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Highway Traffic Two: Collective Behavior

Traffic rates as one of the more annoying experiences of modern culture. Highways have provided some relief from traditional traffic congestion, i.e. that occurring at stop signs and traffic control signals, but highways themselves have spawned new types of congestion.

This article explores that topic, i.e. highway traffic and congestion. This is the second of a two part series.

The first of the series (titled “Highway Traffic One: Collision Avoidance”) delved into one traffic characteristic, namely the maximum traffic flow a highway can sustain at different speeds. We focused on two basic, but fairly universal, determinants of driver behavior. A characteristic driver desires to go as fast as possible while 1) avoiding a ticket and 2) avoiding a rear end collision.

With those determinants, and a little math danielstampa and physics, we built a quantitative model. That model gave a “required following distance” and a “maximum sustainable traffic flow” at each of a number of speeds.

That modeling revealed a paradox. As average speed increased, the sustainable traffic flow also increased. In other words, our model indicated that a highway can sustain a higher traffic flow at moderate speeds (30 to 50 miles an hour) than can be sustained at the typical “heavy” traffic speeds (zero to 20 miles an hour).

Why then does traffic flow drop to the low range under extreme congestion, if the low range provides the worst flow? What forces traffic to drop from highway speeds, i.e. 60 miles an hour, down to a standstill, if a highway’s maximum flow occurs in the 30 to 50 miles an hour range? We likely experience this frequently, particularly as traffic merges at entrance ramps.

The key lies in the dynamic nature of merging traffic. The first article, on maximum sustainable traffic flow, dealt with static, aka constant, conditions. Vehicles traveled at the same speeds, and drivers maintained the same distances between cars. We asked one question – at those constant conditions, what following distance would the characteristic driver set?

Entrance ramps create dynamic, aka, changing, conditions. As cars merge, following distances change, drivers slow and accelerate, and different vehicles have different speeds. These dynamic conditions can push traffic right past the speeds with maximum flow, down to the all too typical highway traffic crawl.

So let’s focus on that phenomenon, of how entrance ramps impact traffic flow. We will do that first qualitatively, just describing what happens, then quantitatively with a bit of mathematical modeling. In doing so, we will obtain a better sense of how the dynamics of entrance ramp merging cause traffic flow to degenerate to such low, and less than theoretically optimum, speeds.

Entrance Ramps: Qualitative Look

Imagine traffic flowing at 60 miles an hour, with cars spaced on average 200 feet apart, with our highway two lanes wide in each direction. From the first part of this series, we found that the characteristic driver had a required following distance at 60 miles an hour, of about 150 feet. Thus absent any disturbances to traffic, our highway can sustain traffic at 60 miles an hour, given the 200 foot spacing, and our drivers should comfortably maintain their highway speed.

Imagine now entrance ramps. We will have two ramps, one entrance ramp into the left lane (not common but certainly occurs) and a second entrance ramp into the right lane.

Now a set of two cars enters (one from each entrance ramp). As they merge into traffic, these entering cars cut the following distances, front-to-front, of the trailing cars behind them on the highway, down to 100 feet. The entering cars in many, if not most, cases are traveling at a speed only a fraction of that of the main highway flow.

As noted above, our modeling (in the first article) calculated a required following distance of just over 150 feet at 60 miles an hour. Given our model reflects how drivers think in real traffic (i.e. the required following distance indicates a driver’s judgment of what is required to avoid a rear end collision), the driver of the directly trailing cars will slow down to increase the following distance. This will be a quick deceleration, since not only will the following distance be insufficient, but the trailing drivers will find themselves quickly closing in on the slower-traveling entering cars.

What occurs then? As this first set of trailing cars slow, a second or so later the next trailing cars slow, and another second later the third trailing cars slow. This sequence of slowing creates a congestion pulse that ripples rearward as each subsequent set of trailing cars slows due to the slowing of the cars in front of them. Now if only two cars are inserted (i.e. one in each of the two lanes), the cars will all sequentially accelerate back up to 60 miles an hour, and the merging causes just a transient backward ripple.

But what if another set of two cars enters behind our first set of trailing cars? The first set of entering two cars creates a backward ripple that slowed the main traffic. This second set of entering cars inserts itself into the ripple, further cutting traffic speeds.

We can see where this is going. What if a third set of cars enters? This third set further cuts down vehicle speeds.

So while the entry of one set of cars causes a transient ripple, we can see that the continual entry of cars increasingly slows traffic. Traffic quickly reaches high congestion, and speed descends downward.

This scenario highlights what causes traffic speeds and flow to descend from a stable level at 60 miles an hour, right past the maximum flow range (i.e. between 30 and 50 miles an hour, where a highway can maintain the highest flows), down to bumper-to-bumper. The cause lies in the sudden and unavoidable discontinuity at the merge point. At that point, merging traffic abruptly cuts following distances, which triggers an abrupt slowing of traffic. Vehicle speeds decrease right past the speed range of maximum flow. Traffic flow can not stabilize in the maximum range since the merge dynamics push speeds down so quickly.

So while the highway overall, if vehicles were all at an ideal speed and separation, could handle more traffic, the abrupt changes at the merge point prevent traffic from settling in at those ideal conditions.

But do entrance ramps present us with an all or nothing situation? For a given set of conditions, will the merging at entrance ramps always produce the same level of slowing and congestion? Or rather can driver behavior improve (or maybe exacerbate) the vehicle speeds and traffic flow at entrance ramps?

Traffic Merging: Impact of Driver Behavior

We have certainly seen, or directly experienced, how events unfold when a vehicle runs out of “runway” on an entrance ramp, and gets stuck, stopped, at the end of the ramp, with no further room to accelerate. In heavy traffic, the driver will find no gaps for entry. Having little choice, the driver will just jut into traffic, at a slow speed, cutting off traffic, and causing oncoming vehicles to slow, in cases severely and suddenly.

But if the driver wasn’t entering from a stop, the oncoming vehicles wouldn’t need to slow so much and so quickly. The faster entry would allow traffic to maintain a higher speed. So from this example we see that driver behavior can affect, possibly significantly, highway congestion.

So let’s look at this. While many different driver behaviors can impact the level of congestion at merge points, we will focus on three major ones. They are:

Speed matching
Velocity priority
Smoothness
Speed matching picks up on the example just mentioned, a vehicle stuck at the end of an entrance ramp. As that stuck car enters, that merging not only cuts the following distance of the vehicle right behind in the main traffic flow, but the low speed of the merging car causes the following vehicle to close quickly. That following car must slow sufficiently to compensate both for the reduction in following distance and the subsequent closing due to the speed mismatch.

If the merging car can match the speed of the main traffic flow, that merging still cuts following distances, but the speed matching means the following car does not close any further. The following car can maintain a higher speed.

Velocity priority relates to which of two variables merging and trailing drivers react more strongly, specifically velocity difference (relative to the leading car) verses following distance (again relative to the leading car).

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