The fastest way through a traffic light

We give a rigorous solution of an optimisation problem of minimizing the expected delay caused by encountering a red traffic light on a road journey. The problem incorporates simple constraints on maximum speed, acceleration and braking rates, and depends on the assumed distribution of the remaining time until the traffic light will turn green, after it is first noticed. We assume that this distribution has a bounded and non-increasing density, which is natural since this holds for the law of the excess time in any stationary renewal process. In two special cases, where this distribution is either Uniform or Exponential, we give a complete characterisation of all possible combinations of phases of maximum acceleration, maximum speed, maximum braking, following an Euler--Lagrange curve, and standing stationary at the traffic light, which can make up an optimal solution. The key technique is to write the problem in terms of a two-dimensional pressure integral, so that the problem becomes analogous to filling a tank with a given quantity of liquid.