Green wave signal timing

Why can retiming a corridor's traffic lights move more cars than adding another lane to it?

A city has a congested arterial with six signals on it. One proposal costs tens of millions and demolishes frontage to add a lane. The other costs a few engineers a fortnight and changes nothing physical — it only alters when each signal turns green relative to its neighbours. Retiming programmes routinely report better travel times than the widening would have bought.

That ought to be suspicious. Timing does not add asphalt. If the road is full, how can rearranging clocks make room?

Ask where a signalised corridor's capacity actually lives. Not on the link — a lane of open road passes vehicles freely. It lives at the stop line, where flow is switched on and off. A queue discharges on green at a fairly stable rate, roughly 1,900 vehicles an hour per lane while moving (recalled, a standard saturation flow figure). But it only moves for its share of the cycle: with a 90-second cycle and 40 seconds of green, real capacity is 1,900 times 40 over 90, about 840 vehicles an hour.

Now the widening argument comes into focus. A new lane on the link adds nothing unless it is carried through the stop line and given green time — and green is the scarce good, because everything the intersection serves shares one cycle. The cross street's green is subtracted from yours. A lane that ends before the stop line, or reaches one whose green cannot grow without starving the cross street, buys storage rather than throughput.

If total green is fixed, only when each green happens is left to change. Take two signals 400 metres apart and a design speed of 50 km/h, which is 13.9 metres per second. A platoon released from the first takes 400 divided by 13.9, about 29 seconds, to reach the second. Set the second green to open 29 seconds after the first and the platoon arrives as it does. Repeat down the corridor and you have a moving window — the green band — travelling at the design speed.

What does that buy, if stop-line capacity is unchanged? Delay falls, obviously. But throughput rises too, for a reason easy to miss: a stopped queue does not discharge the instant the light changes. The first vehicles accelerate, so some seconds of every green run below saturation flow — start-up lost time. A platoon arriving in motion skips that penalty and uses nearly the whole green at full rate.

Test the account against the awkward case: a band in both directions at once. The outbound platoon wants the next signal offset later; the inbound one wants it earlier. The two are compatible only when travel time between signals is near half a cycle, or a whole one. At 13.9 metres per second, half of a 90-second cycle is about 625 metres. So a corridor whose signals sit near that spacing can have a two-way wave, and one with arbitrary spacing cannot — one direction is favoured, usually the peak, and the other made worse. That geometric constraint is a good sign we have the real mechanism, because a folk story about "smoother flow" predicts nothing of the kind.

Three corrections. A green wave does not mean nobody stops — it means vehicles travelling at the design speed, in the coordinated direction, that joined the band, do not stop; drive faster and you arrive before the green, slower and you fall off the back. Second, retiming creates no green: the total in a cycle is conserved, and coordination moves it in time, not in quantity. Third, platoons disperse, because drivers do not hold formation, so the band spreads as it travels and coordination degrades toward nothing as spacing grows. That is a limit rather than a caveat — the technique belongs to closely spaced urban signals, not a rural highway.

Falsification test: the account says the gain comes from arrival phasing at the stop line and from recovering start-up lost time, not from anything on the link. So retiming should leave measured saturation flow at each stop line unchanged while cutting stops and delay, the benefit should vanish where platoons cannot survive the trip between signals, and it should reverse for the uncoordinated direction. All three are checkable, and a story about lanes predicts none of them.

Part of this is contested. Benefits decay as demand drifts, and how fast is argued over. More pointedly, the interested parties disagree about what is being optimised: highway agencies and motoring organisations like green waves because they are cheap and measurable in vehicle-hours; transit operators note that a band tuned to car speed strands buses, which stop between signals and fall out of it; pedestrian advocates note that the long cycles making coordination easier also mean longer waits to cross. Everyone measures honestly. They measure different things.

Where this sits next to its neighbours: induced-traffic.md and phantom-traffic-jams.md bracket this one. Induced traffic asks what happens after capacity improves, and its answer — demand responds to the price of travel — applies to a green wave as much as to a lane, which is why retiming's gains also erode. Phantom jams concern uninterrupted flow, where a jam forms with nothing causing it; that is a link phenomenon, whereas the constraint here is the opposite, an interruption imposed deliberately at a point. The fixed point separating all three is where the bottleneck sits: the stop line here, the link there, the traveller's choice set in the third.

The load-bearing claim is that the binding constraint on a signalised arterial is stop-line green time rather than link lanes. Where a corridor is genuinely link-constrained — long uninterrupted segments, few signals — the widening argument wins and retiming has nothing to offer.

Every term the collection defines is gathered in the glossary.