Traffic Flow Models

Why Model Traffic Flow

Once traffic gets dense, driving stops being a solitary activity and turns strategic — what you do depends on what everyone around you is doing. Two parties act on that traffic, and they want different things.

The single car driver chases an individual optimum: get to the destination quickly, and don’t crash. Both goals are entirely local, and that is the problem. A driver has no way of telling whether a given maneuver — closing a gap, changing lane, braking a little early — helps or hurts the undisturbed flow of traffic as a whole, and therefore whether it helps or hurts the other road users. Intuition is a poor guide here, because the effect of one car on the collective is not something you can see from inside that car.

The police or a traffic control center takes the opposite stance: an overall view, aimed at avoiding jams and accidents across the whole network. It also has real tools to act with — road construction, traffic-guidance infrastructure, flexible lanes that switch driving direction by time of day, traffic lights, and so on.

Neither party can act sensibly without a model of traffic flow. We need one to understand complex situations in the first place, to control traffic in a reasonable way rather than by guesswork, and above all to capture the mutual interaction of traffic density and velocity — both in the big picture, where it shows up as throughput, and in the fine detail, where it shows up as the propagation of stagnations through the stream of cars. That interaction is the heart of the matter: density and velocity are not independent knobs, and the whole chapter is about how they constrain each other.

The model we study is a simple and classical one, due to Lighthill, Whitham & Richards.

Possible Characterizations

A road network is a physical thing, but the mathematics we use to describe it is a choice, and there are several genuinely different ones on the menu. Each turns “cars on roads” into a different kind of object.

Graph or network, with the dynamics via Petri networks. The road network is a graph; the cars in the system are the tokens moving through it, and their dynamics is described by a Petri network (a model of concurrent systems in which tokens move between places according to firing rules).
Cellular automaton, with the dynamics via a rule base. The road is chopped into cells of a fixed length — 7.5 m per cell is the standard choice, roughly the space one car occupies in a jam. Each cell is either empty or holds one car, each car carries an integer velocity (a car with $v = 2$ hops two cells per time step), and the traffic evolves by applying simple local rules to every cell at once.
Queuing network, in the sense of the discrete model. The road network becomes a queuing network: traffic lights and intersections are the elementary queuing systems, and the influx and efflux of cars are stochastic processes. This is exactly the machinery developed for data traffic in computing systems, pointed at cars instead of packets.
Elementary particles. Cars are treated as particles, either deterministically as in molecular dynamics, or stochastically via probabilities of presence, in the spirit of Boltzmann and Schrödinger.
Fluid. The road network is a system of canals, and the overall traffic is a viscous fluid squeezing through it.
Waves. Congestions and traffic lights are read as the propagation of waves, which brings the vocabulary of acoustics into play.

The last two are the ones this chapter follows, and they are really one idea seen twice.

The fluid picture asks us to perform a “liquefaction” of cars. Stop resolving the individual vehicles: instead of a stream of discrete cars occupying a road that narrows from two lanes to one, picture a continuous medium flowing through the same geometry. What was a countable set of cars becomes a density smeared along the road — thick where the cars bunch up, thin where they spread out — and what was a car passing a point becomes a flow through that point. Where the road narrows, the medium squeezes together and the density rises, exactly as a fluid would. We make these quantities precise in the next section.

The wave picture is what that fluid does once it is disturbed. A jam is not a fixed stretch of road; it is a pattern in the density that travels — usually backward, against the direction of the cars themselves — and outlives every individual car caught in it. Treating that pattern as a wave is what lets us say where it goes and how fast.

Analogies Beyond the Road

It is worth knowing how far this pays off, because the same equations turn up in places that have nothing to do with cars. They also describe the expansion of pollution in flow fields such as water or air. And the expansion of a congestion is mathematically analogous to:

the expansion of shock waves — from a jet plane, from gunshots, from the pressure wave after an explosion;
the propagation of fire fronts in forest fires;
the percolation of water into the soil after heavy rain;
the movement of railway cars while shunting.

The reason for the family resemblance is that all of these are, at bottom, the same story: some conserved stuff is carried along, and a disturbance in how densely it is packed travels through it at a speed of its own. Congestion is the version of that story we happen to sit in.

Two situations get covered, in this order: stationary traffic flows, where the traffic is in equilibrium and nothing changes in time, and then dynamical changes and singular disturbances, where it does. Throughout, we use a consistently continuous description, in terms of flow quantities rather than individual cars.

The Role of the Individual

Smearing cars into a fluid should make you suspicious, because cars are driven by people, and people are not fluid. Human (mis-)behavior is variable and adaptable, it is shaped by psychological factors, and it is hardly possible to describe exactly in a mathematical-physical model. We will not attempt to. What saves the continuum model is not that human behavior is simple, but that in the regime we care about, it barely matters.

The influence of the individual depends on the situation, and the two extremes point in opposite directions:

Single-lane road, little traffic, no overtaking. The slowest driver determines the pace for everyone behind. One individual sets the outcome, and average quantities make little sense: an “average velocity” computed over a handful of cars, one of which is dictating the speed of the rest, describes nobody.
Highway, heavy traffic. The single driver has little influence — there are too many cars, too tightly coupled, for any one of them to move the collective. Here averages make a lot of sense: they are what the traffic actually does.

So the focus of modeling and simulation is heavy traffic on multi-lane roads. That is not a convenient dodge — it is precisely the regime in which the individual washes out and a continuous description is legitimate. The single-lane country road with three cars on it is not badly modeled by the theory that follows; it is simply not what the theory is about.

Steady Traffic Flow

Before letting anything change, it pays to look at traffic that has settled: a stream in which nothing varies over time, and nothing varies along the road. That is a strong idealization and we drop it later. But equilibrium is where the vocabulary gets fixed, because there the quantities describing a stream of cars are tied together by relations simple enough to derive by hand — and those relations carry over into the unsteady case as building blocks.

