Queueing Theory

I just saw this tweet and my mind was blown but then I was super skeptical. Can this possibly be true? If it’s true, how? I’m tagging @SteveBloom but maybe we have some amateur queueing theorists who can illustrate why this would be the case.

There’s more in the thread but it doesn’t really explain why. I took a supply chain class in college and I can understand generally how traffic pileups can happen around bottlenecks, but this seems especially extreme.

In any case, this all seems very important to Disney and their lack of understanding of these principles is at work in the parks and hotels.


I think (am pretty sure) that Len and Jim did an entire episode just about queueing theory (which is largely the basis for TP) once? Or maybe I heard Len as a guest on another podcast where he talked about it at length?

It’s pretty damn fascinating.

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Cool, thanks!


Looks like Ben has “borrowed” a classic example. Quick google search results (with explanations):


Nope, not buying it. I don’t know if the formula they talk about is valid or not, but if it is, they are misapplying it.

From the explanation @linsalt posted:
“The waiting times given above assume the model has approached its steady state. That is, the bank has been open long enough for the line to reach a sort of equilibrium.”

That assumption takes it out of the real world, because the vast majority of banks are not open infinite hours.

Most banks are open 9 hours a day at most. At 5.8 customers/hour, that’s 52.2 customers per day. Let’s call it 53 to simplify it.

Let’s assume the worst possible scenario. All 53 customers arrive at the EXACT same moment. Customer 1 waits 0 minutes. Customer 2 waits about 10 minutes. Customer 3 waits about 20 minutes. All the way up to poor Customer 53, who waits all 520 minutes. (Assuming the poor teller sticks around.)

The total wait time for all 53 customers combined is 13,780 minutes. The average wait time is 260 minutes, or 4.3 hours. If any customers arrive at different times (highly likely), it goes down from there.

The bank has to be open over 10.5 hours AND all of the customers still have to arrive at the exact same time to get the average wait up to 5 hours.

P.S. I noticed that using the formula it’s actually 4.83 hours (the original problem says almost 5 hours). That’s still a bank open for over 10 hours with all the customers arriving at the exact same time.


Thank you! I knew something seemed off about this.

The general point about having two tellers is a good one though. And understanding bottlenecks is essentially to having fast-moving queues. There were times at Disney (and at the airport) where I thought they could do with a course on queueing theory.


As long as it’s APPLIED queue theory. :joy:

Maybe Disney only knows about the theoretical bits? Might explain some things. :thinking:


I think they understand queuing theory. They WANT us to wait in line. Len explained it once and I think he heard it from Disney. People don’t value what they don’t pay for. Waiting in a line is paying for it with time. Also shows that other people are interested in it too. So Disney deliberately allows a queue to build.

At first I thought that was the silliest thing ever. Then I realized that I’d just gotten back from my third trip to USF and my first to ride Cat-In-The-Hat, which we absolutely loved! We’d never even noticed it before that. Maybe we thought it was a restaurant, and maybe that was because it never has a line. Only reason we did it the third time was because I made a list of things we hadn’t already done.


Agree that Disney absolutely understands queuing theory. They also understand that the trade off of having people in (at least some) lines is still more valuable to them than the expense of staffing up or building rides with higher capacity. Whether it’s the perceived popularity or that the gate admission is enough vs spending money on food/merch or what, even the lines are still meeting Disney’s goals (even if they aren’t meeting the individual park-goer’s goals).


Interesting point.

I went to Cypress Gardens in August of 2005, after the Central Florida schools had gone back into session. We rope dropped it and when we pulled into the parking lot, we were the first car there. It was a real “welcome to Wally World” moment.

At times we had a staff memeber follow us to which ride we wanted so they ould operate it. We had two 4 year olds and a 2 year old with us. It was literally a perfect day in a theme park.

But I remember it as the sad state fo Cypress Gardens at the time. Because, no waits, no crowds.

I was happy when it got revived as Legoland.


Wow. That sounds both cool and creepy.
And very much Wally World, even to the employee following you around. Except of course they weren’t forced.


When I was there in the 80s, despite there being larger crowds at the time, the rides were handled the same way…basically one ride operator who would jump between rides. This is because the rides themselves were not really a main attraction. You didn’t go to Cypress Gardens for the rides. You went to see the gardens and the shows. The rides just seemed like they were there as a way to appease the young grandkids they would bring along with them who found it boring to look at a wall of bamboo, etc. :slight_smile: (Uh…not that I ever felt that way or anything…)


OMG that is too funny that it has always been that way.

It really was the perfect day with small children. I see how Lego Land is the perfect evolution for that park.

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I do think this is true to a degree.

For example, when I was going into Toy Story Mania, I heard one CM say to another, “C is completely empty.” The other CM replied, “So is B - I’m just going to flood it until it’s full if that’s ok.” To which the first CM said, “that works.”

Translation: They weren’t trying to minimize the wait for us - they were trying to fully utilize one loading dock before they started using the other. This will minimize waits on average, but not for the individual riders who are assigned to the longer line for crowd control purposes.

But there are sometimes when I saw some completely unintentionally bad queue management on my trip. Like when I was at the airport (sorry, I can’t think of a Disney example off the top of my head) and they had two queues going into one TSA agent, and he kept pulling two people from one line for every one from the other.


I would have lost my mind.


@amvanhoose , I agree that the example does not mimic reality and is overstated.

The key to this overstatement is the following presumption made in the example:
“service times are assumed to be exponential with mean 10 minutes”
The author doesn’t really explain or defend this presumption.

I think these comments from the article’s readers emphasis this.

mikemikemikemikemike says:
“You state that each customer takes 10 minutes to service , a customer comes every 10.3 minutes … How the hell is there a 5 hour wait”

Erik points out:
“If the teller gets two fast customers in a row (1 minute each), the teller sits around and waits for 9 minutes for the next customer”

“But if the teller gets two slow customers in a row (20 minutes each), then by the time the second slow customer is finished, there’s two more people in the queue and a fifth one showing up”

“Slow people make for big queues, and they mostly aren’t cancelled out by fast people”.

David Anderson says:
“Service times are probably much closer to Normal than Exponential”
“Long tail of exponential service times is what creates lots of the backlog and excess waiting times”

And from the author of he second article:
“Our lack of intuition about queues has to do with how much the word “average” is hiding”

I was really more interested in the second article and the psychological aspects that are involved.