Schedule-based transit assignment: a new dynamic equilibrium model with vehicle capacity constraints

We propose in this paper a new approach for modelling congested transit networks with fixed timetables where it may happen that there is not enough room onboard to allow all users waiting for a given line on the arriving carrier, so that passengers need to queue at the stop until the service becomes actually available to them. The traditional approach to reproduce this phenomenon within the established framework of diachronic graphs, where the supply is represented through a space-time network, is to introduce volume-delay functions for waiting arcs, which are meant to discourage passengers from boarding overcrowded carriers. However, this produces a distortion on the cost pattern, since passengers who achieve boarding do not suffer any additional cost, and may also cause numerical instability. To overcome these limitations we extend to the case of scheduled services an existing Dynamic Traffic Assignment model, allowing for explicit capacity constraints and FIFO queue representation, where the equilibrium is formulated as a fixed point problem in terms of flow temporal profiles. The proposed model propagates time-continuous flows of passengers on the pedestrian network and time-discrete point-packets of passengers on the line network. To this end, the waiting time pattern, corresponding to a given flow temporal profile of pedestrians who reach a stop to ride a certain line,has a saw-tooth temporal profile such to concentrate passengers on the scheduled runs, while satisfying the constraint that the number of boarding users must not be higher than the onboard residual capacities. An MSA algorithm is also devised, whose efficiency is tested on the regional transit network of Rome.