Congested transit networks: a schedule-based dynamic assignment model with explicit vehicle capacity constrains

For high-frequency services, which are typical of urban areas, the treatment of congestion plays a key role in transit modelling. Congestion has usually been considered implicitly by the use of strictly increasing flow-dependent cost functions aiming at discouraging user boarding on vehicle overcrowded, even if the frontier of the research in this field moves towards considering explicitly vehicle capacity constrains, for which users board the arriving run/line according to its residual capacity. The problem of taking into account congestion has been studied by several authors by using a frequency-based approach and by considering the concept of effective line frequency, which is a fictitious line frequency different from the real one, that is as lower as higher the congestion on the considered line is, trying to simulate the possibility of failing-to-board the overcrowded lines. As in the frequency-based approach services are represented by lines, single vehicles (runs) have not been explicitly considered; it implies an approximation in calculating single vehicle loads that is as relevant as higher the variation of the demand profile (typical of the peak hours) is. This approximation also arises in the case of irregular arrival of vehicles at stops, for which peaks of boarding users at stops cannot be explicitly considered. For this reason, the use of a schedule based approach seems to be more adequate to investigate congestion in high-frequency transit networks by the use of explicit vehicle capacity constrains, as each run with its vehicle capacity can be considered, as well as a generic temporal demand profile can be taken into account. This paper presents a schedule-based dynamic assignment model, which explicitly takes into account vehicle capacity. On the supply side, each run of transit services with its vehicle capacity is explicitly represented both in space and in time through the use of a diachronic network (Nuzzolo et al., 2001). On the demand side, the time-dependent characterisation of origin-destination trips is made by considering users which have desired arrival times (DAT) or desired departure times (DDT), defined by the starting or ending times of their activities; these target times are assumed to have a degree of flexibility (according to trip purpose, e.g. business, study, commuter-work, and so on), for which users can adapt within a certain range their arrival or departure in order to avoid or mitigate congestion effects. The core of the assignment model is the use of a joint departure time and path choice model, in which a path is defined by the choice of an access stop and a boarding run, defined in an explicit space-time dimension (schedule-based approach). The joint departure time and path choice model is based on a mixed pre-trip/en-route choice behaviour, in which some l.o.s. attributes (e.g waiting and travel times) and congestion (fail-to-board experiences), that are defined at single vehicle (run) level, are estimated by day-to-day learning processes (e.g. through exponential filters). In particular, pre-trip choices are relative to departure time and boarding stops, since they are considered before starting the trip and are mainly influenced by past experiences on congestion, while the en-route choice occurs at stops and concerns the decision to board a given arriving run which has a residual capacity and allows arriving at destination. The assignment model is defined through a dynamic process approach in which the within-day network loading procedure allocates users on each run of transit services (explicitly considered as a link of the diachronic network) according to users choices (made on the basis of experiences on l.o.s. and congestion at single run level) and to the residual capacity of arriving vehicles at stops. If congestion arises, the formation and dispersion of queues at stops is solved through FIFO rules, and fail-to-board experiences, as well as the experimented l.o.s. attributes, are part of the learning mechanism for the next-day user choices. Even if theoretical issues of the assignment model are under investigation, some experimental evidences on toy networks show the empirical convergence of the dynamic process towards a solution. An application example of the previously described schedule-based dynamic assignment model on a realistically-sized test network (a part of the transit network of Naples in the middle of Italy) has been carried out to explore the convergence of the assignment model, as well as to test the proposed approach and its potential use to assess effects of congestion and to support transit network operations planning.