Traffic.

It would be a rare individual who has not experienced this artifact of modern culture. Regardless of one’s locale or age, traffic likely ranks among one’s more, if not most, annoying experiences.

The advent of the superhighway several decades ago offered prospective relief from traffic. And to a great extent, superhighways, through elimination of traffic signals, creation of multiple lanes, introduction of acceleration on-ramps, removal of steep grades, smoothing of sharp curves, separation of opposing directions of traffic, and other design steps, have succeeded.

But not completely. Slow traffic still occurs, too frequently, on highways.

Why? We likely have an intuitive feel for why, but let’s dive a bit deeper and use some precision (aka mathematics, though not too complex) to understand the characteristics of traffic. To keep our discussion manageable, we will focus on the road type already mentioned, the superhighway.

We will cover this in two pieces. This article, the first piece, will focus on speed and traffic flow, specifically how much traffic can a highway handle. The second article (titled “*Highway Traffic Two: Collective Behavior*“) will cover how congestion occurs when a highway gets too much traffic.

*Definitions, Terms and Calculation Examples*

We need to start with a few basic terms and definitions. From our experience (and/or driver’s education class), we likely already have a familiarity with these.

- Speed – how fast we are going, normally stated in miles per hour, but here we also need feet per second (i.e. about 1.5 times miles per hour).
- Stopping distance – the distance required to stop a car. Stopping distance consists of two parts, first the reaction time for the driver to begin depressing the brake and second the braking distance the car travels after the brake is engaged.
- Traffic Flow – the rate cars pass a set point. For this discussion, we will express that in vehicles passing per hour, per lane.
- Acceleration/Deceleration – the degree to which we are increasing or decreasing our speed. Gravity accelerates an object about 32 feet per second per second, and full emergency braking with modern anti-locking brakes can just about create up to a one “g” deceleration, depending on the tire and road condition.

We can do some math using these items.

Let’s assume, early in the morning, with traffic light to moderate, cars are moving on the local superhighway at 65 miles per hour, spaced on average 300 feet front-to-front (i.e. from the front bumper of any given car to the front bumper of the directly following car). At 65 miles per hour, that is (about) 100 feet per second. With the cars at 300 feet of separation, we divide the 100 feet per second into the 300 feet of separation, to determine that a car passes (in each lane) about every three seconds. With 3600 seconds per hour, and three seconds per car, we divide the time interval of three seconds into the 3600 seconds, and arrive at a traffic flow of 1200 cars per hour per lane.

This calculation of flow, based on speed and separation, stands as a fairly fundamental relation, so let’s do another other example. In heavy traffic, speeds might be down to 10 miles per hour, with an average front-to-front distance of 45 feet. Now 10 miles per hour equates to 15 feet a second, and with 45 foot spacing, we have a car every three seconds. That again gives a flow of 1200 cars per hour per lane.

Of interest, the flow for the “light” early morning traffic and the “heavy” rush hour traffic equal. So “heavy” traffic here more accurately represents “slow” traffic, since from a traffic flow viewpoint, our two examples give the same number. Thus neither is actually “heavy” or “light” relative to each other.

Deceleration gets a bit trickier, but not too much so. Let’s take two cars, travelling 65 mile per hour, separated by some distance (not critical yet). And the first car slows at a half “g,” or about 15 feet per second per second. The trailing driver takes a second to react, before starting to slow. In that second, the trailing car closes on the leading car by 7.5 feet.

How do we calculate that?

When the lead car starts to slow, both cars are traveling at 100 feet per second. With a deceleration of 15 feet per second per second, the lead car, in the one second of reaction time, slows to 85 feet per second. Given a smooth deceleration, the average speed of the lead car during that second was the average of the initial speed of 100 and the speed after one second of deceleration, or 85 feet per second. That averages to 92.5 feet per second. The trailing car traveled 100 feet during the reaction time, while the lead car traveled only 92.5 feet. This gives a closing distance of the trailing car on the lead car at 7.5 feet.

If the trailing car takes two second to react, the trailing car closes 30 feet in the two seconds of reaction time, i.e. not twice the distance but four times the distance. This occurs because the lead car slows to 70 feet per second in the two seconds. The lead car travels at an average of 85 feet per second (the average of 100 at the beginning and 70 at the end of two seconds), or 170 feet across two seconds. The lead car continued at 100 feet per second for two seconds, traveling 200 feet, bringing it 30 feet closer to the lead car.

