The traffic statics problem in a road network

In this study we define and solve the traffic statics problem in an open diverge-merge network based on a multi-commodity kinematic wave model, whose entropy conditions are given by invariant flux functions derived from macroscopic merging and diverging rules. In this problem, we are interested in finding stationary states on all links when origin demands, destination supplies, and route choice proportions are constant. After discussing the properties of four types of stationary states on a road link and presenting stationary entropy conditions at both the merging and diverging junctions, we derive a system of algebraic equations as necessary conditions for all 16 combinations of stationary states on the two intermediate links. Under different network conditions in road capacities, route choice proportions, and merging priorities, we analytically show that the traffic statics problem always admits stationary solutions, which, however, may not be unique. In particular, such stationary solutions exist even under network conditions when an initially empty diverge-merge network can settle in persistent periodic oscillations after a long time. In the future, we will be interested in discussing the stability property of stationary states and studying the traffic statics problem in general networks. Analytical insights from the simpler traffic statics problems would be helpful for understanding complex traffic dynamics in a road network.