An understanding of stationary traffic patterns in a road network during peak periods is essential for identifying bottlenecks, developing control and management strategies, and improving network design and planning. In this paper we study the existence, stability, and solution of stationary states in the kinematic wave model with constant origin demands, destination supplies, and route choice proportions. Here the dynamics on a link are described by the LWR model, and those through a junction by an invariant model of fair merging and first-in-first-out diverging behaviors. After deriving a system of algebraic equations for stationary states, we propose a map in critical demand levels, with which we prove the existence of stationary states. We show that the map and, therefore, stationary states may be unstable in diverge-merge networks. Then we propose an iteration algorithm to find fixed points and corresponding stationary states in stable networks and demonstrated its validity with an example. This study lays the theoretical foundation for a model of stationary traffic flow alternative to link performance functions and further studies on stationary network problems within the framework of kinematic wave theories.
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