You might be comparing these closing differences to the standard “reaction time” diagrams from driver’s education. Those diagrams will show much larger distances traveled during the driver’s reaction time. However, that situation differs in an important factor – those reaction times relate to a stationary object. For example, relative to a stationary object, a one second reaction time at 65 miles per hour produces a closing distance of 100 feet, not the 7.5 seconds above for two moving cars.

Why do we having two moving cars in our examples? On the highway, essentially all the time, the vehicle in front is moving, and thus closing distances depend not on the absolute speed of our car, but our speed relative to the lead cars in front of us.

__Maximum Sustainable Flow __

Drivers aim to travel as fast as (and in cases faster than) legally allowed. Highway engineers aim to provide for the greatest possible flow for the construction dollars spent.

Let’s investigate this then, i.e. the relation of speed and flow, given that both are critical goals. We will base our investigation on fairly ideal conditions and perform calculations with a fairly basic model. Though we have a simplified approach, our investigation will still contain sufficient descriptive power to highlight key traffic characteristics.

What are our conditions? We want them relatively ideal. So the weather is clear; the drivers travel at a uniform speed; no construction or other traffic constrictions are present; no entrance and exit ramps exist; minimal lane switching occurs; no trucks are present. These are ideal indeed.

How will we model traffic behavior? Given our ideal conditions, driver psychology becomes a main, if not the main, determinant of traffic dynamics. And what motivates our characteristic driver? Most drivers will seek to travel as fast as reasonably possible. So then what does reasonably mean? Reasonably, for the mainstream driver, signifies 1) avoiding a collision and 2) avoiding a ticket. We will translate those two motivations into two actions, specifically our mainstream driver, for our model, will 1) maintain an adequate following distance from the leading car to stop before impacting that car and 2) will travel at a maximum speed of the speed limit plus five miles an hour.

This does leave out here several important driver motivations. For example, we exclude efforts of aggressive drivers to speed the leading car through tailgating; we throw out road rage tactics; we eliminate drivers who either due to too much caution, or due to vehicle limitation, will not or can not maintain the speed limit plus five.

We also, on balance, exclude driver efforts to prevent cars in adjoining lanes from moving over in front of them. We have seen this in actual traffic, and may have done this ourselves. Drivers will tighten the distance to the vehicle in front, or take other actions, to foil attempts of other drivers to change lanes into the space in front of them. While not uncommon in real traffic, our simplified model assumes all vehicles travel at the same speed, so limited motivation exists for lane switching, and thus we will assume limited motivation to block lane switching.

With these ideal, but still informative, assumptions, how do we now calculate the maximum flow for a given speed? Very simply, at a given speed limit, we can increase the flow as long as our drivers can maintain a desired reasonable following distance (i.e. large enough to avoid a collision) while traveling at the speed limit plus five.

So we want a reasonable following distance to avoid a collision. And if we are the drivers, what do we – intuitively, almost subconsciously – consider and calculate to accomplish this? Four things, I would offer:

- Reaction time, i.e. how long we take to start braking after we see a need to slow
- Lead car deceleration rate, i.e. how fast the car in front of us slows
- Trailing car deceleration rate, i.e. how fast we judge we can slow
- Safety margin, i.e. how much extra distance do we want “just in case”

While this list might appear complex and intricate, drivers compute these variables intuitively and holistically. Though most individuals do not study calculus, evolution has provided mankind an innate ability to instinctively perform calculus-like time/distance/speed/acceleration calculations. Eons ago, mankind needed to hunt to survive, and neither man nor beast can hunt successfully absent an intuitive, split-second ability to perform motion calculations. So our ability to drive, as well as do many other activities involving complex motion (sports being a main example) could be said to be due to our ancestors need to eat.

Note we have abandoned the text book recommendations of a following distance of three seconds. That is nice, but if you recall our earlier calculation, at 65 miles an hour, a three second following distance equates to a 300 foot separation, i.e. a football field. That just doesn’t happen. Few maintain such a great distance as traffic volume increases, even at 65 miles an hour.

So we have our basic behavioral considerations for following distance, built on the simple and understandable principle that drivers prefer not to hit the vehicle in front of them. To do some math, we need to convert these qualitative considerations to explicit quantitative inputs. We will use the following assumptions:

- A driver reaction time of 1.5 seconds
- A maximum lead car deceleration rate of one-half “g”, when at 60 miles an hour
- A trailing car deceleration rate slightly faster, specifically 1 foot/second/second faster
- A minimum safety margin of 10 feet at 10 miles an hour
- A scaling factor that increases quantities as speed increases but less than proportional

Let’s review briefly the logic of these assumptions.

