New Jersey High School Team Wins Top Prize for Models that Optimize High-Speed Internet Connectivity

By Lina Sorg

In today’s increasingly digital world, many people take high-speed internet for granted. Individuals with stable internet connections often do not think twice about uploading or accessing social media content, streaming their favorite shows, or video chatting with friends and family. Yet the benefits of high-speed internet extend well beyond entertainment purposes. When many parts of the world shifted to predominately virtual schooling and employment at the onset of the COVID-19 pandemic last year, internet connectivity became more important than ever before.

Those who lack reliable access to high-speed internet are at a significant disadvantage for many daily tasks. It is far more difficult for them to attend online classes and complete assignments, work from home, utilize healthcare portals, participate in civic duties, consume news and information, and so forth. These limitations are especially salient for people in rural and low-income communities, which are often disproportionately affected by connectivity issues. And despite the numerous ways to access the internet—cable, satellites, fiber-optic lines, mobile broadband, etc.—there is no clear path to optimal connectivity; bandwidth requirements depend on region, household type, and usage frequency.

The topic of internet connectivity lent itself well to this year’s MathWorks Math Modeling (M3) Challenge, an annual mathematics competition that is a program of SIAM with MathWorks as its title sponsor. Now in its 16th year, M3 Challenge offers 11th and 12th graders in the U.S. and sixth form students in England and Wales the chance to compete for more than $125,000 in scholarship funds. The contest allows participating teams just 14 consecutive hours to tackle a complex, real-world problem with mathematical modeling and generate a comprehensive report that explains their solutions. All solution papers then undergo three rounds of blind judging by 150 applied mathematicians. This year, the 10 finalist teams virtually presented their solutions to a live panel of judges in late April. An online awards ceremony—in lieu of the live ceremony that is traditionally hosted by Jane Street, a quantitative trading firm in New York City—followed these presentations. Recordings of both the presentations and the awards ceremony are available online.

This year’s problem tasked students with combatting the “digital divide”: the gap between those who benefit from sufficient internet access and those who do not. “The COVID-19 pandemic really brought to light issues with internet connectivity in the U.S. and the U.K. that haven’t been as clear in the past,” Chris Musco of New York University said. “There are many families with insufficient access to the internet who were unable to get their students into online schooling, for example, when communities went into lockdown.”

Musco—who is a 2008 M3 Challenge winning alumnus, co-author of this year’s Challenge problem, and director of judging for the Technical Computing Award—elaborated on the relevance of internet connectivity as an appropriate subject for mathematical modeling and computational thinking. “We’re at a really interesting time for the internet because technology is changing rapidly,” he said. “The past couple of years, we’ve seen 5G technology roll out that is able to provide broadband, connected, wire-internet-like speeds over the air without any need for a wireless connection. This changing technology is clearly changing the conversation around this topic.”

The three-part 2021 Challenge problem first asked students to build a mathematical model to estimate the cost per unit of bandwidth per Megabits per second (Mbps) over the next 10 years for consumers in the U.S. and U.K. Next, they had to create a model to predict a typical household’s need for internet over the course of a year, apply that model to three sample households with varying levels of internet usage, and determine the minimum amount of required bandwidth to cover their total internet needs both 90 and 99 percent of the time. Finally, teams developed an optimal plan for distributing cellular nodes and demonstrated the flexibility of their models in three hypothetical regions.

The champion team from Livingston High School in Livingston, N.J., trained their initial model on the relationships between population density, cost of living, average download speed, and average price per Mbps in the 48 mainland U.S. states. The students found that fixed infrastructure costs are higher in regions with low population densities because the costs are divided between fewer people; the opposite is true in high-density areas. They also noted that internet is more expensive in areas with a higher cost of living, and used average download speed as a measure of the level of infrastructure development that correlates with internet prices. “After collecting data for these three factors, we found that traditional regression techniques would be ineffective and time consuming due to the various skews and nonlinearities in the data,” Leo Stepanewk of Livingston High School said. “We thus utilized random forest regression, a robust machine learning algorithm that is capable of handling and learning complex relationships on its own.”

