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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  danielstampa 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:

 

 

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