Leaving the tub: The nature and dynamics of hypercongestion in a bathtub model with a restricted downstream exit

Yue Bao*, Erik T. Verhoef, Paul Koster

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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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) can describe hypercongestion, but does not explicitly explain the dynamic process that leads to hypercongestion. Earlier studies of hypercongestion on single links have, however, confirmed that these processes are important to consider. The bathtub model is a model that can be used to investigate how hypercongestion can arise in urban areas, when drivers choose their departure times optimally. 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. This is because the travel time losses incurred by travelers in the bathtub are exactly offset by the reductions in travel time losses in exit queues, and thus 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 mere observation of hypercongested speeds does not necessarily mean that there is an efficiency loss from capacity drop at the level of the full system.

Original languageEnglish
Article number102389
Pages (from-to)1-21
Number of pages21
JournalTransportation Research Part E: Logistics and Transportation Review
Early online date12 Jun 2021
Publication statusPublished - Aug 2021

Bibliographical note

Funding Information:
This work is supported by the Fundamental Research Funds for the Central Universities 2021RC243 , the 111 Project B20071, and the funding of the Netherlands Organization for Scientific Research (NWO) as part of the U-SMILE project 438-15-176, which is gratefully acknowledged.


  • Bathtub model
  • Flow congestion
  • Hypercongestion
  • Road traffic congestion


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