Published Feb 7, 2023 (Updated Mar 7, 2023) by Garyoung Lee
Newell, G. F. (2002). A simplified car-following theory: a lower order model. Transportation Research Part B: Methodological, 36(3), 195-205. Download
The Newell’s simplified car-following model begins with a simple principle that a vehicle’s time-space trajectory on a homogeneous highway is identical to its preceding vehicle’s trajectory but with space and time shifts. The exact solution of this model is based on the LWR kinematic wave theory with triangular fundamental diagram which has three parameters: free-flow speed \(u\), wave speed \(-w\), and jam density \(k_j\).
\[x_{i}(t)=\min\{\underbrace{x_{i}(t-\tau)+ \color{orange}{u \tau}}_{\text{free-flow (Y)}},\ \underbrace{x_{i-1}(t-\tau)-\delta}_{\text{congestion (Z)}} \}\]where \(x_{i}(t)\) is the postion of vehicle \(i\) at time \(t\), \(\tau = \frac{1}{wk_j}\) is a time shift (wave trip time) between two consecutive trajectories, and \(\delta = \frac{1}{k_j}\) is a space shift (jam spacing). Here, \(u\tau\) is a deterministic vehicle displacement between \(t-\tau\) and \(t\).
Laval, J. A., Toth, C. S., & Zhou, Y. (2014). A parsimonious model for the formation of oscillations in car-following models. Transportation Research Part B: Methodological, 70, 228-238. Download
Newell’s model has been extended by applying the bounded acceleration model to proudce realistic oscillations and hysteresis. Stochastic desired acceleration has taken into account to incorporate human error with the oscillation. The stochastic vehicle displacement, \(\xi_{i}(\tau)\), replaces the deterministic term of Newell’s original model.
\[x_{i}(t)=\min\{\underbrace{x_{i}(t-\tau)+ \color{orange}{\xi_{i}(\tau)}}_{\text{free-flow (Y)}},\ \underbrace{x_{i-1}(t-\tau)-\delta}_{\text{congestion (Z)}} \}\]Desired acceleration is defined as the acceleration the driver imposes to the vehicle when traveling at a speed \(v(t)\) at time \(t\) under free-flow conditions. The mean desired acceleration is a linearly decreasing function of speed, \(a(v(t)) = (v_c - v(t)) \beta\), where \(v_c\) is a target speed and \(\beta\) is an inverse relaxation time. The standard Brownian motion of \(W(t)\) is used to formulate linear stochastic differential equation as below.
\[\begin{cases} d\xi(t)=v(t) dt, \quad \xi(0)=0 \\ dv(t)=(v_c-v(t))\beta dt + \color{orange}{\sigma dW(t)}, \quad v(0)=v_0 \end{cases}\]By solving this equation, one obtains:
\[\begin{align} &E[\xi(t)] = v_c t - (1- e^{-\beta t})(v_c - v_0)/\beta \\ &V[\xi(t)] = \frac{\sigma^2}{2\beta^3} (e^{-\beta t} (4 - e^{-\beta t}) + 2 \beta t - 3) \end{align}\]If more details are needed, one can refer to the paper.
Yuan, K., Laval, J., Knoop, V. L., Jiang, R., & Hoogendoorn, S. P. (2019). A geometric Brownian motion car-following model: towards a better understanding of capacity drop. Transportmetrica B: Transport Dynamics, 7(1), 915-927. Download
As the standard brownian motion embedded desired acceleration model turned out to be unable to capture capacity drop, the geometric brownian motion is applied instead to address this.
\[\begin{cases} d\xi(t)=v(t) dt, \quad \xi(0)=0 \\ dv(t)=(v_c-v(t))\beta dt + \color{orange}{(v_c - v(t))\sigma dW(t)}, \quad v(0)=v_0 \end{cases}\]Xu, T., & Laval, J. A. (2019). Analysis of a two-regime stochastic car-following model: Explaining capacity drop and oscillation instabilities. Transportation Research Record, 2673(10), 610-619. Download
Xu, T., & Laval, J. (2020). Statistical inference for two-regime stochastic car-following models. Transportation Research Part B: Methodological, 134, 210-228. Download
Xu, T. (2020). Parameter estimation and statistical inference of a two-regime car-following model (Doctoral dissertation, Georgia Institute of Technology).