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1
Modelling of Traffic Flow and Related Transport
Problems
Andreas Schadschneider Institute for Theoretical
Physics University of Cologne Germany
www.thp.uni-koeln.de/as
www.thp.uni-koeln.de/ant-traffic
2
Overview
General topic Application of nonequilibrium
physics to various transport
processes/phenomena

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

Topics

• basic phenomena
• modelling approaches
• theoretical analysis
• physics

Aspects
3
Introduction

• Traffic macroscopic system of interacting
  particles
• Nonequilibrium physics
• Driven systems far from equilibrium
• Various approaches
• hydrodynamic
• gas-kinetic
• car-following
• cellular automata

4
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

5
Asymmetric Simple Exclusion Process
6
Asymmetric Simple Exclusion Process
Caricature of traffic

• Asymmetric Simple Exclusion Process (ASEP)
• directed motion
• exclusion (1 particle per site)
• stochastic dynamics

Mother of all traffic models For applications
different modifications necessary
7
Update scheme
In which order are the sites or particles updated
?

• 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

8
ASEP
ASEP Ising model of nonequilibrium physics

• simple
• exactly solvable
• many applications

• Applications
• Protein synthesis
• Surface growth
• Traffic
• Boundary induced phase transitions

9
Periodic boundary conditions
fundamental diagram

• no or short-range correlations

10
Influence of Boundary Conditions

• open boundaries density not conserved!

exactly solvable for all parameter values!
Derrida, Evans, Hakim, Pasquier 1993 Schütz,
Domany 1993
11
Phase Diagram
Maximal current phase JJ(p)
Low-density phase JJ(p,?)
2.order transitions
1.order transition
High-density phase JJ(p,?)
12
Highway Traffic
13
Spontaneous Jam Formation
space
time
jam velocity -15 km/h (universal!)
Phantom jams, start-stop-waves interesting
collective phenomena
14
Experiment
15
Fundamental Diagram
Relation current (flow) density
free flow
congested flow (jams)
more detailed features?
16
Cellular Automata Models

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

velocity
dynamics Nagel Schreckenberg (1992)
17
Update Rules

• Rules (Nagel, Schreckenberg 1992)
• Acceleration vj ! min (vj 1, vmax)
• Braking vj ! min ( vj , dj)
• Randomization vj ! vj 1 (with
  probability p)
• Motion xj ! xj vj

(dj empty cells in front of car j)
18
Example
Configuration at time t Acceleration (vmax
2) Braking Randomization (p 1/3) Motion
(state at time t1)
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Interpretation of the Rules

• Acceleration Drivers want to move as fast as
  possible (or allowed)
• Braking no accidents
• Randomization
• a) overreactions at braking
• b) delayed acceleration
• c) psychological effects (fluctuations in
  driving)
• d) road conditions
• 4) Driving Motion of cars

20
Realistic Parameter Values

• Standard choice vmax5, p0.5
• 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)

21
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

22
Simulation of NaSch Model
Simulation

• 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

23
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)

24
Fundamental Diagram (vmax1)
vmax1 NaSch ASEP with parallel dynamics

• Particle-hole symmetry
• Mean-field theory underestimates flow
  particle-hole attraction

25
Paradisical States
(AS/Schreckenberg 1998)

• 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?

(can not be reached by dynamics!)
Garden of Eden state (GoE)
in reduced configuration space without GoE
states Mean-field exact! gt correlations in
parallel update due to GoE states
not true for vmaxgt1 !!!
26
Fundamental Diagram (vmaxgt1)

• No particle-hole symmetry

27
Phase Transition?

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

No! Only crossover!!
Exception deterministic limit (p0) 2nd order
transition at
28
Modelling of Traffic Flow and Related Transport
Problems
Lecture II
Andreas Schadschneider Institute for Theoretical
Physics University of Cologne Germany
www.thp.uni-koeln.de/as
www.thp.uni-koeln.de/ant-traffic
29
Nagel-Schreckenberg Model
velocity

1. Acceleration
2. Braking
3. Randomization
4. Motion

vmax1 NaSch ASEP with parallel dynamics
vmaxgt1 realistic behaviour (spontaneous jams,
fundamental diagram)
30
Fundamental Diagram II
more detailed features?
high-flow states
free flow
congested flow (jams)
31
Metastable States

