Asymptotic theory of traffic jams

Abstract

Based on singular perturbation methods, an asymptotic theory of traffic jams of large amplitude is developed in this work. Simple equations describing the form of traffic jams of large amplitude are found. The theory leads to analytical formulas for the characteristic parameters of traffic flow that are independent of road length, vehicle density of the initial traffic flow, or other initial conditions. Analytical investigations have been made showing that, in agreement with earlier numerical results (Kerner and Konhauser, 1994), the boundary flux at which a traffic jam can still exist is equal to the flux in the flow from a jam. The manner in which the shape of a traffic jam evolves due to changes in initial vehicle density is analytically studied. Simple analytical formulas are obtained for parameters of narrow traffic jams capable of forming in a limited interval of vehicle densities. A comparison is also made between results of the present analytical theory of traffic jams, the theory of shock waves in gas dynamics, the classical Lighthill-Whitham theory of kinematic waves (1955), and the recently discovered experimental features and characteristics of wide traffic jams in traffic.