Two Ways to Watch a Stream

Before any quantity is written down there is a choice of viewpoint, and it decides what the mathematics is even about.

In the Euler description, the observer stays outside the system at a fixed location and records what passes by: how dense the traffic is at this point, how fast it moves here, how many cars cross here per hour. The quantities become fields over position and time, and no individual car is identified or followed.

In the Lagrange description, the observer travels along with the matter itself. The system is resolved into particles — for traffic, the single cars — and the motion of each particle is followed through space and time.

The difference is simply where you are standing. Euler is the person on the bridge above the motorway with a clipboard, counting what goes underneath; Lagrange is the person behind the wheel, and the description is written from inside the stream. Both are legitimate, and both are used across physics, but they answer different questions: Lagrange tells you what happened to that car, Euler tells you what is happening at that place.

We take the Euler view for the whole of this chapter. That is what studying traffic flows rather than individual car movements means in practice, and it is the mathematical form of the liquefaction described earlier: once the cars are smeared into a medium, there are no individual trajectories left to follow, only fields.

The Scenario and the Three Quantities

The starting scenario is, at first, very unrealistic, and deliberately so. Every car has the same length $l$ , every car drives at the same speed, and every pair of neighbors keeps the same distance. The flow is steady, stationary and single-lane. Nothing about real traffic looks like this; the point is to have a setting in which the three quantities of interest can be related without any modeling assumption sneaking in.

Normalize the length of the road stretch to $1$ . Then the density is simply the number of cars sitting on it, and the distance between two neighboring cars is

\frac{1 - \varrho \cdot l}{\varrho - 1}

which is worth reading slowly, because both parts of the fraction carry meaning. The $\varrho$ cars occupy $\varrho \cdot l$ of the unit stretch, so $1 - \varrho \cdot l$ is the free space left over. Lining $\varrho$ cars up in a row creates $\varrho - 1$ gaps between them, and by assumption all gaps are equal, so the leftover space is divided into that many equal pieces. The expression stays positive exactly as long as the road is not yet packed solid, that is as long as $\varrho < 1/l$ , and it shrinks to zero as the cars close up bumper to bumper.

Three quantities describe the stream, and each is an average taken over the smeared-out medium rather than a property of any one car.

The traffic velocity $v$ , measured in km/h, is the tempo of the cars: the speed at which the stream moves past a given place.

The traffic density $\varrho$ , measured in cars/km, is the number of cars per section of road. It is the quantity that plays the role of “how tightly the medium is packed” in the fluid picture.

The traffic flow $f$ , measured in cars/h, is the throughput at a checkpoint: the number of cars passing a fixed position per unit of time.

The units already give away how these three fit together. Cars per kilometer times kilometers per hour is cars per hour, so density times velocity has exactly the dimension of flow. The next section shows that this is not a coincidence of bookkeeping but the actual relationship.

A First Relationship: Flow, Density and Velocity

The fundamental relationship between traffic flow, traffic density and traffic velocity — whether the quantities are taken as absolute values or as averages — reads

f = \varrho \cdot v

The argument behind it is a counting argument, and it needs the space dimension to work. Put a checkpoint at position $X$ and watch it for a time span $\tau$ . Every car travels at the mean velocity $\bar v$ , so in that time each one advances by $\bar v \tau$ — a car sitting at $x = 0$ at time $t = 0$ has reached $x = \bar v \tau$ at time $t = \tau$ . Turn that around: the cars that cross $X$ during the span are exactly the ones that started somewhere in the stretch of length $\bar v \tau$ upstream of it, no more and no fewer. That stretch holds $\varrho \cdot \bar v \tau$ cars. Divide by the elapsed time $\tau$ and the flow at the checkpoint is $\varrho \cdot \bar v$ .

This is Little’s law wearing different clothes. That law, in the form filling equals dwelling time times throughput, was derived with no assumption about arrivals or service order at all, and each of its three quantities has a traffic counterpart: throughput corresponds to the flow $f$ , filling corresponds to the density $\varrho$ , and dwelling time corresponds to the inverse velocity $v^{-1}$ . Reading $\varrho = f \cdot v^{-1}$ instead of $f = \varrho \cdot v$ makes the correspondence line up term by term.

The one genuinely new ingredient is the space dimension. In a queuing system a job is simply inside the unit or not, and the dwelling time measures how long it stays; on a road the job has a position, and “how long it stays” turns into “how long it takes to cover the distance.” Everything else is the same piece of bookkeeping.

It is worth being clear about what this relationship does not say. Left on its own, $f = \varrho \cdot v$ is one equation in three unknowns: it constrains the trio but pins nothing down. In particular it does not tell us that dense traffic is slow. Nothing in it forbids a road carrying maximum density at maximum speed, which would be wonderful and is also obviously not how driving works. The fact that velocity collapses as density rises is a separate piece of knowledge about driver behavior, and it has to be added to the model by hand. That is the second relationship.

A Second Relationship: Velocity and Density

The missing ingredient is a characterization of $v$ as a function of $\varrho$ , and its shape is dictated by one observation: $v$ is strictly monotonically decreasing in $\varrho$ . Higher velocity demands larger safety distances, and larger safety distances mean fewer cars per kilometer, so speed and density pull against each other directly.

The two ends of the range fix the curve. In the limit case of maximum density the cars stand bumper to bumper, the distance between them is zero, and nobody moves:

v \to 0 \quad \text{for} \quad \varrho \to \varrho_{\max} = \frac{1}{l}

The maximum density $\varrho_{\max} = 1/l$ is just the statement that a kilometer of road holds $1/l$ cars of length $l$ when they are packed solid. At the other end, on an empty road, there is nobody to be held up by and drivers go as fast as they are willing or allowed to:

v \to v_{\max} \quad \text{for} \quad \varrho \to 0

Note that $v_{\max}$ is a finite ceiling set by the road, the law and the driver, not an unbounded quantity — an empty road does not make cars infinitely fast.