• Empirical results Existence of
• metastable high-flow states
• hysteresis

32
VDR Model

• Modified NaSch model
• VDR model
  (velocity-dependent randomization)
• Step 0 determine randomization pp(v(t))
• p0 if v 0
• p(v) with
  p0 gt p
• p if v gt 0
• Slow-to-start rule

Simulation
33
Jam Structure
NaSch model
VDR model
VDR-model phase separation Jam stabilized by
Jout lt Jmax
34
Fundamental Diagram III
Even more detailed features?
non-unique flow-density relation
35
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

36
3-Phase Theory
free flow (wide) jams synchronized traffic
3 phases
37
Cross-Correlations
Cross-correlation function cc? J(?) / h ?(t)
J(t?) i - h ?(t) i h J(t?)i
free flow, jam synchronized traffic
free flow
jam
synchro
Objective criterion for classification of traffic
phases
38
Time Headway
synchronized traffic

• free flow

density-dependent
many short headways!!!
39
Brake-light model

• Nagel-Schreckenberg model
• acceleration (up to maximal velocity)
• braking (avoidance of accidents)
• randomization (dawdle)
• 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
40
Fundamental Diagram IV

• Empirical results
• Monte Carlo simulations

41
Test Tunneling of Jams
42
Highway Networks

• Autobahn network
• of North-Rhine-Westfalia
• (18 million inhabitants)
• length 2500 km
• 67 intersections (nodes)
• 830 on-/off-ramps
• (sources/sinks)

43
Data Collection

• online-data from
• 3500 inductive loops
• only main highways are densely equipped with
  detectors
• almost no data directly
• from on-/off-ramps

44
Online Simulation

• State of full network through simulation based on
  available data
• interpolation based on online data
  online simulation

(available at www.autobahn.nrw.de)
classification into 4 states
45
Traffic Forecasting
state at 1351
forecast for 1456
actual state at 1454
46
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

47
Defects
Locally increased randomization pdef gt p
shock
Ramps have similar effect!
Defect position
48
City Traffic

• BML model only crossings
• Even timesteps " move
• Odd timesteps ! move
• Motion deterministic !

2 phases Low densities hvi gt 0 High
densities hvi 0 Phase transition due to
gridlocks
49
More realistic model

• Combination of BML and NaSch models
• Influence of signal periods,
• Signal strategy (red wave etc),

Chowdhury, Schadschneider 1999
50
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

51
Finally!
Sometimes spontaneous jam formation has a
rather simple explanation!
Bernd Pfarr, Die ZEIT
52
Intracellular Transport
53
Transport in Cells
(short-range transport)
(long-range transport)

• microtubule highway
• molecular motor (proteins) trucks
• ATP fuel

54
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

55
Microtubule
-

• 24 nm

8 nm
56
Mechanism of Motion

• inchworm leading and trailing head fixed
• hand-over-hand leading and trailing head change

Movie
57
Kinesin and Dynein Cytoskeletal motors
Fuel ATP
ATP ADP P
Kinesin
Dynein

• Several motors running on same track
  simultaneously
• Size of the cargo gtgt Size of the motor
• Collective spatio-temporal organization ?

58
ASEP-like Model of Molecular Motor-Traffic
ASEP Langmuir-like adsorption-desorption
(Lipowsky, Klumpp, Nieuwenhuizen, 2001
Parmeggiani, Franosch, Frey, 2003 Evans,
Juhasz, Santen, 2003)
Competition bulk boundary dynamics
59
Phase diagram

H
S
L


0
1
0

Position of Shock is x1 when SH
x0 when LS
60
Single-headed kinesin KIF1A
KIF1A is a single-headed processive motor.
General belief Coordination of two heads is
required for processivity (i.e., long-distance
travel along the track) of conventional
TWO-headed kinesin.
Then, why is single-headed KIF1A processive?
Movie
61
2-State Model for KIF1A

