Congestion on Multilane Highways

We present a new model for traffic on a multilane freeway (with n lanes). Our basic descriptors are the car density ρ (in cars/mile), taken across all lanes in the freeway, and the average car velocity u (in miles/hour). The flux of cars across all lanes is given by ρ u = ∑ⁿᵢ₌₁ρᵢ uᵢ, where ρᵢ is the car density in the ith lane, and ui the velocity of cars in the ith lane. We shall track only ρ and u and not what is going on in each individual lane. On such multilane freeways, one often observes distinct stable equilibrium relationships between car velocity and density. Prototypical situations involve two equilibria, v=v₁(ρ) > v = v₂ (ρ), \ \ \ 0 ≤ ρ < ρmax, wherev_1(·)$ and $v_2(·)$ are monotone decreasing and satisfy $v_1 (ρₘₐₓ)=v_2(ρₘₐₓ)=0$. The upper curve is typically stable for densities satisfying $0 ≤ ρ ≤ ρ_1$, whereas the lower curve is stable for densities satisfying $ ρ_2 ≤ ρ ≤ ρₘₐₓ$. Our interest is in the situation where $0 < ρ_2 ≤ ρ_1 < ρₘₐₓ$ and $v_2(ρ_2) ≤ v_1 (ρ_1)$. In this paper we present a model that incorporates both equilibrium curves and a simple switching mechanism which allows cars to transit from one equilibrium curve to the other. This switching mechanism, when combined with the continuity equation, produces relaxation or self-excited oscillations in the system, and these oscillations are what interests us here.