Dynamic user equilibrium for departure time choice in the basic trip-based model

Recent research indicates that the trip-based models can perform more accurately for capturing network hyper-congestion than the conventional macroscopic fundamental diagram (MFD) dynamics, especially during transient phases. However, due to the complex mathematical structure of the trip-based models, deriving analytical properties of the dynamic user equilibrium (DUE) of departure time choice with the trip-based models is still a challenge. This paper investigates the DUE problem of departure time choice in an isotropic urban network with identical travelers. Traffic dynamics is captured by the basic trip-based model using a speed-MFD that maps the traffic accumulation to the space-mean speed of the network. Necessary conditions for dynamic user equilibrium with and without inflow capacity constraint are derived, respectively. Under dynamic user equilibrium condition, no traveler can reduce her/his travel cost by changing the departure time. The analysis reveals the significant difference between the basic trip-based model and the conventional MFD models regarding the information support involved in the departure time choice. The derivation does not rely on some common assumptions in the literature such as linear travel cost function, no late arrivals, or linear speed-MFD. The numerical example indicates that the inflow capacity constraint can help prevent two peaks in the departure profile and the vehicle accumulation.