A presentation of Celular Automata (CA) models of traffic flow.

1. PRESENTATION ON REVIEW OF CELLULAR AUTOMATA (CA)
TRAFFIC FLOW MODELS
BY
ANDREW ABAH ONOJA
(SPS/15/MCE/00023)
COURSE
ADVANCED TRAFFIC ENGINEERING
(CIV 8329)
M.ENG – CIVIL ENGINEERING
(TRANSPORT/HIGHWAY ENGINEERING OPTION)
DEPARTMENT OF CIVIL ENGINEERING
BAYERO UNIVERSITY, KANO
3RD June, 2016
PLEASE BE ATTENTIVE!!!

2. • A REVIEW OF CELLULAR AUTOMATA (CA) TRAFFIC FLOW
MODELS
• 1.0 BACKGROUND
• The concept of CA models was originally discovered in the 1940s by
Stanislaw Ulam and John Von Neumann while they were contemporaries
at Los Alamos National Laboratory, USA. Ulam studied the growth of
crystals, using a simple lattice network as his model. At the same time,
John Von Neumann, Ulam's colleague at Los Alamos, was working on the
problem of self-replicating systems. Von Neumann's initial design was
founded upon the notion of one robot building another robot. This design
is known as the kinematic model. Ulam was the one who suggested
using a discrete system for creating a reductionist model of self-
replication. Ulam performed many of the earliest explorations of these
models of artificial life.
• Kinematic is the study of the mathematics of motion without considering
the forces that affect the motion.
• – Deals with the geometric relationships that govern the system
• – Deals with the relationship between control parameters
and the behavior of a system in state space.

3. • It is usually assumed that every cell in the universe
starts in the same state, except for a finite number of
cells in other states; the assignment of state values is
called a configuration. More generally, it is sometimes
assumed that the universe starts out covered with a
periodic pattern, and only a finite number of cells
violate that pattern. The latter assumption is common
in one-dimensional cellular automata.
• Ulam and Von Neumann created a method for
calculating liquid motion in the late 1950s. The driving
concept of the method was to consider a liquid as a
group of discrete units and calculate the motion of each
based on its neighbors behaviors.

4. • Thus was born the first system of cellular automata. Like
Ulam's lattice network, Von Neumann's Cellular Automata are
two-dimensional, with his self-replicator implemented
algorithmically. The result was a universal copier and
constructor working within a cellular automaton with a small
neighborhood (only those cells that touch are neighbors; for
Von Neumann's cellular automata, only orthogonal cells).
Von Neumann gave an existence proof that a particular
pattern would make endless copies of itself within the given
cellular universe by designing a 200,000 cell configuration
that could do so. This design is known as the tessellation
model, and is called a Von Neumann Universal Constructor.
• More recent work has been done by other researchers such
as Zhenke Luo, (2014) at school of Control Science and
Engineering. Shandong University, China.

5. • OVERVIEW OF CELLULAR AUTOMATA MODEL
• In recent decades, growing traffic congestion has become one of the most prior
problems of the society. In populated areas, the existing road networks are not
able to satisfy the demands. The construction of new roads is usually not a
solution and often not socially desired. These reasons together with the great
economical costs lead to new traffic management and information systems,
where, introducing new traffic models gains importance. It was against this
backdrop that there is a wide range of alternative modelling approaches now
available which can be roughly divided into three categories: Macroscopic,
Mesoscopic and Microscopic models.
• Cellullar Automata (CA) are MICROSCOPIC MODELS that are discrete in space
and time and the state variable which in the case of traffic models is usually the
VELOCITY.
• A cellular automaton (pl. cellular automata, abbr. as CA) is a discrete model
studied in computability theory and micro-structure modeling amongst others.
Cellular automata are also called cellular spaces, homogeneous structures,
cellular structures, and iterative arrays.

6. * They consist of a regular grid of cells, each in one of a finite
number of states, such as on and off (in contrast to a
coupled map lattice). The grid can be in any finite number of
dimensions. For each cell, a set of cells called its
neighborhood is defined relative to the specified 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
determines the new state of each cell in terms of the current
state of the cell and the states of the cells in its
neighborhood. 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.