• state 1 strongly bound
• state 2 weakly bound

Hydrolysis cycle of KIF1A
62
New model for KIF1A
1
t
0
1
0
t 1
Brownian
Release ADP (Ratchet)
Hydrolysis
Att.
Det.
63
Phase diagram
0.2 (0.9)
0.1 (0.15)
0.01 (0.0094)
Blue state_1 Red state_2
0.00001 (1)
0.00005 (5)
0.001 (100)
64
Spatial organization of KIF1A motors experiment
MT (Green)
10 pM
KIF1A (Red)
100 pM
1000pM
2 mM of ATP
2 mm
position of domain wall can be measured as a
function of controllable parameters.
Nishinari, Okada, Schadschneider, Chowdhury,
Phys. Rev. Lett. (2005)
65
Modelling of Traffic Flow and Related Transport
Problems
Lecture III
Andreas Schadschneider Institute for Theoretical
Physics University of Cologne Germany
www.thp.uni-koeln.de/as
www.thp.uni-koeln.de/ant-traffic
66
Dynamics on Ant Trails
67
Ant trails
ants build road networks trail system
68
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.

69
Chemotaxis
chemical trace pheromones
70
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

71
Ant trail model
(Chowdhury, Guttal, Nishinari, A.S. 2002)

1. motion of ants
2. pheromone update (creation evaporation)

Dynamics
q
q
Q
f f f
parameters q lt Q, f
(OLoan, Evans Cates 1998)
equivalent to bus-route model
72
Limiting cases

• f0 pheromones never evaporate
• gt hopping rate always Q in stationary
  state
• f1 pheromone evaporates immediately
• gt hopping rate always q in stationary
  state
• for f0 and f1 ant trail model ASEP (with
  Q, q, resp.)

73
Fundamental diagram of ant trails
velocity vs. density
non-monotonicity at small evaporation rates!!
Experiments Burd et al. (2002, 2005)
different from highway traffic no egoism
74
Experimental result
Problem mixture of unidirectional and counterflow

• (Burd et al., 2002)

75
Spatio-temporal organization

• formation of loose clusters

early times
steady state
coarsening dynamics cluster velocity gap to
preceding cluster
76
Traffic on Ant Trails
Formation of clusters
77
Analytical Description

• Mapping on Zero-Range Process

ant trail model
(v average velocity)
phase transition for f ! 0 at
78
Counterflow
hindrance effect through interactions (e.g. for
communication)
plateau
79
Pedestrian Dynamics
80
Collective Effects

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

81
Lane Formation
82
Lane Formation
83
Oscillations of Flow Direction
84
Pedestrian Dynamics

• More complex than highway traffic
• motion is 2-dimensional
• counterflow
• interaction longer-ranged (not only nearest
  neighbours)

85
Pedestrian model
idea Virtual chemotaxis chemical trace
long-ranged interactions are translated into
local interactions with memory

• Modifications of ant trail model necessary since
• motion 2-dimensional
• diffusion of pheromones
• strength of trace

86
Long-ranged Interactions

• Problems for complex
• geometries
• Walls screen interactions

Models with local interactions ???
87
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
88
Static Floor Field

• Not influenced by pedestrians
• no dynamics (constant in time)
• modelling of influence of infrastructure
• Example Ballroom with one exit

89
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

90
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)

91
Lane Formation
velocity profile
92
Friction
Conflict 2 or more pedestrians choose the same
target cell

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

friction constant ? probability that no one
moves
93
Herding Behaviour vs. Individualism
Large kD strong herding

• 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)
94
Evacuation Scenario With Friction Effects
(Kirchner, Nishinari, A.S. 2003)
evacuation time
effective velocity
Faster-is-slower effect
95
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

96
Empirical Egress Times
Tcomp gt Tcoop for w lt wc
Tcomp lt Tcoop for w gt wc
97
Model Approach

• Competitive behaviour
• large kS large friction ?
• Cooperative behaviour
• small kS no friction ?0

(Kirchner, Klüpfel, Nishinari, A. S.,
Schreckenberg 2003)
98
Summary
Various very different transport and traffic
problems can be described by similar models

• 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

99
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,)

100
Collaborators
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

• Cologne
• Ludger Santen
• Ansgar Kirchner
• Alireza Namazi
• Kai Klauck
• Frank Zielen
• Carsten Burstedde
• Alexander John
• Philip Greulich