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

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.