Static traffic assignment with residual queues and spillback

In this paper we propose a novel static traffic assignment model where the network loading component accounts for non-stationary residual queues as well as spillback effects by explicitly adding capacity and storage constraints. This is achieved by deriving a static version of the dynamic general link transmission model that describes average link inflow and outflow rates during a given time period under simplified temporal assumptions. The resulting model adopts the same concave fundamental diagram and first order node model used in state-of-the-art macroscopic dynamic network loading without the need to explicitly describe time. We show that our mathematical problem formulation is an extension of several other models described in the literature, including the traditional capacity restrained static modelling paradigm. The equilibrium problem is formulated as a variational inequality problem while the network loading problem can be formulated as a fixed point problem. We prove that a solution exists to each problem as long as all traffic is loaded onto the network (i.e., no queue spillback into an origin). The model is illustrated via numerical examples on several hypothetical transport networks.