Analysis of fixed-time control

• There exists a unique periodic trajectory of queue length vector, with period T.
• Every state trajectory converges to this periodic trajectory.
• If vehicles do not follow loops, the convergence occurs in finite time.

The paper presents an analysis of the traffic dynamics in a network of signalized intersections. The intersections are regulated by fixed-time (FT) controls, all with the same cycle length or period, T. The network is modeled as a queuing network. Vehicles arrive from outside the network at entry links in a deterministic periodic stream, also with period T  . They 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 link for each movement or phase. Vehicles make turns at intersections in fixed proportions, and eventually leave the network, that is, a fraction r(i,j)r(i,j) of vehicles that leave queue i go to queue j   and the fraction [1-∑jr(i,j)][1-∑jr(i,j)] leave the network. The storage capacity of the queues is infinite, so there is no spill back. The main contribution of the paper is to show that if the signal controls accommodate the demands then, starting in any initial condition, the network state converges to a unique periodic orbit. Thus, the effect of initial conditions disappears. More precisely, the state of the network at time t   is the vector x(t)x(t) of all queue lengths, together with the position of vehicles traveling along the links. Suppose that the network is stable, that is, x(t)x(t) is bounded. Then(1)there exists a unique periodic trajectory x∗x∗, 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.