The Wave Runs Backwards

Part one

The green light that wastes its first two seconds

Start with the thing you have definitely noticed. The light turns green. The first car moves. Then the second. Then the third. By the time the release reaches you, several seconds of your green are gone, and if the queue is long enough you will watch the light cycle back to red without ever having touched the accelerator.

The intuition most people have about this is that it is a discipline problem — that if everyone were paying attention, the queue would move off together and the whole thing would be fixed. That intuition is half right, and the half that is wrong is more interesting than the half that is right.

The light doesn't release the queue. It releases one car.

A signal has exactly one direct customer: the vehicle at the stop line. Everybody else is released not by the light but by the car in front of them. That distinction is the whole mechanism.

Each driver needs some interval between the car ahead began to move and I begin to move. Call it τ. It is not laziness; it is perception, recognition, decision, and the mechanical business of getting off the brake and onto the accelerator. For an alert driver expecting the change it runs around 0.9 to 1.2 seconds. For someone who just looked back up from a phone it can be three or four.

Chain that together and the nth driver in the queue starts moving at roughly n·τ after the light changes. The boundary between "still stopped" and "now moving" travels backwards through the queue at a speed set by two numbers and nothing else:

Put ordinary numbers in. Stopped cars sit about seven metres apart, centre to centre — call it a 4.8 m vehicle and a 2.2 m gap. With τ = 1.2 s that gives 5.8 m/s, or about 21 km/h, backwards.

The simulator below measures it at closer to 16, and the gap between those two numbers is worth a moment, because it is not error. The formula assumes you can move the instant you decide to, and you cannot — the car ahead has to open enough room first. That adds roughly 0.3 seconds per vehicle on top of τ, and the measured wave tracks s / (τ + 0.3) closely across the whole slider range.

Which has a consequence you can check in a few seconds. Drag τ to zero — nobody reacting at all, the whole queue released by the light itself — and the wave does not disappear. It settles at about 22 km/h, set purely by how long it takes for space to appear. There is a floor here that is geometric, not human, and no amount of attention gets underneath it.

That number is not a modelling artefact. Approach it from the opposite direction — the macroscopic kinematic-wave theory, which knows nothing about individual drivers and only sees flow and density — and you get the shockwave speed between a jammed state and a saturated one at about 18 km/h. Field measurements of start-up waves cluster in the same 15–20 km/h band, and they do so across countries, vehicle fleets and decades. It is one of the more stubborn constants in traffic engineering.

It is also the same wave, running the same direction, as the one you meet on a motorway when brake lights ripple toward you out of a jam a kilometre ahead. Traffic's characteristic speeds point upstream. That is the single most counterintuitive fact about the whole subject, and it is why nothing you do with your own car ever seems to help.

Two different reasons you haven't moved yet

Which is what the third lane in the simulator is for. Take the same queue and let it accelerate as a single body — every driver going at the same instant, at the same rate, holding the same spacing. Now nobody is waiting to react and nobody is waiting for room. The wave does not get faster. It stops existing. The queue crosses the stop line as one object.

Lining the three up splits the delay into its two parts:

Reaction time — the thing everybody blames — turns out to be the smaller share. The larger one is geometric. Even with perfect drivers, a car cannot move into space that does not exist yet, and the space arrives one vehicle at a time, travelling backwards. Getting past that does not take sharper attention. It takes the cars being coupled: accelerating together because they are, in some operational sense, a single vehicle.

Which is a description of a train. It is also roughly what a fully automated platoon would be — not humans reacting faster, but vehicles that have stopped reacting to each other at all, because each already knows what the one in front is about to do. The reason that third lane is worth putting on screen is that it shows the ceiling is not set by human reflexes. It is set by whether the queue is a queue or a body.

What it actually costs

Traffic engineers fold this into a quantity called start-up lost time: the seconds at the beginning of a green that produce no vehicles because the queue is still unfolding. The Highway Capacity Manual's default is 2.0 seconds per phase; some agencies use 3.0. Against a saturation headway of about 1.9 seconds per vehicle, that is a little over one vehicle per green, per lane, thrown away.