The simplest curve joining those two endpoints is a straight line, and taking it is the linearization of the velocity-density dependency:

v = v_{\max} \cdot \left(1 - \frac{\varrho}{\varrho_{\max}}\right)

It is a simplification and we will criticize it soon enough, but it satisfies both limits exactly and it makes everything downstream computable in closed form.

Traffic policy would rather turn the relationship around. A traffic planner cares about distances, and therefore wants $\varrho$ as a function of $v$ rather than the other way round — the braking distance is significantly determined by $v$ , so it is the velocity that ought to be the input and the safe density the output. The German poster rule “Abstand halber Tacho” (“distance half speedometer”) is exactly this reading, prescribing a gap of half the speedometer value in meters.

Unfortunately this approach does not lead to reasonable models of driving behavior. What drivers actually do is respond to the traffic they find themselves in, so density is the natural input and velocity the response, which is why the model keeps $v = v(\varrho)$ .

A Third Relationship: The Fundamental Diagram

The two relationships now combine into one. Substituting $v = v(\varrho)$ into $f = \varrho \cdot v$ makes the flow a function of the density alone:

f = f(\varrho) = v(\varrho) \cdot \varrho

and with the linear velocity-density dependency this becomes an explicit parabola.

The parabolic model for the traffic flow follows from the linearized velocity-density relationship:

f(\varrho) = v_{\max} \cdot \varrho \cdot \left(1 - \frac{\varrho}{\varrho_{\max}}\right)

It is a downward-opening parabola with $f \to 0$ for both $\varrho \to 0$ and $\varrho \to \varrho_{\max}$ .

The fundamental diagram is the plot of traffic flow against traffic density, $f$ over $\varrho$ . It is the central picture of steady traffic flow: everything the model has to say about equilibrium can be read off it.

Both vanishing limits make sense on their own terms, and they vanish for opposite reasons. At $\varrho \to 0$ there are cars moving at full speed but almost none of them, so hardly anything crosses the checkpoint. At $\varrho \to \varrho_{\max}$ the road is full of cars but none of them are moving, so again nothing crosses. Between the two extremes the flow must rise and fall back, and the parabola peaks in the middle. Setting the derivative to zero gives

\varrho_{\text{opt}} = \frac{\varrho_{\max}}{2}, \qquad f_{\max} = f(\varrho_{\text{opt}}) = \frac{v_{\max} \cdot \varrho_{\max}}{4}

so in this model the road carries the most cars per hour when it is exactly half full and everyone is driving at exactly half the maximum speed. That the optimum sits at half of everything is a consequence of the linear velocity assumption rather than a fact about roads, and it is the first thing the next section attacks.

f(\varrho) = v_{\max} \cdot \varrho \cdot \left(1 - \frac{\varrho}{\varrho_{\max}}\right)

Drag the marker to move the density. The dashed level line joins the two densities that produce the same flow, and it collapses to a single point only at the peak.

The more consequential reading of the diagram concerns any flow below the maximum. Because the parabola rises and then falls, a horizontal line at some level $f_{lr} < f_{\max}$ cuts it twice. The same throughput is realized by two entirely different states of the road: at the low density $\varrho_l$ , where the corresponding velocity $v_l$ is high, and at the high density $\varrho_r$ , where the velocity $v_r$ is low. Both push the same number of cars per hour past the checkpoint; from the checkpoint’s point of view they are indistinguishable. From inside a car they could hardly be more different.

Equilibrium States

Those two solutions of $f(\varrho) = f_{lr}$ are the two equilibrium states of a road carrying a given flow, and they are different enough to have names.

The free-flow phase is the equilibrium at low density and high velocity — the branch of the fundamental diagram left of $\varrho_{\text{opt}}$ . Cars are far enough apart that they barely interfere with one another.

The congestion phase is the equilibrium at high density and low velocity — the branch right of $\varrho_{\text{opt}}$ . The same flow is achieved by packing many slow cars onto the road instead of few fast ones.

The free-flow phase is plainly the more attractive of the two, and for once everyone agrees. The single driver gets to the destination faster, since the flow being equal says nothing about individual travel time and the velocity is what a driver actually experiences. The traffic planner prefers it too, for the simple reason that drivers are happy in it.

Whether it actually occurs is another question, and the answer depends on how much traffic there is. When traffic is light, yes: everyone revs up, the density stays low by itself, and the road sits comfortably on the left branch. When traffic is heavy, rather not. The demand alone pushes the density past the optimum, and once the road is on the right branch it stays there, because the state is easy to fall into and easy to hold. Keeping a road in free flow at high demand asks every driver to maintain a large gap and a high speed at the same time, which is the more demanding of the two behaviors; drifting into the congested state asks nothing of anybody.

So the better equilibrium does not establish itself. It has to be forced, and that is precisely what traffic controlling is for — traffic lights, tempo limits, and the rest of the infrastructure named earlier. It looks paradoxical that imposing a speed limit can make traffic move better, but the fundamental diagram explains it exactly: capping the speed caps the density that builds up behind it, and holding the road on the left branch of the parabola is worth more than the few km/h it costs.

The Cubic Model

The parabolic model got us this far on one straight line, and that line is where its trouble starts. Writing it out once more,

f(\varrho) = v_{\max} \cdot \varrho \cdot \left(1 - \frac{\varrho}{\varrho_{\max}}\right)

the model is in general unsatisfactory, and often in direct contrast to what is observed on real roads, because of two things it has frozen in place. The maximum flow is frozen at half of the maximum density, and the maximum flow is frozen at half of the maximum velocity. Neither of those is a claim anybody made about traffic; both are arithmetic consequences of having drawn a straight line from $v_{\max}$ down to zero. A real road might well reach its best throughput at 40% or 60% of jam density, and the parabolic model has no way of expressing that. It has only two parameters, $v_{\max}$ and $\varrho_{\max}$ , and both are already spoken for by the endpoints.

