Simulation and Steady State Computation of Queue Length for Signalized Arterial Networks under Fixed Time Control

We consider traffic dynamics for a network of signalized intersections under fixed time control, where the outflow from every link is constrained to be equal to an autonomous capacity function (and hence fixed-time control) if the queue length is positive and equal to the cumulative inflow otherwise. In spite of the resulting dynamics being nonLipschitz, recent work has shown existence and uniqueness of the resulting queue length trajectory if the inter-link travel times are strictly bounded away from zero. The proof, which also suggests a constructive procedure, relies on showing desired properties on contiguous time intervals of length equal to the minimum among all link travel times. We provide an alternate framework to obtain queue length trajectories by direct simulation of differential equations, whose right hand side is the unique solution of an appropriate linear program. Existence and uniqueness is established for piecewise constant external inflow and capacity functions, and the method does not require travel times to be bounded away from zero. Additionally, if the external inflow and capacity functions are periodic and satisfy a stability condition, then there exists a globally attractive periodic orbit. We provide an iterative procedure to compute this periodic orbit with arbitrary precision. The procedure relies on identifying a specific time instant for every link when the queue length transitions from being zero to being positive. Each iteration computes a periodic trajectory and the sequence of these iterates converges uniformly monotonically to the desired periodic orbit.