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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 describing the complete dynamics of the vehicles' positions x_\alpha and velocities v_\alpha. It is assumed that the input stimuli of the drivers are restricted to their own velocity v_\alpha, the net distance (bumper-to-bumper distance) s_\alpha = x_ - x_\alpha - \ell_ to the leading vehicle \alpha-1 (where \ell_ denotes the vehicle length), and the velocity v_ 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 (Ver ...
of each vehicle is characterized by an acceleration function that depends on those input stimuli: :\ddot_\alpha(t) = \dot_\alpha(t) = F(v_\alpha(t), s_\alpha(t), v_(t)) In general, the driving behavior of a single driver-vehicle unit \alpha might not merely depend on the immediate leader \alpha-1 but on the n_a vehicles in front. The equation of motion in this more generalized form reads: :\dot_\alpha(t) = f(x_\alpha(t), v_\alpha(t), x_(t), v_(t), \ldots, x_(t), v_(t))


Examples of car-following models

* Optimal velocity model (OVM) * Velocity difference model (VDIFF) * Wiedemann model (1974) * Gipps' model (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 with ...
(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, tesse ...
(CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length \Delta x and the time is discretized to steps of \Delta t. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form: :v_\alpha^ = f(s_\alpha^t, v_\alpha^t, v_^t, \ldots) :x_\alpha^ = x_\alpha^t + v_\alpha^ (the simulation time t is measured in units of \Delta t and the vehicle positions x_\alpha in units of \Delta x). The time scale is typically given by the reaction time of a human driver, \Delta t = 1 \text. With \Delta t 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 \Delta x 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 5 \Delta x/\Delta t = 135 \text, 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 \Delta x/(\Delta t)^2 = 7.5 \text/\text^2 which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example \Delta x = 1.5 \text, leading to a smallest possible acceleration of 1.5 \text/\text^2. 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