Once they achieved a root mean square error of 0.660, Stepanewk and his teammates applied the random forest regression to the entire U.S. and U.K., adjusted the population densities and costs of living based on the expected percent change in 10 years, and employed a regressed exponential function to calculate future average download speeds. Ultimately, their model predicted a decrease of $0.23 per Mbps in the U.S. and a decrease of $0.57 per Mbps in the U.K. over the next decade.

Next, the team calculated the bandwidth demands for a given household over the course of a year. “The main factors that we considered were age and occupation status, since these variables had the greatest impact on internet usage patterns,” team member Edward Wang said. “The internet usage patterns for each individual consisted of a probability that they would perform a certain internet task at any given time and a range of bandwidth values for that task. Using these patterns, we created Monte Carlo simulations for each household and simulated the bandwidth demand for 1,000 trial weeks.”

These simulations yielded the minimum required bandwidth of predicted demand for three example scenarios. The students determined that 14.5 and 15.5 Mbps is sufficient to respectively satisfy 90 and 99 percent of predicted demand for a couple in their early 30s with a three-year-old; 20.8 and 21.8 Mbps is adequate for a retired woman in her 70s who cares for two school-aged grandchildren twice a week; and 20.6 and 21.9 Mbps meets the demands of three former M3 Challenge participants who are sharing an off-campus apartment while completing their undergraduate degrees and working part time.

Finally, the Livingston team developed a model to optimally distribute 4G and 5G cellular nodes in arbitrary regions. To calculate the locations for node placement, the high schoolers utilized population density to compute a region’s center and concluded that placing nodes at the center of mass would allow them to reach the most users.

“After learning about the intricacies and expenses of 5G, we decided that it would be worthwhile to create a model that only pinpointed areas where 5G would be beneficial,” Sidhant Srivastava of Livingston High School said. “This is where demographics played a role. We created an equation to calculate the minimum average household income for a region to be eligible for 5G. We also calculated the minimum population density for that region to maximize the consumers for a 5G node.” The team deduced that to qualify for a 5G network, a region should have a population density of at least 777 people per square mile and an annual household income of at least $103,689.32. By combining these two elements, the students created a criteria-based model that optimally identified regions for 5G networks and generated an efficient distribution plan for cellular nodes.

The Livingston team—which included Aditya Desai and Charles Yu in addition to Srivastava, Stepanewk, and Wang—took home $22,500 in scholarship funds for their top-notch solution. As they celebrated their impressive earnings, the students considered the ways in which this experience will impact their forays into higher education and future career trajectories. “Partaking in M3 Challenge has further piqued my interest in the intersection of mathematics, business, and technology,” Desai said. “Having experienced the interrelated forces that play a role in the disciplines, I will look to consciously make these connections and apply my problem-solving approach from this problem down the road.”

This is precisely the goal of M3 Challenge, which seeks to expose talented students to the complex facets of mathematical modeling that are not common in standard high school curricula. It introduces participants to relevant and timely topics and presents problems in an unfamiliar way, forcing them to think critically about real issues; quantify and organize data; and represent, analyze, and predict trends in real-world situations. “The open-endedness of M3 Challenge, along with being able to interpret the question in a variety of ways, is so much different from the math education in school,” Yu said. “In school, there’s always a right answer and it’s pretty easy to know if you’re right or wrong. But for M3 Challenge, we had to pitch a lot of ideas and weigh their strengths and weaknesses before settling on one that made the most sense.”

The Livingston team was coached by Cheryl Coursen, a mathematics instructor at the high school who teaches AP Calculus BC and Multivariable Calculus. She tries to routinely incorporate modeling into her courses and praised M3 Challenge for familiarizing students with mathematical applications. “There are no right or wrong answers in life, as it is not a neat and tidy problem situation,” Coursen said. “Every choice has consequences, good and bad. Students need to see and witness this to understand and be able to face life head on, not just in the job force but in their everyday lives.”

Livingston High School’s paper is available online, as is their final presentation.

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Do you have an idea for a real-world problem that would lend itself well to mathematical modeling? A topic that is not well understood or would make a difference to society, the environment, or general quality of life? The M3 Challenge Problem Development Committee is always looking for problem ideas for future competitions and will work with authors to shape their suggestions and locate relevant data. Submit problem drafts or even just rough ideas online, or send them to [email protected]!

Lina Sorg is the managing editor of SIAM News.