758 research outputs found
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
A junction condition by specified homogenization and application to traffic lights
Given a coercive Hamiltonian which is quasi-convex with respect to the
gradient variable and periodic with respect to time and space at least "far
away from the origin", we consider the solution of the Cauchy problem of the
corresponding Hamilton-Jacobi equation posed on the real line. Compact
perturbations of coercive periodic quasi-convex Hamiltonians enter into this
framework for example. We prove that the rescaled solution converges towards
the solution of the expected effective Hamilton-Jacobi equation, but whose
"flux" at the origin is "limited" in a sense made precise by the authors in
\cite{im}. In other words, the homogenization of such a Hamilton-Jacobi
equation yields to supplement the expected homogenized Hamilton-Jacobi equation
with a junction condition at the single discontinuous point of the effective
Hamiltonian. We also illustrate possible applications of such a result by
deriving, for a traffic flow problem, the effective flux limiter generated by
the presence of a finite number of traffic lights on an ideal road. We also
provide meaningful qualitative properties of the effective limiter.Comment: 41 page
On the continuum approximation of the on-and-off signal control on dynamic traffic networks
In the modeling of traffic networks, a signalized junction is typically treated using a binary variable to model the on-and-off nature of signal operation. While accurate, the use of binary variables can cause problems when studying large networks with many intersections. Instead, the signal control can be approximated through a continuum approach where the on-and-off control variable is replaced by a continuous priority parameter. Advantages of such approximation include elimination of the need for binary variables, lower time resolution requirements, and more flexibility and robustness in a decision environment. It also resolves the issue of discontinuous travel time functions arising from the context of dynamic traffic assignment. Despite these advantages in application, it is not clear from a theoretical point of view how accurate is such continuum approach; i.e., to what extent is this a valid approximation for the on-and-off case. The goal of this paper is to answer these basic research questions and provide further guidance for the application of such continuum signal model. In particular, by employing the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1955; Richards, 1956) on a traffic network, we investigate the convergence of the on-and-off signal model to the continuum model in regimes of diminishing signal cycles. We also provide numerical analyses on the continuum approximation error when the signal cycles are not infinitesimal. As we explain, such convergence results and error estimates depend on the type of fundamental diagram assumed and whether or not vehicle spillback occurs to the signalized intersection in question. Finally, a traffic signal optimization problem is presented and solved which illustrates the unique advantages of applying the continuum signal model instead of the on-and-off model
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
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