One vehicle. Not three, not double. If every driver at every signal in a city reacted instantly, you would get roughly 7% more traffic through the same greens — real, worth having, and nothing like the transformation the intuition promises. Hold onto that number — the simulator measures about 7½ — because the rest of this section is about why it is so small, and why that turns out to be the more useful fact.

Why no single fix helps very much

Drag reaction time to zero. The two lanes separate immediately, and the space–time diagram shows why: the diagonal release front collapses toward vertical. Watch the vehicles-per-green figure, though. It moves by about one car — roughly 7½ per cent.

Put it back, and halve the following headway instead. This is the gap you keep once you are moving — a completely different mechanism, governing how tightly the discharging queue packs rather than when it starts. It buys about 7½ per cent.

Put that back, and try the parking gap — in either direction. This one does not cooperate. Tighter is worse, wider is worse, and there is no free improvement sitting in it at all. That turns out to be the most interesting of the three, and it gets its own section below.

Two unrelated improvements, the same answer to within a rounding error. That is not a coincidence, and it is the most useful thing in this section. Queue discharge is limited by whichever constraint is currently worst, so relieving any single one simply hands the job to the next in line. Reaction time stops mattering the moment the cars are packed too tightly to accelerate freely. Following distance stops mattering while everyone is still waiting to be released.

Now do all of them at once — the Automated platoon preset — and the same road carries about a third more traffic. Not because any one change is powerful, but because nothing is left to bind. That is roughly what the literature on cooperative adaptive cruise control reports, and it is why the benefit is so stubbornly non-linear in how many vehicles are automated. A platoon of one is just a car.

The asymmetry runs the other way, and much harder. Improving any single thing gains you seven or eight per cent. Degrading a single thing costs far more: set reaction time to 2.5 s — a driver glancing up from a phone — and throughput falls 38%. The system rewards competence weakly and punishes inattention severely, which is an uncomfortable but reasonably accurate description of driving.

The other half of the question: how far apart you park

There is a second, quieter version of this — the space you leave in front of you when you are stopped. It is worth separating carefully, because two different things are hiding inside one habit.

Start with the mechanism, because it is real and easy to miss. A driver does not pull away when the car ahead moves; they pull away when there is somewhere to pull away to. In the model that is literal. A vehicle parked at exactly its minimum gap has an acceleration of precisely zero — it cannot move at all until the car ahead has made room for it. Park a car length and a half back instead and the room is already there: it launches at nearly full acceleration the moment its driver reacts, with nothing to wait for.

So parking further back really does buy something, and you can watch it. Drag the s₀ slider up and keep your eye on the start-up wave. At a one-metre gap it crawls backwards at about 13 km/h. At six metres it is moving at 24. More than double, and nobody is reacting any faster — the room simply already exists.

And yet the vehicles-per-green figure barely moves, and past a point it gets worse. That is the trade, and it is a clean one: every metre of extra parking gap is a metre that every car behind you must also drive before it reaches the line. The earlier launch and the longer journey very nearly cancel.

Which raises the obvious question — is there a best gap? There is, and what it depends on is not what you would guess.

The quicker your reactions, the further back you should park. That is the exact reverse of the folk picture, in which careful drivers leave room and impatient ones close it up. And it is the same "whichever constraint binds" argument as before. If you react quickly, what is stopping you is the absence of room, so making the room in advance pays. If you react slowly, the room opens up while you are still noticing the light — so parking further back buys you nothing and charges you the extra distance anyway.

At an ordinary 1.2 seconds the optimum lands around two to two and a half metres, which is — to the resolution this model can honestly claim — about what people already do. Human parking habits appear to be roughly tuned to human reaction times. Which is either reassuring or deflating, depending on what you were hoping to find.

The one unambiguous mistake is parking too close. Pull up a metre off the car in front and throughput drops about 7%: you have taken away your own room to launch and got nothing for it. The queue is shorter, admittedly, and that still matters for the reason it always did — a queue that reaches back far enough blocks the junction behind it, and a blocked junction fails in a way that no amount of green time at this signal can repair.

