The Wave The Runs Backwards
The car in front pulls away and, for a moment, nothing happens. Multiply that moment by every car in the queue and you get a wave that runs backwards through the traffic at twenty kilometres an hour, at every light and in every jam. Two simulations from first principles put a number on how much of a city’s stop-and-go is that moment, how much is everything else, and what a faster driver would actually be worth.
Anyone who drives in a city has the same suspicion. The light turns green. The first car goes. The second car does not, not yet; its driver is looking at the phone, or the radio, or simply has not registered that the gap ahead is opening. A second passes, perhaps two, and then it goes. The third car repeats the performance behind it. By the time the eighth car in the queue has begun to move, the light has been green for a good part of its allowance, and the fifteenth car, which was only a hundred metres from the line, watches it turn amber without ever having moved. In Portuguese the whole business has a name, pára-arranca, stop-and-start, and the suspicion is that this, not the roads or the lights or the number of cars, is where the time goes.
The suspicion is worth taking seriously, because it is half right, and the half that is right can be measured. Traffic engineering has a hundred years of counts behind it, and a smaller, more recent body of theory that treats a line of cars as a physical system with a response time. That theory says that every car in a queue adds a fixed delay to the car behind it; that the sum of those delays is a wave, moving backwards through the queue at a speed you can compute; that the same delay decides whether a road with no lights and no obstacles will jam itself; and that both effects run at about twenty kilometres an hour, backwards, for the same reason. This essay works all of that out from scratch, with the numbers, and then puts it next to the other constraints on fluid movement to see which is largest.
Twenty kilometres an hour, backwards
Start with the simplest model of a driver that has any content. Gordon Newell, who spent fifty years on the mathematics of queues, wrote it down in its final form in 2002: a car keeps its front a fixed distance δ behind where the car ahead was a fixed time τ ago.[3] The distance is the spacing of a stopped queue, about seven and a half metres from one front bumper to the next, a car and a gap. The time is the response delay: everything between the car ahead moving and this car moving, which is perception, decision, foot from brake to accelerator, and the engine taking up the slack. Newell’s model has no acceleration, no personality and no physics beyond those two constants, and it is the model behind most of what traffic engineers actually use, because it reproduces the counts.
Now put fourteen cars in a queue and turn the light green. The first driver responds after τ. The second cannot begin until the first has moved, and then responds after τ of their own; the third waits for the second; and so on. The nth car begins to move at nτ after the green, from a position nδ back from the line. Those two facts together say that the moment of starting travels backwards through the queue at a definite speed:
With δ = 7.5 m and τ = 1.3 s, a plausible value for a queued driver who is not paying particular attention, that is 5.8 metres a second, or 21 km/h. The number is not an artefact of the model; it is measured. Start waves at signals, and the shock fronts at the back of jams on motorways, both travel upstream at something between 15 and 20 km/h in every country where anyone has looked, and the reason is always equation (1): a jam spacing divided by a human response time.[9][10]
Figure 1 is that queue, simulated with one refinement: the cars accelerate at a comfortable 1.5 m/s² rather than instantly, and stop accelerating at 50 km/h. Each trajectory bends upward exactly 1.3 s after the one above it, and the dashed line through the bends is the wave. The fourteenth car sits still for eighteen seconds before it has any reason to move. If the light is a typical one, with thirty seconds of green, thirteen cars cross the line and the fourteenth does not.
This is the part of the suspicion that is right, and it is right in a strong form. The delay of a car deep in a queue is not mostly the time it takes cars to accelerate, or the time it takes to drive the distance to the line; it is mostly the chain of response delays in front of it, added one by one. At 1.3 s a car, the tenth car in a queue has thirteen seconds of pure waiting stacked in front of it before the physics of moving even begins. The suspicion that the delays are “people not moving when the car in front moves” is exactly Newell’s τ, and the essay could stop here if the only question were what causes the chain.
