Optimal Queue Length in Torus Networks with Traffic Signals

Inspired by the need for efficient traffic control in urban areas, we study the problem of minimizing the longest queue length in traffic networks. In our model cars only move forward and do not overtake each other and time is divided in rounds of a given length. Cars always cross intersections when they have the green signal and they are at the front of the queue, otherwise they wait. Each car can cross at most one intersection in each round. The only means to control the movement of the cars is by scheduling the green times at the junctions, that allow cars to cross. The objective is to determine the green time schedules to reach a pattern that attains the social optimum, i.e., the minimum possible longest queue length in the network. We study this problem in an oriented torus of dimension n×n where each horizontal and vertical ring has kn cars arbitrarily deployed. We introduce the concept of conflict graphs that represent the propagated negative impact across the network when -through green time assignments the queue length in one segment is reduced. Using these graphs, we provide a lower bound on the longest queue length that any algorithm can attain and show that maximizing the number of cars that cross the intersections in every round can prevent reaching the minimum possible longest queue length. Next, we propose a quadratic running time algorithm that minimizes the longest queue length in each round assuming that the number of cars in each queue is at least the size of the round. Our technique is based on network flooding on the conflict graphs. Finally, we propose an algorithm that guarantees that the cars eventually form a social optimum queue length at the expense of having some green times where no cars cross an intersection.