A modelling note, since this section turns on it. The gap a driver leaves when parking and the minimum gap they keep once rolling are two different numbers, and an earlier version of this simulator used one value for both. That quietly hid the effect: widening the parking gap also widened everyone's following distance at speed, so the launch benefit was buried under a throughput penalty that had nothing to do with parking. They are separate parameters now — the slider is the parking gap, and the in-motion minimum is held at 2 m.

Part two

The lane closure, and the advice that comes with a missing condition

Here is the version of this most of us carry around. There is a sign: right lane closed ahead. The courteous thing, the thing that keeps traffic moving, is to move over as soon as you see it. The drivers who sail past the queue in the closing lane and force their way in at the cones are the reason everyone else is stopped.

That story is intuitive, widely held, and — when traffic is heavy — backwards. What makes it interesting is that the reason it feels right is real. The mistake is in attributing it.

What the agencies actually recommend

Minnesota's DOT runs a public campaign for the opposite behaviour, which it calls the zipper merge: use both lanes right up to the merge point, then take turns, one for one. It credits the practice with cutting backup length by up to 40–50%, removing the speed differential between lanes, and reducing the aggressive last-second manoeuvres that cause work-zone collisions. Several other states have followed. Crucially, MnDOT's own guidance attaches a condition: the zipper merge is for congested conditions, with a volume warrant around 1,500 vehicles per hour. Below that, merging early is the better advice.

That condition is the part that never survives the trip into folk wisdom, and it is why the two pieces of advice you have heard sound contradictory. They are both correct. They apply to different roads.

Three separate quantities, usually confused for one

Almost all of the argument dissolves once you stop treating "traffic" as one thing. There are three, and merge strategy affects them very differently.

1. Throughput — how many cars per hour get past the closure.

   This is set by the bottleneck, and the bottleneck is one lane of road. A work-zone lane carries somewhere around 1,400–1,700 vehicles per hour. No merge strategy changes that number, because no merge strategy adds a lane. If more cars want through than the lane can pass, a queue forms — and it forms whether people merged a kilometre back or at the cones. Merging early does not create capacity. This is the single most common misconception, and the simulator makes it very hard to keep believing.

2. Queue length — how far back the jam physically reaches.

   Here the strategies differ enormously, and the reason is pure geometry. The same number of waiting vehicles stored in two lanes occupies half the road length it does in one. That is the 40–50% figure, and it is not a subtle modelling result — it is arithmetic. It matters because queue length is what determines whether the backup stays inside the work zone's approach or reaches back and swallows the previous interchange, at which point the delay stops being linear and starts being someone else's problem too.

3. Breakdown — whether the bottleneck runs smoothly or in stop-and-go.

   Once a bottleneck breaks down into stop-and-go, its discharge rate drops several per cent below the capacity it had while flowing — the capacity drop. Every forced merge, every vehicle that runs out of road and has to be let in from a standstill, injects a backward wave that pushes the bottleneck toward that state. This is the quantity that merge behaviour genuinely damages, and it is where the folk wisdom is picking up a real signal.

Why the intuition feels so right

Because you have almost never seen a real zipper merge. What you have seen is the mixed case: half the drivers move over early because they were told to, half run to the cones, and there is no shared convention governing what happens at the point of contact.

That mix produces every pathology at once. The closing lane is empty enough that arriving drivers approach at speed and meet a stopped queue — the speed differential that makes work zones dangerous. The drivers who merged early feel cheated, and some of them straddle the lane line to enforce fairness, which converts two lanes into zero. And the drivers who did run to the front arrive with no gap arranged, stop, and have to be admitted from rest — which stops the through lane as well, and sends a wave backwards through everybody who did the right thing.

So the experience is accurate: those late mergers really did just make it worse. What is wrong is the conclusion, because the thing that hurt was not that they merged late. It was that they merged late into a system that had not agreed to it. A zipper merge where everybody zips is orderly. A zipper merge where half the participants are running a different protocol is a fight.

What you should find

At low demand, nothing matters. Drop the demand slider to 900 veh/h. One lane comfortably carries it, no queue forms, and all three strategies produce identical numbers. This is the condition under which merging early is genuinely the better advice — not because it moves more traffic, but because there is no traffic to move and a lane change at 100 km/h is best done early and calmly, far from the cones.

