Traffic Light Sequences
A city grid has lights timed on a 90-second cycle. Model the optimal path across 6 intersections to minimize expected wait time.
A city block has a row of 6 traffic lights, each on an independent 90-second cycle split 45s green / 45s red, with uniformly random phase offsets.
The Challenge
You arrive at the first light at a uniformly random time. Model your journey through all 6 lights. What is the expected total wait time? Can you find a speed (within ±20% of the speed limit) that minimizes your expected wait by "surfing" the light phases?
Each light contributes E[wait] = 22.5s if phases are independent. Naively: 6 × 22.5 = 135s. However, traveling at exactly the "green wave" speed — timed to arrive at each light just as it turns green — can reduce expected wait to near 0. The optimal speed depends on block length L: v* = L/45 m/s. For L = 80m, v* ≈ 1.78 m/s ≈ 6.4 km/h — walking pace syncs you with the lights.