A * reaction time* of 1.5 seconds may be generous (our standard braking distance charts generally show a second or sometimes less). However, in highway traffic, when vehicles are traveling at steady state, the trailing driver must not only see the brake lights of the lead car, but also take a split second to determine the slowing rate.

A * lead car deceleration rate* of one-half “g” is about two-thirds to half of a full braking emergency stop (aka full brake pedal depression to the point of skidding or anti-lock brake engagement). In 99% plus of the time on highways, cars do not undergo full braking stops. So for good or bad, human psychology generally discounts the very low probability events (in this case a full emergency stop of the lead car) and thus our characteristic driver does not base following distance on a full emergency stop by the leading car, but rather on a more gradual slowing.

That we * can stop faster than the lead car* is achievable, given our expectation and assumption that the lead car won’t, and typically doesn’t, go into a full braking stop. Note here a subtle interaction. If we cascaded this assumption, i.e. that a driver can stop one foot per second per second faster than the preceding car, then by 10 to 15 cars back (if each driver stopped faster than their leading car) the deceleration would exceed one “g.” The subtle interaction involves the drivers of these subsequent cars (i.e. third car and beyond) reacting to the stopping of not just the car directly in front of them but also to the stopping of the cars two and three (or more) ahead of them. So our assumption of our trailing car stopping faster than the leading car holds only for the driver directly behind the first car braking.

A * minimum safety margin* provides for contingencies and comfort; we wouldn’t want to plan for our stopping to put us just inches from the bumper of the leading car. If we had that plan, little glitches (we happen to be glancing into a mirror at the trailing car; we are distracted by the passenger beside us dropping their whatever) would send us into a collision. So we add a buffer distance.

We now come to the * scaling factor*, i.e. how to ratio various factors for different speeds. Say we have an intuitive safety margin of 10 feet, at 10 miles an hour, i.e. we want a following distance sufficient so that in the average situation we stop 10 feet behind the lead car at 10 miles an hour. What safety margin do we judge we need at 60 miles an hour?

Well, keeping the safety margin constant at 10 feet (measured from rear bumper of the leading car to our front bumper) seems inadequate. At 60 miles an hour, we travel 10 feet in a tenth of a second. But would we scale up proportionately? Would we plan in the average situation to stop 60 feet back (six times the 10 feet at 10 miles an hour)? Likely not. Two car lengths, about 30 to 35 feet, feels about enough. So we scale up less than proportionately.

Now we run the model, on a computer. This model takes our assumptions, and computes for different speeds the required following distances, and corresponding traffic flows. Let’s see an example situation, e.g. 40 miles an hour, equal to 60 feet a second. For this example, we will have the lead car brake for five seconds, at a deceleration rate of 12 feet per second per second. Where does the 12 rate come from? At 40 miles an hour, the model scales the half “g” (16 feet per second per second) deceleration at 60 miles an hour down to about 12 feet/sec/sec.

So, with all these assumptions and inputs, we run the model and receive the following output.

- We close in on the leading car by 13 feet during our 1.5 second reaction time
- Our car closes another 55 feet for the 3.5 seconds we both are braking
- We brake a second further to slow to the speed of the leading car, closing 8 more feet
- We end up at our desired safety margin of 32 feet when both cars stop braking

We total these piece parts (i.e. 13 + 55 + 8 + 32) to obtain a required following distance of 108 feet, measured from the back of the leading car to the front of our car. Now for traffic flow calculations we need to add in the length of the leading car. We will assume that to be 15 feet. The resulting front-to-front required following distance becomes 123 feet.

As mentioned before, this math simply represents in numbers the result of what a driver determines intuitively. Drivers know a lag occurs between when the leading car in front of them begins stopping, and when they start stopping. They also know that when they begin stopping, the lead car has already slowed to a lower speed, while they are still at the original speed. From experience and innate abilities, they mentally compute a following distance to compensate for the reaction time lag, and the slowed speed of the lead car. We have split that intuition into mathematical piece parts, but that does not imply real drivers compute following distances this way.

We now calculate the traffic flow. We divide the required following distance by our speed (i.e. 123 feet divided by 60 feet a second) to find a car spacing of just about two seconds. Our maximum sustainable traffic flow becomes 1750 cars an hour per lane, calculated by dividing 3600 seconds in an hour by our spacing of just over two seconds.

Now let’s look at the results for a range of speeds. For each speed, the list below gives the required following distance and the maximum sustainable traffic flow.

