Predictions, written before any simulation ran

Twenty-two vehicles occupy a circular road, with a compressed orange cluster and wider green gaps.

The claim under test

Setup

What the theory says (computed before any simulation)

Predictions for the simulation

What would count against margin as the concept

Limits known in advance

The claim under test

Where is the automation in a car driven through a city? What is automation? Is there a better phenomenal concept than automation to explore and explain the contingent relations that make motor vehicles in a city work?

The proposal is margin: the room a loop has to be late or wrong before something fails. Every loop in a ride, from wheel slip to the driver's foot to a signal plan, has a lag (how late it acts) and a margin (how late it can afford to be). A city of cars works while each lag stays inside its margin, and the contingency is that the margins are shared: one driver's lag spends the margin of the driver behind. If that is right, whether a line of cars flows or jams should be decided by lag against margin, whoever or whatever is driving.

The test bed is Sugiyama et al.'s (2008) experiment: 22 cars on a 230 m circle, where a jam formed with no bottleneck.

Setup


Ring	230 m, 22 cars of 4.5 m (Sugiyama et al. 2008); no overtaking
Headway rule	V(s) = clamp((s − 2 m)/T, 0, 15 m/s): the speed a gap s allows at margin T
Human	a = κ(V(s) − v) + β(v_lead − v), everything perceived τ seconds late; gains scaled to the lag, κ = 0.1/τ and β = 0.45/τ (near the best possible, see below); default lag τ = 1.0 s, margin T = 1.2 s; acceleration noise 0.15 m/s² with a 1 s correlation time
ACC	the same law with Milanés & Shladover's (2014) commercial ACC gains, k₁ = 0.23 s⁻², k₂ = 0.07 s⁻¹ (κ = k₁T, β = k₂), margin 1.1 s, no lag, no noise
Calm	drives at its leader's speed smoothed over 20 s, pulled weakly (0.02 s⁻¹) toward a 2.5 s headway; 0.2 s lag; a = 0.5 s⁻¹ × (target − v)
Everyone	acceleration within [−8, +2] m/s²; a braking reflex when the anticipated closing needs more than 2.5 m/s² (the perceived gap extrapolated forward by the lag, as in Treiber et al.'s human driver model); integration step 0.05 s
Start	uniform flow at the ring's equilibrium for its mix of drivers; for all-ACC rings, one car nudged back 1 m

Measure	Definition
stop	a car below 0.5 m/s; a stop ends when it passes 1.0 m/s
stop-and-go	at least one stop in the window
speed spread	standard deviation of the 22 speeds, sampled every second, averaged over the window
jam speed	the shift (per second) that best aligns the speed field around the ring with itself 5 s later, over minutes 10–20; negative is backward against the road
contact	a gap reaching zero (the simulation clamps it and counts it)
mean speed	all speeds averaged over the window

What the theory says (computed before any simulation)

A driver's speed answers its leader's through a transfer function G(s). A line of cars is string stable when no disturbance grows as it passes back through the line, |G(iω)| ≤ 1 for all ω. A ring's eigenvalues are where the product of the G around the ring equals 1.

Result	Value
Smallest stable margin, best possible gains	2.06 × the lag
Same, for a driver who ignores closing speed (β = 0, no response to looming)	3.06 × the lag
Same, for the human gains used here	2.08 × the lag, at every lag from 0.2 to 1.4 s, on the 22-car ring as on a long line
Default human (lag 1.0 s, margin 1.2 s) on the ring	growth 0.062 s⁻¹ (e-folding 16 s) at 0.414 rad/s (period 15 s)
ACC, no lag	stable only for margins ≥ 2.66 s; at 1.1 s, growth 0.103 s⁻¹ (e-folding 10 s)
k calm drivers among default humans	growth 0.062, 0.023, −0.006, −0.032, −0.050 s⁻¹ for k = 0, 1, 2, 3, 4: two are needed
Same with humans at margin 1.6 s	0.027, −0.002 s⁻¹ for k = 0, 1: one is enough
Uniform-flow speed with 0, 1, 2, 4 calm drivers	3.30, 3.14, 3.00, 2.75 m/s (their longer headway costs speed)

Time rescaling leaves the linear problem unchanged, so the required margin is exactly proportional to the lag. An independent computer-algebra check reproduces the 2.079 s, the 2.660 s and the default growth rate.

Predictions for the simulation

Ten seeds per condition unless stated.

ID	Prediction	Confidence	Fails if
P1	Phantom jam. Default humans stop and go: some car stops within 10 minutes.	85%	2 or more of 10 seeds reach 10 minutes without a stop
P2	The jam belongs to no car. Over minutes 10–20 the jam moves backward at 8–20 km/h, every car stops at least 5 times, and no single stop lasts longer than 30 s.	55%	in any seed: jam speed outside −20 to −8 km/h, a car with fewer than 5 stops, or a stop longer than 30 s
P3	Margin ≈ 2 × lag survives the nonlinear ring. Sweep lag 0.2–1.4 s (step 0.2) × margin 0.4–3.2 s (step 0.2), 5 seeds, 20 minutes. At every lag, the smallest margin at and above which no seed stops in minutes 10–20 lies within 0.2 s of 2.08 × lag rounded up to the grid (0.6, 1.0, 1.4, 1.8, 2.2, 2.6, 3.0 s).	60%	any lag off by more than 0.2 s
P4	Removing the lag is not enough. 22 ACC cars at margin 1.1 s stop and go within 5 minutes of the 1 m nudge. At margin 3.0 s the nudge dies away (speed spread at minute 20 below half its maximum over minutes 0–2) and no car stops.	75%	no stop by 5 minutes at 1.1 s, or a stop or no decay at 3.0 s
P5	Dose of calm, from the start. Calm drivers placed evenly among default humans from t = 0, 20 minutes: with 1, some car stops in at least 7 of 10 seeds; with 2, no car stops in at least 7 of 10; with 4, no car stops in all 10.	55%	any of the three missed
P6	Calm after the jam. Default humans for 10 minutes, then k of them switch to calm. With k = 1, cars still stop in minutes 20–30 in at least 8 of 10 seeds; with k = 4, no car stops in minutes 20–30 in at least 8 of 10.	50%	either missed
P7	Smoothness pays for its margin. In P6 with k = 4, the mean speed over minutes 20–30 (averaged over seeds) is higher than the all-human ring's over the same minutes, although four calm drivers lower the uniform-flow speed from 3.30 to 2.75 m/s.	50%	it is lower
P8	No contacts. No gap reaches zero in any P1, P5 or P6 run.	60%	any contact

Predictions, written before any simulation ran?

The claim under test?

Setup?

What the theory says (computed before any simulation)?

Predictions for the simulation?

What would count against margin as the concept?

Limits known in advance?

Predictions, written before any simulation ran

The claim under test

Setup

What the theory says (computed before any simulation)

Predictions for the simulation

What would count against margin as the concept

Limits known in advance