Rush Hour Displacement

$N$ commuters all want to leave for work at the same preferred time. The road has a fixed capacity: up to $C$ commuters per time slot travel quickly, but when more than $C$ try to leave in the same slot, everyone in that slot experiences extra delay proportional to the overload. Each day, commuters observe yesterday’s travel times and shift their departure by one slot toward a less congested option with some probability. Much to their disappointment, the rush hour never disappears. Instead it: flattens slightly (spreading across more slots), but shifts its peak position over successive days, and reaches a new quasi-equilibrium that may be no less congested than the original, just at a different time. The intuition is that any slot that becomes less congested immediately attracts new commuters from adjacent overloaded slots, refilling it. Individual optimization is self-defeating in aggregate. The simulation in this tutorial shows emergent dynamics: The arrival distribution begins…