Microscopic traffic flow model

Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. 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. 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 α {displaystyle x_{alpha }} and velocities v α {displaystyle v_{alpha }} . It is assumed that the input stimuli of the drivers are restricted to their own velocity v α {displaystyle v_{alpha }} , the net distance (bumper-to-bumper distance) s α = x α − 1 − x α − l α − 1 {displaystyle s_{alpha }=x_{alpha -1}-x_{alpha }-l_{alpha -1}} to the leading vehicle α − 1 {displaystyle alpha -1} (where l α − 1 {displaystyle l_{alpha -1}} denotes the vehicle length), and the velocity v α − 1 {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: In general, the driving behavior of a single driver-vehicle unit α {displaystyle alpha } might not merely depend on the immediate leader α − 1 {displaystyle alpha -1} but on the n a {displaystyle n_{a}} vehicles in front. The equation of motion in this more generalized form reads: 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 Δ x {displaystyle Delta x} and the time is discretized to steps of Δ t {displaystyle 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: (the simulation time t {displaystyle t} is measured in units of Δ t {displaystyle Delta t} and the vehicle positions x α {displaystyle x_{alpha }} in units of Δ x {displaystyle Delta x} ). The time scale is typically given by the reaction time of a human driver, Δ t = 1 s {displaystyle Delta t=1{ ext{s}}} . With Δ t {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 Δ x {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 5 Δ x / Δ t = 135 km/h {displaystyle 5Delta x/Delta t=135{ ext{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 Δ x / ( Δ t ) 2 = 7.5 m / s 2 {displaystyle Delta x/(Delta t)^{2}=7.5{ ext{m}}/{ ext{s}}^{2}} which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example Δ x = 1.5 m {displaystyle Delta x=1.5{ ext{m}}} , leading to a smallest possible acceleration of 1.5 m / s 2 {displaystyle 1.5{ ext{m}}/{ ext{s}}^{2}} .