Optimal route selection by passengers on a public transportation system is strongly influenced by congestion. For highly congested networks such as the one in Santiago, the effect of waiting time may become as relevant as travel time, and drastically change the picture with respect to an uncongested situation. Optimization models are not well suited to analyze such situations, which are better formulated in terms of equilibrium models. In contrast with the case of private transportation --- where congestion mainly affects the travel times and one may still find an equivalent nonlinear network flow optimization problem --- public transportation has an essentially different nature as congestion has to do with the increase in "waiting times" experienced by passengers at "bus stops". In this talk we discuss the static equilibrium model for passenger assignment in congested transit networks presented in the work of Cominetti and Correa. We begin by analyzing the common-line problem under congestion, providing an explicit description of the set of equilibria and revealing some unexpected features such as the coexistence of several optimal strategies at equilibrium, the fact that an increase in congestion need not induce larger equilibrium times, or still that the equilibrium may fail to be Pareto and there exist assignments where every user gets better off. We also show how the common-line model may be used as a building block to formulate a network model consisting of a set of common-line problems linked by flow conservation. Finally, we discuss a recent result from Cepeda which provides an alternative characterization of the network equilibria and opens the way to practical computations for large scale networks. The latter will be further developed in the talk by Michael Florian.

In order to determine how passengers use a transit network, it is necessary to establish their route choice. For heavily congested transit networks, these choices depend strongly on the waiting times at bus stops. It is difficult to develop waiting time models that depend on the passenger flows in the entire network. The models developed in this work provide a good estimation of the average waiting times of passengers that wait at stops in order to board transit vehicles. The best of these stop models incorporates the effect of alighting passengers, the on-board passengers and the passengers waiting to board given a limited capacity of the transit vehicle. Also, we show that other models, that consider exclusively the boarding flow at stops and the mean residual capacity of the buses, can substantially underestimate the waiting time.

The basic theoretical results reported in the first two papers of this session have been embedded in a computational procedure that has been used in several applications. The results obtained as well as the computational times are presented using networks originating from Santiago, Chile and the South East region of England.