7. A TYPICAL CELLULAR
AUTOMATA (CA) PATTERN IN CHINA
• CA models are
promising models
for their ability to
simulate detailed
phenomena (each
individual vehicle) in
traffic which yields
to an accurate
representation of
traffic flow.
• Free flow
• Synchronize flow
• Wide moving jam
• Retarded
acceleration
• Timely breaking

8. • Among microscopic traffic flow models, cellular automata
(CA) models have the ability of being easily implemented
for parallel computing because of their intrinsic
synchronous behaviour.
• For simulating a complex behaviour in the CA models, it
used a set of simple rules specifying the evolution of the
system, which are discrete in both space and time
variables.
• The simplicity of these models makes them
computationally very efficient and can be used to simulate
large road networks in real-time or even faster.
• CA models are capable of capturing micro-level dynamics
and relating these to macro-level traffic flow behaviour.

9. • Typically, the space discretization in CA model is chosen
such that each cell is occupied by at most one car. Then
the length of one cell corresponds to approximately 7.5m
in reality, i.e., the length of a car plus the average
headway to the next car in a dense jam or be empty.
• This is the Nagel – Schreckenberg (NaSch) Model which is
defined on a one-dimensional array of L -"cells" under
open or periodic boundary conditions.

10. Suppose xn and vn denote the location and the
speed of vehicle, n, respectively, and vmax
represents the maximum velocity of a vehicle.
dn = xn + 1 - xn – 1
The expression above is the distance from the
vehicle, n to its front vehicle. Then each
vehicle can move with an integer velocity. An
update of vehicle state in the CA model
involves with the following four consecutive
steps.

11. • Step 1: Acceleration
• If vn < vmax , then increase the velocity of vehicle, n by one
unit,
• Vn = min(Vn + 1, Vmax}.
• Step 2: Deceleration
• This is usually due to effect of neighbouring cars, but in
some cases other realistic factors will play a role in
decelerating. Such factors can include: pavement
condition, pedestrians obstructions and distractions and
weather (in places of high snowy weather like China)
• If dn < vn , then the speed of vehicle, n is decreased to dn
• i. e. Vn = min (Vn, dn)

12. • Step 3: Randomization
• If vn > 0, then the velocity decreases by one unit with
a probability, p which is accounting for conditions
that the velocity decreases due to the influence of
other uncertain factors, such as pedestrians
obstructions and distractions.
• i. e. Vn = max (Vn – 1, 0).
• Step 4: Movement
• The vehicle updates its location with the velocity
determined by Steps 1-3,
• i. e. xn = xn + vn

13. The simple interpretations to the rules include:
Step 1: Expresses the desire of the drivers to move
as fast as possible (or allowed). Often vmax corresponds to
a speed limit which is the same for all the cars.
Step 2 Reflects the interactions between
consecutive vehicles. It guarantees the absence of
collisions in the model, Critics: (But this is not completely
possible in most realistic context). Here the velocity of
the preceding car is not taken into account. This is already
sufficient to reproduce the basic properties of real traffic.
In order to obtain a good agreement with the ‘fine-
structure’ of empirical data a more sophisticated braking
rule is necessary.

14. • Step 3: This looks rather simple but incorporates many effects
that play an important role. It introduces an asymmetry between
acceleration and deceleration. The acceleration is delayed since a
car will accelerate only with probability 1 – p, if possible. On the
other hand, deceleration processes are enhanced. If a car has to
brake due to another car ahead (Step 2) with probability p, it will
even brake further in Step 3. Another effects incorporated in Step 3
are natural fluctuations. Even on a free road a driver will not keep a
strictly constant velocity which will then show small fluctuations.
• Step 4: All cars will move with their new velocity as
determined in the first three steps.
Similar concept has already been employed in the case of mixed
car/truck traffic in which cars are "shorter" vehicles and trucks are
"longer" vehicles (Nagel and Schreckenberg1992), this is an example
of mixed traffic flow representation in an environment with a two-
lane bi-directional road segment.

15. • Conclusion:
• The Concept of CA has really broaden my
knowledge in the aspect of microscopic
modelling.
• But more research work need to be done to
encompass more traffic realities, such as
effect of weather as usually experienced in
large cities like Abuja, Nigeria.

16. THE END
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