The fix is to buy freedom by adding parameters.

The cubic model for the traffic flow replaces the linear velocity-density dependency with a quadratic one, giving a cubic polynomial for the flow:

f(\varrho) = v_{\max} \cdot \varrho \cdot \left(1 - \alpha\varrho - \beta\varrho^2\right)

with two free parameters $\alpha$ and $\beta$ that are fixed by imposing model conditions.

Some of the conditions we want are satisfied by the shape of the ansatz itself, before any parameter is chosen. Since the bracket is multiplied by $\varrho$ , the flow vanishes on the empty road,

f(\varrho) \to 0 \quad \text{for} \quad \varrho \to 0

and differentiating shows the same construction fixes the slope there as well,

f'(\varrho) \to v_{\max} \quad \text{for} \quad \varrho \to 0

which says the first few cars on an empty road each add their own full $v_{\max}$ worth of throughput, since none of them is slowed by the others. Those two come for free.

The remaining conditions have to be imposed, and they are what $\alpha$ and $\beta$ are for. At the jam density the road must again carry nothing,

f(\varrho) \to 0 \quad \text{for} \quad \varrho \to \varrho_{\max}, \qquad v(\varrho_{\max}) = 0

and these two are the same requirement seen twice, since $f = v \cdot \varrho$ and $\varrho_{\max} \neq 0$ means one vanishes exactly when the other does. That is one condition. The second is the one that buys back what the parabola lost, and it comes in two interchangeable forms: either the flow peaks at a density we prescribe,

f'(\varrho_{\text{opt}}) = 0

or the peak reaches a height we prescribe,

f(\varrho_{\text{opt}}) = f_{\max}

Two free parameters, two conditions to pin them down. The point of the exercise is that $\varrho_{\text{opt}}$ and $f_{\max}$ are now inputs to the model rather than outputs forced by its algebra.

Solve the two conditions for

\alpha

and

\beta

Take the first form of the second condition, so that $\varrho_{\max}$ and $\varrho_{\text{opt}}$ are both prescribed. The velocity implied by the cubic ansatz is $v(\varrho) = v_{\max}(1 - \alpha\varrho - \beta\varrho^2)$ , so $v(\varrho_{\max}) = 0$ reads

\alpha\varrho_{\max} + \beta\varrho_{\max}^2 = 1

Differentiating $f(\varrho) = v_{\max}(\varrho - \alpha\varrho^2 - \beta\varrho^3)$ gives $f'(\varrho) = v_{\max}(1 - 2\alpha\varrho - 3\beta\varrho^2)$ , so $f'(\varrho_{\text{opt}}) = 0$ reads

2\alpha\varrho_{\text{opt}} + 3\beta\varrho_{\text{opt}}^2 = 1

Two linear equations in $\alpha$ and $\beta$ . Eliminating $\alpha$ gives

\beta = \frac{\varrho_{\max} - 2\varrho_{\text{opt}}}{\varrho_{\max} \cdot \varrho_{\text{opt}} \cdot \left(3\varrho_{\text{opt}} - 2\varrho_{\max}\right)}, \qquad \alpha = \frac{1 - \beta\varrho_{\max}^2}{\varrho_{\max}}

The numerator of $\beta$ is worth a look. It vanishes exactly when $\varrho_{\text{opt}} = \varrho_{\max}/2$ , and then $\beta = 0$ and $\alpha = 1/\varrho_{\max}$ , which is the parabolic model back again. So the cubic contains the parabola as the special case “optimum at half of jam density”, and its extra parameter is precisely the freedom to move that optimum somewhere else.

Fitting the Cubic to Real Data

Different choices of $\alpha$ and $\beta$ produce visibly different courses of $v$ and $f$ , and that is the whole point: the two parameters are handles for fitting the model to real traffic observations and to whatever model conditions the situation imposes. In practice one specifies $\varrho_{\text{opt}}$ together with $f'(\varrho_{\text{opt}}) = 0$ , or specifies $f_{\max}$ together with $f(\varrho_{\text{opt}}) = f_{\max}$ , and solves for the pair.

Held against measured data from a two-lane German Bundesautobahn, the result is instructive in both directions. The measurements do not form a curve at all — they form a broad cloud, dense and narrow near the empty-road end and fanning out badly at higher densities, because at a given density real drivers pick quite different speeds. A fitted cubic tracks the middle of that cloud better than the parabola does, since it can put the peak where the data puts it instead of at the halfway mark.

Not all of the fitted courses are “legal”. A curve can satisfy every condition listed above and still be nonsense as a traffic model, and a cubic in particular has enough freedom to misbehave inside the very interval we care about. Nothing in the conditions forces $f(\varrho) \ge 0$ or $v(\varrho) \ge 0$ across the whole of $[0, \varrho_{\max}]$ , and nothing forces $v$ to stay monotonically decreasing. A cubic fitted to the cloud can therefore dip below the axis somewhere before $\varrho_{\max}$ and curl back up to meet it, which the algebra is perfectly happy with and which claims that at a certain density cars drive backward.

That is the sanity check the conditions do not perform for you. The physically meaningful requirements — non-negative flow, non-negative and monotonically decreasing velocity on the whole interval — have to be verified on the fitted curve afterward, not assumed from the fact that the fit converged.

Separating the Two Phases

Even a well-behaved cubic is still expandable, and the sharpest criticism of it is that the two phases do not differ enough. Both the parabola and the cubic make the free-flow and congestion branches near mirror images of one another, and real traffic is not symmetric like that.