Two seconds a car
The better question is what the chain costs, and for that there is a hundred years of counting. In 1947 Bruce Greenshields and his colleagues at the Yale Bureau of Highway Traffic stood at intersections with stopwatches and timed the cars crossing the line after a green.[1] The first car took about 3.8 s from the green; the second crossed 3.1 s after the first; then 2.7, 2.4, 2.2, 2.1, and from about the sixth car onward a steady 2.1 s each. Those numbers, with minor updates, have been in every traffic engineering handbook since. The Highway Capacity Manual, the reference that road authorities use to size junctions, puts the steady value at 1.9 s a car, a saturation flow of 1,900 cars an hour of green in a lane, and charges the first few cars a start-up lost time of about two seconds for being slow to get going.[2]
The model reproduces that sequence without being told about it. With τ = 1.3 s the simulated headways at the line run 4.5, 2.6, 2.3, 2.2, 2.1, 2.0, 1.9 and then settle at 1.86 s: the same decaying shape as Greenshields, with the long first headways coming from the cars still accelerating, and the steady value coming from equation (1) read the other way. Once the wave has passed, every car is following the one ahead at the same speed, and crosses the line a fixed time after it:
At 50 km/h the second term is half a second; the rest of the two seconds a car is the driver. The simulated saturation flow is 1,935 cars an hour of green, two per cent from the manual’s base value, and the start-up lost time comes out at four seconds, which is Greenshields rather than the manual. That is one free parameter, τ, fitted to nothing, landing on both of the classic measurements. It is reasonable to conclude that the response delay of a queued driver really is somewhere near 1.3 s, and Figure 2 shows what would happen if it were something else.
The curve is the whole argument in one line. The lane’s throughput in a green is inversely proportional to τ plus half a second, and nothing else about the drivers matters much. A driver who responded in 0.7 s instead of 1.3, which is roughly the difference between someone watching the car ahead and someone glancing up from a phone, would raise the saturation flow from 1,900 cars an hour to 2,850, and a thirty-second green would pass twenty cars instead of thirteen. A driver at 2 s, which is not rare, brings it down to 1,400 and ten cars. The junction has not changed; the cars have not changed; the same green is worth half as much or half as much again, depending only on how long each driver takes to notice that the car ahead has gone.
There is a limit to the argument, and it is the other term. Even at τ = 0, a queue that starts one car at a time cannot pass more than one car every δ/v, about 0.54 s at 50 km/h, or some 6,600 an hour; and no human driver, however alert, follows at half a second. A perceptual and mechanical floor of around 0.7 s puts the ceiling for human drivers at roughly 2,800 an hour of green, and the observed 1,900 is two thirds of it. The remaining third is the cost of inattention, and it is spent at every green, at every junction, all day.
It is fair to ask whether 1.3 s describes anyone. Most queued drivers probably respond in about a second; a minority, on the phone or simply elsewhere, take three or four. The model can be given that mixture directly, one delay per driver, and Figure 5 is the result. The lane turns out to feel only the average. A queue in which nine drivers respond in 1.1 s and the tenth in 3 s discharges exactly as if everyone took 1.3 s: 1,950 an hour, the manual’s number again, thirteen cars a green instead of fifteen. The reason is in the arithmetic of the chain. A slow driver’s excess is passed back, undiminished, to every car behind them, because a late starter opens a gap but cannot exceed the free speed to close it; where they sit in the queue changes nothing until they are behind the last car that would have made the green anyway. A single 4 s driver among 1.1 s drivers costs two cars a green from the second position, and two from the twelfth. What the spread does change is the count: with the same average, the number through a given green varies by one or two cars with where the slow drivers happen to land, and at a busy junction one or two cars is the difference between a queue that clears and one that rolls over into the next cycle.
The other reading of Figure 5 is the steep one. Among attentive drivers, the slow minority costs more, not less: one in ten at 3 s takes an alert queue from twenty cars a green to seventeen, three cars for one driver in ten, where the same minority costs a 1.1 s queue a car and a half. The better the rest of the queue, the more each slow driver is worth, and the more a city would gain from finding them.
Nothing lights up when a car pulls away
Why 1.3 s and not 0.7? Part of it is ordinary reaction time. Marc Green’s survey of driver perception-brake times, the time from a hazard appearing to the foot on the brake, found 0.7 to 0.75 s for a driver who is expecting something, 1.25 s for one who is not, and 1.5 s or more when the hazard is genuinely surprising.[5] A queued driver is, in principle, expecting the green. In practice, having stopped, they have handed the task to the environment; the environment will tell them when to move, and they are free to look elsewhere, which most of them do.
