Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling
Lecture 17 : Traffic Flow Modeling Analogies
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Inference

  1. Density $ k$ is constant along characteristic lines
  2. Characteristic lines are straight lines emanating from the boundaries of $ x-t$ plane
  3. The slope of the characteristic line is
    $\displaystyle \frac{dx}{dt} = f(k) + k.\frac{df}{dk} \equiv \frac{dq}{dk}$      

    ie., Characteristic curve has the slope equal to the tangent of the flow density curve.
  4. When two Characteristic lines intersect (ie., 2 $ k$ values at a given x,t) shock waves are generated; and characteristic line terminate.
  5. Shock wave represent mathematical discontinuity ie., abrupt changes to $ k,q,v$.
  6. Speed of the Shock wave is ratio of the time storage rate to space storage rate; that is:

    $\displaystyle v_w = \frac{q_d -q_u}{k_d -k_v}.$