Microscopic traffic flow model

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 ${\displaystyle x_{\alpha }}$ and velocities ${\displaystyle v_{\alpha }}$. It is assumed that the input stimuli of the drivers are restricted to their own velocity ${\displaystyle v_{\alpha }}$, the net distance (bumper-to-bumper distance) ${\displaystyle s_{\alpha }=x_{\alpha -1}-x_{\alpha }-\ell _{\alpha -1}}$ to the leading vehicle ${\displaystyle \alpha -1}$ (where ${\displaystyle \ell _{\alpha -1}}$ denotes the vehicle length), and the velocity ${\displaystyle v_{\alpha -1}}$ of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

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

In general, the driving behavior of a single driver-vehicle unit ${\displaystyle \alpha }$ might not merely depend on the immediate leader ${\displaystyle \alpha -1}$ but on the ${\displaystyle n_{a}}$ vehicles in front. The equation of motion in this more generalized form reads:

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

Cellular automaton (CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length ${\displaystyle \Delta x}$ and the time is discretized to steps of ${\displaystyle \Delta t}$. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:

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

${\displaystyle x_{\alpha }^{t+1}=x_{\alpha }^{t}+v_{\alpha }^{t+1}\Delta t}$

(the simulation time ${\displaystyle t}$ is measured in units of ${\displaystyle \Delta t}$ and the vehicle positions ${\displaystyle x_{\alpha }}$ in units of ${\displaystyle \Delta x}$).

The time scale is typically given by the reaction time of a human driver, ${\displaystyle \Delta t=1{\text{s}}}$. With ${\displaystyle \Delta t}$ 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 ${\displaystyle \Delta x}$ 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 ${\displaystyle 5\Delta x/\Delta t=135{\text{km/h}}}$, 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 ${\displaystyle \Delta x/(\Delta t)^{2}=7.5{\text{m}}/{\text{s}}^{2}}$ which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example ${\displaystyle \Delta x=1.5{\text{m}}}$, leading to a smallest possible acceleration of ${\displaystyle 1.5{\text{m}}/{\text{s}}^{2}}$.

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 real-time or even faster.

• Microsimulation