With the “classic” method, I’m referring to the single-line method where everyone is waiting in one big line that passes by all the items. This method works well at low arrival rates. For instance at 0.3 people per second, here’s the state after 100 seconds (see full video) My simulation script produces these videos as a side effect: with random emojis for the food and people. I hope you enjoy it, because there will be a lot more! Focusing on the classic method for now: it turns out to be “unstable” as soon as the arrival rate goes up by just a little bit, in this case 0.7 people per second (see full video)
You can see the people accumulating on the far left side waiting for their turn. The line of people in fact just keeps growing and growing. It turns out that this method has a critical maximum capacity of something between 0.3 and 0.7 people per second. What do I mean with that? Every queue system has some upper capacity that determines the highest possible arrival rate. Above that, the queue keeps growing longer and longer because fundamentally the arrival rate is higher than the rate at which people are exiting the system.
There’s no easy way to estimate the capacity of the system other than varying the arrival rate and seeing at what point the system breaks down. We do that by simulating the system at different rates. Let’s consider some more methods next!
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