Tuesday, March 19, 2013

154> Fractals!

One thing concerns me about the typical vision of a PRT system is the idea that should consist entirely of a network of one-way loops that would be expanded by adding new loops contiguously.  This seems to me to be an oversimplification, and generally not the most efficient design in terms of system cost or travel time.  Yet even bi-directional routing can be thought of as a loop, of sorts – just one that has been squeezed along its length. So how could the paradigm of one-way loops be lacking?

First of all, let’s take an idealized example where the density of potential riders is completely even and the track forms a perfect grid of squares. (round the block loops)  Clearly there will be a bias towards higher ridership on routes crossing the middle than at the outer edges.  So what do we do?  Is the middle subject to wait times or are the outer routes paid for, but underutilized?  Add to this the fact that the middle will probably have more potential riders anyway, and you can see that the problem is just that much worse.

Beyond that, the assumption of an ever-expanding network of contiguous loops tends to ignore the uneven distribution of important destinations.  Although cities have a central business district that needs to be the heart of a PRT system, after that there are seemingly no rules for what should come next.  A spider web design is better than the grid, since the converging radial routes enable a somewhat denser concentration toward the middle.  But cities don’t grow evenly and the transportation bottlenecks can be far from downtown, an inconvenient fact that would-be PRT providers will surely downplay.  After all, there is plenty to be done downtown, so why worry that far ahead?  I say, “Inconvenient” because really dealing with it increases the complexity of a system immensely.  In my opinion the solution will have to include some combination of multiple speeds, multiple vehicles sizes, and multiple, parallel tracks- like there are multiple lanes on a freeway.  Anything less, and the system will be one that can only solve some of the problem in some of the situations, and that fact may be baked-in by the design and therefore forever unfixable.  

If there is only one speed, all track must be routed without any sharp turns if the system is to be fast enough to be useful for anything more than very short trips.  This means acquiring right-of-way on nearly every corner or bypassing needed, but troublesome routes.  Then there is the matter of multiple vehicle sizes.  Here I am referring to GRT.  Certain routes, say to an airport , stadium or “park&ride” lot  might benefit from a larger shuttle-type service rather than only individual smaller vehicles.  This would help with the rush hour demand for routes that many passengers share in common, without requiring extra track or vehicles just for a few hours each day.  The third option, multiple parallel tracks, is also for this situation.  One of the great things about PRT is that it can be designed to have very inexpensive track, which makes such possibilities more palatable.  I think following the familiar highway model, where there is both a high-speed express and a more local feeder aspect might be particularly advantageous.  Such an investment paves the way for future loops along the way, and could often be done using existing highway easements.     

In either of these last two cases it is assumed that there are enough destinations along the way (or at either end) to justify a PRT-compatible track rather than say, an ordinary bus.  Having GRT sharing the track reduces what would otherwise be a requirement for more parallel tracks, but parallel tracks, if spread apart by a few blocks, would provide enhanced access all along the way.  Like I said, there should be some combination, if at all possible, of these three methods.  This may complicate the system design, but it makes the system more versatile and therefore ultimately a more compelling value.    

So how are we to determine the relative merits of the various systems and routing schemes?  I think there are probably some mathematical formulas that describe the general problem and give some ballpark ratios and other guidelines that might be useful as a starting point.  Such formulas seem to follow the general principles of fractals, something that occurred to me when Nathan Koren ttp://www.podcar.org/blogs/nathan-koren/, in this excellent two-part (similarly themed) post, used a leaf as an example of a transportation system.

Imagine a growing town building new roads outward into surrounding countryside.  Along each new road are natural “sweet spots” to develop housing, warehousing, retail, etc.  Any closer, and land is too expensive.  Any farther out, and the commute is too far. Maximizing the usefulness of the new road and access to these areas can be accomplished with simple branching.  Now you have twice as many sites with the same travel time without needing separate roads.  Branch the branches and now you have four. Branch once more and you have eight.  Indeed, radially emanating roads must diverge if they are to access any reasonable portion of the ever widening land mass anyway.

