DETERMINISTIC TRAFFIC
MODELS
USING
ONE DIMENSIONAL
CELLULAR AUTOMATA
1
.
Aravindan
A
nbarasu
[3rd
year,
B.E (
ECE
)
]
Kumaraguru Colleg
e of Technology
,
Coimbatore, India
Mobile
:
+
91
9597706300
M
ail:a.aravind1136@gmail.com
Abstract
The paper
briefly discusses
two deterministic
traffic cellular automata
models
,
wolfram
rule
184 and
deterministic fukui

ishibashi
traffic cellular automata.
The
paper with
an
introduction
to cellular automata which is used
fo
r various kinds of modeling also
explains
one dimensional
binary state cellular automata
and wolfram numbers in
detail.
The
paper also describes
the basics of traffic flow modeling us
ing cellular automata,
wolfram
rule 184, d
eterministic fukui

ishibashi
traffic cellular automata
models
and their
application to automated freeway traffic situations.
Keywords:
C
ell
ular Automata, Wolfram number, W
olfram rule 184,
D
eterministic
Fukui

Ishibashi
model.
Introductio
n
A Cellular Automaton (CA) is an infinite, regular lattice of simple finite state machines
that change their states synchronously, according to a local update rule that specifies the
new state of each cell based on the old states of its neighbors.
CA's ar
e
discrete dynamical systems
and are often described as a
counterpart
to
partial differential equations
, which have the capability to describe
continuous
dynamical
systems. The mea
ning of
discrete
is
that space, time and properties of the automaton can
have only a finite, countable number of states. The basic idea is not to try to describe a
complex system from "above"

to describe it using difficult equations, but simulating
this
system by interaction of cells following easy
rules. It
is a
discrete
model studied in
computability theory
,
mathematics
,
physics
,
complexity science
,
theoretical biology
and
microstructure modeling.
T
he essential p
roperties of a CA
A
regular n

dimens
ional lattice
(n is in most cases of one or two dimensions),
where each
cell
of this lattice has a discrete state,
A
dynamical
behavior
, described by so called
rules
. These rules describe the state
of a cell for the next time step, depending on the states
of the cells in the
neighborhood
of the cell.
Terms
Cell
:
A single element of a cellular space, the smallest unit of the space.
Cellular Space
:
A lattice space made up of cells, each of which is in one of
several
predefined states. This is what most peo
ple nowadays call a Cellular
Automaton, but I
find the distinction useful.
Cellular Automaton
:
A structure built in a cellular space, an automaton built
out of
cells. “CA” is an abbreviation that is often
used for these terms.
Local Rule
:
The rule governin
g the transition between states

The definition
of a cell’s
finite state machine. It’s called “local”, because it only uses the
Neighborhood as
its
input.
Neighborhood
:
The surrounding
cells that influence
its
next state. The
choice of
neighborhood influen
ces the
behavior
of the cellular space.
Configuration
:
A snapshot of all cell states, representing a single point in time.
When
talking
about a configuration, it’s usually the starting point or a result
of running a
cellular space.
Generation
:
One step in
the evolution of a cellular space, an intermediate configuration.
Passage of time in a cellular space is measured in generations.
Simple cellular automata
Cellular automata consist of a regular grid of
cells
, each in one of a finite number of
states
, such as "On" and "Off". The grid can be in any finite number of dimensions. For
each cell, a set of cells called its
neighborhood
(usually including the
cell itself) is
defined relative to the sp
ecified cell.
An initial state (time
t
=0) is selected by assigning a
state for each cell. A new
generation
is created (advancing
t
by 1), according to some
fixed rule (generally, a mathematical function) that deter
mines the new state of each cell
in terms of the current state of the cell and the states of the cells in its neighborhood. For
example, the rule might be that the cell is "On" in the next generation if exactly two of
the cells in the neighborhood are "On"
in the current
generation;
otherwise the cell is
"Off" in the next generation. Typically, the rule for updating the state of cells is the same
for each cell and does not change over time, and is applied to the whole grid
simultaneously, though exceptions
are known.
One dimensional

