Sunday, June 22, 2014

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.

Then, 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.

Rotaries 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.

Notice 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.

If 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.

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