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This paper proposes a macroscopic model to describe the equilibrium distribution of passenger arrivals for the morning commute problem in a congested urban rail transit system. We use a macroscopic train operation sub-model developed by Seo, Wada, and Fukuda to express the interaction between the dynamics of passengers and trains in a simplified manner while maintaining their essential physical relations. The equilibrium conditions of the proposed model are derived and a solution method is provided. The characteristics of the equilibrium are then examined both analytically and numerically. As an application of the proposed model, we analyze a simple time-dependent timetable optimization problem with equilibrium constraints and reveal that a ‘capacity increasing paradox’, in which a higher dispatch frequency increases the equilibrium cost, exists. Furthermore, insights into the design of the timetable are obtained and its influence on passengers' equilibrium travel costs is evaluated.

Morning commute in congested urban rail transit system: a macroscopic model for equilibrium distribution of passenger arrivals