The Dynamic System of the Traffic Assignment Problem: Part III. Incorporating Traffic Dynamics

In the two preceding papers, we established the so-called J-system as the dynamic system of the traffic assignment problem for static case and developed corresponding numerical solution methods. In this paper, we present J-system formulation of the dynamic traffic assignment problem. Here we first extend the definition of J-functions of First-In-First-Out violation for a dynamic traffic network. Then we propose both discrete and continuous J-systems, whose steady states are defined as (dynamic) Jequilibria (JE) and stable steady states as (dynamic) user equilibria (UE). Then we present finite difference method and perturbation-based method intrinsic to J-system for solving JE and UE. For a simple network we show that the finite difference method gives convergent solutions measured by J-index of convergence and that the perturbationbased method yields UE solutions. This finishes our initial discussions of the dynamic system of the traffic assignment problem. We expect these studies to have fundamental and profound impacts on studying transportation systems and other social systems with user equilibria.