A single-lane road tunnel carries traffic modelled by the deterministic ( )…
PHYSICS · Subjective · Fermi University Physics · Computational Physics · Thermal, Statistical & Complex Systems · Monte Carlo and lattice simulation of physical systems
Many Parts, One Behaviour, Ch Emergence and self-organisation: collective order from local rules
Problem
A single-lane road tunnel carries traffic modelled by the deterministic (ps=0) Nagel-Schreckenberg cellular automaton: each cell is 7.5m long, one time step is 1s, and every car obeys the parallel update (accelerate towards vmax, respect the gap ahead, move). The maximum speed is vmax=4 cells per step. Consider the long-time state of N cars on a ring of L cells (periodic boundary, no junctions). Cars may have different integer gaps and speeds. (a) Show that no configuration, at any density, can have a flow above min(ρvmax,1−ρ) (density ρ=N/L in cars per cell, flow in cars per step past a point). Given that the long-time deterministic state reaches this bound, state the two regimes and the critical density ρ∗ at which they meet. (b) Evaluate ρ∗ and the maximum flow for vmax=4. Convert the maximum flow to vehicles per hour for this single lane. (c) At the density that gives the maximum flow, find the average spacing between cars in metres, and hence the density in vehicles per kilometre. (d) The long-time state need not contain stopped cars at all, but suppose that in the congested regime (ρ>ρ∗) the ring holds a queue of stopped cars packed nose to tail, whose front car has at least one empty cell ahead and whose outflow is unobstructed. Explain why the front edge of such a queue moves back exactly one cell per time step, whatever vmax is, and convert this speed to kmh−1.