This follows the rules of a type of fractal geometry known as the Lindenmayer system, (L-system) which is seen thoughout biology and is efficient for movement of blood, plant nutrients, etc.  The math for the above example is simple.  Go a distance, split, go half as far, split, go half as far, split, etc. (I cut the "trunk" to save space)  Add a bit of randomness and you can create forms which look like photographs from a botany book.

Want to live on a cul de sac?  This “H tree” (left) gives everyone the piece and quiet of their own dead end, equidistant from the main road.  To the right is what is called a quadric cross, which represents a three-way split at right angles. 

Here is a different kind of fractal geometry at work:  Let’s consider, for a moment, the merchant.  Here we have the same desire for cheap property, but it is coupled with the need for exposure to customers.  Clearly the intersections formed by branching roads are particularly advantagous in this regard.  But another plus would be the presence of other businesses, to help draw customers.  This too, can be described in mathematical terms with more fractal geometry, this time with what is called a “Diffusion-limited aggregation.” (DLA)  Here, particles (businesses) randomly migrate from a source, but not too far, only to plant themselves on an edge (2D) or surface. (3D)  This is not unlike coral growth, and can be seen in satellite views of cities, especially aspects like pavement coverage vs. green areas.  In three dimensions, constrained by city blocks, an effect much like crystal formation is seen in the growth of groups of multistory buildings.  Again, these similarities are not merely coincidental. They are the result of similar natural, measurable forces.  
What is interesting about the combination of DLA and L-system effects in city growth is that together they generate satellite communities.  This formation is easy to observe by anyone who takes a farm road out of town.  It usually starts with a gas station/convenience store at a rural intersection.  Soon an eatery or an auto repair garage follows.  As more businesses join the group, land values rise, creating a climate for land speculation and further development.  Much, much later the resultant communities create a traffic nightmare for the host city by interfering with the radial flow of vehicles during rush hour.  In the typical spider web roadway configuration the radial strands that serve the central area are inherently at odds with the concentric routing that serves traffic between neighboring outlying communities.  This creates pockets of traffic congestion that are far from downtown but are still sorely in need of something like PRT.  The classic remedy has been to build a freeway with overpasses over the main crossing roads, cutting the outlying communities in half.
These fractals only go so far in describing the problem facing people tasked with designing PRT routing, because of the subject that I brought up first, which is loops.  After all, notably absent in the earlier discussion about a “sweet spot” was the obvious way to get the most bang for the buck. That is to have the branches loop back upon themselves.  Fractal forms can include loops as well, as in the leaf below. Note the classic fractal multi-scale self-similarity in the tendency towards branching at right angles.
Below is a fractal of loops representing growth along an east/west corridor.  I have included some secondary development, (shown in red) representing the value of shortcuts.  The next step would be to connect the outlying areas directly to form an outer loop.
The “rules” outlined above do more than help explain the uneven geographical distribution of potential PRT traffic.  They also illustrate the different capacity requirements for the track itself.  I drew the L-system tree with separate lines on the “trunk” to illustrate the simple fact that there simply cannot be equal traffic between it and the branches.  The quadric cross example also shows the relative traffic increases (line thickness) toward the center.  Either the branches are at a fraction of capacity or the trunk is overburdened.  As I stated earlier, the idea of simply handling peak loads with massively parallel loops is iffy at best.  But multiple lane trunk lines or GRT have drawbacks as well.
The conventional thinking used to be that starter downtown circulator loops could be added to until eventually there is a network that fulfills the total needs of the covered areas.  There is still some truth to this, but it certainly isn’t the whole story.  Cutbacks in government transportation spending have forced us to examine any and all inefficiencies, including any underutilized track or vehicles.  In the end we will probably end up using some combination of “all-of-the-above.”  I am not sure how useful fractal modeling can actually be in practice, but the subject certainly seems worth pondering.  Surely there is a fractal form whose shape is the result of a mathematical modeling of various forces that shape our cities, and therefore our transportation needs.  One thing is for sure. We shouldn’t be surprised when we find that the most cost effective and expandable PRT solutions mimic nature more than a checkerboard, or use many of the same techniques that have proved effective in moving people via our current network of roads.