Analysis of Fixed-Time Control

TL;DR

This paper models a network of intersections with fixed-time control as a queuing system, proving the existence, uniqueness, and convergence to a periodic steady state under stability conditions.

Contribution

It provides a mathematical analysis of fixed-time traffic control, establishing the existence and convergence to a unique periodic trajectory in a deterministic queuing network.

Findings

Existence of a unique periodic trajectory under stability.

Convergence of all trajectories to this periodic solution.

Finite-time convergence if vehicles do not follow loops.

Abstract

A network of signalized intersections is modeled as a queuing network. The intersections are regulated by fixed-time (FT) controls, all with the same cycle length or period, $T$. Vehicles arrive from outside the network at entry links in a deterministic periodic stream, also with period $T$, make turns at intersections in fixed proportions, and eventually leave the network. Vehicles take a fixed time to travel along each link, and at the end of the link they join a queue. There is a separate queue at each intersection for each movement. The storage capacity of the queues is infinite, so there is no spill back. The state of the network at time $t$ is the vector $x(t)$ of all queue lengths, together with the position of vehicles traveling along the links. The state evolves according to a delay-differential equation. Suppose the network is stable, that is, $x(t)$ is bounded. Then (1) there exists a unique periodic trajectory $x^*(t)$, with period $T$; (2) every trajectory converges to this periodic trajectory; (3) if vehicles do not follow loops, the convergence occurs in finite time. The periodic trajectory determines the performance of the entire network.