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 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).

Consider two different merging drivers, both entering at slightly less than the speed of the main traffic flow.   One driver focuses more closely on the velocity difference.  Since traveling more slowly than the lead car, this one driver accelerates slightly upon entering the highway, increasing speed to that of the main flow, while letting the slight temporary speed difference build an increased following distance.

The second driver reacts, alternately, to the short following distance.  Since that distance has dropped well below the required distance, this driver, instead of accelerating, slows down to immediately lengthen the following distance.

We can clearly see the differing impact.  The first driver, by accelerating, keeps traffic moving, while the second driver, by slowing, triggers the following cars to slow.

Note however, velocity priority may not always be best.  If merging cars enter at very low speed, then a velocity priority causes trailing cars to slow to that low speed, instead of gradually compressing following distances to maintain speed.  So one approach does not fit all situations.

Smoothness means just that, how gradually, or alternately how abruptly, a driver responds to changing conditions.

At first look, one might conclude that fairly quick reactions would allow traffic to flow faster.  However, faster reactions can turn out to be counter-productive.  Why?  Strong, quick responses can cause a driver to over shoot their target for speed or following distance, or both.

An example helps.  Let’s say a driver sees that at a given point their following distance exceeds what they judge needed.  They accelerate quickly and strongly.  But in congestion conditions change often, and as the driver accelerates the leading car slows.  The quick acceleration, combined with the slowing of the car in front, causes the trailing car to close too quickly on the leading car, creating too short a following distance.  The trailing car driver now brakes quickly and strongly.  We can see where this leads.   The strong reactions cause continual speeding and slowing as the driver over shoots the speed and following distance needed.

Entrance Ramps:  Quantitative Look

Let’s model what we have just described.

Now in the first article, we assumed, and could assume given the goal of the modeling, that every driver traveled at the same speed and maintained the same following distance.  We could thus model one car, since that one car could represent all the cars.

Here, for entrance ramps, we decidedly can not assume conditions remain similar across vehicles and across time.  The merging cars trigger continual changes in vehicle speeds, distances and acceleration/deceleration.  And it is these very changes we desire to study and understand.

Our model must thus track each car, at each instant, for multiple variables, no small task.  To keep the model understandable, then, we will focus on the core interaction, the merging, and have just a one lane entrance ramp merging into a one lane highway.  True, actual entrance ramps can have more than one lane, and actual highways almost always have more than one lane.  The extra lanes, however, primarily add a different phenomenon, lane switching, which does influence merging impacts, but in a secondary way.  Our simplified one lane highway and one lane entrance ramp, while not all encompassing, will still provide sufficient scope to explore our focus, entrance ramp merging.

So how will we start?  We need some initial conditions, simple enough to comprehend but representative of actual traffic.  We will thus start the main traffic speed at 60 miles an hour, with 200 foot front-to-front distances.  The model will insert a merging car between each of the cars in the main traffic flow, at a speed at a percent (that we can vary) of the main highway traffic.  For “required following distance,” we will use the equations and relations from the modeling in the first article.  The model will have 160 vehicles, 80 on the main highway and 80 merging sequentially.

We now run the model, stepping sequential through time increments of about three-quarters of a second (with that increment representing how often a driver can adjust to changing conditions).  For each time increment, the model calculates each vehicle’s speed and location, as well each driver’s reaction to current conditions. 

The driver’s reaction consists of how much they accelerate, or brake.  Critically, we can vary that reaction, since as noted above it is just that driver reaction we want to study.  So the model permits variation in the “velocity priority” from very low to very high, and in the “smoothness” from very mellow to very aggressive.  And as just noted, the model permits variation in the entry speed of merging.

