (despite their simplicity):

• Spontaneous jam formation• Metastability, hysteresis• Existence of 3 phases (novel correlations)

Simulations of networks faster than real-time possible

• Online simulation• Forecasting

Finally!

Sometimes „spontaneous jam formation“ has a rather simple explanation!

Bernd Pfarr, Die ZEIT

Intracellular

Transport

Transport in Cells

(long-range transport)

(short-range transport)

• microtubule = highway• molecular motor

(proteins) = trucks• ATP = fuel

Molecular Motors

DNA, RNA polymerases: move along DNA; duplicate and transcribe DNA into RNA

Membrane pumps: transport ions and small molecules across membranes

Myosin: work collectively in muscles

Kinesin, Dynein: processive enzyms, walk along filaments (directed); important for intracellular transport, cell division, cell locomotion

Microtubule

24 nm

m10~

8 nm

- +

Mechanism of Motion

inchworm: leading and trailing head fixed

hand-over-hand: leading and trailing head change Movie

• Several motors running on same track simultaneously

• Size of the cargo >> Size of the motor

• Collective spatio-temporal organization ?

Fuel: ATP

ATP ADP + P Kinesin

Dynein

Kinesin and Dynein: Cytoskeletal motors

ASEP-like Model of Molecular Motor-Traffic

(Lipowsky, Klumpp, Nieuwenhuizen, 2001 Parmeggiani, Franosch, Frey, 2003 Evans, Juhasz, Santen, 2003)

q

D A

ASEP + Langmuir-like adsorption-desorption

Competition bulk – boundary dynamics

Phase diagram

　

　

　

　

3/1/ ad

０

SL H

Position of Shock is x=1 when SH 　　　 x=0 when LS

wx0 1

L

HS

cf. ASEP

General belief: Coordination of two heads is required for processivity (i.e., long-distance travel along the track) of conventional TWO-headed kinesin.

KIF1A is a single-headed processive motor.

Then, why is single-headed KIF1A processive?

Single-headed kinesin KIF1A

Movie

2-State Model for KIF1A

state 1: “strongly bound”

state 2: “weakly bound”

Hydrolysis cycle of KIF1A

K KT

KDPKD

ATP

P

ADP

d

Bound on MT

Brownian& Ratchetmotion on MT

hydrolysis12

New model for KIF1A

ー ＋1 0 0 2 0 1 21 1 2 10

Brownian, ratchet

Attachment

2,1

2,1 Detachment

t

t +1

1

2

h2

1

s

1

f2

2 2

b

2

BrownianRelease ADP（ Ratchet ）

Hydrolysis

0

1

1

0

a d

Att. Det.

]ms[ -1h

]ms[ -1a

0.01 (0.0094)

0.1 (0.15)

0.00001 (1) 0.001 (100)

0.2 (0.9)

Blue: state_1Red: state_2

])[mMol(ATP

])nMol[A1(KIF0.00005 (5)

x

t

Phase diagram

position of domain wall can be measured as a function of controllable parameters.

Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005)

KIF1A (Red)

MT (Green)10 pM

100 pM

1000pM

2 mM of ATP2 m

Spatial organization of KIF1A motors: experiment

Andreas Schadschneider

Institute for Theoretical Physics

University of Cologne

Germany

www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic

Modelling of Traffic Flow and

Related Transport Problems

Lecture III

Dynamics on

Ant Trails

Ant trails

ants build “road” networks: trail system

Chemotaxis

Ants can communicate on a chemical basis:

chemotaxis

Ants create a chemical trace of pheromones

trace can be “smelled” by other

ants follow trace to food source etc.

Chemotaxis

chemical trace: pheromones

Ant trail model

Basic ant trail model: ASEP + pheromone dynamics

• hopping probability depends on density of pheromones• distinguish only presence/absence of pheromones• ants create pheromones• ‘free’ pheromones evaporate

q q Q

1. motion of ants

2. pheromone update (creation + evaporation)Dynamics:

f f fparameters: q < Q, f

Ant trail model

q q Q

equivalent to bus-route model (O’Loan, Evans Cates 1998)

(Chowdhury, Guttal, Nishinari, A.S. 2002)

Limiting cases

f=0: pheromones never evaporate

=> hopping rate always Q in stationary state

f=1: pheromone evaporates immediately

=> hopping rate always q in stationary state

for f=0 and f=1: ant trail model = ASEP (with Q, q, resp.)

Fundamental diagram of ant trails

different from highway traffic: no egoism

velocity vs. density

Experiments:

Burd et al. (2002, 2005)

non-monotonicity at small

evaporation rates!!

