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.

Introduction to Buying Website Traffic

From the moment the Internet appeared, lots of users used it to seek ideas, solutions or entertainment. So the important question is: how can you bring these people to your site? There are lots of manners for you to do this: promote your website, optimize it for search engine ranking, advertise on it, link to other sites or purchase traffic.

But is purchasing traffic a good idea?

This question can be answered with both yes and no. You will want and need to purchase real traffic to your website, but at the same time you should know that purchasing traffic for your website can be quite risky. When you are purchasing targeted traffic in order to increase your exposure or generate sales, you do not have a guarantee that will actually obtain sales from the traffic you’ve bought. But you still can hope to change the traffic into sales if your site has what the potential buyer is looking for.

Cheap

Purchasing untargeted traffic is a very cheap source of traffic, but it is extremely unpredictable. You can’t expect a lot of conversion out of it. Thus, this is a great solution to increase the potential rank of your website, but at the same time it can lead to massive loss. Make sure you don’t purchase fake traffic. This can be obtained through spam or bots. The last thing you need is to purchase traffic that was forced. For instance, the pop ups can drive a person crazy every time he or she opens a site. Those who open a site that contains a pop up usually choose to leave because they get distracted and annoyed by this. This is a sure way to drive people away.
What traffic is good for you?

There are various manners in which one can obtain paid traffic. But all the methods revolve around the quality traffic. Of course, the organic traffic would be more than welcomed, but we all know that it might take too long to get it. In fact, playing with SEO techniques is not the easiest thing in the world. That is if you want to make money fast. However, if you purchase site traffic, you will get it instantly and you will not have to pray to the Google god in order to make your site matter. This is the main reason why purchasing site traffic is extremely popular.

Let’s not examine the popular methods of buying traffic just yet. Before getting into that, it is vital to keep in mind the following phrase: return on investment (or ROI). Purchasing traffic is definitely risky. Any type of advertising has its risks. You will be paying to get traffic, so you will have to get a profit. The paid traffic is extremely expensive and cannot be used for branding (internet marketers or business owners know what I am saying). But we will get to that soon.

For now, let’s talk about the primary ways to get traffic (the paid type).

Media Buys

For this you will have to know who your target audience is and where to reach them. Lots of people use this method in order to purchase banner space for their sites.

It is true that the AdWords program allows you to buy banners on the network, but there are lots of options when it comes to this topic: ad and social networks, co-registration offers, or direct buys. The last option can bring you high traffic sites in your niche and can help you select the ones that fit your advertising needs.

PPC Advertising

This is the most influential traffic because it comes from accurate searches. Lots of persons are interested in what you have to offer. Facebook, Google AdWords or Yahoo Search are just 3 of the PPC programs which permit buying a space for ads, depending on a clear list of keywords. This way you can control the delivered traffic and tweak the existent campaigns in order to increase the ROI. Remember that there are a few advertisers that place the name of their company in the title of the text dedicated for branding. However, if you want to make the best out of your ROI, concentrate on keywords related to product names.

CPV Traffic

The cost per view method is quite cheap, but at the same time it is a very complicated paid traffic source that doesn’t bring back an important return. You don’t have a lot of options in this case and you have to pay per every impression. Which is not the case for the cost per thousand method. In this case, traffic is delivered under the form of page ads (full pop unders or overs) to those who have previously agreed to receive it. They do this by adding different software with or without their knowledge. However, keep in mind that this type of traffic is still new and it has little competition. So it doesn’t hurt to look.

Mobile advertising

And last but not least, the mobile marketing technique has become quite popular, since lots of people browse the Internet from the phone. If you want to try this method, check AdMob Network from Google. This has started small and it is now an important source. So if you handle an offline business, verify the text message marketing and you will see that the response rate will be higher.

There are lots of options when it comes to purchasing web traffic. Only you can decide which one is more suited for you. Keep in mind the important elements: the ROI and the conversions. Try new stuff and go with the options that work!

City’s Traffic Light Camera Appeal Denied

Finally, a small victory for the little guy! It comes as no surprise to anyone that red light traffic cameras are an increasing annoyance to everyone except the municipalities who use them and the private companies that make a fortune off of installing and then maintaining them. Like many other traffic laws in Florida, traffic cameras serve one purpose and one purpose only – to continue to line the pockets of local politicians. Yes, we all know that proponents of red light traffic cameras espouse that their purpose is to advance public safety. Well, I am not buying it. As a traffic ticket attorney, I have seen too many instances where drivers have gotten traffic tickets when they weren’t even moving. Additionally, the knowledge that these cameras are the livelihood of private companies who have a major financial interest in advocating their use bolsters my skepticism.

