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 formula_1 and velocities formula_2. It is assumed that the input stimuli of the drivers are restricted to their  own velocity formula_2, the net distance (bumper-to-bumper distance) formula_4 to the leading vehicle formula_5 (where formula_6 denotes the vehicle length), and the velocity formula_7 of the leading vehicle. The equation of motion 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 formula_9 might not merely depend on the immediate leader formula_5 but on the formula_11 vehicles in front. The equation of motion in this more generalized form reads: 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 formula_13 and the time is discretized to steps of formula_14. 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 formula_17 is measured in units of formula_14 and the vehicle positions formula_1 in units of formula_13). The time scale is typically given by the reaction time of a human driver, formula_21. With formula_14 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 formula_13 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 formula_24, 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 formula_25 which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example formula_26, leading to a smallest possible acceleration of formula_27. 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.