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A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium

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Publication details

JournalAnnals of Operations Research
DatePublished - Dec 1995
Issue number1
Number of pages21
Pages (from-to)59-79
Original languageEnglish


Suppose that a road network model is given, together with some given demand
for travel by (say) car and that the demand for travel varies with time of day but not from day to day. Suppose that this demand is given in the form of specified total outflow rates from each origin headed towards each destination, for each origindestination pair and for each time of day, and that some initial time-dependent routeinflow rates, meeting the given demand, are given. Finally, suppose that within-day time is represented by a continuous variable. This paper specifies a natural smooth day-to-day route-swapping procedure wherein drivers swap toward less expensive routes as day succeeds day, and shows that under reasonable conditions there is an equilibrium state of this dynamical system. If such a collection of route-inflows has arisen today, say, then there is no incentive for any route-inflow to change tomorrow, in the sense that at each moment of today each of today's route-inflows is already on a route which today yielded the smallest travel cost. Such a set of "no-incentive-tochange" route-inflows is called a dynamic equilibrium, or a dynamic user-equilibrium, and may be regarded as a solution of the dynamic equilibrium traffic assignment problem. Thus, the paper introduces a smooth day-to-day dynamic assignment model and, using this model, shows that there is a dynamic user-equilibrium in a continuous
time setting. The paper briefly considers the day-to-day stability of the route-swapping process, also in a continuous setting. Finally, the paper gives a simple dynamical example illustrating the stability of the route-swapping process in a simple two-route network when there is deterministic queueing at bottlenecks.

    Research areas

  • Dynamical traffic assignment, , user-equilibrium, , dynamical system, , Lyapunov

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