A cellular automaton model for freeway traffic

(1)

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A cellular automaton model for freeway traﬀic

Kai Nagel, Michael Schreckenberg

To cite this version:

Kai Nagel, Michael Schreckenberg. A cellular automaton model for freeway traﬀic. Journal de

Physique I, EDP Sciences, 1992, 2 (12), pp.2221-2229. �10.1051/jp1:1992277�. �jpa-00246697�

(2)

aassification Physics Abstracts

02.50 02.70 o3AoG

A cellular automaton model for freeway traffic

Kai

Nagel(~)

and Michael

Schreckenberg(~)

(~) Mathematisches Institut, Universitit _zu I~6ln, Weyertal 86-90, W-sore I~6ln 41, Germany (~) Institut fir Theoretische Physik, Universitit

zu K61n, Zfilpicher Str. 77, ^W-sore ^I~6ln 41, Germany

(Received

3 September 1992, accepted 10 September

1992)

Abstract. We introduce

a stochastic discrete _automaton model _to simulate freeway traffic.

Monte-Carlo simulations of the model show _a transition from laminar traffic flow to start-stop-

waves with increasing vehicle density, _as is observed in real freeway traffic. For special _cases

analytical results _can be obtained.

1. Introduction.

Fluid-dynamical approaches

to traffic flow have been

developed

since the 1950's

[ii.

In recent

times,

the methods of nonlinear

dynamics

_were

succesfully applied

to these

models, stressing

the notion of _a

phase

transition from laininar flow to start-stop-waves ^with

increasing

_car

density

_[2]. Automatic detection of stronger fluctuations _near this critical

point

has

already

been used to install better traffic control systems ⁱⁿ

Germany

_[3].

On the other

hand,

boolean stimulation models for

freeway

traffic have been

developed

_{[4, 5].}

For lattice _gas automata, it is well known that boolean models _can simulate fluids [6]. We show that indeed _our boolean model for traffic flow has _a transition from laminar to turbulent

behavior,

and _our simulation results indicate that the system ^reaches _a

possibly

critical _state

by

itself in _a bottleneck situation

(reminiscent

of self

organizing criticality [7]).

^This

point

together

^with an extension to multi-lane traffic will be the

subject

of further

investigations

[12].

The outline of this _paper is _as follows: At the

beginning,

_we describe _our model

(Sect. 2)

and discuss its

phenomenological behavior, especially

the transition

(Sect. 3).

In section 4 results for the bottleneck situation _are

presented.

Section 5 contains _a discussion of the results and compares them to values in

reality.

Sections 6 and 7

give

_a short

conflusion/outlook.

A

preliminary

account of the model is

given

in [8].

(3)

2222 JOURNAL DE PHYSIQUE I N°12

2. The model.

Our

computational

model is defined _on _a one-dimensional _array of L sites and with _open _or

periodic boundary

conditions. Each site may either be

occupied by

_one

vehicle,

_or it _may be empty. Each vehicle has _an

integer velocity

^with values between _zero and _vmax. For _an

arbitrary configuration,

_one

update

of the system ^consists ^of ^the

following

four consecutive steps, ^which are

performed

in

parallel

for all vehicles:

I)

Acceleration: if the

velocity

_v of _a vehicle is lower than _vmax and if the distance to the next car ahead is

larger

than _v ₊

I,

the

speed

is advanced

by

_one _[v

- v

+11.

2) Slowing

down

(due

to other

cars):

if _a vehicle at site I _sees the next vehicle _at site I +

j (with j

<

v),

it reduces its

speed

to

j

I [v -

j

_11.

3)

Randomization: with

probability

_p, the

velocity

of each vehicle

(if greater

^than

zero)

is decreased

by

_one [v _- v 11.

4)

^Car ^motion: êach ^vehicle îs âdvanced v sites.

