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Queueing Theory

Queueing Theory vs Waitwhile's Queue Management Practice

Waitwhile may strike some as a novel product focused on alleviating the frustration of lines and long wait times. And that frustration is real.


Liza Mock

Fri Nov 05 2021 • 5 min read

Queueing Theory vs Waitwhile's Queue Management Practice

Americans spend 37 billions hours a year waiting in lines. And while Waitwhile is working hard to reduce that number, we do a lot more than wait time management. For instance, our app provides brick and mortar businesses the ability to build useful repositories of behavioral data to help improve their customer experience.

That said, the heart of our business is reducing wait times and we’re always working to evolve our core model. In the works is a new system which will be even better at understanding the different variables at play for different businesses. The theory that enables us to evolve has a long and fascinating history. In fact, there’s an entire field of study dedicated to it. It’s called Queueing Theory.

The History of Queueing Theory

Queueing Theory arose in the early 20th century as a direct result of new technology - the telephone. Around 1908, a young math teacher, Agner Erlang, met Johan Jensen, an amateur mathematician and the Chief Engineer of the Copenhagen Telephone Company (CTC), at the Danish Mathematicians’ Association. The two struck up a conversation and then a friendship, and soon Mr. Erlang was working for the CTC trying to resolve issues connected with phone systems getting overwhelmed by demand - humans do, after all, like to talk.

At the time, phone calls were routed manually by human operators who would connect the caller to the call receiver by means of jack plugs and cord boards. The challenge lay in determining the number of circuits needed to provide an acceptable level of telephone service - basically to keep phone users from waiting “on hold” for too long before being connected. In addition to the number of circuits, Erlang also wanted to know what volume of calls a single telephone operator could process in a given period of time.

Over the course of eleven years, Erlang published three seminal papers that basically created the field of queueing theory. In 1909, he wrote "The Theory of Probabilities and Telephone Conversations" which showed that the Poisson distribution (a probability calculation) can be directly applied to telephone traffic. In 1917, he delivered the "Solution of some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges", which contains his classic formulas for call loss and waiting time.

Finally, in 1920, he published "Telephone waiting times," the foundational work on the subject. Erlang’s studies, formulas, and theories were so important to the field of telecommunications that to this day the international unit of telephone traffic is known as "the Erlang.”

Of course there’s a second aspect to queueing theory that’s equally important, which is the psychological side – how people experience waiting. Qiuping Yu, an assistant professor of Operations Management and Business Analytics at the Georgia Institute of Technology, has done extensive research on the subject, resulting in some remarkable findings. First, she found that providing wait time estimates actually reduces wait times. Why? Because inevitably, once they know how long it’s going to take, some people don’t want to wait. Once they leave, the wait times get shorter for everyone else.

Another finding, which Disney theme parks are known for, is that under promising and over delivering by telling people waits are longer than they actually are, leads to a better customer experience. Interestingly, while getting to the front of the line sooner than expected only makes people slightly happier, having to wait longer than was promised makes people significantly more unhappy. As Yu writes, “The penalty for under-delivery, in fact, was up to seven times larger than the reward for over-delivery.”

Finally, and perhaps most interestingly is the discovery that people who end up waiting longer than expected will take more time when it’s their turn. A study was conducted where calls were made to a bank call center. Callers had some say over how long they spent on a call. Those who waited longer, ended up taking more time once they got through. The researchers theorized that customers who waited longer than anticipated spent more time on calls either because they were complaining or felt like they’d earned more time by waiting. This made them feel justified to ask for additional services.

No matter what angle you look at queueing theory from - be it the mathematical concept of optimization or the psychological effect of waiting - it’s a fascinating field. So how do we think about queueing? Let’s take a look.

Waitwhile’s queueing theory

Our queue management system draws from the concepts created by Erlang. Here are the basics.

In order to accurately estimate wait times, Waitwhile collects wait time data from each and every guest’s visit. We can also feed in other variables that affect wait times – such as staff member, guest service, and party size. The goal is to determine the average time it takes for one guest to move up one spot in line, factoring in the type of service they’re getting, the size of their party, etc. Once we know this, we can estimate wait times for anyone, no matter their place in line. The more data we gather, the more accurate our estimates become.

The system basically has two key inputs: services and resources. Let’s take a hair salon as an example. A hair salon offers many different services - from haircuts to beard trims to shaves to colorings to straightenings - and on and on. But a beard trim takes a lot less time than a coloring. Factoring in those variables is critical to understanding how long a salon chair will be occupied and thus how long the next person in line will have to wait.

The resource, on the other hand, is each stylist. And it could be that hair stylist #1, we’ll call her Joi, is faster with her snippers than hair stylist #2, we’ll call her Leanne. So if both stylists are offering the same service, Joi will finish faster and thus have an open chair sooner. Being able to track how long each stylist takes will also determine how soon the next customer can be seated.

Not every business will have access to key inputs. For instance, at a hair salon you usually know that the person is coming in for a coloring as opposed to a beard trim which makes the wait estimation easier to predict. At a restaurant, the business has very little insight into what the customer’s plans are once they’re inside. One table for two might order salads and sparkling water while another table for two might start with a cocktail, order appetizers, get a main course, and then have dessert and coffee. The restaurant doesn’t know up front which type of customer they’re going to get. That doesn’t mean that restaurants can’t get good estimations of wait times, rather it means that inputs are more limited. The main point here is that the number of variables that affect wait times are huge. Which is why mathematicians were required to crack the problem and build formulas to account for all those variables in the first place. If managing wait times was simple, Waitwhile wouldn’t have a problem to solve.

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