Departure time choices in the morning commute with a mixed distribution of capacity

Qiumin Liu, Rui Jiang, Wei Liu, Ziyou Gao

Research output: Journal article publicationJournal articleAcademic researchpeer-review


Incidents and other random factors may create variations to the transportation system and thus result in stochastic road capacity during the travel period. The realized capacity on a given day (i.e., an average value over the travel period) changes from day to day. For instance, existing empirical studies indicate that incident capacity reduction can be approximated as a continuous random variable. This study examines the morning commute problem with a road bottleneck whose capacity is constant within a day but changes stochastically from day to day. This study extends existing stochastic bottleneck model studies by considering a more general distribution of the bottleneck capacity. In particular, the capacity of bottleneck is at the designed value under good external conditions and degrades into a smaller value within an interval (i.e., the capacity degradation range) under adverse external conditions (a “mixed” distribution). Commuters’ departure time choices follow the Wardrop's first principle in terms of their mean individual travel cost. Given the considered distribution of capacity, additional equilibrium departure/arrival patterns not identified by the literature have been identified and examined. How the mean travel cost and the mean of total travel time may vary with the capacity degradation probability and the level of capacity degradation have been analyzed. The impacts of the width of degraded capacity range have also been investigated to quantify how the results are affected if the degraded capacity is assumed as the mean value (rather than a degraded capacity range). Our results indicate that with less severe capacity degradation, the mean travel cost always decreases and improving the capacity under the “worst condition” can be more effective than improving the capacity under the “best adverse condition”. However, it is not always the case for reducing the total travel time, and the mentioned measures may exacerbate the system's congestion, especially when the capacity degradation rarely occurs. Under a given mean capacity, the mean travel cost would be underestimated if the capacity degradation range is ignored. We also compare system performance considering different capacity distributions. It is found that the “mixed” capacity distribution defined in this study outperforms the binary capacity distribution in terms of evaluating the departure/arrival pattern and the mean travel cost. This study enhances our understanding on the morning commute problem under capacity uncertainty.

Original languageEnglish
Article number104011
JournalTransportation Research Part C: Emerging Technologies
Publication statusPublished - Feb 2023


  • Capacity distribution
  • Mean travel cost
  • Stochastic bottleneck model
  • User equilibrium

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Automotive Engineering
  • Transportation
  • Management Science and Operations Research


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