In traffic flow theory, Newell’s car-following model is a method used to determine how vehicles follow one another on a roadway.
The density on the roadway can be determined using the spacing between vehicles and is computed simply the equation: kA = 1/sA Geometric relations from the fundamental diagram can be used to calculate the density as well, given by the equation: kA = (kj w)/(vA+w) In the time-space diagram, the trajectories of the leading (top) and following (bottom) vehicle are separated by the distance δ and time τ.
Using relationships between the previous equations, variables τ and δ can be solved for: τ = 1/(wkj) δ = 1/kj Thus, τ and δ are constants defined by the wave speed and jam density, independent of the speed of the leading vehicle and the traffic state.
The path of vehicle i, a function of time, can be determined using the equation: xi(t) = min(xAF(t), xAC(t)) Position of vehicle i under free-flow conditions: xiF(t) = xi(t-τ) + vf * τ Position of vehicle i under congested conditions: xiC(t) = xi-1(t-τ) - δ Under real-world conditions, a hypothetical following driver may drive improperly, resulting in deviations from the time-space trajectories proposed under Newell’s model.
Time-space trajectories from data collected on roads and highways can be compared to its respective Newell’s car-following model trajectory to determine whether a driver is cautious or aggressive.