We revisited the global traffic light optimization problem through a cellular automata model, which allows us to address the relationship between the traffic lights and car routing. We conclude that both aspects are not separable. Our results show that a good routing strategy weakens the importance of the traffic light period for mid-densities, thus limiting the parameter space where such optimization is relevant. This is confirmed by analyzing the travel time normalized by the shortest path between the origin and destination. As an unforeseen result, we report what seems to be a power-law distribution for such quantities, indicating that the travel time distribution slowly decreases for long travel times. The power-law exponent depends on the density, traffic light period, and routing strategy, which in this case is parametrized by the tendency of agents to abandon a route if it becomes stagnant. These results could have relevant consequences on how to improve the overall traffic efficiency in a particular city, thus providing insight into useful measurements, which are often counter-intuitive, which may be valuable to traffic controllers that operate through traffic light periods and phases.
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