The larger part is that the environment tells them badly. Braking is announced: a car that brakes shows two red lights, a discrete event, the kind of signal the visual system detects in a tenth of a second. Starting is not. A car pulling away from you produces no light, no sound worth hearing, and no colour change; the only information is that its image on your retina is getting smaller, and the rate at which it shrinks, in the first second of a car accelerating at 1.5 m/s² from seven metres away, is tiny. David Lee showed in 1976 that drivers time their braking on precisely this quantity, the rate of expansion of the image ahead, and a looming object produces it in abundance.[6] A receding one, from a standing start, barely produces it at all. The signal that would let a driver respond to the start is the weakest signal the car ahead ever gives.
Then there is the light itself, which most queued drivers watch instead of the car ahead, so that the first car’s response is to the green and every other car’s is to a car that has already had its τ. Then the mechanics: brake to accelerator, the engine off idle, an automatic gearbox taking up drive, a second or so between the decision and the wheels turning. Then, for some drivers, a deliberate wait: the belief that it is safer to let a gap open before moving, which converts one response delay into two. Each of these is small. Together they are 1.3 s a car, and the queue adds them up.
Jams with no cause
So far the light has been the reason for the stop, and the response delay has only governed the start. But stop-and-go traffic happens where there is no light: on ring roads, on motorways, on any stretch of road carrying enough cars, a wave of stopped traffic appears, moves backwards against the flow, and dissolves, with nothing at its head, no accident, no merge, no lane closed. Drivers call these phantom jams. In 2008 Yuki Sugiyama and his colleagues produced one on demand: twenty-two cars on a 230 m circular track in a car park, told to drive at a steady 30 km/h and keep their distance, no instructions beyond that.[8] Within a minute the even spacing had broken; within two there was a cluster of stopped cars going round the ring backwards at about 20 km/h, and it stayed for as long as the experiment ran.
The theory of this was worked out in 1995 by Masako Bando and a group of Japanese physicists that included Sugiyama.[7] Each driver has a preferred speed for each spacing, a function V(s) that is zero at the jam spacing and rises to the free speed, and relaxes toward it with a response time τ: the acceleration is (V(s) − v)/τ. That is the whole model. A line of such cars at equal spacing is an exact solution; the question is whether a small disturbance to it grows or dies, and the linear analysis gives a clean answer. The uniform flow is stable if and only if
The right side is how sharply the preferred speed changes with spacing; it is largest at moderate spacings, where a metre of gap is worth a lot of speed, and small at either end. The left side is the driver. Slow drivers at moderate density violate the inequality, and when they do, any disturbance whatever, one car braking briefly, one gap slightly narrower than the others, grows into a jam. Figure 3 is that happening.
The simulation is sixteen cars on Sugiyama’s ring, in the Bando model, with the same τ = 1.3 s as the signal and a preferred-speed function chosen so that the model’s drivers want 40 km/h when the road is clear. The ring is set up almost uniformly; the disturbance is a random tenth of a metre in each initial spacing. After two minutes there are two stop-and-go waves running backwards round the ring at 19 km/h, each car passing through both of them on every lap, stopping dead in each. With τ = 0.7 s and everything else identical, the same sixteen cars settle into a smooth circulation at their equilibrium speed and stay there. Nothing about the road has changed. The difference between a jam and no jam is six tenths of a second in the driver.
That the wave runs at 19 km/h is the same physics as the start wave at the light, and worth pausing on. Inside a jam the cars are at jam spacing; at its downstream edge they leave one at a time, each τ after the one in front, so the edge moves upstream at δ/τ. At its upstream edge cars arrive and stop, and the arriving car stops one spacing behind the one that stopped before it, again one τ later. A jam is a queue at a light that has lost its light: the same chain of responses, the same wave, the same 20 km/h, and it will run against the traffic for as long as the density upstream keeps feeding it.
Where the minutes go
The last thing the ring can give is the curve traffic engineers care about most, the fundamental diagram: flow against density. Run the ring at every density from nearly empty to nearly full, wait for the transients to die, and count cars past a point. The dashed line in Figure 4 is what the road would carry if every car held its preferred speed for its spacing: a hump that rises from zero, peaks at about 1,800 cars an hour near 58 cars per kilometre of lane, and falls back to zero at the jam density of 133. The dots are what the road actually carries.
At 0.7 s the drivers follow the curve over the top: the road delivers its capacity. At 1.3 s they follow it up to 53 cars per kilometre, exactly where inequality (3) fails, and there they leave it. The flow at 70 per km is 1,340 an hour instead of 1,690, a fifth less, delivered in lurches; the road never reaches its own peak, because the density at which the peak lives is a density at which its drivers cannot keep the flow smooth. This is the general result, and it is the second half of the suspicion: slow response does not only lengthen the start at a light, it lowers the ceiling on what a road can carry at all, and it does it exactly in the range of densities where the road would otherwise be at its most productive.
