title: Microscopic traffic flow model
text: # 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) $s_{\alpha }=x_{\alpha -1}-x_{\alpha }-\ell _{\alpha -1}$ to the leading vehicle $\alpha -1$ (where $\ell _{\alpha -1}$ 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:
  
      ${\ddot {x}}_{\alpha }(t)={\dot {v}}_{\alpha }(t)=F(v_{\alpha }(t),s_{\alpha }(t),v_{\alpha -1}(t),s_{\alpha -1}(t))$
  
  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:
  
      ${\dot {v}}_{\alpha }(t)=f(x_{\alpha }(t),v_{\alpha }(t),x_{\alpha -1}(t),v_{\alpha -1}(t),\ldots ,x_{\alpha -n_{a}}(t),v_{\alpha -n_{a}}(t))$
  
  ### Examples of car-following models
  
  * Optimal velocity model (OVM)
  * Velocity difference model (VDIFF)
  * Wiedemann model (1974)
  * Gipps' model (Gipps, 1981)
  * Intelligent driver model (IDM, 1999)
  * DNN based anticipatory driving model (DDS, 2021)
  
  ## 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 updated rules of the form:
  
      $v_{\alpha }^{t+1}=f(s_{\alpha }^{t},v_{\alpha }^{t},v_{\alpha -1}^{t},\ldots )$
      $x_{\alpha }^{t+1}=x_{\alpha }^{t}+v_{\alpha }^{t+1}\Delta t$
  
  (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 real-time or even faster.
  
  ### Examples of cellular automaton models
  
  * Rule 184
  * Biham–Middleton–Levine traffic model
  * Nagel–Schreckenberg model (NaSch, 1992)
  