What will the model tell us?  Many (many) traffic characteristics, but we will focus on four key items.  These four items relate closely to the frustration level drivers feel in highway congestion:

  • Lost distance, i.e. how much farther back does the 160th car fall due to congestion
  • Average minimum speed, i.e. what is the lowest speed on average for each vehicle
  • Acceleration intensity, i.e. how much acceleration/braking occurs
  • Time at less than 40 miles an hour, i.e. how much time across all the cars in the model

Let’s take a sample run.  Merging cars will enter at 80% of the highway speed, and drivers will exhibit a moderate priority on velocity, and a moderate smoothness.  We run the model for ten minutes (model time, so about 800 time increments; the model itself requires only a second real time.)  We find the following:

  • The 160th car losses 18,600 feet, over three and a half miles
  • Each driver accelerates or brakes quickly, on average for about 86 seconds
  • On average, each driver experiences a slowing, at least once, to 20 miles an hour
  • Drivers collectively experience 3 hours at 40 miles an hour or lower

Some comparison points will help.  In the ten minutes, at 60 miles an hour, absent the congestion, a vehicle will travel 10 miles, or about 52,800 feet.  So the 160th car lost about a third of the normal distance, and cars beyond that (not modeled) will lose more.  Accelerate or brake quickly means to do so at greater than 50% of the maximum braking or acceleration allowed in the model, and the 86 seconds should be compared to the total 600 seconds of the model run.

Could the drivers do worse?  Yes, with a lower priority on velocity, but aggressive acceleration and braking, still with the 80% merging speed, we find the following:

  • The 160th car losses 31,500 feet, almost six miles
  • Each driver accelerates or brakes quickly, on average for about 125 seconds
  • On average, each driver experiences a slowing, at least once, to 8 miles an hour
  • Drivers collectively experience just over 5 hours at 40 miles an hour or lower

Can then do better?  Yes, with a strong priority on velocity, but gradual acceleration and braking, still with the 80% merging speed, we find the following:

  • The 160th car losses only 13,000 feet, a bit over two miles
  • Each driver accelerates or brakes quickly, on average for only about 18 seconds
  • On average, each driver experiences a slowing, at least once, to 28 miles an hour
  • Drivers collectively experience about 2 hours 20 minutes at 40 miles an hour or lower

These results reveal amazing differences in congestion severity for different collective driver behaviors.   Thus, with the advantage of a relatively favorable merge speed (i.e. the 80% factor), driver behavior, specifically attention to velocity differences and gradual acceleration/braking, can reduce congestion.

What if the situation involves unfavorable merge speeds, for example a merge speed of only 30% of the traffic flow?  While driver behavior can ease congestion some, under any driver behavior congestion remains high.

  • The 160th car always losses at least 25,900 feet, almost 5 miles
  • Traffic always slows to 11 miles an hour or less for at least one point, sometimes zero
  • The collective delay always reaches four hours or more

Slow merge speed scuttles traffic flow so negatively that no particular set of driver responses can prevent traffic from descending, at some point, to a crawl.  So if merging drivers practice “poor” behavior, i.e. slow merge speeds, driver behavior in the main traffic flow can not  significantly offset that.

In contrast, as seen above, if merging drivers achieve a good merge speed (the 80% rates as good, in fact almost as good as a 100% merge speed) driver behavior in the main flow greatly impacts the level of congestion.

Practical Steps

While possibly interesting (i.e. the relation of driver behavior to congestion), can anything actually be done to alter or align that driver behavior to relieve congestion?  Is there hope?  The answer is yes, traffic engineers, to a degree, can coax drivers in ways to improve traffic flow.

Entrance Ramps Signal Controls – Given that merging traffic in general, and poor merge speed in particular, contribute greatly to congestion, controlling merging via traffic signals can partially reduce congestion.

We likely have seen such traffic signals.  These signals don’t stop traffic like a typical traffic light, but rather meter it, spacing merging cars or groups of cars several seconds apart.  This gives each car sufficient time and room to accelerate to highway speeds (i.e. getting to our model 80% and avoiding the 30%).  Ramp signals also spread out the overall flow of merging traffic to prevent short-term backups that can degenerate into larger congestion.

