In the modeling of signalized junctions, a signal control is typically expressed as an on-and-off control in continuous time, or as a binary variable in discrete time. One way of approximating such signal control is through a continuum approach where the on-and-off control variable is replaced by a 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; to what extent is this a valid approximation. The goal of this paper is to answer these basic research questions and provide guidance for application, by rigorously analyzing a network model consistent with the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1955; Richards, 1956). In particular, convergence of the on-and-off signal model to the continuum model is investigated in regimes of diminishing signal cycles. The authors also provide numerical analysis on the approximation error when the signal cycle is not infinitesimal. The convergence results and approximation accuracy are dependent on the type of fundamental diagram employed and whether or not spillback occurs in a network.
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