"Linear Density"

BostonUrbEx

BostonUrbEx

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Long story short, in completing my thesis, I was looking to make some measurements of "linear density". ie: on average, how many people live along one 1 mile line? The primary purpose was in determining feasibility of utilities and showing how under free market conditions, suburban - and especially rural - users must pay significantly more (at least, for initial installation).

So basically, I took the square mileage of Boston, and got the square root of that. Without checking, it was something like a linear density of 120 citizens per linear mile. Thus, if you stretched out a transmission line for one mile, on average, you would hit 120 customers.

Do you think my desire to calculate linear density by taking the square root of density per square mile is correct? I tried to search for works by other people, but frankly, it looks like I'm the first person to even think of it. Perhaps its simply because my statistic is just useless? What do you think?

Anyway, in case your curious: making some general assumptions, the average rural customer must bear 30 times the cost of electrical installation than that of an urban dweller. Now, of course, this really has very, very little bearing today. But my thesis revolves around how New Deal policies set the foundation of suburban sprawl which would sweep the country post-war.

Anyway, I tried to keep this brief and yet still understandable.
 
BostonUrbEx said:
Long story short, in completing my thesis, I was looking to make some measurements of "linear density". ie: on average, how many people live along one 1 mile line? The primary purpose was in determining feasibility of utilities and showing how under free market conditions, suburban - and especially rural - users must pay significantly more (at least, for initial installation).

So basically, I took the square mileage of Boston, and got the square root of that. Without checking, it was something like a linear density of 120 citizens per linear mile. Thus, if you stretched out a transmission line for one mile, on average, you would hit 120 customers.

Do you think my desire to calculate linear density by taking the square root of density per square mile is correct? I tried to search for works by other people, but frankly, it looks like I'm the first person to even think of it. Perhaps its simply because my statistic is just useless? What do you think?
Click to expand...
It seems a worthy first try. I would suggest two possible other models to consider:

String of Pearls: Should the math look more like a string of pearls--radii around service points, and you assume that density within each radius is "like" the average for the metro area and therefore the per-linear mile is just the sum of the population in the "pearls" along the line?

Strip of Paper: If service points along the service line is effectively continuous is there an "effective width". Effectively Continuous would be if the service points were closer together than (say) half of their effective radius, at which point the line's effective width is just 2 times the radius.

Something strikes me as very good, but not quite "complete" with taking the square root. Maybe I don't know enough about what the service network looks like? Spokes on a wagon wheel? Overlapping hexagons (like cell service)? A perfect cartesian grid? One long line? There's probably an understanding of the topology of the network that pushes the model in favor of squares, strips, pearls, or a metro-wide circle. Can you be of any help on that?
 
Given that the utilities typically follow streets, the solution depends on the block size.

For example, suppose you have a 1x1 square mile city with 10,000 people.

If you have 8 blocks to the mile, then you have, (8+1)^2 = 81 miles of road, corresponding to an average density of 123.5 people / mile. TNow, suppose you have the blocks in one dimension be half the length of the blocks in the other dimension. Then you have 247 people / mile.

According to page 6 (pdf page 9) of http://www.cityofboston.gov/transportation/accessboston/pdfs/front.pdf, Boston has 785 miles of roadway. Using a population of 625,087, this works out to 796.29 people / mile.

It may make more sense, for the sake of argument, to use the number of people who are typically in the city on a given day instead of the population.
 
I don't think it's quite right to just take the square root, that seems to discard too much information. Also the units don't seem quite right: sqrt(people / sq mile)?

I saw a paper once which measured observed cost of providing water infrastructure at different levels of density, and it showed a U-shape where the most efficient provision seemed to be between 200 and 400 people per acre. I'll see if I can come across it again.
 
Anyway, thank you all for your input. I'm going to delve more into this.

I must agree that the equation certainly seems way too simplified and leaves alot out. But I'm really looking for some sort of "all else being equal" equation for averages.

Also, Cozzyd, could you explain your equation a little more? I'm not sure I understand? And where does the 1 come from (8+1)?
 
It may have been hectares, actually. Sorry about that.
 
BostonUrbEx said:
Anyway, thank you all for your input. I'm going to delve more into this.

I must agree that the equation certainly seems way too simplified and leaves alot out. But I'm really looking for some sort of "all else being equal" equation for averages.

Also, Cozzyd, could you explain your equation a little more? I'm not sure I understand? And where does the 1 come from (8+1)?
Click to expand...

Well, for example, if you have two blocks, you have three streets:
==== 1st st
1st block
==== 2nd st
2nd block
==== Third st

So if you have 8 blocks / mile that corresponds to 9 (=8+1) streets in both directions.
 
Matthew said:
I don't think it's quite right to just take the square root, that seems to discard too much information. Also the units don't seem quite right: sqrt(people / sq mile)?
Click to expand...

Good point: never take the square root of people.
 
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