Above capacity, throughput converges and queue length does not. Push demand past about 2,000 veh/h and run all three. The throughput bars land within a few per cent of one another — they are all measuring the same single lane — while the queue bars separate. In this model the zipper's queue comes out roughly a quarter shorter than the early-merge queue, and its average delay about a third lower. Field studies report larger queue reductions, up to 40–50%. The direction is the robust part; the magnitude is the model's.

Cooperation pays, but only where the outcome is still in doubt. Set the strategy to zipper, put demand near 2,000 veh/h, and move the cooperation slider from 0 to 100%. Average delay falls by about a sixth. Now push demand to 3,000 and try the same thing: almost nothing happens. That is not a bug. Once demand is far beyond what one lane can carry, the bottleneck is saturated however politely anyone behaves, and the queue grows at the difference between arrival and capacity. Courtesy decides whether a marginal bottleneck breaks down. It cannot rescue a hopeless one.

And the mixed case is not clearly the worst — in this model. Which is worth saying plainly, because the argument would be tidier if it were. With merging spread along the road, the mixed case sometimes posts the highest throughput of the three; at high demand it posts the worst delay. It is the most erratic of the three rather than the most uniformly bad. The real case against it, which is strong, rests on the speed differential between a stopped lane and a moving one, the crash risk that creates, and the drivers who straddle the line to police the queue. None of that is in this model, and I have not put it in to make the result come out the way the argument wants.

And there is a best place to zip. Pick Set the merge point on either panel and move the distance slider. Aim for the last ten or twenty metres and the model starts producing drivers who reach the cones, find nobody will let them in, and stop dead — the stranded counter climbs, and it climbs faster when fewer drivers are willing to yield. Back off to fifty or a hundred metres and that failure disappears while the throughput actually improves by a few per cent, because the merge is now a negotiation between two moving vehicles rather than a standing car trying to launch into a moving queue. Go much beyond two hundred metres and it degrades again for a different reason: the merge point becomes a second bottleneck upstream of the first.

So the practical version is narrower than "merge late" and much narrower than "merge early". Use both lanes, and aim to be in the through lane by about a hundred metres before the cones — moving, alongside a gap, not fighting for the last car length. That is what a zipper actually is, and it is the reading that survives the simulator.

What survives all of that is the part the geometry guarantees. Above capacity a queue must exist, and the only real question is how much road it occupies. Stored in two lanes it is half as long as stored in one — and queue length, not throughput, is what decides whether the backup stays inside the work zone's approach or reaches back and takes out the interchange behind it.

Where the model is generous to the zipper: real drivers are worse than these ones. There is no road rage in here, nobody straddles the lane line to police the queue, and heavy vehicles — which change everything about gap acceptance — are absent. Where it is harsh: real zipper merges benefit from signage and enforcement that this model has no way to represent. The direction of the result is robust; treat the exact percentages as the model's, not the world's.

Bonus

The arithmetic of five miles an hour

Both of the situations above are cases where a small individual gain and a large collective cost sit on opposite sides of the same decision. There is a third, and it is the one almost everybody makes several times a day, so it is worth doing the arithmetic properly.

Twenty miles at sixty takes twenty minutes. Twenty miles at sixty-five takes eighteen minutes and twenty-eight seconds. The entire prize is ninety-two seconds — and that is the ceiling, not the expectation. One red light you would otherwise have missed, one slow left turn, one moment behind a bus, and it is gone before you arrive.

What you are buying it with

The standard tool here is Nilsson's power model, which relates a change in traffic speed to a change in crash outcomes: fatal crashes scale with roughly the fourth power of speed, fatal-or-serious with the third, all injury crashes with the second. Those exponents come from decades of before-and-after studies of speed-limit changes, and Elvik's later meta-analyses put the fatal exponent slightly higher still, near 4.9.

Sixty-five over sixty is a ratio of 1.083. Raised to the fourth power it is 1.38. So the ninety-two seconds cost you something like a 38% increase in the risk of a fatal crash for the duration of the trip.

Worth being precise about what that model is: it relates changes in the mean speed of traffic on a road to changes in crash counts on that road. Applying it to one driver's personal decision is an extrapolation. The individual-level evidence is, if anything, harsher — Kloeden's case-control study of real crashes in 60 km/h zones, with alcohol controlled and matched control vehicles, found the risk of a casualty crash roughly doubling for every 5 km/h above the limit.

