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Theme park cast member stories: Doing the math behind theme park wait times

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Published: February 7, 2011 at 12:32 PM

Theme park lines are heaven to a math geek like me.

No, I don't like waiting in them more than anyone else does. But I find the math behind theme park queues fascinating.

Hogwarts Castle at Universal Orlando
Crowds queue for Harry Potter and the Forbidden Journey at Universal's Islands of Adventure theme park

It seems simple, at first. If more people during any given hour get in line for an attraction than that attraction can handle in that hour, some of those people are going to have to wait.

And if you know how many people get in that line, as well as how many riders that attraction can handle per hour, you can do the math to figure out the average wait time.

[Math geek stuff: The average wait time for a theme park attraction equals the number of people who get in line during an hour minus the attraction's hourly capacity. Take that number and divide it by the hourly capacity. Then multiply that result by 60 - for minutes in the hour. The result is your average wait time during that hour. See, I told you it was simple! :^) ]

But experience theme park operators know that the physical shape and orientation of a attraction's queue can affect wait times, as well. I thought about these issues after reading a piece by John Seabrook in the New Yorker last week, about the dangers of waiting crowds.

Seabrook details some of the disasters that happened when organizers failed to plan properly how to queue waiting crowds: One dead at a Walt-Mart in New York, nine at a concert in Denmark, 95 people at a football match in England.

As a former theme park employee, I find it unconscionable that anyone would organize an event sure to draw a large crowd, and not plan for how those people will get into the event. Whether you create a physical queue or a virtual one (such as Disney's FastPass system, or advance reservations for restaurants or museum exhibits), organizers simply must have a plan for handling people while they wait their turn.

Experienced theme park operators have become masters of crowd control, and anyone planning a major event - from a rock concert to an after-Thanksgiving sale, would do well to learn from them.

First, you must create a space in which people can wait. Ideally, the space will be designed in such a way that the people who arrive first are admitted first. First come, first served.

The simplest way to do this is with what we call a serpentine queue. That's because it looks like a long snake, with people lines up single file, in queues that go up and down long parallel rows.

That queue works great if you are going to load people onto an attraction one at a time. But what happens when you need to load an entire theater of theme park guests? Filling a 500-seat theater by pulling people one at a time out of a serpentine queue would take 15-20 minutes, time enough to run one extra performance for most theme park shows. Remember, hourly capacity is everything in reducing wait times. If you're giving up half your performances every hour because you need that time to load the theater, you've cut your capacity in half, doubling everyone's wait time.

That's why theme park shows have pre-show waiting areas. It's just not to entertain you while you wait. It's so that operators can group you into a single mass, the size and shape of the theater, so that the crowd can slide from the pre-show area into the theater in less than five minutes.

Saving that 10-15 minutes every show cycle allows the park to run more performances per hour, reducing wait times and allowing park guests to enjoy more shows each day. But that wouldn't happen without efficient pre-show queue design. (And a fondness for doing the math!)

I encountered this challenge while working as lead on the Tom Sawyer's Island rafts. Our hourly counts were horrible at the beginning of that summer. Part of the problem was the amount of time it was taking us to load the rafts.

Guests would wait for rafts in fenced waiting areas. There was a gap in the fence through which people would walk onto the dock and then onto the raft. The trouble was, people would crowd around that gap, creating a much smaller version of that crushing effect that Seabrook detailed in his New Yorker article.

No one's health was a risk with such relatively small crowds. But it did slow down the flow of guests onto the raft, enough to cause us to take closer to 10 minutes for each round-trip cycle, instead of the five minutes we could do with more efficient loading.

So, with the help of a few of the raft drivers, we redesigned the waiting areas, moving benches around to narrow the area right in front of the raft to the width of the gap in the fence. Each raft's capacity was 55 people, so we moved enough benches in place to define a space that would hold about that many people. This way we could slide that group of 55 from the wait area and onto the raft like a pre-show crowd into a theater. Behind them, the size of the wait area opened up. But by keeping the "pinch point" behind the next group of riders, the crowding around it didn't slow the loading of the rafts, allowing us to move more rafts per hour and reduce the overall average wait time for everyone.

