# Microscopic traffic flow model

Microscopic traffic flow models are a class of scientific models 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) $s_\alpha = x_{\alpha-1} - x_\alpha - l_{\alpha-1}$ to the leading vehicle $\alpha-1$ (where $l_{\alpha-1}$ denotes the vehicle length), and the velocity $v_{\alpha-1}$ of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

$\ddot{x}_\alpha(t) = \dot{v}_\alpha(t) = F(v_\alpha(t), s_\alpha(t), v_{\alpha-1}(t))$

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:

$\dot{v}_\alpha(t) = f(x_\alpha(t), v_\alpha(t), x_{\alpha-1}(t), v_{\alpha-1}(t), \ldots, x_{\alpha-n_a}(t), v_{\alpha-n_a}(t))$

## Cellular automaton models

Cellular automaton (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 update rules of the form:

$v_\alpha^{t+1} = f(s_\alpha^t, v_\alpha^t, v_{\alpha-1}^t, \ldots)$
$x_\alpha^{t+1} = x_\alpha^t + v_\alpha^{t+1}$

(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, $\Delta t = 1 \text{s}$. 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 $5 \Delta x/\Delta t = 135 \text{km/h}$, 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 $\Delta x/(\Delta t)^2 = 7.5 \text{m}/\text{s}^2$ which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example $\Delta x = 1.5 \text{m}$, leading to a smallest possible acceleration of $1.5 \text{m}/\text{s}^2$.

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 realtime or even faster.