Ramp controls, while useful, provide only moderate relief.  Traffic on the main highway improves incrementally, in theory and often (but not always) in practice, but the improvement becomes offset in part by the delays drivers experience waiting behind red lights on the ramp signals.  Also, merging volume where two main highways cross (and where merging traffic volumes generally render ramp signals impractical) can backup traffic so severely that ramp signals at upstream local roads provide no gain.

HOV and Similar Restricted Lanes – Just like ramp signals, we have likely experienced these, i.e. special lanes for buses and/or high occupancy cars, or which are reversible to match rush hour traffic direction.  A twist on these lanes includes charging tolls, including variable tolls, to influence traffic flow.

In cases where these restricted lanes repurpose existing lanes, achieving some benefit generally depends on people changing to buses or cars pools, thereby reducing the number of cars.  Otherwise, these restricted lanes provide offsetting benefits, i.e. those individuals in a bus or multi-occupant car go faster, while single occupant vehicles go slower.

Note in some cases restricted lanes can create a benefit even without individuals switching commuting modes, by maintaining existing bus and car pool participation.  If buses and car pools did not have a privileged lane, individuals may revert back to single occupancy in a car.

For new highway construction, the added lanes often become specialized lanes.  The new construction can readily include advanced signaling, variable toll collection, specialized access ramps and other features to achieve maximum flow, and serendipitously good revenue collection.

Automated and Autonomous Vehicle Control Systems – With some presumptuousness, I will label this the engineer’s dream solution (note I am an engineer by background).  These systems relieve the driver from control (i.e. takes the wheel out of their hands) and use centralized and distributed algorithms and processors, plus real-time data collection, along with internal vehicle electronics and external highway sensors and transceivers, to guide individual vehicles and overall traffic via computer control. 

As a typical example, these systems could and would group cars into platoons with inter-car spacing of just a few feet, and guide the platoons down the highway at typical highway speeds.  The potential?  If we look at the first of these articles, we see that at 45 foot spacing, ultimately achievable by these systems, a highway can handle up to 8,000 cars per lane per hour, an enormous increase in flow.  Achieving only half that capacity would still provide great flow improvements.

However, while such systems represent exquisite engineering challenges, and promise elegant and extraordinary engineering solutions, these systems traditionally have posed equally extraordinary problems.  These include cost (including public funding, which brings in politics), complexity (real traffic poses intricate and pesky nuances), implementation (revamping miles of highway for sensors and controllers), public acceptance (drivers like to stay in control), and vehicle equipment (auto manufacturers generally resist adding modules to cars which provide a public good but increase the car’s cost to the individual).

But developments not related to such systems have opened up the possibilities.  What are these developments?  They are many and multiple, including the rapid emergence of GPS devices, the explosive expansion of cellular networks, the continued increase in on-board vehicle computers, and most recently, the penetration and, importantly acceptance, of vehicle driver assistance modules.  The later, for example, can, without driver intervention, parallel park the vehicle, pre-tension seat belts, adjust headlights, start brake application, give blind spots warnings, detect collision threats, differentially apply braking to avoid skids, and on and on.

These developments provide breakthroughs on which to build area wide vehicle control systems.  GPS provides positioning and thus highways will need many fewer sensors.  With the driver assistance modules, drivers will be gaining acceptance of autonomous vehicle control, and the vehicles themselves will increasingly contain the necessary automated control systems.  Given its now ubiquitous presence, cellular provides an infrastructure for communicating with vehicles and between vehicles.

A decade or more ago, creating area-wide autonomous and automated vehicle control would require creating all the piece parts from the ground up, against possible skepticism from the public, concern from politicians and likely resistance from manufacturers.  Now the piece parts are to a greater or less extent appearing unaided.  These developments by themselves don’t represent a system, but do make creation of the system and its implementation a conceivable and realistic possibility.

So next time in traffic, envision a world say a decade from now where you will peruse the news or the video of interest on your internet eyeglasses or vehicle heads-up display while the traffic-controller-in-the-sky whisks you along smoothly but quickly down the highway.

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