In the free-flow phase, cars genuinely do not interfere with each other. Adding one more car to a nearly empty road costs nobody any speed, so the flow increases almost linearly with density and the velocity curve starts out nearly flat — density has almost no effect on how fast anyone drives, right up until the gaps get tight. In the congestion phase, interference rises steadily and the flow does not fall linearly; instead, velocity collapses quickly once the interference sets in, and traffic is effectively at a standstill well before the road is packed bumper to bumper. A polynomial with a symmetric hump cannot produce that flat start, that sharp break, and that long flat tail at once.

The ansatz that can is a power law:

f(\varrho) = v_{\max} \cdot \varrho \cdot \left(1 - \left(\frac{\varrho}{\varrho_{\max}}\right)^{\alpha}\right)^{\beta}

The two exponents do the two jobs. The inner exponent $\alpha$ controls how long the velocity curve stays flat before it starts to fall, and the outer exponent $\beta$ controls how hard it drops once it does and how flatly it approaches zero afterward. Setting $\alpha = \beta = 1$ collapses the whole thing back to the parabolic model, so this is again a strict generalization of what came before.

Here the parameters are not fixed by prescribing conditions but read off the data directly, typically by the least-squares method — choosing the $\alpha$ and $\beta$ that minimize the sum of squared vertical deviations between the model curve and the measured points. Fitted this way, the curve threads the measurement cloud far more convincingly than the cubic does: it stays pinned to $v_{\max}$ across the low-density band where the data really is flat, bends over sharply in the middle where the data does, and then hugs the axis for the rest of the range. The near-standstill is reached long before $\varrho_{\max}$ , which is exactly what a jammed road looks like and exactly what a polynomial fit kept getting wrong.

Unsteady Situations

Everything so far has assumed the traffic has settled. Now we let it move. The density is allowed to vary in time and in space, which is considerably more realistic, and the question becomes: how does the flow evolve? We want the answer as a continuous quantity,

f = f(x, t)

describing not the movements of the single cars but the traffic flow as a whole. The Euler viewpoint is what makes this phrasing possible at all: $x$ is a position on the road and $t$ is a moment in time, and $f(x,t)$ is what an observer standing at $x$ would count at time $t$ .

What We Assume to Get There

Three simplifying assumptions carry the derivation, and it is worth naming them before they get used silently.

First, a reasonable time scale and length scale can be chosen in advance. The continuum picture only makes sense if “the density at $x$ ” means an average over a stretch long enough to contain many cars but short enough to still be local, and likewise for time. We assume such scales exist and have been fixed.

Second, and most consequentially, the steady-state relationship still holds:

f = f(\varrho) = v(\varrho) \cdot \varrho

A traffic model is quasi-stationary when the steady-state flow-density relationship $f = f(\varrho)$ is assumed to hold at every location and every instant, even while the density is changing. Equivalently: cars adapt to a new traffic situation without any time lag.

That is a strong assumption, and it is easiest to see what it costs by picturing where it fails. When a traffic light turns green, the density in front of the first car drops instantly, and the quasi-stationary model says every driver in the queue is already traveling at the velocity that the new density prescribes. Real drivers take a second to react and several more to accelerate; the queue empties as a wave of reaction times, not all at once. The situation right after a change of the traffic lights is therefore exactly what the assumption leaves out. Everything happening on a longer time scale than one driver’s reaction is fair game.

Third, a continuous description is possible at all — no trouble regarding smoothness, so density and flow may be treated as continuous and differentiable functions wherever we need to differentiate them. This one turns out to be the assumption traffic itself likes least, and later in this chapter we watch it break down on purpose.

With those in place, we can deduce continuous relationships.

A Conservation Law

The derivation follows the same recipe as the heat equation: set up a balance on a finite region, express the same change in two different ways, and let the region shrink to a point. Nothing about velocity is needed for it — the entire argument is bookkeeping about cars.

Take a closed test track running from $x = a$ to $x = b$ . Closed means there are no entrance or exit ramps anywhere along it, which is the conservation assumption: cars cannot appear or vanish in the middle. Traffic is unidirectional, entering at $a$ and leaving at $b$ . Three continuous quantities live on the track: the density $\varrho(x,t)$ at location $x$ at time $t$ , the flow $f(x,t)$ at location $x$ at time $t$ , and the number of cars on the track $n(t)$ at time $t$ .

The first relationship is immediate. The number of cars on the track at any moment is the density summed along it:

n(t) = \int_{x=a}^{x=b} \varrho(x,t)\,\mathrm{d}x

Differentiating with respect to time gives the rate at which that number changes. From here on we use subscript notation for partial derivatives — $\varrho_t$ means $\partial\varrho/\partial t$ , $f_x$ means $\partial f/\partial x$ — which keeps the expressions readable once several of them appear in one line:

\frac{\partial}{\partial t}n(t) = n_t(t) = \int_{x=a}^{x=b} \varrho_t(x,t)\,\mathrm{d}x

Now compute the same rate a second way, and this is where conservation enters. Since no car can join or leave in the middle of the track, the only things responsible for changes of $n(t)$ are the flows at the two ends: cars stream in at $a$ and out at $b$ . The count grows at exactly the rate at which the inflow exceeds the outflow,

\frac{\partial}{\partial t}n(t) = n_t(t) = f(a,t) - f(b,t) = -\int_{x=a}^{x=b} f_x(x,t)\,\mathrm{d}x

where the last equality is the fundamental theorem of calculus applied along the track: the difference of $f$ between the two ends is the integral of its spatial derivative, and the minus sign appears because the difference is taken in the order $a$ minus $b$ rather than $b$ minus $a$ .

