  Modelling passenger assignments in public transport networks is a fundamental
task for city planners, especially when deliberating network infrastructure
decisions. A key aspect of a realistic model for passenger assignments is to
integrate selfish routing behaviour of passengers on the one hand, and the
limited vehicle capacities on the other hand. We formulate a side-constrained
user equilibrium model in a schedule-based time-expanded transit network, where
passengers are modelled via a continuum of non-atomic agents that want to
travel with a fixed start time from a user-specific origin to a destination. An
agent's route may comprise several rides along given lines, each using vehicles
with hard loading capacities. We give a characterization of (side-constrained)
user equilibria via a quasi-variational inequality and prove their existence by
generalizing a well-known existence result of Bernstein and Smith (Transp.
Sci., 1994). We further derive a polynomial time algorithm for single-commodity
instances and an exact finite time algorithm for the multi-commodity case.
Based on our quasi-variational characterization, we finally devise a fast
heuristic computing user equilibria, which is tested on real-world instances
based on data gained from the Hamburg S-Bahn system and the Swiss long-distance
train network. It turns out that w.r.t. the total travel time, the computed
user-equilibria are quite efficient compared to a system optimum, which
neglects equilibrium constraints and only minimizes total travel time.
