Rotaries v Intersections, An Exercise in Analysis

(Picture from here.)

This is the sort of thing my brain does. Which probably tells you more about me than you want to know. Indulge me. Next time I’ll talk about space drives. I promise.

I live up here in Massachusetts. When I moved up here, back in the Cretaceous,  I encountered something called a “rotary.” It’s also called a “roundabout”, “circular intersection”, “traffic circle” and other names. Some of which are even printable. They took a little getting used to but I managed. The problem with Mass drivers isn’t rotaries; it’s that they, and the state police, view traffic rules as mere suggestions or revenue sources. No one takes them seriously.

ConcordRotaryThen, in the 80s, there was a big push to get rid of rotaries. It is a certain truth in life that no intersection is so completely devoid of merit that a Massachusetts traffic engineer can’t make it worse. Perfectly functional rotaries were replaced by obvious inventions of the devil. At left is a picture of what happened to the Route 2 Rotary in Cambridge. The black circle is where the rotary used to be. That mess is controlled by four barely functioning stoplights.

Get caught in this thing and kiss an hour of your life good-bye.

So, I’m driving into work today and not happy about it. To alleviate my boredom I took a back way in through Lincoln. The back way goes through a 5-points. This may be a term most people are not familiar with. It’s a relic of my sordid youth in Alabama. A 5-points is a 5 way intersection. And, of course, there was little or no understanding by the drivers of who has which turn. I got through that and mused on intersections all the way in.

It came to me that intersections have an ascending complexity rule. It’s an easy one to formulate. Let’s consider the simplest intersection, a straight line with a stop sign. No cross streets.

Since the road is bi-directional, two drivers have to be considered. So each driver has exactly one choice. This is a complexity level of two. The next kind of intersection is three points. Each driver has to consider two choices. This gives us a rule of:

C = R * (R-1), where R is the number of roads coming into the intersection and C is the complexity level.

This number goes up fairly quickly. For R = 3, the C value is 6. For R = 4, the C value is 12 and for my favorite, the 5-points, the C value is 20. If you graphed this it would look like an ascending curve.

Rotary v IntersectoinRotaries have a different rule. Each entry at a rotary has, in effect, an R of 2. The choice is limited by 1) spreading out the intersections across the rotary and 2) determining that direction of travel in a rotary is one way. This turns the rotary into a series of R-2 intersections. The complexity of the entire rotary could be considered a sum of the R-2 intersections.

So, in this case, if we have a four way intersection, we have a complexity level of 8 as opposed to the R-4 straight intersection complexity level of 12.

Rotary v Intersectoin-graphNotice that while the curve for  a linear intersection is an ascending curve,  the increasing complexity of the rotary is linear. This means we can add complexity to a rotary with much less impact than adding the same complexity to a linear intersection. However, it also says that for a small intersection, the complexity of a linear intersection is less or equal to a rotary.

However, rotaries were often the victim of their own efficiency. One could get on a rotary faster than one could get off. This caused nightmarish congestion. The Brits came to our rescue and redesigned the rotary in the 1960s. The big change was the addition of a precedence rule: vehicles in the rotary have priority to those outside the rotary.

(This, by the way, was always the rule in Massachusetts but drivers often had difficulty understanding it. The problem driver precedence, I think, was one reason rotaries fell out of favor.)

But there is a hidden complexity in the rotary. The 4-way intersection is intended to be traversed one at a time. Time is not a factor except as measured by the impatience of other motorists.

In the rotary, however, vehicles are on the move. Time is a significant factor. The rotary has to be sized that a given vehicle has enough time to get on and off the rotary. Ideally, this is done without much slowing down.

Rotary v Intersectoin-graph 02If we take 30 mph as an ideal speed of traversal, that’s 44 feet/second. Let’s allow 3 seconds at a given intersection in the rotary just to have room. That’s 132 feet between intersections. A simple two road rotary would have to be 264 feet in circumference or 84 feet in diameter.

I’m not sure how to evaluate this numerically. If 30 mph is the proper speed than as the size of the rotary increases or decreases, the complexity must increase or decrease. The number of choices remains the same but the time in which to make them is a variable.

When I was visiting my home town in Missouri I saw what must be the smallest rotary imaginable. It could not have been more than 20 feet in diameter. Two cars draped across the center would hang off both ends. This little thing had four roads coming into it. If you didn’t hang hard on the steering wheel, you would drive off the road. But I digress.

This is the sort of analysis engineers do on all sorts of things. There’s a reason that data going across a network is called “traffic.” Many of the original network topologies use road metaphors. One of the famous problems of mathematics, solved by Euler in 1735, is how to determine the optimum path to traverse seven bridges in Konigsberg. That was the beginning of graph theory. Which was the start of that map program in your smart phone.

Which could help you navigate one of the Massachusetts rotaries.




Rotaries v Intersections, An Exercise in Analysis — 12 Comments

  1. There is a major “rotary” (I’m not from Massachusetts) in Washington, DC, that goes around Dupont Circle, a park at the intersection of Connecticut and Massachusetts Avenues and several streets. Most of the roads leading into it have traffic signals and it has signals at several points on the circle. To enter that circle at any time is unpleasant; at rush hour it can take a large chunk of your life just to continue on Connecticut or Mass. That is, it is both scary and a bottleneck — it’s stop and start at the best of times. At least the traffic signals make it possible to walk across it with only normal risk to your life (DC drivers see pedestrians as an obstacle to be removed even when they have the right-of-way).