binary state cellular automata and Wolfram number
One

dimensional

binary state CA that uses the nearest neighbors to determine their next
state is called elementary cellular automata. There are only 2
^8
= 256 elementary CA,
and i
t is quite remarkable that one of them is computationally universal.
Elementary CA
was
experimentally investigated in the 1980’s by S.Wolfram. He
designed a simple naming scheme that is still used today. The local update rule gets fully
specified when one
gives the next state for all 8 different contexts:
111
→
b7
110
→
b6
101
→
b5
100
→
b4
011
→
b3
010
→
b2
001
→
b1
000
→
b0
The Wolfram number of this CA is the integer whose binary
expansion is
b7b6b5b4b3b2b1b0.
Elementary CA
is
known by their Wolfram num
bers.
For example, elementary CA number 102 has local update rule
111→0
110→1
101→1
100→0
011→0
010→1
001→1
000→0
In this rule
,
a b c→ b + c (mod 2) and it will be referred to as the xor

CA.
Space

time diagram for elementary cellular automata
A space

tim
e diagram is a pictorial representation of the time evolution of the CA where
consecutive configurations are drawn under each other. Horizontal lines represent space,
and time increases downwards.
For example, the space

time diagram of the xor

CA starting
from a single cell in
state 1 is the self

similar
Rule 102
Figure1: the space

time diagram of the xor

CA starting from a single cell in state 1 is the
self

similar
T
raffic models
Road layout and
the physical environment
When applying the cellular automaton analogy to vehicular road traffic flows, the physical
environment of the system represents the road on which the vehicles are
driving. In a
classic single

lane setup for traffic cellular automat
a, this layout con
sists of a one

dimensional lattice that is composed of individual cells (our descrip
tion here thus focuses
on unidirectional, single

lane traffic). Each cell can either be empty, or is occupied by
exactly
one vehicle;
the
term
single

ce
ll
models
are
used
to
describe these systems.
Because
vehicles move
from one cell to another, TCA models are also called
particle

hopping
models.
Deterministic
traffic models
This traffic model is defined as a one dimensional array with
L
cells with closed
(periodic)
boundary conditions. This means that the total number of vehicles
N
in the
system is
maintained constant. Each cell (site) may be occupied by one vehicle, or it may be empty.
Each cell corresponds to a road segment with a length l equal to the
average
headway in a
traffic jam. Traffic density is given by
k
= N/L.
Each vehicle can have a
velocity from 0
to
v
max
.
The velocity corresponds to the number of sites that a vehicle
advances in one
iteration. The movement of vehicles through the cells i
s determined by a
set of updating
rules. These rules are applied in a parallel fashion to each vehicle at each iteration. The
length of iteration can be arbitrarily chosen to reflect the desired level of simulation detail.
The choice of a sufficiently smal
l iteration interval can thus be used to
approximate a
continuous time system. The state of the system at
iteration
is
determined by the
distribution of vehicles among the cells and the speed of each vehicle in
each
cell.
The
following notation to characte
rize each system state:
x
i
:
position of the ith vehicle,
v
i
:
speed of ith vehicle, and
g
i:
gap between the ith and the (
i+1)th vehicle (i.e., vehicle immediately ahead)
and is
given by
g
i
= x
i+1

x
i
–
1.
A typical dis
cretisation
scheme assumes ΔT = 1 s and ΔX = 7.5 m, corresponding
to speed
increments of
ΔV = ΔX/
ΔT = 27 km/h.
Therefore, vehicles assume the discrete
speeds v
0
= 0km/h,
v1
=
27km/h, v
2
= 54km/h and so on.
The spatial discretisation
corresponds
to the average length
a conventional vehicle occupies in a closely packed jam,
i.e

7.5m (and
as such, its width is neglected), whereas the time discretisation is based on
a
typic
al driver's reaction time and
implicitly assume that a driver does not react to events
between two
consecutive time steps.
Wolfram's rule 184 (CA

