Microscopic traffic flow models are a class of
scientific models
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted ...
of
vehicular traffic dynamics.
In contrast, to
macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles.
Car-following models
Also known as ''time-continuous models'', all car-following models have in common that they are defined by
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
describing the complete dynamics of the vehicles' positions
and velocities
. It is assumed that the input stimuli of the drivers are restricted to their own velocity
, the net distance (bumper-to-bumper distance)
to the leading vehicle
(where
denotes the vehicle length), and the velocity
of the leading vehicle. The
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verla ...
of each vehicle is characterized by an acceleration function that depends on those input stimuli:
:
In general, the driving behavior of a single driver-vehicle unit
might not merely depend on the immediate leader
but on the
vehicles in front. The equation of motion in this more generalized form reads:
:
Examples of car-following models
*
Optimal velocity model (OVM)
*
Velocity difference model (VDIFF)
*
Wiedemann model (1974)
*
Gipps' model
Gipps' model is a mathematical model for describing car-following behaviour by motorists in the United Kingdom.
The model is named after Peter G. Gipps who developed it in the late-1970s under S.R.C. grants at the Transport Operations Research Gr ...
(Gipps, 1981)
*
Intelligent driver model
In traffic flow modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided wit ...
(IDM, 1999)
*
DNN based anticipatory driving model (DDS, 2021)
Cellular automaton models
Cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
(CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length
and the time is
discretized to steps of
. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:
:
:
(the simulation time
is measured in units of
and the vehicle positions
in units of
).
The time scale is typically given by the reaction time of a human driver,
. With
fixed, the length of the road sections determines the granularity of the model. At a complete standstill, the average road length occupied by one vehicle is approximately 7.5 meters. Setting
to this value leads to a model where one vehicle always occupies exactly one section of the road and a velocity of 5 corresponds to
, which is then set to be the maximum velocity a driver wants to drive at. However, in such a model, the smallest possible acceleration would be
which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example
, leading to a smallest possible acceleration of
.
Although cellular automaton models lack the accuracy of the time-continuous car-following models, they still have the ability to reproduce a wide range of traffic phenomena. Due to the simplicity of the models, they are numerically very efficient and can be used to simulate large road networks in real-time or even faster.
Examples of CA models
*
Rule 184
Rule 184 is a one-dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems:
* Rule 184 can be used as a simpl ...
*
Biham–Middleton–Levine traffic model
The Biham–Middleton–Levine traffic model is a self-organization, self-organizing cellular automaton microscopic traffic flow model, traffic flow model. It consists of a number of cars represented by points on a lattice with a random starting p ...
*
Nagel–Schreckenberg model (NaSch, 1992)
See also
*
Microsimulation
References
{{DEFAULTSORT:Microscopic Traffic Flow Model
Road traffic management
Mathematical modeling
Traffic flow