Through

the steps one to four _very

general properties

of

single

lane traffic _are modelled _on the basis of

integer

valued

probabilistic

cellular automaton rules [9, 10].

Already

this

simple

model shows nontrivial and realistic behavior.

Step

3 is essential in

simulating

realistic traffic flow since otherwise the

dynamics

is

completely

deterministic. It takes into account natural

velocity

fluctuations due to human behavior _or due to

varying

external conditions. Without this

randomness,

_every initial

configuration

of vehicles and

corresponding

^velocities reaches very

quickly

a

stationary

pattern ^which is shifted backwards

(I.e. opposite

the vehicle

motion)

one site _per time step.

The Monte Carlo simulations have

mainly

been carried out with the choice of _vmax

= 5 for

reasons stated below. The model has been

implemented

in

FORTRAN, using

a

logical

_array

for the

positions

of the _cars and _an

integer

_array for the velocities. For the 'realistic case' of _vmax

= 5 _we made the

following

observations: in the

interesting regime

of _a

relatively

low

density

of

occupied

sites

(usually only

about _one

fifth),

_an

implementation using

IF-statements has been about five times faster than _an

implementation using only

boolean variables and

operations (necessary

for

multispin coding

for 32 _cars at

once).

As the

expected gain

of

multispin coding

would therefore

give only

_a factor of about

six,

_we

postponed

the work _on _a bitwise

implementation.

The

speed

_on _an

IBM-RS/6000

320H workstation _was about 0.2

site-updates

_per microsec-

ond,

which is about _a factor of 7 slower than the

speed

of _a

non-vectorized, multispin coding Ising

^model

[I ii.

A

significant

part ^of ^the Monte-Carlo simulation _runs have been carried out

on an iPSC-860

Hypercube using

_up to 16

nodes,

with

only slight

modifications of the _prc-

gram. The model's

speed

_on _one _processor of the

Hypercube

_was about the _same _as _on the

workstation.

3. Waific _on _a circle

(closed system).

In this section _we present results from systems with

periodic boundary

conditions

(thus

sim-

ulating

^traffic in _a closed

loop

"as in _car races" but

only

_on _a

single lane).

As the total number N of _cars in the circle _cannot

change during

the

dynamics,

it is

possible

to define _a

constant system

density

N number of _cars in the circle

~ L number of sites of the circle ^' ^~~~

(4)

space (road)

...4...,...6...6

...4...6...

.6...6...4..

6...6...6...:

...4...4...,...6...

.,...6...4...:...6...

.6...3....4...,...,...,...,.4...

...6...,.,,3...4.,,...,,.,...4,,....,...

~ ...6...,.3...6...,..,..,...,...6..,...

~

,...,,...,.,.6.,....4...4...,..,...4....

...4...6...6...4

Qd ...6...4...4...

...6...6...6...

...6...4...6...

...3....6...4...

...4...4...4...

...6...4...6...

...6...6...4..

...6...4...

...4....4...

...3....

...3.

Fig.

I. Simulated traffic at _a

(low)

density of o.03 _cars per site. Each _new line shows the traffic lane

after _one further complete velocity-update and just before the _car motion. Empty sites _are represented

by _a dot, sites which _are occupied by _a _car _are represented by the integer number of its velocity. At low densities, _we _see undisturbed motion.

which is

usually

not

possible

in

reality. Thus,

in order _to mimick real

conditions,

_we measured densities

(= occupancies) p~

_on _a fixed site I

averaged

_over _a time

period

T:

v=y

io+T

^~ n;(f) (2)

where

n;(t)

₌

0(1)

if site I is empty

(occupied)

at time step t. For

large

T _we have lim

p~

= p.