Now the budget. Here is what stands between a car and fluid movement through a city, in roughly the order of how much each costs, for the ordinary case of a signalised arterial that is busy but not gridlocked.
The ranking is honest but it should not be read as an acquittal of the chain. The red light is the largest term because it is a fixed cost, and fixed costs are what signal timing is for; the traffic engineers have spent a century on green splits, offsets and coordination precisely to shrink it. The chain is different. It is a per-car cost, and it multiplies: a queue that would have cleared in one green at 0.7 s takes two greens at 1.3 s, and every car that rolls over to the second green pays the full red again. When a signal is near saturation, the response delay is the thing deciding whether the queue grows or shrinks each cycle, and that is the regime in which cities spend their rush hours. The suspicion had the mechanism right and the ranking slightly wrong, and it is the mechanism that matters, because it is the one with a lever on it.
What a faster driver is worth
There are three ways to shrink τ, and traffic engineering is using all of them. The first is the driver. A driver who watches the car ahead rather than the light, and better still the car two ahead, effectively responds to the wave before it reaches them; the models with this kind of anticipation in them are markedly more stable, and the drivers who do it are the ones you notice pulling away with the car in front as if attached to it.[15] The mechanism of that response is nothing more than attention, and the cost of not having it is a third of every green.
The second is the machine. Adaptive cruise control responds in about a second; the cooperative version, in which each car receives the acceleration of the one ahead by radio rather than watching it recede, has been driven at 0.6 s gaps in real traffic, and would put the start wave at 45 km/h and the saturation flow well above 3,000.[14] A queue of such cars starts almost as one, the way a train does, because the train has a coupling and the platoon has a radio, and both have τ near zero. Less ambitiously, in 2018 a single autonomous car among twenty human ones on Sugiyama’s ring, driving nothing more clever than a steady speed, damped the stop-and-go wave for the whole ring: one car in twenty that does not amplify the disturbance is enough to stop it growing.[13]
The third is not to shrink τ but to stop paying it so often, which is what coordinated signals, the green wave, do: a platoon released from one light arrives at the next while it is green, and starts once per trip rather than once per junction. Where that works it removes both the red and the chain at once, and it is why the same street can feel fluid at eleven at night and hopeless at six, with the same lights and the same drivers, differing only in whether the platoons are still platoons when they arrive.
So: the car in front moves and, for a moment, the next one does not. The moment is about 1.3 s. It sets the speed at which the queue unfreezes, the number of cars a green is worth, the density at which a road jams itself, and the speed at which the jam travels. It is not the largest item in the budget of an urban trip; the red light is. It is the second, it is the one that multiplies when the road is busy, and it is the only one that is entirely in the hands of the person who has just looked up from the phone and noticed the gap.
Where this could go
Measure τ in Lisbon. A phone on a tripod at any signalised junction, filming a green, gives the crossing times of the queue and hence the headway sequence; the model fits τ to it in a line. Twenty greens at three junctions would say whether 1.3 s is the local number, and whether it differs by time of day, by lane, or by how many drivers are visibly holding phones.
Recovering the loss. Figure 5 assumes a late starter cannot make up the time, because nobody in the model exceeds the free speed. Real late starters often accelerate harder, and the car behind them may close part of the gap before the line. A model with a recovery rule would say how much of a slow driver’s excess the queue actually gets back, and whether the answer differs at a 30 s green and a 60 s one.
Two ahead. The multi-anticipation models deserve their own run on the ring: how much stability does watching the second car buy, and does it also raise the saturation flow at the signal, where the second car is the one that shows the wave coming?
The physical floor. The 0.7 s floor in this essay is a perception number. The mechanical part, brake release to wheels turning, is measurable on any car and is different for a manual, an automatic and an electric motor, and the electric motor has almost none of it. Fleet electrification may shrink τ without anyone deciding to.
This piece was written collaboratively with Claude Fable 5.1 (Anthropic): human specification, editorial direction and critical review; machine synthesis, drafting, computation and figure generation.
