DETERMINISTIC TRAFFIC MODELS

rumblecleverAI and Robotics

Dec 1, 2013 (3 years and 6 months ago)

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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