Although the constitutionality of using these red light traffic cameras has been challenged in the court systems ever since their inception, recent news gives drivers, and traffic ticket attorneys, hope that they may be on their way out. On October 15, a three-judge panel for the Fourth District Court of Appeals denied the appeal of the City of Hollywood to have an earlier decision overturned. The previous matter supported Eric Arem’s claim that the issuance of citations by a private company is not permissible under Florida state law, which was then supported by the appeals panel reinforcing that state law does not permit private companies to issue traffic citations. Yet individual municipalities hire private companies, one Arizona-based company in particular, to install and maintain red light traffic ticket cameras. This results in a fiscally symbiotic relationship for both parties.

Drivers are surely elated at this development, but it does pave the way for many unanswered questions. Surely, one question that is in the forefront of most drivers’ minds is whether or not traffic tickets will still be issued in South Florida as a result of red light traffic cameras. It is not an easy question to answer because, although some towns have done away with traffic cameras, this most recent ruling pertaining to Hollywood is not yet carved in stone. According to Hollywood spokesperson, Raelin Storey, there is still a possibility that there may be a rehearing or it can be appealed before the Florida Supreme Court.

“This case has the potential to impact a number of cities that contract with (… the Arizona company),” Storey said. “If the administration of the program has to change dramatically, we would, of course, have to evaluate whether we can continue to afford to operate it.”

The determination made by the appellate court ruling states, “Such outsourcing to a third-party… for red light camera violations is contrary to the plain wording of the Florida statutes.” This “plain wording” provision prevents cities and municipalities from applying their own interpretation of law, thereby ensuring uniformity throughout the state in the application of traffic laws and the fines and penalties that arise from traffic violations.

This is far from the first time that the use of red light traffic cameras has been a matter of controversy. Back in June, a court ruled that several other Florida towns side-stepped Florida state law by the use of traffic light cameras prior to July 1, 2010 when the State Legislature approved the use of these cameras. The continued push to repeal or restrict the use of these cameras has met with resistance by those who support them and by safety studies being impeded in the State Legislature.

This recent ruling may have been aimed at Hollywood, but other towns such as Hallendale Beach and Hialeah have already done away with the unpopular red light traffic cameras. In light of all of this controversy, it’s not surprising to learn that many other cities are actively trying to circumvent similar issues from occurring. Additionally, the list of South Florida towns that are suspending the use of red light traffic cameras pending further action by the courts continues to grow.

“We have to be prudent,” Palm Beach County Attorney Denise Nieman said regarding that county’s decision to temporarily suspend their traffic camera program.

Unfortunately for drivers, each city still gets to choose whether or not to use these traffic cameras. Although the argument for doing so is that they reduce accidents, the fine for committing a red light camera violation is $158. With over 900 traffic cameras installed in Florida, most of which are in South Florida, this practice has generated over 750,000 traffic tickets and more than $119 million in fines.

A large portion of that revenue goes to the company that installs the cameras and generates the traffic tickets. This makes it obvious that this is more about being a profit-generating business than about any genuine interest in reducing accidents. After all, what better way to cover budgetary shortfalls than under the guise of concern for the public welfare? Of course, the private company’s website also touts how they are improving safety for the public’s own good, but you will be hard pressed to see anything posted there as to how lucrative this business has become for them.

In what would appear to be a frantic effort, the Arizona company that provides these traffic cameras to Hollywood claims that it can change the way they issue the traffic tickets. Why wouldn’t they scramble to come up with an alternative? In 2013 and 2014 alone, this particular company has gleaned a large percentage of roughly $28,000,000 that the State of Florida has paid to the traffic camera vendors.

Even though the debate regarding the legality and efficacy of traffic cameras continues to rage, no other precedent exists so this recent is ruling is now case law, at least for the time being. As such, it will hopefully pave the way for more municipalities to consider the legal ramifications of having the cameras installed and maintained. As long as they continue to be profitable, it’s likely that local governments will try to justify their existence. Eliminating the profit margin by doing away with third-party issued traffic tickets and fighting these erroneous traffic tickets in court are the best way to ensure a fairer playing field when red light traffic tickets are issued. In many cases, it is possible that refunds are due to motorists who have received red light traffic camera citations.

If you feel that you are one of those many drivers who have received an unjust red light traffic ticket, give us a call for a free consultation. We will be happy to review your traffic ticket with you and continue to work to protect your right and the rights of other drivers.

Highway Traffic One: Collision Avoidance

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.

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.

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.

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.

  1. We close in on the leading car by 13 feet during our 1.5 second reaction time
  2. Our car closes another 55 feet for the 3.5 seconds we both are braking
  3. We brake a second further to slow to the speed of the leading car, closing 8 more feet
  4. 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.

Florida Traffic Tickets: Public Safety or Big Business?

