You don’t want to wait in a theme park? We hired a mathematician to tell you the tips and tricks!

For weeks, you looked forward to the nice summer day where you would take the train to Disneyland Paris with your friends. Once you entered the park, you inspected the map and saw that there are 3 must-do thrill rides: **Big Thunder Mountain Railroad, Indiana Jones and the Temple of Peril** and **Hyperspace Mountain**; the three most-visited and stunning roller coasters of the park. (If you never visited Disneyland you can see these attractions in the pictures below. Don’t they look super exciting?) You decided to ride these attractions in the order they appeared on the map. Unfortunately, half-way through the day you realized that this was not the best strategy.

You ended up waiting for hours, since all other visitors had the same strategy.

But how long are you prepared to wait before you can enter a thrill-ride that only brings a couple of minutes of excitement?

5 minutes? 15 minutes?

The waiting times in a popular theme park, like Disneyland Paris, can be more than one hour! Nevertheless, this doesn’t imply that you need to accept your fate. Next time you visit Disneyland Paris you can use the following tricks to beat the waiting lines and get more attractions done than the other visitors

Let’s introduce Lia, she will be your personal guide for an optimal day of Disney fun and she will reveal some methods to reduce your waiting times at the various attractions. Lia is a young queueing theorist, which means that she is a mathematician who studies waiting times and queue lengths in various settings. Her specialty, reducing waiting times in theme parks.

## Why do mathematicians like Disneyland?

First of all, why is Disneyland so interesting for Lia? Well, Disneyland and other theme parks are mathematical version of heaven for queueing theorists.

Every attraction itself can be seen as a system consisting of a waiting line (queue) and the roller coaster itself. Visitors join the end of the waiting line and leave the waiting line once they board the attraction. If Disneyland would consist of only one roller coaster, it would be rather easy for Lia to calculate how long your average waiting time would be and so the number of times you could return to this attraction in one day. This could be done by carefully observing the single attraction and counting the number of visitors in the park. Luckily theme parks are much larger than this, what makes Lia’s job much more interesting and challenging.

On top of the high number of different attractions, most of them operate according to different schedules.

For instance, one train of the Big Thunder Mountain can pick up 30 visitors simultaneously and each ride will always take exactly 3 minutes and 56 seconds. On the other hand, Mickey Mouse can meet only one family at a time and there will probably be some variability in between families. Suppose that the family in front of you has 7 children and all of them want an autograph and a picture with the most famous mouse in the world…

Human behavior also plays an important role when queueing theorists want to analyze the waiting times at the different attractions. Although you might expect that young adults only visit the thrill rides, no one will deny you and your friends the access to the Carrousel of Lancelot (the local merry-go-round). Therefore, you cannot strictly divide al visitors into different groups based on their preferences and behavior.

So, the total number of attractions, their different service schedules and the (unpredictable) behavior of the visitors make it enormously more complex for Lia to oversee the whole park and get a grip on all the queue lengths and waiting times, that is exactly why Disneyland is mathematical version of heaven for queueing theorists.

## Good Preparation is half the battle

Lia her first advise is:

*Spend the days before your trip behind your computer*

The current waiting times for each attraction are available on various websites.

A couple of days before Lia travels to Paris, she monitors the waiting times of all the attractions, and carefully analyses this information. For instance, she constructs diagrams, like the one on the right, that indicate the average waiting time at the Big Thunder Mountain for each moment of the day.

Lia immediately observes that the peak moments of this attraction are around 13 and 17 o’clock, these are the moments she would avoid this attraction. This is a trend she notices for multiple attractions, so why not schedule your lunch or souvenir shopping at these moments? Moreover, it appears that the waiting time is reasonably small right after the park opened and somewhere around 15 o’clock.

But your aim is to do more than one attraction. How could you schedule the remainder of your day as efficient as possible?

## More than one waiting line

Lia her second advise is:

*Go to Disneyland alone and jump the queue.*

Does this confirm that mathematicians, like Lia, are lone wolfs and antisocial? Of course not, but Lia encourages you to make use of two waiting line constructions the Disney engineers developed. They are also aware of the long waiting lines so they came up with “single-rider” and “fastpass” lines next to the regular waiting line. Both lines are there to tackle different problems.

First, if you are the only one brave enough to go for a ride on the Hyperspace Mountain it might be beneficial to opt for the single-rider queue. The single-rider queue is a separate line with people who want to ride the attraction alone. Every time a seat remains open because other visitors in the regular line want to ride the attraction together, the first person from the single-rider queue can take it.

But how much can you gain from this? Lia made some calculations and simulations and concludes that it is highly beneficial to join the single-rider queue if there are not many visitors present. For instance, there are 960 people waiting in the regular line and 10 people in the single-rider line at the Hyperspace Mountain, that has room for 24 visitors simultaneously. On average you can board the ninth train if you had chosen the single-rider queue, but you can only board the 47^{th} train if you queued at the regular line. So you can board the attraction 38 trains earlier if you opt for the single rider line!

How was Lia able to draw this conclusion?

Well, she knows that approximately every minute one of the in total five trains will start its ride. So if the plate above the entrance of the regular line indicated that you have to wait 40 minutes, this implies that there are roughly 40×24=960 visitors in this line since each train picks up 24 visitors. Among these visitors, 30% are pairs, 30% consists of trios and 30% consists of groups of four, five or six visitors. With this information, Lia can make simulations that mimic the actions of the Disney employee that directs the visitors to the train.

First you ask how big is the group of visitors at the end of the waiting line . Then you check if there are enough empty seats in the train. If this is the case, they are allowed to board the train. If the group is too large, all remaining seats are filled with single riders. This gives Lia the opportunity to estimate which train you would board if you joined the regular or the single-rider line. The difference is enormous as you can see in the diagram below. Next to the setting described above, Lia conducted the simulations also for a setting with a shorter regular line with an estimated waiting time of 20 minutes. The difference might be smaller but it is still worth joining the single-rider queue!

A second invention by the Disney engineers is the fastpass. As Lia could already see from the waiting time data, there are some peak moments where a lot of the visitors want to ride the attractions. It would be beneficial for all visitors if these peaks were less high and if all visitors were nicely divided among the different attractions all day long. Therefore the fastpass was invented, these tickets are distributed as long as supplies last at a ticket machine in front of selected attractions. They are free of charge for all visitors and have a particular time slot indicated on them. During this time slot you can return to the attraction, and you will be rewarded: you are allowed to join the fastpass line. You will receive ‘priority’ over the other visitors in the regular line, which reduces your waiting time to a couple of minutes.

For instance, for the Big Thunder Mountain, there are two separate platforms where visitors can board one of the in total five trains that are on the track. One of the platforms is used for visitors in the regular line and the other platform is uniquely occupied by visitors with a fastpass. Then it is as if there are exactly two copies of the same attraction, where one of them has a shorter waiting line. Of course, that is the line that also Lia prefers!

So next time you visit Disneyland, or any other theme park like the Efteling, Walibi or Phantasialand, think about Lia and her fascination of waiting lines. Do not blindly follow the crowd, but remember the different waiting line constructions and how you could use them to your own advantage!