Waiting in Line
By Jackie Swift
Most of us would rather avoid waiting in lines, so in an effort to make waiting more pleasant, many businesses have adopted virtual queues. Using an app, we can take our place in the line at the restaurant or hair salon and still do other things while we wait. Are virtual queues, however, any faster or more satisfying than physical ones?
“What is the effect of giving people the virtual option?” asks Jamol Pender, Cornell University College of Engineering, who studies the dynamics of queueing networks and applies his research to very practical problems. “When you compare the observable or physical queue where people can see everything that’s happening to the virtual or unobservable one, you find some interesting things.”
Virtual Queues versus Physical Lines
Pender and a colleague, independent researcher Otis B. Jennings, observed that people would often leave the virtual queue but forget to log out of the app. In essence, they’d still effectively be in the queue, which caused subsequent users to expect a longer wait than would ultimately transpire. Valuable information about the queuing system was being lost, and this had the effect of making the apps imprecise.
When one ride had a low wait time, a few minutes later an influx of people joined that line. Once that line was too full and the line for the other ride was shorter, we’d see a flow of people heading the other way.
Moreover, users would then choose not to join the queue because they perceived it as too long. Pender and Jennings believed the system would function more accurately if system managers could anticipate the real-time abandonment of the queue by users.
“We asked whether it’s possible to give an approximation of how many people currently in the system would eventually leave without service,” Pender says.
The researchers derived a new formula that shows the number of people who would ultimately abandon a virtual queue depends quadratically on the queue length. In addition, not taking into account the number of people who will leave the queue, the unobservable system always appears to be larger than the observable one. Yet, after subtracting those people who will leave, the unobservable system is always smaller.
“So what you see isn’t necessarily what you experience,” Pender says. “In the unobservable case, you see a lot more people, but your experience is better than you expect because some of those people will leave without you realizing it. In the observable system, people tend to abandon the queue up front, so you’ll see a shorter line, but the wait may end up longer.”
The Human Behavior Connection between Waiting for a Restaurant Table and Waiting in Traffic
Pender and Jennings applied math to the problem of virtual queue abandonment and came up with the number of people who will leave given a certain queue length, but they’d like to go further. “We want to study specific human behavior, so we can tell restaurants, for example, exactly how much patience their customers have when it comes to waiting for a table,” says Pender. “Or maybe we could say something like ‘wealthy people tend to have less patience than less wealthy people,’ so a fancy restaurant should use this distribution to approximate the waiting time and a less fancy one should use another distribution.”
Pender has looked at other systems and the impact that information on delayed queue length has on the outcomes. In one project, he collaborated with Cornell colleagues Richard H. Rand, Mathematics/Mechanical and Aerospace Engineering, and Elizabeth Wesson, postdoctoral associate, Center for Applied Mathematics, to study the effects of traffic delay times on the behavior of drivers crossing the George Washington Bridge to enter New York City from New Jersey. The bridge has an upper and lower level with signs that give the traffic wait times for both levels. The researchers came up with probabilistic models showing that the delay in updating the wait time information has a big impact on traffic congestion outcomes. Drivers will continue to follow the signs to the level showing less wait time even after congestion there has become critical. Once the wait time is updated to reflect reality, traffic will surge over to the other level until the same outcome causes it to move back again to the first level.
“In a traffic setting you can see these oscillations form as you delay the time that people are actually observing,” Pender says. “But these types of oscillations actually show up in all kinds of waiting situations.”
What Waiting in Queue at Disneyland Revealed
Disneyland provided the setting for Pender to pursue further research into oscillation. He and ORIE undergraduate student Samantha Nirenberg ’18 and ORIE graduate student Andrew Daw ’20 analyzed the effect of queue waiting times on rides at the amusement park. “We looked at the wait times reported for park rides on the My Disney app,” Pender explains. “We isolated a section of the park that had two rides that were equally liked by people, and we looked at the length of the lines for each ride versus the reported wait times. When one ride had a low wait time, a few minutes later an influx of people joined that line. Once that line was too full and the line for the other ride was shorter, we’d see a flow of people heading the other way. So there was that oscillation again.”
They were able to find that the oscillation wasn’t fully caused by delayed information in the My Disney app but also because Disney rounded the waiting times up in increments of five minutes. “If the wait time for a ride was two minutes, for instance, the app would round up to five minutes, and if the wait were eight minutes, the app would round up to 10 minutes,” Pender says. “We observed that this type of rounding also induces these oscillations. And then there’s also the distance time to get to the ride; the queue length for a ride will have changed by the time a person reaches it. If you view this as delay, then the delay plus the rounding causes even more oscillation.”
Pender and his collaborators built a probabilistic model that describes the situation at Disneyland. “We can clearly show change would have some benefit,” Pender says. “If Disney changed the way they communicated waiting time to increments of one minute instead of rounding to increments of five minutes, people would expect the waiting time to move up or down by one minute every minute. They would see the wait’s trajectory. They would funnel toward the ride with the lower wait and that wait time would go up, while the ride with the higher wait time would have a negative slope. This would even out the oscillations. The information we give people is important, and we have to understand that it has an impact on how people behave. Changing the wait time increments from five to one minute intervals seems like a meaningless thing, but it actually can yield drastically different results.”
Pender especially enjoyed working on the Disneyland project with Nirenberg and Daw. “It took all three of us to get it done,” he says. “It’s a very nice example of collaborative effort between all levels of the university. We all contributed in different ways and our collective experiences at Disney made a huge difference in the quality of the work.”
Originally published on the Cornell Research website. All rights are reserved in the images. If you’d like to reproduce the text for noncommercial purposes, please contact us.