The figures are computed, not traced. The script scripts/traffic.py runs two models. The signal is Newell’s car-following model with a bounded acceleration of 1.5 m/s², a free speed of 50 km/h, a jam spacing of 7.5 m and an explicit response delay, the lead car starting one delay after the green; Figures 1 and 2, the headway sequence and the throughput figures in the text are read off it directly, for delays from 0.3 to 2.5 s. Figure 5 gives each driver their own delay, 3 s for a random share of the 24-car queue and 1.1 s or 0.7 s for the rest, and averages 120 placements per point. The ring is the optimal-velocity model of Bando et al. on a 230 m ring, with the preferred-speed function V(s) = v0 [tanh((s − 10)/8) − tanh((7.5 − 10)/8)] / [1 − tanh((7.5 − 10)/8)] for s above 7.5 m and zero below, v0 = 40 km/h, and the sensitivity 1/τ. That model on its own lets cars overlap at slow response times, a known defect, so a hard stop is added: a car never closes to less than the jam spacing behind the car ahead. Figure 3 is one run with sixteen cars; Figure 4 averages the flow over the last 150 s of a 420 s run at each density, for two response times. The jam speed is the median, over all pairs of consecutive cars stopping, of the distance between their stopping points divided by the time between them. The stability boundary in Figure 4 is inequality (3) evaluated on V. The numerical output is in docs/traffic-results.json. The Greenshields headways and the Highway Capacity Manual values are quoted from the sources and were not used to fit anything.
Authored by: Luis Matos Ferreira — Physicist, Developer, Writer
- Greenshields, Schapiro & Ericksen, Traffic Performance at Urban Street Intersections, Technical Report 1, Yale Bureau of Highway Traffic, 1947. The headway sequence 3.8, 3.1, 2.7, 2.4, 2.2, 2.1 s is the one reproduced in the traffic engineering textbooks since.
- Transportation Research Board, Highway Capacity Manual, 6th edition, 2016, chapter 19: base saturation flow rate 1,900 passenger cars per hour per lane, start-up lost time 2.0 s.
- Newell, “A simplified car-following theory: a lower order model”, Transportation Research Part B 36, 195 (2002).
- Lighthill & Whitham, “On kinematic waves II: a theory of traffic flow on long crowded roads”, Proceedings of the Royal Society A 229, 317 (1955); Richards, “Shock waves on the highway”, Operations Research 4, 42 (1956). The continuum theory of which Newell’s model is the car-by-car version; the backward wave speed is its shock speed.
- Green, “‘How long does it take to stop?’ Methodological analysis of driver perception-brake times”, Transportation Human Factors 2, 195 (2000).
- Lee, “A theory of visual control of braking based on information about time-to-collision”, Perception 5, 437 (1976).
- Bando, Hasebe, Nakayama, Shibata & Sugiyama, “Dynamical model of traffic congestion and numerical simulation”, Physical Review E 51, 1035 (1995). The stability condition is their equation for the linearised model; the identification of the sensitivity with an inverse response time is theirs.
- Sugiyama, Fukui, Kikuchi, Hasebe, Nakayama, Nishinari, Tadaki & Yukawa, “Traffic jams without bottlenecks: experimental evidence for the physical mechanism of the formation of a jam”, New Journal of Physics 10, 033001 (2008).
- Treiber & Kesting, Traffic Flow Dynamics: Data, Models and Simulation, Springer, 2013, especially the chapters on the optimal-velocity model, its collision problem, and the empirical propagation speed of jam fronts of about 15 km/h.
- Kerner & Rehborn, “Experimental properties of complexity in traffic flow”, Physical Review E 53, R4275 (1996), for the measured upstream speed of wide moving jams on German motorways.
- Webster, Traffic Signal Settings, Road Research Technical Paper 39, HMSO, 1958. The uniform-delay term is C(1 − λ)² / 2(1 − λx) with C the cycle, λ the green fraction and x the degree of saturation; 14 s is C = 90 s, λ = 0.45, x = 0, and 21 s is x = 0.8.
- Cassidy & Bertini, “Some traffic features at freeway bottlenecks”, Transportation Research Part B 33, 25 (1999).
- Stern et al., “Dissipation of stop-and-go waves via control of autonomous vehicles: field experiments”, Transportation Research Part C 89, 205 (2018).
- Milanés, Shladover, Spring, Nowakowski, Kawazoe & Nakamura, “Cooperative adaptive cruise control in real traffic situations”, IEEE Transactions on Intelligent Transportation Systems 15, 296 (2014).
- Lenz, Wagner & Sollacher, “Multi-anticipative car-following model”, European Physical Journal B 7, 331 (1999).
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