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The Laboratory Berlin Gummi Queue: A Mathematical Approach

Hosted every month, but seeing its busiest months during Folsom Europe and Easter, the Gummi rubber party at Laboratory Berlin is the place to go for most rubber heads. On busy nights, it’s however known for its insane queue. Everyone seems to have a favorite idea of how to arrive: some suggest early, some suggest late, and nobody seems to agree. On busy nights Gummi hosts easily over a thousand people. This is easy to notice, since everyone is numbered on the wrist, starting with 1, and by the end of the evening, you see people with numbers over one thousand. Adding to the stress of going to Gummi, the doors are only open for two hours (from 22:00 to midnight), and entry is not guaranteed if the queue is not cleared by midnight. In order to examine this problem from a simple mathematical point of view, let’s make some basic assumptions for the sake of simplicity. These might not reflect real life conditions, but they are necessary in order to generate some graphs. To make it more interesting, let’s also assume that by midnight there are 200 people left out, so the total number of rubberists on a given night is 1200.

Assumption #1: Rubberists’ arrival times to the queue at Gummi follow a normal distribution

Also known as the Gaussian distribution, the normal distribution is a useful simplification for phenomena, although its applications to life sciences are limited, where the log-normal distribution is used more often. In the context of Gummi, the assumption means that the mass of people’s arrival times is centered around an average (which can be changed for scenario planning purposes) and how far out people’s arrival times are spread out is reflected in the standard deviation. Using example values of an average of 9:20pm and a standard deviation of 50 minutes, and sectioned into 10-minute chunks, the arrival of 1200 people would look like this:

Assumption #2: Gummi processes rubberists at fixed speed

In order to calculate the length of the queue at any point in time, we need to know how people are let in. As there is generally a fixed number of people handling arrivals, it’s fair to simplify Gummi to a First-In-First-Out (FIFO, not to be confused with fist-in-fist-out) queue with a fixed speed. Assuming Gummi processes a total of 1000 rubberists in 120 minutes while the doors are open, this means a speed of about 8.3 drones per minute. Calculating the queue length is now simple: the queue only increases until 10pm, and after that is affected by both people arriving and people being processed at fixed speed. Note that in our assumption, at midnight when the doors close there are still about 200 people in the queue.

Calculating total wait time

Calculating your individual wait time is now easy. It is:

```(minutes before opening when you arrived) +
(queue length on arrival) / people processed per minute```

The graph below shows the wait time for any arrival time, although it doesn’t take into account that if you arrive too late, you won’t actually get in:

With these example values, it’s easily seen that the waiting time between 8pm and 10:40pm does not significantly change, and with these assumed numbers, people arriving after 10:40pm are actually not getting in. The numbers used in the example are fictitious. However, plotting the graphs with different arrival time averages and standard deviations does not significantly change the flatness of the waiting time curve.

So when should I arrive at Gummi?

Based on this simplified analysis, the arrival time generally does not matter that much. Arriving way too early guarantees fast access after doors open, but you end up spending most of your time in a non-moving queue. Arriving after opening will get you to the end of a long queue, which psychologically might be more stressful. Arriving much later than the opening includes the risk of not being let in at all. Perhaps the Kiwi physicist Ernest Rutherford said it best:

“If your experiment needs a statistician, you need a better experiment.”