On the existence of stationary states in general road networks

• Formulate stationary states as solutions of a system of algebraic equations.
• Decouple a general road network into links with artificial origins and destinations.
• Define the critical demand level in effective demands and supplies.
• Derive a map in critical demand levels for a general road network.
• Prove the existence of stationary states with Brouwer’s fixed point theorem.

Our daily driving experience and empirical observations suggest that traffic patterns in a road network are relatively stationary during peak periods. In numerous transportation network studies, there has been an implicit conjecture that stationary states exist in a network when origin demands, route choice proportions, and destination supplies are constant. In this study, we first rigorously formulate the conjecture within the framework of a network kinematic wave theory with an invariant junction model. After defining stationary states, we derive a system of algebraic equations in 3-tuples of stationary link flow-rates, demands, and supplies. We then introduce a new definition of junction critical demand levels based on effective demands and supplies. With a map in critical demand levels, we show that its fixed points and, therefore, stationary states exist with the help of Brouwer’s fixed point theorem. For two simple road networks, we show that the map is well-defined and can be used to solve stationary states with a brute-force method. Finally we summarize the study and present some future extensions and applications.