(3)

T-m

The

time-averaged

flow # ^between ^I ^and I + I is defined

by

f

"

~~~l ~ ~i>I+I(f) (4)

where

n;,;+i(t)

₌ I if _a _car motion is detected between sites I and _{I + I.}

With these

definitions,

it _was easy to

perform

_many simulations with different densities _p, thus after relaxation to

equilibrium getting

data for the

commonly

used fundamental

diagrams

which

plot

^vehicle flow # _vs.

density

_p.

To be

specific,

_we _start with _a random initial

configuration

of

cars with

density

_p and

velocity

0 and

begin

the collection of data after the first to time steps, where _we took to = 10_x L.

In

figures

I and 2 _we sho,v

typical

situations _at low and

high

densities. Whereas _we find laminar traffic at low

densities,

there _are

congestion

clusters

(small jams)

at

higher densities,

which

are formed

randomly

due _to

velocity-fluctuations

of the _cars. If _one follows

(in Fig. 2)

_one

(5)

space (road)

...4...6...6...6...OO.1...4...6

..4...6...4...6...2...00..2...6...

..,.,.4.,,.,.,,..6.,...,,,...,.4,...3...0.Ol....2...6..

4...,4,,.,,...4...,.6.,,...01,1,2,,...3.,..,.,.,.,,.,

....6...,,6..,.,...6...O.1.0.2..3...,,.4...

...6.,...,...4...,,,,,,,.4...,....,,.1..00...3,..3.,...,.6,,,,...

.4...4,...4...6...1.00...3...3...6..

..,,.4...,,,6,,,...4...,,..3....000...2...3...

.,,,...,.4.,...6...4...0.001.,...,3,...3...

...4...6...4...0.00.2...4....4...

4...4...6...2...I.Ol.,,2.,...,.,3....

,..,4,..,...,..4...,...,,.,.2.,I..0i.2.,..3,.,...,.4,

...6...6...0,I.0.2.,2,..,.3...,...

...6...6...0..00...I,.2,...3....,...,

,.4,,..,...,..4,...,,.,...3.,,,I.,00,,.,2,.,3.,,....3.,...

l~ .,.,..6...,..4.,.,...,...,...2..0,00,...2....3...3...

f~ ..6...,....4....,....,....4.,...0i.00,,...3....,4,.,,...4,.,

.- .6...4...6...,.6...0.000...3...4...

+~ ...4....6...6...4....1.001...3...4...

4...,...6.,,..6.,..,...4,...,...,..1.000.I...,..,...4,...,

....6...4...6...,6...,,..0000..I...,...,....,..6....

...4...6..,.,,4.,.,...4...0000.,.2,...,... ...6...4...6...0.O001...2...

...4...,...4...,1..0.000.2...2...

.6...,.,.4...2...0.1.000...3..,...3...

,,..,,6,...,,...,.,.6.,...I.O..0001...4...4..,...

..6...6...,...2...01..001.2...4...6...

...6...4...,...,1.0.0.0i.1..3...,...6.,...4...

....6...6...6...,...,.00.0.O.I.1....3...4...6.

...4...6...6...0i,I.O.,0,I,...3...4....

...4...6..,..,.2...I.O.00.,1..2.,...4...,...6

...,...6...4....2..0i.0i...2...3...6...

6,.,.,.,,.,...,...4...,,,..2..0i,00.2....3.,..3,...4...

...4...6...00.000...3.,...3....4...,...6...

Fig.2.

Same picture _as figure I, but at _a higher density of o.I _cars per site. Note the backward motion of the traffic jam.

individual _car

coming

from the

left,

_one _sees that the _car _comes in with _a

speed varying

between four and five and then has to stop ^due to the

congestion

cluster. There it stays stuck

in the _queue for _a certain time with _some slow

advances,

and _can accelerate to full

speed

after

having

left the cluster at its end. So the cluster represents _a

typical

start-stop-wave _as found in

freeway

traffic

(cf. Fig. 3).

We present ^the fundamental

diagram

of _our model in

figure

4. Whereas the line indicates the results of

averaging

_over ^10~ time steps, the dots represent _averages _over

only

^10~ time steps and _may be

compared

with results from real data

(Fig. 5).

It _can

clearly

be _seen that

a

change-over

takes

place

_near _p

= 0.08. Further simulations show that the

position

and the form of the maximum of

q(p) depend

_on the system size.

(Simulations

without randomization do not show _a

dependence

