Parking lot math puzzle

This decision has faced all of us who drive hundreds of times. You approach a store with a large parking lot from the end of the parking lot. You can play it safe and take a space towards the back where there is a smattering of cars, gamble and go towards the front, hoping to get one recently abandoned, or take the middle road. If you reach the front of the store, then let's say you turn around and then just take the first parking space you see.

To simplify the problem, let's assume just a single lane with parking spaces on just one side. I'm open to other assumptions, but let's say all the other cars start at the front of the store and go away from it, seizing the first space the find. To just say anything, let's assume the number of cars that enter and exit this situation per unit of time follows a Poisson process (meaning the time between cars entering, and time between exits, follows an exponential distribution). You can make any specific assumptions you like, but if you want me to suggest some, how about the mean number of cards entering per hour is 60 and each car parks for exactly one hour.

Let's say you enter the parking lot after this store has been open for a very long time so the situation as reached an equilibrium. The question is what is a good basic strategy for when to seize a parking space? The goal should be to minimize expected time to get to the store. We need some assumptions on speed. Shall we say the car goes at 10 MPH and walking speed is 2 MPH.

This is a little like the 100 secretaries problem. The link rewords it to the cleanest outhouse problem, the math is the same.

I do not know the answer by the way. My personal strategy is rather conservative in which I park when I get to the point where there is a mix of about 50/50 empty to occupied spaces. A basic strategy might to be pass x parked cars and then to take the next available space. This idea probably was planted by the 100 secretaries problem.

So, any thoughts? Has anyone been asked this before? Has anyone thought about it before?

If the lot filled with folks who shop about the same amount of time, it means that “lumps” of spaces located near each other will appear when folks leave. I would look for, and take one of these spaces.

In actual practice, this works well for me at Costco, as I plan to show up about half an hour after opening, when most of the doorbuster folks have finished their shopping and are pulling out.

In your original problem, do the employees use the same lot?

I generally just try to park in the easiest or most convenient spot. Usually it's a bit further from the door, but who cares. If it's busy, then I'll park in the nearest spot (to the front) that I can see. Although I often get juked out by motorcycles or those weird little golf-cart-like "cars".

As a practical matter, I think the answer depends on how well you park.

If possible, I take the last immediately obvious available spot with nothing on either side. I’m terrible at parking.

Quote: Ayecarumba

In your original problem, do the employees use the same lot?

In the interests of simplicity, we can either ignore them or assume they follow the same behavior as the shoppers.

I typically go where I think nobody has parked. Back of a lot, top of a parking garage. Cannot assume a car travels at any speed in real life, it's just constant stopping having to react to other drivers. Driving without interruption and walking in is a safe bet I take every time. Experience has told me a good spot is more difficult to get than it's worth.

I want to minimize the time spent searching for my car when I get out, so parking it near a landmark is a consideration. However, with only one lane in your problem statement, that's not a big issue.

So far, no one is being very helpful or mathematical..

I'm about ready to buy a Smart car or atleast something smaller to go to the casino. These damn casino garages are tight. I am also tired of people dinging my truck. I usually try to park where I will be by myself. However, I could park in and empty back row only to return and find someone parked next to me. The rest of the back row spaces would still be empty.

Quote: MaxPen

I'm about ready to buy a Smart car or atleast something smaller to go to the casino. These damn casino garages are tight. I am also tired of people dinging my truck. I usually try to park where I will be by myself. However, I could park in and empty back row only to return and find someone parked next to me. The rest of the back row spaces would still be empty.

My 20 year old car is for sale. It could get dinged and you wouldn't know, nor would you care!

Quote: RogerKint

My 20 year old car is for sale. It could get dinged and you wouldn't know, nor would you care!

Is this an open invitation for us to go down there and ding up your car? Sweet. Let me know what kinda car you have and where it's at.

You stole my car, parked yours in my garage, and we just happen to both have the same car make, model, year, AND color? Well good. I'll go out there right now and tear it to pieces. That'll show you!



Btw I'll buy your car for $20, if you're interested. But I want a free lifetime warranty.

Quote: RS

Is this an open invitation for us to go down there and ding up your car? Sweet. Let me know what kinda car you have and where it's at.

You stole my car, parked yours in my garage, and we just happen to both have the same car make, model, year, AND color? Well good. I'll go out there right now and tear it to pieces. That'll show you!



Btw I'll buy your car for $20, if you're interested. But I want a free lifetime warranty.

Add another $3.50 and we may have a deal.

I guess nobody is interested in the mathematics of this. *sigh*

I suppose an easier problem would to be what is the basic strategy to getting the best parking spot with only one pass through.

Wizard, when you were a college professor, did you always have that one bad kid, like RS and MaxPen, sitting in the back of the room interrupting class and distracting the well-behaved smart kids, like me?

