This my first post here, so forgive me if the style is not to spec yet.
The street I live on has the following parking structure:
- On the 15th of every month the cars switch sides they are legally able to park on.
- On one side the houses have driveways, so the chunks in-between the driveways creates 'natural' street parking spots for cars to park. For this half of the month all the possible street parking spots are perfectly filled as a result.
- The other side is just a continuous strip of approx. $273$ft with no driveways. When the cars switch to this side, and the first car parks randomly along the strip (often does), it creates a ripple effect where given they rarely park in the exact 'perfect' spot on the strip, it creates odd spacing as the rest of the cars fill in and one car gets squeezed out of the possible max number of spots the strip could fit.
- The best option would be for
car1to park at the very end of either side of the strip, and each subsequent car to follow orderly after or before... but human nature, people don't do this. So the randomness that people park leaves the strip poorly optimized to max the # of cars that fit.
What I'm trying to solve for:
- What is the probability that the first car parks in the perfect spot (multiple options) such that it does not disturb the optimization of the strip for street parking?
- What is the probability that all subsequent cars fill in the strip in such a way that the max # of cars get a spot?
The work I have done (apologies, no formal math background):
Total length of the block strip $(L) 273$ft
Fire hydrant $(F) 16$ft
Avg car length $(Cl) 14$ft
Max spots with 0 clearance room to get in and out:
$$(L - F) / Cl = 18$$
Clearance needed to enter + exit spot: $4$ft assuming $2$ft per side
Total needed clearance: $$18 \times 4 = 72$$
New total space: $$L - F - 72 = 184$$
New total cars space: $$184 / Cl = 13$$
$13$ cars is typically the # on the strip when every spot is filled, so the rough math checks out.
Variables to define:
$P1$ = position of first car
$P2 - P13$ = position of all other cars
$S1$ = spot $1$
$Sn$ = a near unlimited number of possible spots if the increment is tiny of course... but not practical for a car. I think for this unit we would want to use something like $1$ft - ie. I don't think a 1ft difference in a spot space will dramatically alter the outcome of being able to fit $13$, but I'm not sure.
If $1$ft: $L - F = 257$ of total space, and if each spot increment is $1$ft, then you have $S1 - S257$ and thus $P1$ has $257$ options from $S1 - S257$.
The way I'm simply visualizing this now is $P1$ has $S257$ possible spots, it then reduces the number of possible options for $P2$ based on which it took up, and/or how many it occupies.
Ie. $P1$ in $S248$ would take up the $14$ft for the car $+ 2$ft on either side for clearance, so it would eat up $18$ more $S's$ going an equal amount forward so $S248 - S257$ and $S248$ back to $S239$.
Diagram
Let the road be an under score __
Fire hydrant an $F$
Car an $X$
So without cars we have:
________F______
As cars start going in we have:
_X_X_X_F_X_X_ (could have more cars)
Next Steps
How can this be modeled to figure out which of the spots from $S1 - S257$ are actually 'perfect' such that it fits all $13$ max possible cars? And then how to simulate $P1$ coming in at random to all the options from $S1 - S257$ and then $P2 - P13$ neatly fitting in as well? I.e. answering the questions up top.