Highlights ► The continuous-time point-queue model is an ODE with a discontinuous right-hand side. ► The original point-queue model may lead to negative queue length during discretization. ► Two modified point-queue models are proposed to ensure physical realism. ► Both modified point-queue models are superior to the original model. ► Convergent discretization schemes are developed to solve the modified models.
Abstract Extending a recent work () pertaining to a simple single-bottleneck model, this paper is the first of a two-part research wherein we undertake a mathematically rigorous study of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs) (). The first step in this comprehensive research is to gain a thorough understanding of some continuous-time point-queue models, which will be used as the building block of a computationally tractable model for the continuous-time DUE problem that we will study in detail in the accompanying paper (). Starting with the original point-queue model introduced by , we summarize some desirable properties that a continuous-time point-queue model should possess, and show that one of these properties—the nonnegativity of the queue lengths—is violated by Vickrey’s original model. As a remedy to this drawback of Vickrey’s model and with the goal of extending it to a continuous-time setting, we introduce two continuous-time point-queue models and show that they satisfy the properties we propose. Discretizations of the continuous-time models are discussed and construction of numerical trajectories is presented; convergence of such trajectories as the time step approaches zero is established; regularity of a solution to the continuous-time problem is clarified, and numerical results are presented.
Continuous-time point-queue models in dynamic network loading
Transportation Research Part B: Methodological ; 46 , 3 ; 360-380
2011-11-10
21 pages
Article (Journal)
Electronic Resource
English
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