So the same quantity $n_t(t)$ has been computed from two genuinely different vantage points. One observer walks the whole track and integrates how fast the density is changing at every point along it. Two other observers stand still at $a$ and at $b$ and do nothing but an access-and-escape balance on the cars crossing them. Neither observer sees what the other sees, and both are right.

Excursus: densities and sum quantities. The step from $\varrho(x,t)$ to $n(t)$ is an instance of a pattern worth naming, because it recurs everywhere a continuous description is used. Given a density $N(x,t)$ spread over a region, the corresponding sum quantity $M(t)$ is what you get by integrating it across the region,

M(t) = \int_{x=a}^{x=b} N(x,t)\,\mathrm{d}x

In the special case of a density that does not vary at all, the integral collapses to a product,

M(t) = M = N \cdot (b-a), \qquad M_t(t) = 0

which is the flat-sheet picture: a constant height over the interval, and an area that never changes. The general case replaces the flat sheet with a surface rippling over the interval, and $M(t)$ is the area under the slice at the current time — a number that now moves as the ripples do. Density is the local, pointwise quantity; the sum quantity is the global one, and integration is the bridge in one direction while differentiation is the bridge back.

The Traffic Equation

Setting the two expressions for $n_t(t)$ equal gives

\int_{x=a}^{x=b} \varrho_t(x,t)\,\mathrm{d}x = -\int_{x=a}^{x=b} f_x(x,t)\,\mathrm{d}x

or, collecting everything on one side,

\int_{x=a}^{x=b} \bigl(\varrho_t(x,t) + f_x(x,t)\bigr)\,\mathrm{d}x = 0

The decisive observation is about the scope of this statement. It holds for every point in time $t$ , and for every test track $[a,b]$ — nothing in the derivation used a particular choice of endpoints, only that the stretch between them had no ramps. An integral that vanishes over every conceivable interval is a much stronger statement than one that vanishes over some interval, and it forces the integrand itself to vanish everywhere.

Why “zero over every interval” forces the integrand to be zero

Suppose the integrand were nonzero somewhere, say positive at a point $x_0$ . Being continuous, it stays positive on some small interval around $x_0$ . Choose that interval as the test track. Integrating a strictly positive function over it produces a strictly positive number, contradicting the statement that the integral vanishes over every test track. The same argument runs with signs reversed if the integrand is negative somewhere. So it can be nonzero nowhere.

This is why continuity was assumed in the first place: without it, the integrand could be nonzero at isolated points and the integral would not notice.

Therefore, with sufficient differentiability,

\varrho_t(x,t) + f_x(x,t) = 0 \quad \forall x,t

and this can be written entirely in terms of the density. Since the quasi-stationary assumption gives $f = f(\varrho)$ , the spatial derivative $f_x$ unfolds by the chain rule into the derivative of the flow with respect to density times the derivative of the density with respect to position.

The traffic equation is the conservation law for cars written as a partial differential equation in the density,

\varrho_t(x,t) + f_x(x,t) = 0 \quad \forall x,t

or equivalently, using $f = f(\varrho)$ and the chain rule,

\varrho_t(x,t) + f_\varrho(\varrho(x,t)) \cdot \varrho_x(x,t) = 0 \quad \forall x,t

It is a nonlinear PDE — nonlinear because the coefficient $f_\varrho(\varrho(x,t))$ multiplying $\varrho_x$ depends on the unknown $\varrho$ itself.

Two things about it are worth fixing in mind before we start using it.

It contains nothing but the conservation of cars. No term in the equation creates density or destroys it, so a change of density is never a source of extra flow and a change of flow is never a source of extra density; the two are tied together purely by the balance, and whatever leaves one place arrives at another.

And it has exactly one unknown: the density. The flow does not need its own equation, because once $\varrho(x,t)$ is known everywhere the flow follows immediately from $f = f(\varrho)$ . That is the payoff of the quasi-stationary assumption — it turned a system in two unknown fields into a single equation in one.

Reading the Traffic Equation

Two remarks about the equation are worth making before we put it to work.

The first is that it is not a complete model on its own. The dependency $f = f(\varrho)$ is a modeling assumption and has to be supplied from outside, and every choice we made earlier is a legitimate candidate: the parabolic model, the cubic model, the power-law ansatz. The traffic equation says how density is transported; it says nothing at all about how drivers behave. Swap in a different $f(\varrho)$ and the same equation predicts different traffic.

The second is what kind of equation it is. Structurally the traffic equation is a quasilinear 1st-order PDE — it is linear in the derivatives $\varrho_t$ and $\varrho_x$ , but the coefficient in front of $\varrho_x$ depends on the unknown. In the language of the phenomena it produces, it is a simple example of a nonlinear hyperbolic wave equation, the family that describes general wave-propagation phenomena. That family membership is the formal reason why the analogies collected at the start of this chapter — shock waves, fire fronts, percolation — hold as tightly as they do: those systems are governed by relatives of this very equation.

In simple cases the traffic equation can even be solved analytically, by the method of characteristics — following curves through the $(x,t)$ plane along which the density stays constant, so that the PDE collapses into an ordinary differential equation along each curve. The whole of the next section is, in effect, an account of what those curves do and why they eventually collide.

Signal Velocity

The coefficient sitting in front of $\varrho_x$ has a name and a meaning, and it is the single most useful quantity in the chapter.

The signal velocity is the derivative of the flow with respect to the density,

\frac{\partial}{\partial \varrho}f = f_\varrho

It is the speed at which the information about a change in density — a disturbance — propagates through the traffic.

Two things about it want emphasizing straight away.

It really is a velocity, and the units say so. The flow is measured in cars/h and the density in cars/km, so the derivative of one by the other has the unit (cars/h) divided by (cars/km), which is km/h. Nothing about the definition mentions motion, but dimensional analysis leaves no choice about how to read it.

