James M. Greenberg, Department of Mathematical Sciences, Carnegie Mellon University

Congestion on Multilane Highways

On the multilane freeways one often observes distinct stable equilibrium relationships between auto velocity and density. Prototypical situations involve two equilibria

$\begin{displaymath} v=v_1(\rho) > v = v_2 (\rho) \ \ \ , \ \ \ 0 \leq \rho < \rho_{\rm max} \end{displaymath}$

where $v_1(\cdot)$ and $v_2(\cdot)$ are monotone decreasing and satisfy $v_1 (\rho_{\rm max})=v_2(\rho_{\rm max})=0$ . The upper curve is typically stable for densities satisfying $0 \leq \rho \leq \rho_1$ whereas the lower curve is stable for densities satisfying $\rho_2 \leq \rho \leq \rho_{\rm max}$ . Our interest is in the situation where $0 < \rho_2 \leq \rho_1 < \rho_{\rm max}$ and $v_2(\rho_2) \leq v_1 (\rho_1)$ .

In this paper we present a model which incorporates both equilibrium curves and a simple switching mechanism which allows drivers to transition from one equilibrium curve to the other. What we observe with this model are relaxation oscillations seen in congested traffic; i.e. periodic waves separating fast and slow moving traffic which propagate upstream.

Joint work with A. Klar and M. Rascle.

Location: PPB 300
Time: 1:30 P.M.

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