I always strove to be the intelligent bad kid. It was never sufficient to be merely bad. I had to be bad and clever! (There are others on this site who seem to share this goal!) Sometimes some gal had the nerve to prove she was more intelligent and clever than I was. Boy, did that tick me off!

Insofar as the parking problem goes, I'm more interested in getting a "fat" parking space than a close parking space. I strive to find a place where the car on the left of the space parked left-of-center and the car on the right of the space parked right-of-center. (this is not intended to be a political discussion.) Such cars are unlikely to ding mine --- or me theirs.

Three adjacent empty spaces are not as good, because even if I park exactly in the center of the middle space, I have no control over how competently my not-here-yet neighbors will park.

Should I find such a fat space, I'm delighted and I take it immediately. My delight more than compensates for a few extra minutes of travel time. My car has a few dings --- but I prefer it doesn't get any more.

I try to park far enough away to not have not-yet-here neighbors to worry about. Mainly because I still have to worry about backing out, which I do terribly.

If I am at any store where I'd need a cart, I always park as close to a cart return as possible. It's kind of pointless to park in the closest spot when you'd need to walk half the parking lot, or back into the store, to put your cart back.

People who leave carts laying around the parking lot should be shot on sight.

Quote: Wizard

I guess nobody is interested in the mathematics of this. *sigh*

I suppose an easier problem would to be what is the basic strategy to getting the best parking spot with only one pass through.

I'm not sure how much you could simplify the original prompt, personally I had to throw my hands up at this part:

"Let's assume the number of cars that enter and exit this situation per unit of time follows a Poisson process (meaning the time between cars entering, and time between exits, follows an exponential distribution) Let's say you enter the parking lot after this store has been open for a very long time so the situation as reached an equilibrium."

Also do we need to know or assume the size of the parking lot?

Quote: gamerfreak

Also do we need to know or assume the size of the parking lot?

I think I'll give up on a good mathematical discussion of the problem, so will address this.

The signs generally say "Please return cart here" wherever there is a place for them, which kind of implies it's optional. Some stores don't have many such corals so you're left wandering around the parking lot with an empty cart, looking like a dofus. Just playing the devil's advocate here. I personally return them.

However, let me ask you this. Suppose you're in a hurry. The closest shopping cart coral is 100 feet away. However, 10 feet away are several shopping carts pushed together and hogging up a parking space. Assuming you shove your cart in the last cart the full way, it won't inconvenience anybody as there is room for plenty more carts before the row gets so long it sticks out in the aisle.

BTW, where I lived in Baltimore you couldn't bring the carts into the lot at grocery stores. There were barriers to prevent it. Or maybe that is just because I lived in a bad part of town (Woodlawn).

Would the people with tiny cars PLEASE stop pulling all the way into the parking spot so that, even from a short distance, it doesn't look empty! You're shattering the hopes and dreams of many fellow humans.

Quote: Wizard

I think I'll give up on a good mathematical discussion of the problem, so will address this.

The signs generally say "Please return cart here" wherever there is a place for them, which kind of implies it's optional. Some stores don't have many such corals so you're left wandering around the parking lot with an empty cart, looking like a dofus. Just playing the devil's advocate here. I personally return them.

However, let me ask you this. Suppose you're in a hurry. The closest shopping cart coral is 100 feet away. However, 10 feet away are several shopping carts pushed together and hogging up a parking space. Assuming you shove your cart in the last cart the full way, it won't inconvenience anybody as there is room for plenty more carts before the row gets so long it sticks out in the aisle.

BTW, where I lived in Baltimore you couldn't bring the carts into the lot at grocery stores. There were barriers to prevent it. Or maybe that is just because I lived in a bad part of town (Woodlawn).

Some places have electronic wheel locking mechanisms that don't allow you to go far. Not fun if you are hauling ass with a cart that's front heavy and it suddenly stops.

Some places like the 99c stores make you put .25 in a slot box attached to the cart, its only returned to you when you return the cart to the cart line(so annoying).

I always wondered if there's money to be made returning abandoned shopping carts people take off the property.

Quote: AxelWolf

I always wondered if there's money to be made returning abandoned shopping carts people take off the property.

While you young wippersnappers were in your warm cozy casinos with your fancy abandoned multipliers, I was out in the snow vulturing shopping carts and returning them uphill (both ways)

Quote: AxelWolf

I always wondered if there's money to be made returning abandoned shopping carts people take off the property.

HAH, I think about this every time I am at Aldi, which is a grocery store that does the quarter thing. If I was homeless with no way to earn money, I'd follow people to their car in the least creepy way possible and ask if I could return their cart for them. I bet you could make enough money to eat doing this.