on the system

size.) However,

it is not clear from these

pictures

where _to locate _an exact transition

point.

For _an

analytical

treatment of the circular

traffic,

_one chooses _as

starting point

the _case vmax = I, ^where ^the ^situation

simplifies considerably.

Here step _one of the four

update

_steps is

trivial since _every _car _can accelerate in

only

one time step to its maximum

speed

_vmax ₌ I and

therefore,

before step two, every car has

speed

I. The

question

in the

update procedure

then is

simply

if the next site is free

(step two)

and if the

speed

is

(randomly)

set to _zero

(step three).

(6)

space (road)

fi

~,

Fig.3.

Space-time-lines

(trajectories)

for _cars from Aerial Photography

(after

[16]). Each fine represents ^the movement of _one vehicle in the space-time-domain.

Simulation

I

l~

0~ ^~

_{_§~~~}

~

r....~~.~

fi

3

~

/

.: .:.

Z

_.

~ ^"'~~ ^~~

0~ ^°~.

Clq

w

~

oj :

o i

iS o

~

~0

₂ ₄ ₆ ₈

density [ears _per site]

FigA.

Traffic flow _q

(in

_cars _per time

step)

_vs. density

(in

_cars _per

site)

from simulation results

(L

=

lo~).

Dots _are _averages _over lo0 time steps, the line represents _averages over lo~ time steps.

(7)

Real Traffic

_

~ ~~

~ ²⁰⁰⁰ _oj _&

O

o~~o)°~(,

$$ ~.,

fl"'.

~j

~~#~

~ 15QQ

'~.)~i~lJl

QJ _.: ?.' ";,"~ .$..

Q_ Z_, _z to

..'I'.'~i,

m ^' ;:'>j;,,1,=

~ > '--W'-~

Qj 'I

O

20

ccupancy [Xl

Fig.5.

Traffic flow _q

(in

_cars _per

hour)

_us. _occupancy

(in

_cars _per

hour)

from measurements in reality. Occupancy is the percentage ^{of the} ^road which is covered by vehicles

(after

[17]).

For

speeds larger

than I _an additional parameter for the current

speed

is needed which

gives

serious difficulties for

analytical

treatments.

However,

even in this _case _a kind of mean-field

approximation

is

possible

and the results will be

reported

elsewhere

[12].

For direct calculations the easiest _way to formulate the

dynamics

is _on the basis of _a master

equation

with continuous time and random

sequential update.

This makes of _course _a difference to the

parallel updating (which

_can

simply

be _seen

by simulating

the two different

updates)

but the results should be

qualitatively

similar and the randomization parameter _p

plays

a

particular simple

role for random

sequential update.

With the _use of the _more familiar

spin

variables _a;

= +I with +I for

occupied

and -I for empty

sites,

the transition

probability W(-a;,

_-a;+i

la;,«;+i)

from

(«;, a;+i)

to

(-a;, -a;+i) (I.e.

_a _car _moves from I to I +

I) simply

reads

W(-«;,-«;+il«;,«;+i)=(I-P) @ ₍₅₎

With

periodic boundary

^conditions ^this ^transition

probability

_ensures the conservation of the total

magnetization £;

_«; in the system. The master

equation

for the

probability P((«;),t)

to find

configuration (a;)

at time t is

given by

~jl'

⁼

~ _W(-a;,

-a;+i

la;, a;+i) P(ia;, «;+i,11

;

+

~

_~i~(°I>

_°"+1(

°i>

~°"+l) l'((~°i>

_~°i+I

f)). (6)

;

From this formula it _can

easily

be _seen that the factor I

p) only gives

_a

simple scaling

factor of the time axis.

Therefore,

in continuous

time,

the systems with different _p _< I _are

equivalent

in the _sense that

they

have the _same

equilibrium

distribution.

Analyzing

^this

equilibrium

(8)

distribution _one finds

that, given

_a fixed total

magnetization,

_every

configuration

of the

spin

is

equally probable. Therefore,

in this

simple situation,

_one has

q = p

(i p) (7)

which is

just

the

probablity

