  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 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 in a
network. 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 one.