The number that makes it concrete

Percentages are abstract. This is not. Put a hazard in the road at exactly the distance where the driver doing the limit can stop — reaction time, then brakes, halting with nothing to spare. The driver doing five over reaches that same point having used more of it up thinking, with less of it left to brake in.

They do not stop. At sixty versus sixty-five they arrive at the hazard still doing about thirty miles an hour. Not five miles an hour faster than a stop — thirty. Speed does not subtract from the front of your stopping distance; it subtracts from the end, where all the energy is. That is the whole reason the exponent is four and not one.

And the part that hits your wallet

Above about fifty, aerodynamic drag takes over and fuel economy falls away steeply. The Department of Energy’s measurements across 74 vehicles put the loss at 12.4% going from 50 to 60 mph, another 14% from 60 to 70, and another 15.4% from 70 to 80 — which is why they express it as “every 5 mph over 50 is like paying another 29 cents a gallon.”

Divide the extra fuel by the time it bought and you get the real exchange rate. On the default trip it lands around seven dollars an hour — you are buying your own time back at below minimum wage, and paying in petrol before you count anything else. Electric cars are cheaper per mile but proportionally worse at this: with no idling losses to amortise, a larger share of the energy goes straight into pushing air, and the penalty for speed is steeper.

Run the distance slider out to a two-hour drive and watch that figure. It does not move. Both the fuel you burn and the time you save scale with distance, so the exchange rate between them is a property of the two speeds and nothing else. A long journey does not make speeding a better deal — it just buys more of the same bad one. Nine minutes on a two-hour drive still costs you the same seven dollars an hour, and still multiplies the risk of every mile of it.

Taken together

Both problems are the same problem

Put the three side by side and the shared structure is hard to miss.

Each one is a case where an individual gain is small, immediate and obvious, and the corresponding cost is larger, delayed, and paid by somebody else — or by a version of you that has not happened yet. Two seconds of inattention at a green light. A car length gained at a lane closure. Ninety-two seconds on a twenty-mile commute. Each is nearly nothing on its own, which is exactly why each is so easy to take.

Two of them are literally the same physics. The release front at the signal and the shockwave from a forced merge both travel upstream at close to twenty kilometres an hour, for the same reason — jam spacing divided by reaction time — and neither cares which road it is on. The third is not a wave at all, but it has the same shape of argument: the quantity you are buying grows linearly and the quantity you are spending grows with the fourth power.

And in all three the fix is coordination rather than effort. Nobody in any of these simulations needs to drive harder, faster, or with less margin. The gains come from everyone doing the same predictable thing — which is precisely what a road full of strangers is worst at, and precisely what makes a shared convention worth more than any individual’s cleverness about it.

If there is a practical version, it is short. Close up and pay attention at the light, because the queue behind you is longer than the one you can see. Use both lanes at a closure and merge in turn at the merge point, and — the part that matters most — let the other car in, because a merge that has to happen from a standstill is the one that stops everybody. And on the open road, notice that the ninety seconds you are chasing are already spent: you are buying them back at seven dollars an hour in petrol, and underwriting them with a risk that rises four times faster than the speed does.

The most useful takeaway is probably still the smallest one. When you are stuck, the cause is not in front of you and it is not behind you. It passed through your position a while ago, travelling the other way, and what you are experiencing now is its wake.

Appendix

What's actually running

Longitudinal motion in both simulators is the Intelligent Driver Model (Treiber, Hennecke & Helbing, 2000) — a continuous car-following model where each vehicle's acceleration depends on its own speed, the gap ahead, and the rate at which that gap is closing. Default parameters: desired time headway 1.2 s, maximum acceleration 1.8 m/s², comfortable deceleration 2.5 m/s², minimum jam gap 2.0 m, vehicle length 4.8 m. Integration is ballistic at 1/60 s.

A driver carries two different minimum gaps: the one they leave when parking behind a stationary car, which is the slider, and the one they keep once rolling, held at 2 m. Separating them is what makes the parking-gap question answerable at all — with a single value, widening the parking gap also widens every following distance at speed, and the two effects are impossible to tell apart.