that _a _car has _a free site in front of it. The property ^that _every

configuration

of the _cars is

equally probable

is

certainly

not true for _vmax _> 0. The

parallel update gives

_a different function

q(p)

but it is also

symmetric

with respect to p =

1/2 _[12].

4. Waific in _a bottleneck situation

(Open system).

For this

section,

we

apply

different

boundary conditions, leaving

the rest of the model _un-

changed:

. When the leftmost site

(site I)

is empty, we occupy it with _a _car of

velocity

0. As

our traffic is

going

from left to

right,

_one _may

imagine

_a bottleneck situation where _a saturated twc-lane-street feeds _a street of

only

_one lane

(which

_we

simulate).

. At the

right

^side

(I.e.

the end of the

street),

_we

simply

delete _cars _on the

rightmost

six

sites,

thus

producing

_an _open

boundary. (This

simulates the

beginning

of _an

expanded (four-lane) freeway.)

Our simulations included

grid length

_up to 10000 sites with durations _up to 5 _x 10~ time steps. After

relaxation,

the model shows traffic at _a

density

of <pc> ₌ 0.069 + 0.002 and _a flow of <qc> = 0.304 + 0.001.

Again

the situation here

simplifies considerably

for _vmax

= I. The _case of random _sequen- tial

update

is

exactly equivalent

to _an

asymmetric

exclusion model

investigated

in detail in

references

[13,

14] ^where ^the

equilibrium

distribution in _a system with _open boundaries is calculated. The _case _tx

=

fl

₌ I

(in

the notation of Refs.

[13, 14]) corresponds

then _to _our bottleneck situation. It _can be _seen that in this

simple

_case the system drives itself _to the _state of maximum flow at p =

1/2.

However this seems not to be _a

general

property also valid for

larger

_vmax > I

(cf. Fig. 6).

5.

Quantitative comparison

with realistic tra%c.

In this

section,

_we make _some

rough arguments concerning

the

length

^scale and time scale of the simulation model. The easiest

approach

to scale the model is the claim that in _a

complete jam

each _car

occupies

about 7.5m of

place,

which is thus the

length

of _one site.

As the _average

velocity

in free traffic of 4.5 sites _per time step should

correspond

to _a

velocity

of about 120

km/h (in Germany),

_we arrive

naturally

at _a time for _one iteration of

7.5 ~3 x4.5 ^"~~~

/(120/3.6)

^~

(8)

site time-step m

m

(I

second _per

time-step)

which _agrees with [5].

A second

possibility

is to scale the model

by

the

position

of the maximum in the fundamental

diagram.

From traffic measurements, this maximum is found at about _p

r-

(30

vehicles _per lane and

kilometer)

₌

(0.225 vehifles/7.5 m),

which is

by

_a factor of about 2

higher

than the

position

of the maximum in the

scatterplot

for _our model.

Similarly, freeways

have _a maximum

capacity

of about

(2000

vehicles _per hour and

lane)

=

(0.56

vehicles _per

second).

As _our maximum of the flow is

only

0.32 vehicles _per time step,

(9)

Flow-density

relation: High ^Resolution near rho=0.08 0.34

j~

~ ~~ 'V5/p.5/Xe4/T8e5.d' O

.)

_0.32

~ ~ _~

~

1~0.31 ^~ ^~~ ^° ^° °

o

~

i

~

0.3

~

)

0.29

~

(

_0.28

f

o 0.27

~ ~

0.26

0.06 0.08 0.1 0.12 0.14 0.J 6

av. density rho [# of _cars per site]

Fig.6.

Interesting part

(near

the

change-over)

of figure 4

(only

long time

averages).

The _cross indicates the point from the bottleneck situation (I.e. _open system with maximal input of _cars from the left and _an

absorbing

boundary at the right, _see

text);

tips of the _arrow indicate the

error of

