A partial differential equation formulation of Vickrey’s bottleneck model, part I: Methodology and theoretical analysis

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Title: A partial differential equation formulation of Vickrey’s bottleneck model, part I: Methodology and theoretical analysis
Authors: Han, K
Friesz, TL
Yao, T
Item Type: Journal Article
Abstract: This paper is concerned with the continuous-time Vickrey model, which was first introduced in Vickrey (1969). This model can be described by an ordinary differential equation (ODE) with a right-hand side which is discontinuous in the unknown variable. Such a formulation induces difficulties with both theoretical analysis and numerical computation. Moreover it is widely suspected that an explicit solution to this ODE does not exist. In this paper, we advance the knowledge and understanding of the continuous-time Vickrey model by reformulating it as a partial differential equation (PDE) and by applying a variational method to obtain an explicit solution representation. Such an explicit solution is then shown to be the strong solution to the ODE in full mathematical rigor. Our methodology also leads to the notion of generalized Vickrey model (GVM), which allows the flow to be a distribution, instead of an integrable function. As explained by Han et al. (in press), this feature of traffic modeling is desirable in the context of analytical dynamic traffic assignment (DTA). The proposed PDE formulation provides new insights into the physics of The Vickrey model, which leads to a number of modeling extensions as well as connection with first-order traffic models such as the Lighthill–Whitham–Richards (LWR) model. The explicit solution representation also leads to a new computational method, which will be discussed in an accompanying paper, Han et al. (in press).
Issue Date: 1-Mar-2013
Date of Acceptance: 1-Jan-2013
URI: http://hdl.handle.net/10044/1/63145
DOI: https://dx.doi.org/10.1016/j.trb.2012.10.003
ISSN: 0191-2615
Publisher: Elsevier
Start Page: 55
End Page: 74
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
Continuous-time Vickrey model
Ordinary differential equation
Partial differential equation
Explicit solution
The Lax-Hopf formula
1507 Transportation And Freight Services
0102 Applied Mathematics
Logistics & Transportation
Publication Status: Published
Online Publication Date: 2013-01-08
Appears in Collections:Faculty of Engineering
Civil and Environmental Engineering

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