Same road, same random traffic: three ways to share one conflict zone. Built from a Blender 5.2 model, run by a micro-simulation.
Blue: throughput (cars/h, last 10 min). Amber: mean delay of the last 300 cars. Delay = trip time minus the time an unobstructed car would need.
Runs all three methods at six demand levels (equal flow on all four arms, your turn shares; the signal plan is re-optimised with Webster's cycle for every level). Each point is 15 simulated minutes after a 5 minute warm-up.
Webster. Straight-only symmetric traffic on a two-phase plan. The saturation flow and effective green are measured from a queue discharging in the simulator, then Webster's closed-form delay is compared with the simulated mean delay. Little's law. L = λW must hold for every method.
Cars from different directions cross the same patch of asphalt, the conflict zone. Any control method is a rule for who may use which part of the zone, and when. Capacity is limited by how much of the zone's space-time is wasted.
A queue discharges at the saturation flow s = 3600 / h (h = headway of the queue, about 2 s). With effective green g and cycle C, the capacity is c = s g / C and the degree of saturation is x = q / c. Webster's delay per car:
Here L counts the 3 s amber and 2 s all-red of each phase. Left turns are permissive: they wait for a gap in oncoming traffic and may sneak through at the end of green, which is why lefts hurt a single-lane signal so much.
The conflict zone is cut into 12 x 12 tiles of 1 m. Each approaching car asks the intersection manager for a slot: its planned arrival time T and speed profile give the time the car's body occupies each tile it touches, [T + τ(s_in) - δ, T + τ(s_out) + δ], with a safety buffer δ = 0.35 s. The request is granted only if those intervals are disjoint from every interval already booked (first come, first served); otherwise the car slows to try a later slot. Cars that cannot keep their schedule cancel and re-book. No stop lines, no phases: cars interleave through the zone in space and time.
Entering cars give way to circulating traffic and need a time gap larger than a critical gap t_c (about 2.2 s here). Entry capacity falls as circulating flow rises; a standard fit is q_e = q_0 e^(-q_c t_c / 3600) (Siegloch / Tanner family). Speeds are capped by curvature, v ≤ sqrt(a_lat R).
Every car obeys the Intelligent Driver Model:
Throughput: cars that completed the junction in the last 10 simulated minutes, scaled to cars/hour. Delay: trip time from 100 m before the line to 20 m after the zone, minus the unobstructed time at the speed limit (13.9 m/s, with the lower speed the geometry forces in the zone). Zone occupancy: car footprint area inside the zone divided by the zone area (12 x 12 m box, or the 7 m wide ring road), time-averaged. Close calls: pairs of cars whose centres come within 2 m in the zone (a counting aid, not a crash model).
(a) Webster's formula above, with s and g measured from the simulator. (b) Little's law L = λ W: the time-average number of cars on the network equals arrival rate times mean time in the network. It is an identity of any queueing system, so a bookkeeping bug in the simulator would show up as a mismatch.
One lane per approach, right-hand traffic, no pedestrians, buses, yellow-light running or driver errors; the reservation system assumes perfect sensing, instant communication and every car obeying its slot (that is what "perfect autonomous" means here); signal timings are fixed, not actuated; the roundabout is small and tight, so its speeds are low. Real results depend heavily on lane layout, demand pattern and turning shares, so treat the comparison as a model of the mechanisms (lost time, head-of-line blocking, gap acceptance, space-time packing), not as a design tool. Not affiliated with any research group or vendor; the reservation idea follows the published "Autonomous Intersection Management" concept in spirit, with this simulator's own simplified scheduler.