Quote: RogerKint

While you young wippersnappers were in your warm cozy casinos with your fancy abandoned multipliers, I was out in the snow vulturing shopping carts and returning them uphill (both ways)

Good one!

Quote: gamerfreak

HAH, I think about this every time I am at Aldi, which is a grocery store that does the quarter thing. If I was homeless with no way to earn money, I'd follow people to their car in the least creepy way possible and ask if I could return their cart for them. I bet you could make enough money to eat doing this.

Once when I was at the airport a guy was about to return his cart, to get a quarter back. I offered him $1 to just give it to me, because the rental fee is around $4. He said "no."

Quote: Wizard

I guess nobody is interested in the mathematics of this. *sigh*

I suppose an easier problem would to be what is the basic strategy to getting the best parking spot with only one pass through.

Personally, I try to minimize my time driving around the lot, and in particular avoid driving on the lane connectors closest to the store - you often have to wait for walking customers there.

I think your speed estimates of driving in the parking lot, 10mph, is too high. Approaching the area where I plan on trying to park might be that fast, but then in order to detect a spot and slow down and turn into it, and to avoid pedestrians, 10mph is too fast.

All that said, my strategy is to enter the lot somewhere far away from the store, usually at least two lanes away from a lane lined up with the entrance, particularly towards the left or right outsides of the lot, then as I advance towards the store, park in the last open spot that I can see from where I am. That "last open spot" can change as I get closer to it.

That way, I think I am minimizing the total amount of time driving + walking to get into the store, at the expense of, sometimes, a longer walk than if I drove around looking for the closest space available.

Mathematically, I think there are too many unknowns, given my personal experience of variable waiting times at and around the ends of the lot closest to the store.

Quote: Dalex64

All that said, my strategy is to enter the lot somewhere far away from the store, usually at least two lanes away from a lane lined up with the entrance, particularly towards the left or right outsides of the lot, then as I advance towards the store, park in the last open spot that I can see from where I am. That "last open spot" can change as I get closer to it.

Yeah, that is pretty much what I do but never actually thought about it. I dislike parking space hunting and don't really mind walking.

On a related note, does it bother any of the men on the forum when their wives/GFs turn into a parking lot wing man when you're driving and trying to find the ideal space? Yes, I'm sure they are trying to be helpful, but I find it very annoying. Mrs. Wizard always wants to get as close as possible and doesn't mind when a space doesn't have much elbow room because the cars next to it are either enormous pick up trucks or they didn't park in the middle of their space while crowding yours. I put a high value on a space with no neighbors but if it must have neighbors, small cars owned by people who know how to park in the middle of the space.

Quote: Wizard

On a related note, does it bother any of the men on the forum when their wives/GFs turn into a parking lot wing man when you're driving and trying to find the ideal space? Yes, I'm sure they are trying to be helpful, but I find it very annoying.

If she's only doing this in parking lots, consider yourself blessed.

My wives don't do that, but it seems like any time I'm driving someone and I park, they get all bitchy about me parking "too far away". Like bruh, sure, I could'a parked 20 feet closer, but I'd'a had to wait for that car to pull out just so I can park between a damn bus and some 20 y/o POS parked diagonally.

Life hack -- Don't park next to beat up cars with scratches, dings, etc. My theory is they're less likely to be aware of their surroundings and either hit you when they leave or at least slam their door into yours.

Another life hack, if you're looking for a new place to buy or rent, look at the average type of car in the neighborhood or the complex, that'll give you a good idea of how much crime and other shenanigans there is in that area. If it's got newer looking cars that are washed and clean, that's good. If they're older cars, beaten up, etc. etc. then not so good.

I did a very quick simulation involving 3600 cars parking ranomly in a 60h period and looked at various strategies between 22:00 on the first day and igoring the last two hours. Cars spent a range of 30mins, 40mins...90 mins rather than all 60 mins.
There were 100 parking slots and I assumed it took 2 seconds to drive a slot and 10 seconds to walk one.

You started looking at slot X and either parked in the first one or went all the way and turned round to find the nearest slot. I assumed no cars arrived or departed while you did this.

I'm guessing the disadvantage of looking earlier is you park fairly often but have to walk quite a way, whereas starting to look later means you find some of the nicer spots even though sometimes you have to turn round. If you can walk it in 5 secs then the best strategy is about slot 40-45.

Slot start looking	Average time taken
5	605
10	584
15	575
20	568
25	557
30	558
35	558
40	568
45	582

Quote: charliepatrick

I did a very quick simulation involving 3600 cars ...

Good stuff, thanks Charlie. I'm glad somebody found it interesting. I'd be interested to see the average time taken if the strategy is pass x parked cars and then take the first available spot. Kind of like the Best Secretary strategy.