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 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 }-\ell _{\alpha -1}} to the leading vehicle α − 1 {\displaystyle \alpha -1} (where ℓ α − 1 {\displaystyle \ell _{\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:
x ¨ α ( t ) = v ˙ α ( t ) = F ( v α ( t ) , s α ( t ) , v α − 1 ( t ) , s α − 1 ( t ) ) {\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 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:
v ˙ α ( t ) = f ( x α ( t ) , v α ( t ) , x α − 1 ( t ) , v α − 1 ( t ) , … , x α − n a ( t ) , v α − n a ( t ) ) {\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))}
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) Rakha-Pasumarthy-Adjerid model (RPA model) Fadhloun-Rakha model (FR model)
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 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 updated rules of the form:
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