Speed Limit plus Five Required Following Distance Maximum Flow

(in mph) (front-to-front, in feet) (cars/hr/lane)

10 45 1175

20 73 1475

30 99 1625

40 123 1750

50 139 1925

60 153 2100

70 167 2250

70 if 45 8200

70 if 315 1175

Let’s get a conceptual sense of these results. As the average speed increases, the maximum sustainable flow, assuming our ideal conditions, also increases. Note, consistent with our scaling assumption, that the maximum flow does not increase in proportion to the increase in speed, i.e. at 50, 60, and 70 miles per hour we do not achieve flows five, six and seven times those at 10 miles per hour. We achieve a lower multiple.

The last two entries give a perspective on the scaling of flow with speed. If the required following distance increased proportional to speed (so that at 70 that distance was seven times the 45 feet at 10 miles an hour), then the required following distance increases to 315 feet. We already mentioned that from our own driving or just observation, as traffic increases, in real life drivers do not maintain a football field following distance at 60 and 70 miles an hour. Alternately, if the required following distance didn’t increase at all with speed (so that at 70 that distance remained at 45 feet), then the front-to-front spacing distance becomes unnerving and unsafe, i.e. beyond the comfort zone, and most importantly skill, of most drivers to avoid a rear end collision.

Our model gives a flow within these two extremes, i.e. flow increases with speed, but less than a proportional scale up.

__Perturbations__

Our discussion above assumes ideal conditions and uniform driver behavior. That simplification highlighted how a fairly universal driver motivation (i.e. drive quickly but leave a reasonable following distance) influences the traffic a highway can sustain at different speeds.

But actual traffic conditions are not ideal and driver behavior is not uniform. How do actual traffic and road conditions typically deviate from our model above?

- Driver variability – Drivers will inherently travel at speeds above and below the average, and at following distances more or less than our “typical” driver above.
- Maximum driving speeds – Even absent trucks, a subset of vehicles/drivers (fully loaded vans, mechanically deficient cars, risk averse drivers) will travel at a maximum speed less than the speed limit plus five.
- Road variability – even absent bad weather and exit/entrance ramps, and even with the best design, highways have inclines, curves, cross-winds, sun conditions, bridge abutments, worn road surfaces, and so on.

So even setting aside trucks (always present) and bad weather (that when present becomes a large determinant of traffic flow), significant deviations exist from ideal and these will decrease flow. We experience this consistently. Drivers traveling faster the average will change lanes to pass slower drivers, and their lane changing will often trigger slowing in the lane into which they are moving. Two inherently slower drivers who align side-by-side will create a partial block. Sun just over the horizon in line with traffic direction will cause drivers to slow, as will inclines. And so on.

Critically, these deviations impact flow at high speeds more than slow speeds. 40 miles an hour is below almost any driver’s maximum speed, while 70 may be above a sizeable percentage. At 20 miles an hour, drivers may differ a car length on their required following distances, but at 60 miles an hour, drivers may differ by 100 feet. A bit of road surface wear means nothing at 10 miles an hour, but may be jarring at 50 miles an hour.

How does this impact the relation of speed to maximum sustainable flow? In general, while these deviations from ideal decrease the maximum sustainable flow at all speeds, they impact maximum flow a much greater percentage at the upper speed (50 miles and hour and more).

In actual traffic, then, unlike the ideal model, maximum flow reaches some peak and then decreases with increasing average speed.

__Wrap Up__

To the degree this caught your interest, as you drive in the future, observe the separation, i.e. following distances, of free-flowing highway traffic. To the degree our modeling here reflects essential elements of real traffic, we will find that freely flowing traffic exhibits a certain minimum spacing of cars.

Now certainly the maximum spacing is unlimited, i.e. at 2 AM cars may be a thousand feet apart. But as traffic flow increases, the spacing will diminish, and in our observations, we should notice that the spacing does not drop to less than a certain amount, with that amount varying with average speed.

And since we drive in real traffic, not under our ideal conditions here, as traffic flow increases, perturbations – vehicles traveling slower or faster than average, slight inclines in the road, sun directly into traffic, drivers changing lanes, and so on – constantly arise. Ripples crop up in the traffic flow, creating sporadic and even continuing congestion, even without trucks, bad weather, on ramps or lane constrictions.

But perturbations serve as only a secondary trigger of congestion. Again based on our collective driving experience, we can point to a primary source of congestion, and that is the merging of traffic at entrance ramps. The second article will move to that subject, congestion from entrance ramp merging.