Experimental result

(Burd et al., 2002)

Problem: mixture of unidirectional and counterflow

Spatio-temporal organization

formation of “loose clusters”

early times steady state

coarsening dynamics:

cluster velocity ~ gap to preceding cluster

Traffic on Ant Trails

Formation of clusters

Analytical Description

Mapping on Zero-Range Process

ant trail model:

})1(1{)1()( // vxvx fqfQxu (v = average velocity)

phase transition for f ! 0 at 2qQ

qQc

Counterflow

hindrance effect through interactions (e.g. for communication)

plateau

Pedestrian

Dynamics

Collective Effects

• jamming/clogging at exits• lane formation • flow oscillations at bottlenecks• structures in intersecting flows

Lane Formation

Lane Formation

Oscillations of Flow Direction

Pedestrian Dynamics

More complex than highway traffic

• motion is 2-dimensional• counterflow • interaction “longer-ranged” (not only nearest neighbours)

Pedestrian model

Modifications of ant trail model necessary since

motion 2-dimensional:• diffusion of pheromones• strength of trace

idea: Virtual chemotaxis

chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“

Long-ranged Interactions

Problems for complex

geometries:

Walls ’’screen“ interactions

Models with local interactions ???

Floor field cellular automaton

Floor field CA: stochastic model, defined by transition probabilities, only local interactions

reproduces known collective effects (e.g. lane formation)

Interaction: virtual chemotaxis (not measurable!)

dynamic + static floor fields

interaction with pedestrians and infrastructure

Static Floor Field

• Not influenced by pedestrians• no dynamics (constant in time)• modelling of influence of infrastructure

Example: Ballroom with one exit

Transition Probabilities

Stochastic motion, defined by

transition probabilities

3 contributions:• Desired direction of motion • Reaction to motion of other pedestrians• Reaction to geometry (walls, exits etc.)

Unified description of these 3 components

Transition Probabilities

Total transition probability pij in direction (i,j):

pij = N¢ Mij exp(kDDij) exp(kSSij)(1-nij)

Mij = matrix of preferences (preferred direction)

Dij = dynamic floor field (interaction between pedestrians)

Sij = static floor field (interaction with geometry)

kD, kS = coupling strength

N = normalization ( pij = 1)

Lane Formation

velocity profile

Friction

Friction: not all conflicts are resolved! (Kirchner, Nishinari, Schadschneider 2003)

friction constant = probability that no one moves

Conflict: 2 or more pedestrians choose the same target cell

Herding Behaviour vs. Individualism

Minimal evacuation times for optimal combination of herding and individual behaviour

Evacuation time as function of coupling strength to dynamical floor field

(Kirchner, Schadschneider 2002)

Large kD: strong herding

Evacuation Scenario With Friction Effects

Faster-is-slower effect

evacuation time

effective velocity

(Kirchner, Nishinari, A.S. 2003)

Competitive vs. Cooperative Behaviour

Experiment: egress from aircraft (Muir et al. 1996)

Evacuation times as function of 2 parameters:

• motivation level

- competitive (Tcomp)

- cooperative (Tcoop )

• exit width w

Empirical Egress Times

Tcomp > Tcoop for w < wc

Tcomp < Tcoop for w > wc

Model Approach

Competitive behaviour:

large kS + large friction

Cooperative behaviour:

small kS + no friction =0

(Kirchner, Klüpfel, Nishinari, A. S., Schreckenberg 2003)

Summary

Variants of the Asymmetric Simple Exclusion Process

• Highway traffic: larger velocities

• Ant trails: state-dependent hopping rates

• Pedestrian dynamics: 2-dimensional motion

• Intracellular transport: adsorption + desorption

Various very different transport and traffic problems can be described by similar models

Applications

Highway traffic:• Traffic forecasting• Traffic planning and optimization

Ant trails:• Optimization of traffic• Pedestrian dynamics (virtual chemotaxis)

Pedestrian dynamics:• safety analysis (planes, ships, football stadiums,…)

Intracellular transport:• relation with diseases (ALS, Alzheimer,…)

Collaborators

Cologne:Ludger SantenAnsgar KirchnerAlireza NamaziKai KlauckFrank ZielenCarsten BursteddeAlexander John Philip Greulich

Thanx to:

Rest of the world:

Debashish Chowdhury (Kanpur)

Ambarish Kunwar (Kanpur)

Vishwesha Guttal (Kanpur)

Katsuhiro Nishinari (Tokyo)

Yasushi Okada (Tokyo)

Gunter Schütz (Jülich)

Vladislav Popkov (now Cologne)

Kai Nagel (Berlin)

Janos Kertesz (Budapest)

Duisburg:

Michael Schreckenberg

Robert Barlovic

Wolfgang Knospe

Hubert Klüpfel

Torsten Huisinga

Andreas Pottmeier

Lutz Neubert

Bernd Eisenblätter

Marko Woelki