    I’m sure some traffic engineer can use your math and demonstrate the efficiency of these horror-novel creations, but I just find they increase my stress level (which is already high when I’m driving in DC or Massachusetts). (Or urban Texas, for that matter, but we don’t have a lot of traffic circles — we have overpassses and flyovers.) The scariest ones I ever saw were in England, not because they were badly engineered (I assume they weren’t) but because you had to go around them to the left. I wasn’t driving and I was petrified, though the driver was English and therefore had no problem.

  2. A few years ago, they replaced some of the worst traffic light intersections with roundabouts in my area. Oh, the complaints! Oh, the howling! “They look so weird!” “I can’t figure out how it works!” “They look so dangerous!”

    But here’s the thing: traffic moved faster. At the roundabout, there’s very little waiting, if any, and the cars keep moving smoothly. It’s a wonder. The department of transportation ran several studies and came up with a verdict: the roundabouts do indeed keep traffic moving more efficiently.

    “They look so weird!” “I can’t figure out how it works!” “They look so dangerous!”

    People howl and complain as they continue moving smoothly through the roundabouts. =That’s= what’s really weird.

  3. Ehm, uh, no. That is, Euler’s problem wasn’t an optimization issue, rather it was an abstract problem as to whether something (i.e., going for a walk which crosses each bridge exactly once) can be done or not. The answer is no, and thus topology was born.
    PS I will now sit on my hands so as not to try and explain linear programming.

  4. The Brits are so addicted to roundabouts that one in Winchester is the size of a manhole cover with only 2 entrances and exits, but marked on the map as big as the ones on the M1 with 8 entrances and exits. I passed it 3 times before I figured that one out.

    Once I figured them out, I like roundabouts. They work. And if you encounter a big complex one you can get in the inside lane and go around a second or third time until you find the right exit.

  5. I learned to drive in Massachusetts (and I still recall the rule for who has precedence at a four-way intersection, although it does not appear to work that way in any other state in the union). Roundabouts in the western part of the state work (or used to work) somewhat better than in the east–on the principle that the closer you get to Boston, the worse the driving gets, I think.

    When I first drove in England I feared the roundabout: driving on the wrong side of the street, and going in circles? But I found it ultimately a much more sensible way to get through an intersection, where there is sufficient space to permit it.

    And paranthetically: when did they get rid of the Route 2 rotary? I thought that needed permission from God.

  6. Ithaca NY had numerous pitchfork intersections (5 or more roads). As one approached the trick was to watch the other drivers eyes. If the nearest one showed signs of a flinch, then the dominant driver floored the accelerator. The most interesting one was on the main approach to the hospital and had an unsignalled railway crossing in addition to a 6 road pitchfork.

  7. I saw what must be the smallest rotary imaginable

    Nope. Google “mini roundabout” and look at the images. I would say they have popped up like blisters all over Britain, but they’re flatter than that. Phyl’s “manhole cover” is pretty much exact. Sometimes they come in slews, like a spray of acne, and you have to wind your way between them.

  8. Ah, the Route 2 ex-rotary in Cambridge. That’s no fun even if you hit it at a time of light traffic, like Sunday afternoon departing from a Boston con.

  9. The rest of the world makes a distinction between “traffic circle” and “roundabouts.” Dupont Circle and others are “traffic circles”. These were very bad. But the roundabouts were really born in the 1960s with new rules. The new roundabouts seem to work well.

    I misspoke about Euler’s problem. It was not intended to determine an optimum path. But it was the beginning of topology and graph theory and they are definitely used to determine optimum pathas.

  10. Dupont Circle is surely an anomaly, designed by demons from Hell. It is organized in such an un-intuitive way that there are daily accidents. The only thing to be said about it is that it has been that way for so long, most residents are used to it.

  11. A few years ago they started adding roundabouts to my hometown in the Albany, NY area. First they put in one, then a couple more popped up, and now they’re freaking everywhere. There is a stretch of road that has no fewer than four within a couple of miles of each other. It’s possible to bypass two of them if you don’t feel like using them but eh, they work well enough. Then they overhauled a poorly-designed intersection that had been that way for as long as I’ve been alive, probably longer. It used to be a four-way, four-plus-lane-in-each-direction stop light. Now they’ve added a bridge in one direction, and a roundabout underneath, with one traffic light in one direction so that the people on the roundabout underneath can get onto the bridge. It was incredibly confusing at first, as evidenced by the lady driving the wrong way who almost hit me head-on one night… But I think most people have gotten the hang of it by now.

    I work on a sort-of giant roundabout too. The buildings are on two loops, the outer and the inner, and each goes in a different direction. There have been a lot of incidents of people driving the wrong way on those loops even though they’re signed one-way… I think in NY we just have trouble with reading (and a lot of the wrong-way drivers happen to be elderly). While in Western MA, where I currently live, no one pays attention to traffic rules to begin with. You’re ten feet away from me driving 55mph? I think I’ll just pull out slowly from this side street in front of you with no turn signal and not speed up at all. Happens *every day*.

    The roundabouts have their pros and their cons, but I think the thing I like the best is that even while they don’t necessarily prevent collisions, the accidents that do happen are at much slower speeds than would have been seen at a four- or five-way stop, and so there are fewer fatalities. And, if traffic is flowing smoothly, I hardly ever have to stop when entering one. I guess that’s a win, right?