A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation

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Title: A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation
Authors: Han, K
Friesz, TL
Yao, T
Item Type: Journal Article
Abstract: The Vickrey model, originally introduced in Vickrey (1969), is one of the most widely used link-based models in the current literature in dynamic traffic assignment (DTA). One popular formulation of this model is an ordinary differential equation (ODE) that is discontinuous with respect to its state variable. As explained in Ban et al., 2011 and Han et al., 2013, such an irregularity induces difficulties in both continuous-time analysis and discrete-time computation. In Han et al. (2013), the authors proposed a reformulation of the Vickrey model as a partial differential equation (PDE) and derived a closed-form solution to the aforementioned ODE. This reformulation enables us to rigorously prove analytical properties of the Vickrey model and related DTA models. In this paper, we present the second of a two-part exploration regarding the PDE formulation of the Vickrey model. As proposed by Han et al. (2013), we continue research on the generalized Vickrey model (GVM) in a discrete-time framework and in the context of DTA by presenting a highly computable solution methodology. Our new computational scheme for the GVM is based on the closed-form solution mentioned above. Unlike finite-difference discretization schemes which could yield non-physical solutions (Ban et al., 2011), the proposed numerical scheme guarantees non-negativity of the queue size and the exit flow as well as first-in-first-out (FIFO). Numerical errors and convergence of the computed solutions are investigated in full mathematical rigor. As an application of the GVM, a class of network system optimal dynamic traffic assignment (SO-DTA) problems is analyzed. We show existence of a continuous-time optimal solution and propose a discrete-time mixed integer linear program (MILP) as an approximation to the original SO-DTA. We also provide convergence results for the proposed MILP approximation.
Issue Date: 1-Mar-2013
Date of Acceptance: 1-Feb-2013
URI: http://hdl.handle.net/10044/1/62309
DOI: https://dx.doi.org/10.1016/j.trb.2012.10.004
ISSN: 0191-2615
Publisher: Elsevier
Start Page: 75
End Page: 93
Journal / Book Title: Transportation Research Part B: Methodological
Volume: 49
Copyright Statement: © 2012 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords: Social Sciences
Science & Technology
Engineering, Civil
Operations Research & Management Science
Transportation Science & Technology
Business & Economics
The Vickrey model
Discrete-time solution
System optimal dynamic traffic assignment
Mixed integer linear program
1507 Transportation And Freight Services
0102 Applied Mathematics
Logistics & Transportation
Publication Status: Published
Online Publication Date: 2013-02-17
Appears in Collections:Faculty of Engineering
Civil and Environmental Engineering

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