Anyone experienced working with crowds comes to learn that, in a large group, people become liquid. You know how a fire hose can pour a lot more water than a little garden hose? Theme park queues are that way, too. A wide queue, such as at Disney World's Pirates of the Caribbean, can handle a lot more volume through it that a single-file queue, such as Big Thunder Mountain.

But eventually, we'd need the crowd to narrow to single file as we assigned them into specific rows on the boat. Ever notice how on some wide Disney queues, such as Disneyland's Haunted Mansion and the left queue at Disney World's Pirates, there's a U-turn just before you arrive at the loading area? That U-turn functions as a funnel, narrowing the flow of guests to a single file and making life much easier for the cast or team member at the loading position.

Overlook at last detail, and multiple parties will crowd the loader at once, each appealing to go before the others, forcing the load operator to calm everyone down and, literally, straighten things out.

Which, of course, slows the flow of guests onto the ride, reducing the hourly capacity and increasing everyone's wait time.

The math behind this stuff can get heavy duty. In college, I took a math class at Northwestern's Kellogg Graduate School of Management that included a section on queuing problems as part of a discussion of what the instructor called "mechanism design theory." (The instructor went on to win a Nobel Prize for this stuff. Like I said, heavy duty.)

But you don't need to be a Nobel laureate, or even a math geek, to appreciate how a theme park can move thousands of people through dozens of queues every hour of every operating day of the year. Just get in line, keep up with the folks in front of you, and please, be nice to the folks around you as you wait your turn.

You can read more of Robert's stories about working at Walt Disney World at themeparkinsider.com/stories.

Readers' Opinions

From Joshua Counsil on February 7, 2011 at 12:50 PM
All great points, Robert. As one of our engineering projects in the undergrad, we were asked to propose a solution to a problem you've seen on campus. I noticed the crowds around one of the Tim Horton's coffee shops were not only slowing service down at the shop, but also disrupting the flow of people to their classrooms. I approached the solution similarly to the one you implemented for Tom Sawyer's boats, keeping in mind the efficiency of Pirates' queue line.
From M. Ryan Traylor on February 7, 2011 at 2:19 PM
Disney and Universal are the master-queuers in both efficiency and theme. This even carries over to the Mickey & Friends parking garage at Disneyland. Funneled from toll plaza to single lane to parking lane. The only wait here depends on how the guest is paying (Parking pass, Credit/Debit, exact cash, cash w/change).

I for one hope that if the Dodger decide to build a parking garage at their stadium, they hire the Disney team.

From Scott Supeck on February 7, 2011 at 4:06 PM
Great stuff to pay attention to while on line. I think one of the best queue designs has to go to Disney's Expedition Everest. The layout and intense theming not only move guests through at a natural pace, but also creates the storyline of the coming ride. Nice observations, Robert.
From David Sutter on February 8, 2011 at 6:43 AM
Ill bite! But here come the red flag, It works with a level playing field of guests. Who walk up wait their turn. And low and behold, FAST PASS Lane. Which still bothers me to no end ive waited my time in the line and here comes throng of fast pass holders who get the jump on us waiting. No park has yet to perfect that system unless its a dual attraction. Example Space Mountain WDW Magic Kingdom. AS we all know theres the alpahia side and the omaga side. Ones ued for fast pass the other those who didnt get one or have one for another attration. Works well. No dont jump me guys. But If a park has a attraation thats got more then one side or show area good deal. If not. Lets do the math.
From 68.205.193.158 on February 8, 2011 at 5:38 PM
I loathe the fastpass system with every bone in my body. If you are planning on going on only one or two rides with fastpasses per day, it works great, you get fastpasses for those two and hardly have to wait in line. But if you plan on going every ride in the park that uses fastpass, your going to have to wait in some standy-by lines. stand-by lines that are going to have twice as long of a wait than if there was no fastpass. throughtout the course of the day you probably dont save any time at all, and may even lose time if you are hit with bad luck.
From Myrina A on February 10, 2011 at 10:17 PM
Disney has reported a death on one of their carousel rides. This is a part of the quarterly report on in-park injuries. The Disney carousel death was one of three major injuries at the park in Oct, November and Dec.. The 3 accidents were to park visitors over the age of 71.

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