  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 et al (2017a,b) 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 through analytical discussion and numerical
examples. 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" exists such that a higher dispatch
frequency can increase the equilibrium cost. Furthermore, insights into the
design of the timetable are obtained and the timetable influence on passengers'
equilibrium travel costs are evaluated.
