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

TL;DR

This paper introduces a novel simulation framework for traffic flow in signalized arterial networks, enabling steady state computation without requiring travel times to be bounded away from zero, and provides an iterative method to find stable periodic traffic patterns.

Contribution

It develops an alternative delay differential equation approach for queue length simulation, proving existence and uniqueness without travel time restrictions, and offers a convergent iterative method for periodic orbit computation.

Findings

The proposed method does not require travel times to be bounded away from zero.

Existence and uniqueness of solutions are established for piecewise constant functions.

An iterative procedure converges to a stable periodic traffic pattern.

Abstract

We consider traffic flow dynamics for a network of signalized intersections, where the outflow from every link is constrained to be equal to a given capacity function if the queue length is positive, and equal to the minimum of cumulative inflow and capacity function otherwise. In spite of the resulting dynamics being discontinuous, recent work has proved 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 delay differential equations, where link outflows are obtained from the provably unique solution to a linear program. Existence and uniqueness of the solution to the proposed model for traffic flow dynamics is established for piecewise constant external inflow and capacity functions, and the proposed 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. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. The update for a given link is based on the periodic trajectories computed in the previous iteration for its upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit.