Congested Equilibria in Large-Scale Traffic Networks: Existence, Stability and Robustness through Chemical Reaction Network Analogues

Discrete fluid-like models such as the Cell Transmission Model (CTM) have proven successful in modeling traffic networks. In general, these models employ discontinuous dynamics or nonlinear terms to describe phenomena like shock waves and phantom jams. Given the complexity of the dynamics, it is not surprising that the stability properties of these models are not yet well characterized. Recent results prove the existence of a unique equilibrium in the free flow regime for certain classes of networks modeled by the CTM; however, these results restrict network demands and hold only for acyclic network topologies. Further, it is of interest to understand network behavior in congested regimes, since practical networks are often congested. We propose a new modeling paradigm for traffic networks, where an analogy between discrete fluid-like traffic models and a class of chemical reaction networks is constructed by suitable relaxations of key conservation laws in the CTM. Using this analogy, we provide structural conditions on the network graph topology for the existence of equilibria in congested regimes. Drawing upon entropy-like Lyapunov functions from chemical reaction network theory, we prove that the network admits multiple stable and robust congested equilibria.