Continuity of the Effective Path Delay Operator for Networks Based on the Link Delay Model

This paper is concerned with a dynamic traffic network performance model, known as dynamic network loading (DNL), that is frequently employed in the modeling and computation of analytical dynamic user equilibrium (DUE). As a key component of continuous-time DUE models, DNL aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays. The DNL procedure gives rise to the path delay operator, which associates a vector of path flows (path departure rates) with the corresponding path travel costs. In this paper, we establish strong continuity of the path delay operator for networks whose arc flows are described by the link delay model (Friesz et al., 1993). Unlike result established in Zhu and Marcotte (2000), our continuity proof is constructed without assuming a priori uniform boundedness of the path flows. Such a more general continuity result has a few important implications to the existence of simultaneous route-and-departure choice DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed.Comment: 12 pages, 1 figur

A variational approach for continuous supply chain networks

We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896–920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433–460] and [D. Armbruster, C. De Beer, M. Fre- itag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104–114]. A model was proposed for such a network by [S. G ̈ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. Fu ̈genschuh, S. Go ̈ttlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490–1507; S. Go ̈ttlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559], which demonstrates the modeling and computational advantages of the variational approach

Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance

This paper analyzes dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure times and routes. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelers’ tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates. This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to the model formulation, analysis of existence, solution characterization, and numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for normal DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms with convergence results are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency

A Link-based Mixed Integer LP Approach for Adaptive Traffic Signal Control

This paper is concerned with adaptive signal control problems on a road network, using a link-based kinematic wave model (Han et al., 2012). Such a model employs the Lighthill-Whitham-Richards model with a triangular fundamental diagram. A variational type argument (Lax, 1957; Newell, 1993) is applied so that the system dynamics can be determined without knowledge of the traffic state in the interior of each link. A Riemann problem for the signalized junction is explicitly solved; and an optimization problem is formulated in continuous-time with the aid of binary variables. A time-discretization turns the optimization problem into a mixed integer linear program (MILP). Unlike the cell-based approaches (Daganzo, 1995; Lin and Wang, 2004; Lo, 1999b), the proposed framework does not require modeling or computation within a link, thus reducing the number of (binary) variables and computational effort. The proposed model is free of vehicle-holding problems, and captures important features of signalized networks such as physical queue, spill back, vehicle turning, time-varying flow patterns and dynamic signal timing plans. The MILP can be efficiently solved with standard optimization software.Comment: 15 pages, 7 figures, current version is accepted for presentation at the 92nd Annual Meeting of Transportation Research Boar

Lagrangian-based Hydrodynamic Model: Freeway Traffic Estimation

This paper is concerned with highway traffic estimation using traffic sensing data, in a Lagrangian-based modeling framework. We consider the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) in Lagrangian-coordinates, and provide rigorous mathematical results regarding the equivalence of viscosity solutions to the Hamilton-Jacobi equations in Eulerian and Lagrangian coordinates. We derive closed-form solutions to the Lagrangian-based Hamilton-Jacobi equation using the Lax-Hopf formula (Daganzo, 2005; Aubin et al., 2008), and discuss issues of fusing traffic data of various types into the Lagrangian-based H-J equation. A numerical study of the Mobile Century field experiment (Herrera et al., 2009) demonstrates the unique modeling features and insights provided by the Lagrangian-based approach.Comment: 17 pages, 7 figures, current version submitted to Transportation Research Part