Two-dimensional Continuum Modeling Approach to Transportation Problems

Ho, H.W.; Wong, Stephen C. P.

doi:10.1016/s1570-6672(07)60002-6

Cited by 46 publications

(29 citation statements)

References 77 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Replacing this function into the conservation equation (13a) we obtain that 16) and this holds for all t ∈ [0, T ).…”

Section: Model In Motionmentioning

confidence: 99%

See 1 more Smart Citation

Magnetworks: How Mobility Impacts the Design of Mobile Ad Hoc Networks

Silva¹,

Altman²,

Debbah³

et al. 2010

2010 Proceedings IEEE INFOCOM

View full text Add to dashboard Cite

International audienceIn this paper we study the optimal placement and optimal number of active relay nodes through the traffic density in mobile sensor ad hoc networks. We consider a setting in which a set of mobile sensor sources is creating data and a set of mobile sensor destinations is receiving that data through multihop wireless paths. We make the assumption that the network is massively dense, i.e., there are so many sources, destinations, and relay nodes, that it is best to describe the network in terms of macroscopic parameters rather than in terms of microscopic parameters. A simple one-dimensional scenario is used to introduce the problem. We solve the two-dimensional scenario where the mobility of the nodes is deterministic and when it follows the brownian mobility model

show abstract

“…Replacing this function into the conservation equation (13a) we obtain that 16) and this holds for all t ∈ [0, T ).…”

Section: Model In Motionmentioning

confidence: 99%

“…We impose that no information is leaving the considered domain D, in equation (13b) and from equation (15) this condition translates into ∇ϕ · n = 0 From equation (16) and last condition we obtain the following system…”

Section: Model In Motionmentioning

confidence: 99%

Magnetworks: How Mobility Impacts the Design of Mobile Ad Hoc Networks

Silva¹,

Altman²,

Debbah³

et al. 2010

2010 Proceedings IEEE INFOCOM

View full text Add to dashboard Cite

show abstract

“…If V j is differentiable then, under suitable conditions, it is the unique solution of (14). In the case that V j is not everywhere differentiable then, under suitable conditions, it is the unique viscosity solution of (14) (see [3,9]).…”

Section: User Optimization and Congestion Independent Costsmentioning

confidence: 99%

Continuum equilibria and global optimization for routing in dense static ad hoc networks

Silva¹,

Altman²,

Bernhard³

et al. 2010

Computer Networks

View full text Add to dashboard Cite

We consider massively dense ad hoc networks and study their continuum limits as the node density increases and as the graph providing the available routes becomes a continuous area with location and congestion dependent costs. We study both the global optimal solution as well as the non-cooperative routing problem among a large population of users where each user seeks a path from its origin to its destination so as to minimize its individual cost. Finally, we seek for a (continuum version of the) Wardrop equilibrium. We first show how to derive meaningful cost models as a function of the scaling properties of the capacity of the network and of the density of nodes. We present various solution methodologies for the problem: (1) the viscosity solution of the Hamilton-Jacobi-Bellman equation, for the global optimization problem, (2) a method based on Green's Theorem for the least cost problem of an individual, and (3) a solution of the Wardrop equilibrium problem using a transformation into an equivalent global optimization problem.

show abstract

“…The continuum model for steady‐state user equilibrium problems, with route choice behavior considered, has been developed (Ho and Wong, 2006; Wong et al, 1998) which has a wide range of applications to, for example, the multi‐commodity cost‐flow relationship (Wong, 1998; Wong et al, 2004), market share determination (Wong and Yang, 1999), elastic market externalities (Yang and Wong, 2000), the cordon‐based congestion pricing problem (Ho et al, 2005), the combined multiclass, distribution and assignment model (Ho et al, 2006), the combined discrete and continuum model (Wong, 1994), and the housing problem (Ho and Wong, 2007). For the dynamic macroscopic modeling of pedestrian flow problems, Hughes (2002) provided a systematic framework without explicitly considering the user equilibrium concept and without providing a numerical solution procedure.…”

Section: Introductionmentioning

confidence: 99%

High-Order Computational Scheme for a Dynamic Continuum Model for Bi-Directional Pedestrian Flows

Xiong

Zhang

Shu

et al. 2010

Computer-Aided Civil and Infrastructure Engineering

Self Cite

View full text Add to dashboard Cite

In this article, we present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bi-directional pedestrian flows. We first review the dynamic continuum model for bi-directional pedestrian flows. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax-Friedrichs difference scheme with firstorder Euler forward time discretization, the third-order WENO scheme with third-order total variation diminishing (TVD) Runge-Kutta time discretization, and the fast sweeping method, and demonstrate how to apply them to the model under study. We present a comparison of the numerical results of the model from the first-order and high-order methods, and conclude that the high-order method is more efficient than the first-order one, and they both converge to the same solution of the physical model.

show abstract

Two-dimensional Continuum Modeling Approach to Transportation Problems

Cited by 46 publications

References 77 publications

Magnetworks: How Mobility Impacts the Design of Mobile Ad Hoc Networks

Magnetworks: How Mobility Impacts the Design of Mobile Ad Hoc Networks

Continuum equilibria and global optimization for routing in dense static ad hoc networks

High-Order Computational Scheme for a Dynamic Continuum Model for Bi-Directional Pedestrian Flows

Contact Info

Product

Resources

About