184)
The simplest deterministic traffic model is wolfram’s rule 184.
Figure2: simplest deterministic traffic model is wolfram’s rule 184
The shaded portion represents the presence of a vehicle
.
The un

shaded portion represents the absence of a vehicle
This has the physical meaning
that a vehicle (Black Square) moves to the right if its neighboring
cell is empty.
In this model max velocity is 1.
For a TCA
model,
the
previous actions as a set of rules t
hat are
consecutively
applied to
all vehicles in the lattice
.
Rule 1
Acceleration
and braking
v
i
(t)
←
min
{
g
i
(t
−
1), 1
}
Rule 2
Vehicle
movement
x
i
(t)
←
x
i
(t
−
1) +
v
i
(t)
Rule
1, sets the speed of the ith vehicle, for the current updated
configuration of the
system; it states that a vehicle always strives to drive at a speed
of 1 cell/time step, unless
its impeded by its direct leader, in which case
g
i
(t
−
1) =0 and the vehicle consequently
stops in order to avoid a collision. The second rule
2
is not actually a ‘real’ rule; it just
allows the vehicles to advance
in the system.
Figure3: Typical time

spac
e diagrams of the CA

184 TCA model
Typical time

space diagrams of the CA

184 TCA model. The shown closed

loop
lattices each contain 300 cells, with a visible period of 580 time steps (each vehicle is
represented as a single colored dot).
Left:
vehicles dri
ving a free

flow regime with a
global
density
k
= 0.2 vehicles/cell
.
Right:
vehicles driving in a congested regime
with
k
= 0.75
vehicles/cell. The congestion waves can be seen as propagating in the
opposite direction of
traffic; they have an eternal life
time in the system. Both time

space diagrams show a fully
deterministic system that continuously repeats itself.
Deterministic Fukui

Ishibashi TCA (DFI

TCA)
In 1996, Fukui and Ishibashi constructed a
generalization
of the prototypical CA

184
TCA
model
. A
lthough their model is essential
ly a stochastic one
,
this paper
discusses
its
deterministic version. Fukui and Ishibashi’s idea
was two

fold: on the one hand, the
maximum speed was increased from 1 to vmax
cells/time step, and on the other hand,
vehicles w
ould accelerate
instantaneously
to the highest possible speed. Corresponding to
the definitions of the rule set of a
TCA
model;
the CA

184’s rule R
ule1
changes as
follows:
Rule1
:
acceleration and braking
v
i
(t)
←
min
{
g
i
(t
−
1), v
max
}
Just as before, a veh
icle will now avoid a collision by taking into account the size
of its space gap. To this end, it will apply an instantaneous deceleration: for example, a
fast

moving vehicle might have to come to a complete stop when nearing the end of a
jam, thereby
abru
ptly
dropping its speed from vmax to 0 in one time step.
Figure4:
Left:
Several (k, v
s
) diagrams for the deterministic DFI

TCA,
Right:
several of the
DFI

TCA’s (k, q) diagrams
Left:
S
everal (k, v
s
) diagrams for the deterministic DFI

TCA, each for a di
fferent V
max
∈
{
1
, 2, 3,
5
}
. Similarly to the CA

184, the global space

mean speed remains constant, until
the critical density is reached, at which point vs. will start to diminish towards zero.
Right:
several of the DFI

TCA’s (k, q) diagrams, each having
a triangular shape (with
the slope of the congestion branch invariant for the different vmax).
The mean speed remains constant at v
max
till critical density after which the mean speed
becomes (1

k)/k.
Figure5:
Left:
the (k, v
s
) diagram for the determini
stic CA

184,
Right:
the (k, q) diagram
for the same TCA model
Left:
the (k, v
s
) diagram for the deterministic CA

184, with now vmax
→
+
∞
.
Right:
the (k, q) diagram for the same TCA model, resulting in a critical density k
c
= 0.
Conclusion
The models discussed here are deterministic models. They are basic models that work in
a deterministic way and do not include the effects of sudden
braking or idling. There are
various
stochastic models like Nagel

Schreckenberg TCA
to deal with such cases. T
he
basic models discussed here
can be applied only to automated freeway systems.
Reference
[
1
]
.A deterministic traffic flow model for the two

reg
ime approach
–
by Ceder, A
[
2
]
.Self

organization and a dynamical transition in traffic

flow models

by
Ofer Biham
and
A. Alan Middleton
[
3
]
.
A
cellular automaton model for freeway traffic
–
by Kai Nagel and Michael
Schreckenberg
[
4
]
. A
set of effective coordination number (12) radii for the

wolfram structure elements
–
by
S. Geller
[
5
]
.
Traffic Flow
Theory

by
Sven Maerivoet
,
Bart De Moor
[
6
]
. Statistical mechanics of cellular automata by

Stephen Wolfram
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