Generalized CTM

Generalized CTM is an extension of the cell transmission model that would approximate the hydrodynamic model for an equation of state that allows backward waves with speed $w \leq v$ (see Fig. 1 ). This is a realistic model, since on many occasions speed of backward wave will not be same as the free flow speed.

**Figure 1:** Flow-density relationship for the generalized CTM
$\begin{figure} \centerline{\epsfig{file=qfFlowDensityGeneral.eps,width=8 cm}} \end{figure}$

$\displaystyle y_i(t)=min~[n_{i-1}(t), Q_i(t), w/v\{ N_i(t) - n_i(t)\}]$

(1)

A small modification is made in the above equation to avoid the error caused due to numerical spreading. Equation 1 is rewritten as

$\displaystyle y_i(t) = min~[n_{i-1}(t), Q_i(t),\alpha\{Ni_(t) - n_i(t)\}]$

(2)

where, $\alpha = 1~\mathrm{if~n_{i-1}(t)~\leq~Q_i(t)}$ , and $\alpha = w/v~\mathrm{if~n_{i-1}(t)~>~Q_i(t)}$ .