  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.