And it is not the car velocity $v$ . This is the point at which intuition needs correcting most, because the two are easy to conflate and they behave completely differently. Cars only ever drive forward, so $v \ge 0$ always. The signal velocity carries no such restriction: it can be positive, negative or zero. What propagates at $f_\varrho$ is not matter but news — the information that the density here has changed — and news can travel upstream against the cars carrying it.

The signal velocity is always at most the car velocity. This follows in one line from the product rule. Since $f(\varrho) = v(\varrho)\cdot\varrho$ ,

f_\varrho = v(\varrho) + \varrho \cdot v'(\varrho)

The density $\varrho$ is non-negative, and $v'(\varrho) \le 0$ because velocity decreases with density, so the second term is never positive and $f_\varrho \le v$ . Information about a disturbance therefore never overtakes the cars — it can at best keep pace with them, and generally falls behind or runs backward.

Walking the density from empty road to standstill shows what the sign of $f_\varrho$ means in practice, and the story it tells is not a comfortable one.

Empty road. Here $\varrho \to 0$ , so $v \to v_{\max}$ and $f \to 0$ , and the correction term $\varrho \cdot v'(\varrho)$ vanishes, leaving $f_\varrho = v$ . Signal and cars travel together at the same speed. Whatever disturbance a car causes — an unnecessary brake, a slow overtake — is carried downstream with that car and removed from the system along with it. The other participants never notice anything.

Increasing density and flow. The correction term grows, and disturbances start being removed slower than the cars that caused them. The gap between $f_\varrho$ and $v$ is exactly the amount by which the disturbance lags behind its author, and what lags behind gets handed to the drivers coming up from the rear. Following cars are now affected by something the car in front did.

At maximum flow. Here $f_\varrho = 0$ , by definition of the maximum: $\varrho_{\text{opt}}$ is where the fundamental diagram peaks and its derivative vanishes. The signal velocity is zero, so a disturbance does not travel at all — it stays exactly where it was created while the cars stream through it, and every single car that arrives runs into it. This is catastrophic, and it happens in the ideal case of maximum flow of all cases. The state we spent the whole steady-flow section identifying as optimal is precisely the state in which disturbances become permanent fixtures of the road.

Beyond it. As the density increases further the flow decreases again, the fundamental diagram slopes downward, and $f_\varrho$ turns negative. Information now goes backward, upstream, against the direction of travel. That is congestion, and for high densities the signal velocity becomes extremely negative: a jam front races back along the queue far faster than any car in it is moving forward, which is why the jam you are stuck in was created by something that happened kilometers ahead of you and minutes ago, and why it survives long after that something is gone.

Tangent and Secant on the Fundamental Diagram

All of this can be read off a single picture, and once seen it is hard to unsee. On the fundamental diagram $f(\varrho)$ , pick a density and look at two lines through the point on the curve above it.

The signal velocity is the slope of the tangent at that point. That is just what a derivative is: $f_\varrho$ is the rate at which $f$ changes as $\varrho$ changes, which is the steepness of the curve there.

The traffic velocity is the slope of the secant from the origin to that point. That one comes from the fundamental relationship. The secant from $(0,0)$ to $(\varrho, f(\varrho))$ has slope $f(\varrho)/\varrho$ , and since $f = \varrho\cdot v$ that ratio is exactly $v$ .

v(\varrho) = \frac{f(\varrho)}{\varrho} \quad \text{(secant)}, \qquad f_\varrho(\varrho) = \frac{\partial f}{\partial \varrho} \quad \text{(tangent)}

So the two velocities are two different slopes at the same point, and reading the diagram is now a matter of comparing them. Near $\varrho = 0$ the curve leaves the origin in a straight line, so the tangent and the secant coincide and $f_\varrho = v = v_{\max}$ . As the density grows the curve bends over, the tangent flattens faster than the secant does, and the gap between them opens up — that is the signal falling behind the cars. At the peak the tangent is horizontal while the secant still climbs, so $f_\varrho = 0$ while $v$ is still comfortably positive. Past the peak the tangent tips downward, and it tips further and further as $\varrho \to \varrho_{\max}$ until $f_\varrho \ll 0$ , while the secant merely flattens toward zero. The secant, being a chord from the origin to a point on a non-negative curve, can never have negative slope — which is the geometric version of the statement that cars never drive backward.

Taking sample densities of 65, 95 and 125 cars/km on a parabolic diagram makes the three regimes concrete: a gently positive tangent below the optimum, a gently negative one just above it, and a steeply negative one well into the congestion phase. All three secants stay positive and all three grow shallower from left to right.

The parabolic model, worked out

For the parabolic model the two slopes can be written down explicitly. Differentiating

f(\varrho) = v_{\max}\cdot\varrho\cdot\left(1 - \frac{\varrho}{\varrho_{\max}}\right)

gives the signal velocity

f_\varrho(\varrho) = v_{\max}\cdot\left(1 - \frac{2\varrho}{\varrho_{\max}}\right)

while the traffic velocity is the linearization we started from, $v(\varrho) = v_{\max}(1 - \varrho/\varrho_{\max})$ . Comparing the two, the signal velocity is the same straight line with twice the slope: it reaches zero at $\varrho_{\text{opt}} = \varrho_{\max}/2$ , where the traffic velocity is still $v_{\max}/2$ , and it bottoms out at $-v_{\max}$ on the completely jammed road, where the traffic velocity is zero.

So in this model, a jam front on a fully congested road travels backward exactly as fast as free traffic travels forward.

The End of a Tailback

The first phenomenon to explain with all this is the one every driver has met: the moment the traffic in front of you turns from moving to stopped, with no warning and no visible cause.