Four million – that is approximately how many traffic tickets the state of Florida issues every year. FOUR MILLION! Think about what a huge number that is. That averages out to almost 60,000 traffic citations for every county in the state, although the amount number issued differs tremendously from county to county. Some counties issue a great deal more traffic citations than others based upon population and how stringent the law enforcement is in that specific county, but no matter how you look at it, that equals a huge amount of revenue for the state.

Keep in mind that most of these tickets are the result of speeding, therefore, they have widely varying fines depending upon the county you are in when you get the speeding citation and by how much you have exceeded the speed limit. When you combine the millions of traffic citations written and factor in the speeding ticket variable, this creates quite a prolific industry. In fact, although traffic citations are a multi-billion dollar industry on a national scale, one television investigation determined that from early 2011 through early 2012, the state of Florida alone received more than $101 million in traffic fines. This is just the amount that the state got and doesn’t include whatever portion of the traffic fines that was divided up with the cities or counties in which they were written.

Law enforcement agencies use much of these funds to hire more enforcement officers to keep the wheels of this financial machine moving, and it’s not just the law enforcement agencies who write traffic citations that realize financial rewards either. When you consider the number of entities that benefit from this revenue, it is easy to see that this is a profit-motivated system that thrives off of what is often just human error. Public and private agencies, just like law enforcement, get a piece of this financial pie. These agencies include court systems, city and state governments, insurance companies, and traffic ticket camera companies. Here just a few of the ways that these profits are used:

• The budget of the Clerk of Court in the areas in which the citation was written is frequently affected by the amount of fines and penalties it receives.

• The state anticipates a certain amount of revenue from traffic citations which is also uses to balance its budget.

• Insurance companies use these traffic citations as a means to classify someone as a “high risk” driver to justify raising insurance rates which is the catalyst for automobile insurance becoming a multi-billion dollar industry. This is certainly great motivation to claim to support safety programs and donate various speed detection devices to police agencies. The millions of dollars they may spend on these devices is just a small percentage of the profits they stand to gain by more traffic citations being issued. Additionally, although insurance companies often claim that enforcing traffic laws is in the interest of public safety, there is really very evidence to substantiate that there is a correlation between infrequently getting a traffic ticket and posing a greater risk of causing a traffic accident.

• Traffic ticket camera companies are, in my mind, the most odious of all of these as they are private companies that are strictly for-profit and cannot effectively argue that their interests are in “public safety.”

• Let’s not forget the ancillary beneficiaries such as the companies who make speed detection devices, traffic schools, and so on.

With all of these groups looking to make financial gains off of hapless drivers, it is not surprising that traffic tickets are just one more multi-million dollar Florida business. That’s right – a business, not a public safety concern. It’s a business that, at $150 or more per traffic citation, proliferates so greatly that many municipalities try to obfuscate the true numbers about how much profit is involved. Not only do they then have to share less of the funds received, but it helps to try to keep average drivers from becoming outraged at how much money they are bilked out of through traffic ticket practices that can often be less than ethical.

One such example of questionable ethical behavior on the part of law enforcement is when someone is stopped for a traffic violation, but is arrested on a greater charge such as marijuana possession. In such a case, the issuance of the traffic ticket would not be included in traffic violation statistical data. Anyone with a rudimentary understanding of statistics quickly realizes that this practice skews the data that is presented regarding traffic citation information.

In effort to keep this tremendous profit-driven scheme in motion, the legislature has to pass laws that allow cops to stop drivers for any arbitrary reason they choose. Something as seemingly insignificant as “improper lane change” is legal grounds to pull someone over and allow the police officer to look for other possible reasons to write citations. When you couple this with other factors such as the legal system projecting the image that anyone with a traffic ticket or two as a bad driver and a menace on the roadways, it acts as justification to keep those ever-increasing fines to be continuously rolling in. If our lawmakers can keep these fines to a level that will curtail the majority of ticketed drivers from fighting back against the system, they can continue to expect this golden goose to keep producing to their benefit. They do so knowing that most drivers who receive traffic citations are at a disadvantage when fighting these tickets, which makes the inequity involved in the entire traffic law process patently clear.

Yes, it is true that even attorneys get their portion of funds created by the issuance of traffic tickets. Specifically, traffic ticket attorneys make our living by representing drivers who receive traffic citations. I do, however, believe that most do so for the same reason that we at the Traffic Ticket Team do – because we feel that the average driver is taken advantage of by the system because it is a system that is set up so that the odds are stacked against anyone who gets a traffic citation. This is why, unless you are quite savvy in the courtroom, it is usually beneficial to you to hire a traffic ticket attorney. We are here to make sure that you have every advantage possible when fighting a traffic ticket, so give us a call at 954-967-9888 for a free consultation.