Module 3 : Microscopic Traffic Flow Modeling
Lecture 14 : Car Following Models
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In computer, implementation of the simulation models, three things need to be remembered:

  1. A driver will react to the change in speed of the front vehicle after a time gap called the reaction time during which the follower perceives the change in speed and react to it.
  2. The vehicle position, speed and acceleration will be updated at certain time intervals depending on the accuracy required. Lower the time interval, higher the accuracy.
  3. Vehicle position and speed is governed by Newton's laws of motion, and the acceleration is governed by the car following model.

Therefore, the governing equations of a traffic flow can be developed as below. Let $ \Delta{T}$ is the reaction time, and $ \Delta{t}$ is the updation time, the governing equations can be written as,

$\displaystyle v_n^t$ $\displaystyle =$ $\displaystyle v_n^{t-\Delta t}+a_n^{t-\Delta t}\times \Delta t$ (1)
$\displaystyle x_n^t$ $\displaystyle =$ $\displaystyle x_n^{t-\Delta t}+v_n^{t-\Delta t}\times \Delta t+\frac{1}{2}a_n^{t-\Delta t}\Delta t^2$ (2)
$\displaystyle a^t_{n+1}$ $\displaystyle =$ $\displaystyle \left[\frac{\alpha_{l,m}{(v^{t}_{n+1}})^m}{{(x^{t-\Delta T}_n-x^{t-\Delta T}_{n+1})^l}}\right](v_n^{t-\Delta T}-v^{t-\Delta T}_{n+1})$ (3)

The equation 1 is a simulation version of the Newton's simple law of motion $ v = u + at$ and equation 2 is the simulation version of the Newton's another equation $ s = ut+\frac{1}{2}a{t^2}$. The acceleration of the follower vehicle depends upon the relative velocity of the leader and the follower vehicle, sensitivity coefficient and the gap between the vehicles.