Departure Time and Mode Choice for the Morning Commute in a Highway/Railway Network

Every morning commuters travel from a residential area to a workplace by using their private cars on a bottleneck-constrained highway or selecting the regularly dispatched capacity-constrained train on railway. We depict commuters' departure pattern by private car or train respectively, thus derive the conditions and property of a dynamic equilibrium for ensuring that the total private cost is equal for any commuter at any used time and by any traffic mode. Numerical results are presented to test the equilibrium conditions and resolve the equilibrium solution.