Leaving the Tub: the Nature and Dynamics of Hypercongestion in a Bathtub Model with a Restricted Downstream Exit

Hypercongestion is the situation where a certain traffic flow occurs at a combination of low speed and high density, while a more favorable combination of these could produce the same flow. The macroscopic fundamental diagram (MFD) allows for such hypercongestion, but does not explicitly describe the dynamic process leading up to hypercongestion. Earlier studies of hypercongestion on single links have, however, confirmed that such dynamic processes are important to consider. The bathtub model is one class of model that can be used to investigate how hypercongestion can arise in urban areas, when drivers can choose their departure times. This paper investigates equilibrium outcomes and user costs under the realistic assumption that there is finite capacity to exit the bathtub, without which it would be hard to explain why hypercongestion would not dissolve through shockwaves originating from the bathtub exit. We find that when the exit capacity of the bathtub is lower than the attempted equilibrium exit flow from the bathtub, no additional inefficiencies arise due to hypercongestion in the bathtub, as the travel time losses incurred in the bathtub translate into exactly offsetting reductions in travel time losses in exit queues and the capacity of the full system is not affected. In contrast, when the exit capacity is higher than the equilibrium exit flows from the bathtub in the central part of the peak period, hypercongestion in the bathtub produces the additional inefficiencies known from the conventional textbook description. Our results thus show that the observation of hypercongested speeds does not necessarily mean that there is an efficiency loss from capacity drop at the level of the full system.