Simulate the traffic equation with the parabolic flow-density model on a stretch of about ten kilometers, over about ten minutes, and start it from a mean density $\bar\varrho > \varrho_{\text{opt}}$ — the road is already on the congested branch. Give the initial density a smooth sinusoidal profile, so nothing anywhere is sharp: just a gentle swell of slightly denser and slightly thinner traffic. What develops out of that perfectly smooth start is an end of a tailback: a density front that steepens as it goes, and that travels with the signal velocity $f_\varrho(\bar\varrho)$ against the driving direction.

The reason is that the different density levels along the front do not travel at the same speed. Look at the flank where the density is rising, take the sample values from the fundamental diagram, and read off the signal velocity at each:

at the peak density (say 125 cars/km), deep in the congestion phase, the signal velocity is highly negative, so this level races backward;
at the minimum density (say 65 cars/km), below the optimum, the signal velocity is slightly positive, so this level creeps slowly forward;
at $\varrho_{\text{opt}}$ itself the signal velocity is exactly zero, so that level does not move at all.

Every level of density is transported at its own speed, and on a rising flank the fast-moving high-density information sits ahead of the slow-moving low-density information in the direction of travel of the front. So the high levels catch up with the low ones. The flank gets steeper with every minute, and the smooth sine profile turns into something very different.

A density jump, also called a shock, is a discontinuity in the density profile that forms when the signal velocities on a flank cause density levels to overtake one another. Once formed, the jump propagates as a unit, moving with the signal velocity of the average density $f_\varrho(\bar\varrho)$ across it.

For the sample values above with $\bar\varrho = 95$ cars/km, that means the finished jump moves at $f_\varrho(95)$ , which is negative: the tailback end backs up along the road toward the cars that have not reached it yet.

This is where the smoothness assumption breaks — and it breaks on its own. The derivation of the traffic equation needed $\varrho$ to be differentiable, and the initial profile was as smooth as could be asked for. Nothing external tore it. The equation’s own dynamics steepened the flank until the slope became infinite, and past that moment there is no differentiable solution left to speak of.

That is not a flaw in the model, it is the model reporting something true: a jam front really is a near-discontinuity, a few car lengths wide on a road ten kilometers long. Handling what happens afterward is exactly what the theory of shock waves in nonlinear hyperbolic equations is for, and it is why the congestion analogy with sonic booms and pressure fronts is a mathematical identity rather than a metaphor.

Driving Off at a Traffic Light

The signal velocity is not an abstraction; we can experience it, or rather suffer from it, at every red light.

Standing traffic at a light is at maximum density, and the moment the light turns green that density has to be resolved. It does not resolve instantly or all at once: it resolves over time, front car first, as an abrupt density drop that works its way backward through the queue. The reason it is so slow is exactly the signal velocity. Maximum density means maximum negative signal velocity, so the news “you may now drive” travels backward through the queue at the slowest rate the model allows — which is why the tenth car in line waits so long for a signal that was given all at once.

This is also where controlling has real leverage. The intervention possibilities are the location of the traffic lights, the synchronization of consecutive traffic lights so that a platoon released by one arrives at the next as it turns green, and tempo limitations that keep the density in between from climbing back to where the resolution is slow again.

Formation and Dissolution of Congestions

The second phenomenon is the construction-site bottleneck, and it can be worked through with numbers.

Take a two-lane road under the parabolic model, running at maximum flow at time $t = t_0$ :

t = t_0, \qquad f_{t_0} = \frac{v_{\max} \cdot \varrho_{\max}}{4}

At time $t = t_1$ one lane is blocked — a construction area, an accident, anything. The whole capacity question is now about how the two lanes’ worth of traffic merges into one, and there are two very different answers.

With optimal merging, the maximum capacity and flow of the free lane are preserved. Losing a lane bisects the maximum density the road can hold, and since the peak flow in the parabolic model is proportional to $\varrho_{\max}$ , bisecting the density bisects the formerly maximal flow:

t = t_1, \qquad f_{t_1} = \frac{v_{\max} \cdot \varrho_{\max}}{8}

Half the flow for half the lanes. That is the best outcome available, and it is what a zip merge is trying to achieve.

With usual merging, everybody wants to change lane as fast as possible. All the cars from both lanes crowd into the remaining one at once, so the density of that lane becomes twice as high, which makes it maximal. At maximum density the velocity is zero, and the result is a dead-lock: nothing moves at all, and the flow that was supposed to be halved is instead destroyed. The difference between the two outcomes is entirely behavioral, and it is the clearest case in the chapter of individual decisions that look locally rational adding up to a collective disaster.

When the second lane clears up again, the process runs in reverse and produces a density jump downward — the same discontinuity as the tailback end, with the sign flipped.

A Density Drop Spreads Out

The tailback showed a rising flank steepening into a jump. The mirror image is a falling flank, and it does the opposite.

Consider a density decrease along a road section — for instance the point where a speed limit is lifted. Ahead of that point the traffic is free to spread out, so the density drops from a high plateau to a low one, and correspondingly the velocity rises from a low plateau to a high one. Reading the signal velocity at the two ends of the transition, $f_\varrho$ is small on the high-density side and large on the low-density side, since $f_\varrho$ decreases as density increases.

The traffic situation then moves along with $f_\varrho$ , and because the two ends of the transition move at different speeds, comparing the profile at $t = 0$ with the profile at $t = T$ shows the transition region has widened. The low-density end runs ahead quickly while the high-density end lags, and the gap between them stretches: the intersection area becomes larger because the signal moves slower in the case of high density.

So the two flanks behave in exactly opposite ways, and the sign of the density gradient decides which. A flank where density rises in the driving direction compresses until it becomes a shock, sharp and violent and permanent. A flank where density falls in the driving direction spreads out, softening and lengthening until nothing sharp is left of it. Jams form abruptly and dissolve gradually, and that asymmetry — the one every driver complains about — is not psychology. It is a consequence of the fundamental diagram curving downward, so that denser traffic always carries its information more slowly than thinner traffic does.