IDM on its own has no reaction time — vehicles respond instantly. The start-up delay is layered on top as an explicit release rule: a stopped driver may accelerate only τ seconds after whichever cue was actually blocking them, the signal or the vehicle ahead. This is what produces the wave, and setting τ = 0 is what makes the impossible comparison lane possible.

Saturation headway and start-up lost time are measured the way a field engineer would: regress stop-line crossing time against queue position over the saturated part of each platoon, read the headway off the slope, and extrapolate back to vehicle zero for the lost time. The wave speed is a second regression, of release position against release time. Nothing displayed is a stored constant.

The merge model adds gap acceptance in both directions — a merging driver needs room in front and behind — with the required gap shrinking as they run out of road, and the through-lane driver's required gap shrinking if that driver is one of the cooperative ones. Arrivals are Poisson from a seeded generator, so the three strategies are compared against an identical vehicle-by-vehicle sequence.

Where the calibration sits. The signal discharges at a saturation headway of about 2.2 s, against the Highway Capacity Manual's ideal of 1.9 — this model is roughly 15% pessimistic, mostly because its vehicles are still accelerating as they cross the line rather than arriving at cruising speed. The work-zone lane carries around 1,500 veh/h, which sits inside the usual 1,400–1,700 field range. Read the percentage differences between scenarios rather than the absolute numbers; the former are far more robust than the latter.

Deliberately absent: heavy vehicles, lane-width and shoulder effects, grade, weather, turning movements, pedestrians, signal coordination between adjacent junctions, and driver heterogeneity beyond the cooperative/uncooperative split. Any of these would move the numbers. None of them would reverse the two results the post rests on: that the start-up wave costs about a vehicle per green, and that merge strategy moves queue length far more than it moves throughput.

The speed section uses Nilsson’s power model for crash outcomes (exponents 4, 3 and 2 for fatal, fatal-or-serious, and all injury crashes), with the caveat stated in the text that it is a road-level rather than a driver-level relationship. Stopping distances assume a coefficient of friction of 0.8 — dry asphalt, decent tyres — and the reaction time you set on the slider. The petrol curve is fitted to the Department of Energy’s figures for 74 vehicles (−12.4% from 50 to 60 mph, −14% to 70, −15.4% to 80) with a 1/v term for idle and accessory load; the electric curve is fitted to a constant-speed test measuring 224.7 Wh/mi at 50 mph and 366.2 Wh/mi at 80. Both default to a representative vehicle, not yours.

Who wrote this

Claude did — the prose, the three simulators, and the analysis. I set the questions, argued with the answers, and asked for most of the things that make it worth reading: the third release lane at the signal, manual control of the light, the side-by-side merge comparison, the slider for where to actually zip, and the entire speed section.

The part I want to be clear about is that it did not simply agree with me. I came in believing that merging early is the courteous, traffic-saving thing to do and that late mergers cause the jams. The evidence and the model both say otherwise above a certain volume, and the piece says so rather than flattering the premise. When I refined the point — use both lanes, but do not fight for the last car length — that turned into a slider and a sweep, and the model backed it with numbers neither of us had before. Several claims in earlier drafts were wrong and got corrected when the simulation contradicted them; a few results here are messier than the argument would like, and they are reported messy.

Every quantitative claim above has a test behind it. If the models drift, the numbers stop matching and the suite fails, which is the only reason I am willing to put my name near figures I did not derive myself.

Sources

• MnDOT — Zipper Merge
• MnDOT — When to use the late merge (zipper) system (PDF; volume warrant and conditions)
• ITRE, NC State — field study of zipper merges in North Carolina
• Lost time (traffic engineering) — start-up lost time and HCM defaults
• Intelligent Driver Model
• Elvik et al. — Speed and road accidents: an evaluation of the Power Model (TØI 740/2004)
• Kloeden et al. — Travelling speed and the risk of crash involvement (case-control, individual vehicle speeds)
• US DOE FuelEconomy.gov — driving more efficiently
• US DOE Fact #982 — fuel economy versus speed across 74 vehicles