The fastest way through a traffic light

TL;DR

The paper tackles the problem of minimizing the expected travel time to a destination when approaching a red traffic light at distance d under velocity and acceleration constraints, with the red duration T following distributions with bounded, nonincreasing densities. It casts the problem as a variational one with an action functional, introduces a novel pressure/tank interpretation to identify the optimal trajectory, and proves existence of a unique optimiser in the general case. It then solves two tractable instances—Uniform and Exponential residual times—by deriving explicit EL-based trajectories and phase structures, including a surprising discontinuous braking switch in the Exponential case. The approach yields a clear, mechanistic view of optimal control under uncertainty and provides phase diagrams linking initial velocity and distance to the optimal sequence of motion modes, with potential extensions to energy-aware objectives and more complex distributions.

Abstract

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.

Figures (13)

• Figure 1: A velocity–time diagram for the Uniform case showing $t_1 \le t_2 < t_3 < t_4 \le q$ for a particular $v_0$. The green lines denote $\dot v=-\alpha$ isobars. The red boundaries of the tank represent the constraints, while the blue dash-dotted lines illustrate potential optimal trajectories with various volumes $d$ of water.
• Figure 2: A velocity–time diagram for the Exponential case. The red boundaries of the tank are as in the Uniform case. The green curves denote the Euler--Lagrange isobars followed by $\beta$ segments. The blue dash-dotted lines illustrate potential optimal trajectories with various volumes $d$ of water. Note the change from Euler--Lagrange to $\beta$ occurs at velocity $v_c$, which does not depend on $d$. In general, the Euler--Lagrange curve is not tangent to the $-\beta$ segment where they meet.
• Figure 3: Plot of $F(t)$, its derivative $f(t)=F'(t)$, and the associated Euler--Lagrange curve $v_{\text{E--L}}(t)$.
• Figure 4: The four possible shapes of positive tent
• Figure 5: The four possible shapes of negative tents

Theorems & Definitions (49)

• Definition 3.1: $(\alpha,\beta)$-Lipschitz Function
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6
• proof
• Theorem 3.7