p. The _error of _q is too small to be visible _on this scale. The simulation clearly indicates that the

self-organ12ing bottleneck state does not correspond to the maximum of the flow, _as it is the

case for

Vmax =1.

our model time step ^should

cot-respond

to

0.32/0.56

_m 0.5

seconds,

thus

being by

_a factor of two lower from the value

presented

above.

A fourth

possibility

_uses the value of the

velocity

of the

back-travelling

start-stop-waves, where _a value of about 15

kin/h

_m 4.2

m/s

has been measured _on

freeways.

As the maximum

capacity

of _our simulated system ^is ^about ^0.3 cars per time step, the maximum

speed

of the

backpropagating

_wave is 0.3 sites

(m

2.25

m)

_per time step

(I.e.,

about every third time step a new car arrives at the back of the traffic

jam).

This would fix _one model time step at

2.25/4.2

_m 0.7

seconds,

thus

yielding

a value between those of the first and of the third method.

Thus all these estimates agree in order of

magnitude:

time steps

correspond

to seconds.

Another

interesting

result of the last argument is that start-stop-waves

might intrinsically

be

a

queueing phenomenon

where

only

the formation takes

place dynamically:

_once _a

congestion

cluster of

length

L has

formed,

there is _a

probability

_{p~~~ per} time step that _a _new _car arrives at the

end,

and _a

probability

_p~~P _per time step

(determined

in _a nontrivial _way

by

the

dynamics

of the

model)

that _a

car

departs

at the front of the

jam (cluster).

As p~~P at _one cluster dictates p~~~ for the next

cluster,

all clusters act like _queues at the critical threshold where _p~~~

= p~~P.

But the formation of the flusters _as well _as the

self-organization

of the critical value of

p~

take

place

without external control.

One should note

that, contrarily

to

intuition,

the parameters ^for a discrete traffic flow model

are

relatively

fixed _once _one accepts "one _car _per lattice site in _a

jam".

The

parameters

^used

for the simulations _seem

relatively

reasonable.

6. Conclusion.

It has been shown that _a discrete model

approach

for traffic flow is not

only computationally

advantageous

_[5,

4],

^but ^that ^it ^contains some of the

important

aspects ^{of the}

fluid-dynamical

(10)

approach

to traffic flow such _as the transition from laminar to

start-stop-traffic

in _a natural _way

(see particularly

the end of the last

section). Thereby,

it retains _more elements of individual

(though statistical)

behavior of the

driver,

which

might

lead _to better usefulness for traffic simulations where individual behavior is concerned

(e.g. dynamic routing).

An additional

interesting

observation is that sand

falling

down in _a

long

and _narrow tube shows _very similar

behavior,

I-e-

"start-stop-waves" originally

caused

by

fluctuations in the

velocity

of the

freely moving particles.

In the _case of sand these fluctuations _are due to

dissipation

at the

boundary [15].

Acknowledgements.

We want to thank A. Bachem for initiation _to the theme

and, together

with R.

Kiihne,

P. Konhiuser and M.

Rbdiger,

for

helpful discussions,

D. Stauffer for numerical and method-

ological help,

and H-J- Herrmann for

interesting

^hints. ^ZAM at the Jiilich Research Center

provided computing

time _on the

newly

installed Intel

iPSC/860,

and R. Knecht

helped

in

using

it. This work _was

partly performed

within the research _program of the

Sonderforschungsbereich

341 K61n-Aachen-Jiilich

supported by

the Deutsche

Forschungsgemeinschaft.

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