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Analytics Magazine

Five-Minute Analyst: Probabilistic parking problems

July/August 2014

Harrison SchrammBy Harrison Schramm, CAP

Few things make me more conflicted than parking lots. On a personal level, I loathe the whole parking activity. It brings out what I think is the worst behaviors of humankind: hoarding, brinksmanship, scarcity mentality, irrational objective functions… and now you see why as an O.R. professional I love parking lots: because they are so interesting to study.

At the corner of Hades Street and Styx Ave. is (at least to me) the world’s worst parking lot. Here’s the set-up: There is an upper level with metered parking. The meter has a two-hour limit at a rate of $1.25/hour, but pressing a silver button on the meter sets the time to 60 minutes if the meter is currently less than 60 (see Figure 1). This makes parking here free to most visitors. The lower level is a standard parking garage, which has a flat $2 per hour fee which can be validated by the two “anchor” stores, making it essentially free for most patrons as well. While this is light and exploratory, there is serious work going on with parking problems [1].

In the sterile world of figures and mathematics, this sounds like a reasonable way to run a parking lot, and patrons who miss the upstairs free parking will simply renege and take the lower level free parking. In reality, people “mob” the upstairs portion in search of “free parking.” My assistant and I had observed this behavior over a number of weeks, and we were interested in learning about the time parked cars spent in the lot, with an eye for simple metrics such as expected wait time for a parking spot or the expected number of cars “trolling” for a slot. This interest became action (the key for any analysis), and we chose 6:30 p.m. on a Thursday evening – a time that we knew the parking lot would be full – to collect data from the meters, which is displayed for anyone who wishes to see.

Figure 1: A “smart meter” in a parking lot. This meter has a button next to the coin lot that may be pressed for a free hour of parking. Coins may be added for additional time, up to two hours.
Figure 1: A “smart meter” in a parking lot. This meter has a button next to the coin lot that may be pressed for a free hour of parking. Coins may be added for additional time, up to two hours.

What we found was surprising.

We expected to see uncorrelated parking lot data. We did not expect to find many over-time parking spots. I hoped that the data would be exponential – which would lead to nice, clean analysis. What we discovered was, well, a mess.

Of the 100 parking spots surveyed, 25 percent were “flashing” or over-time (violation). Of the parking spots that were not over-time, six showed times over one hour, implying that the persons parked there had in fact put money in the meter. We are completely discarding the possibility that someone would park in a spot that had been previously occupied but was not vacated, i.e., showing up with 30 minutes remaining on meter and not pressing the button/inserting coins. I had hoped that the sojourn times would be exponentially distributed, but that is a case that is pretty difficult to make with this dataset (see Figure 2).

Figure 2: Histogram of raw parking meter data. Note the tri-modal nature of the data. “Overtime,” i.e., flashing parking meters are represented by -1 in the red-shaded oval and constitute the large bar at the origin of the graph. Known paid parking meters are at the right and have a blue oval.
Figure 2: Histogram of raw parking meter data. Note the tri-modal nature of the data. “Overtime,” i.e., flashing parking meters are represented by -1 in the red-shaded oval and constitute the large bar at the origin of the graph. Known paid parking meters are at the right and have a blue oval.

Now, we don’t actually know how many patrons have paid, or how many have simply run over. However, there are 100 parking spots considered, and of these, six currently have clocks over one hour. We can (crudely) estimate [2] the true number of paid parking spots by realizing that we are observing the last hour of what may be a two-hour process. Therefore, we think approximately 12 parking spots have been paid for at any given time.

Figure 3: Histogram of parking time remaining, less than 60 minutes. Approximately six of these data points are actually spill over from “paying” customers.
Figure 3: Histogram of parking time remaining, less than 60 minutes. Approximately six of these data points are actually spill over from “paying” customers.

Yes, But What Does it all Mean?

So in one sense, the distributions of the data are irrelevant; there are 100 parking spots on average, and the average time that a parking spot is occupied is some time greater than 27 minutes. If we make the (not bad!) assumption that the parking spots that run over are occupied for 90 minutes, then the average occupancy is 43 minutes. In a lot with 100 spots, this means that on average, one spot comes open every 30 seconds. This doesn’t sound so bad. If we treat the system as a queue, and use the (observed) steady state cars waiting of three, we can place a rough lower estimate [3] that a new car arrives every 30 seconds looking for a parking spot, and that they have between a 15 percent and 25 percent chance of finding an open spot. These crude estimates, however, do not agree very well with observation, because they neglect the “blocking” effect of other cars waiting for spots to open up. A better analysis of this parking lot would involve simulation, which would go beyond our intent.

The World’s Worst Parking Lot?

Because of the behavior of the drivers while trolling for a parking spot, it might be considered the world’s worst parking lot. Enforcement of the parking policy might help because it would decrease the sojourn times of the cars parked in the lot, but there is no guarantee, and – more importantly – no direct incentive for the parking lot owners to do so. This is because the number of “free” parking spots is fixed, and once they are filled, they are filled, regardless of by whom. From the lot manager’s point of view, it doesn’t matter if they are “long” or “short” parkers. In fact, the rate structure is such that short parkers are slightly more lucrative for the parking lot owner than parking above ground.

In conclusion, it’s probably a bit of literary hyperbole to imply that this is the world’s worst parking; I’m sure there are others that are much worse. This is because I like to make short trips to this area and visit the locations that don’t validate parking, and I really don’t like the risky behaviors aggressive parkers participate in. On the upside, there’s time to write 12 articles in a single push of the button!
I’d be interested in hearing real contenders for the “World’s Worst Parking” lot.

Update: Between the original draft of this article and its publication, the parking lot in question began installing an electronic system to help customers determine how many spots were available before entering the parking “queue.” It has yet to be determined if it will change the behaviors of the parking lot. Look forward to an update in a future column!

Harrison Schramm (harrison.schramm@gmail.com) is an operations research professional in the Washington, D.C., area. He is a member of INFORMS and a Certified Analytics Professional (CAP).

NOTES & REFERENCES

  1. Fabusuyi, Hampshire, Hill and Sasauma, 2014, “Decision Analytics for Parking Availability in Downtown Pittsburgh,” Interfaces, INFORMS, Hanover, Md.
  2. This is just an estimate. More delicate techniques may be applied.
  3. Using the M/M/1 queuing model to find the “lower” or optimistic estimates, and the M/G/1 queuing model to find the upper estimate.

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