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 employ a macroscopic train operation sub-model developed by Seo et al. (2017a,b) to express the interaction between dynamics of passengers and trains in a simplified manner while maintaining their essential physical relations. We derive the equilibrium conditions of the proposed model and discuss the existence of equilibrium. The characteristics of the equilibrium are then examined through numerical examples under different passenger demand settings. As an application of the proposed model, we finally analyze a simple time-dependent timetable optimization problem with equilibrium constraints and show that there exists a capacity increasing paradox in which a higher dispatch frequency can increase the equilibrium cost. Further insights into the design of the timetable and its influence on passengers equilibrium travel costs are also obtained.
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