In July 2024, the UK’s National Health Service launched a new mobile app that promised to ease emergency room visits. It displayed real-time wait times at local hospitals, allowing patients to decide where to go for faster treatment. It was hailed as a breakthrough in digital health services. The app aimed to turn confusion into clarity and convert wasted time into smarter choices.
But what happened next surprised even the designers.
Soon after the app went live, some emergency departments reported sudden spikes in patient volume. These were followed by longer wait times, particularly at the hospitals that had just been listed as having the shortest queues. It appeared that too many people were using the app to avoid long waits, only to crowd the very places they were trying to beat.
This pattern was not just anecdotal. It reflected a deeper issue uncovered by three researchers in France, who developed a mathematical model around this exact phenomenon.
Their conclusion was both simple and unsettling. The act of checking queue length, especially when done by many people at once, might be making the system worse for everyone.
The Hidden Cost of Information
The modern world thrives on visibility. From Google Maps’ traffic heatmaps to ride-hailing surge indicators, we live in a world where information is assumed to lead to efficiency. The idea seems intuitive. If we know what lies ahead, we can plan smarter and act faster, helping us avoid mistakes.
But in their paper titled “Queues with Inspection Cost: To See or Not to See?”, researchers Jake Clarkson, Konstantin Avrachenkov, and Eitan Altman question this assumption.
They modeled a system in which individuals arriving at a queue could choose among three actions:
- Join blindly without knowing the current wait
- Balk blindly and walk away
- Pay a small inspection cost to view the current queue length before deciding
That inspection cost might be something minor, such as using mobile data while roaming, waiting for a slow app to load, or experiencing mental fatigue during an already stressful situation. For some, it could also mean privacy concerns or cognitive overload.
Surprisingly, the researchers found that when many people start inspecting the queue, especially when the cost is low but not zero, the overall performance of the system begins to degrade.
This is not just a theoretical oddity. It represents a real-world paradox where attempts to optimize individual experiences can produce worse results for the group.
The Journey of Discovery
At the center of this research is the M/M/1 queue. This well-known model describes a system with a single server, where arrivals and service times follow exponential distributions. It is widely used in modeling service desks, customer lines, and data networks. But Clarkson, Avrachenkov, and Altman added a unique twist.
They expanded the classical model by introducing strategic decision-making. The foundation for this work came from Naor’s 1969 model. In that setup, customers could see the current queue length. They would then join the line only if the number of people ahead was below a certain threshold.
This threshold is calculated as: ne = ⌊Rμ / cw⌋.
Where R is the reward for being served, μ is the service rate, and cw is the cost of waiting per time unit.
In Naor’s world, everyone sees the queue and acts with perfect information. But in the real world, information often comes with effort. Whether through data usage, app loading times, or cognitive fatigue, accessing queue details is rarely free. This observation led the researchers to introduce a critical modification: the option to inspect the queue at a cost (CI).
With this addition, customers now faced three choices upon arrival. They could join blindly, balk blindly, or pay a small inspection cost to see the queue before making a decision. This transformed the system from a simple yes-or-no setup into a layered decision process. It reflected how people actually behave in environments like hospitals or theme parks.
The team calculated equilibrium probabilities for each possible action. Depending on the relationship between the reward, cost of waiting, and inspection cost, different behaviors emerged. In some situations, everyone inspected. In others, nobody did. Often, the result was a mix, with some customers inspecting while others skipped it.
The authors also developed utility functions to analyze the strategies. They identified the expected utility for those who inspect, denoted as UI, and for those who join blindly, as UJ. The difference between them, UJ − UI, helped define whether inspection was worthwhile.
A key insight came from observing how both UI and UJ changed as congestion increased. As more people joined the queue, the expected utility for each new arrival decreased. This mirrors real-world experience. When a line gets too long, even the benefit of knowing its length might not be enough to make joining worthwhile.
To make the findings clearer, the researchers mapped out equilibrium zones. These maps visually showed how different values of R and CI influenced the likelihood of inspection. This added a new level of depth to the mathematical framework. It also gave the study strong practical relevance, helping to explain how even small changes in inspection cost could shift crowd behavior on a large scale.
This Mathematical Model Explains Digital Congestion
The turning point in the study came from analyzing how these strategic behaviors affected social welfare. This measure calculates the total benefit for everyone in the system, subtracting the total waiting and inspection costs.
One might assume that increasing the reward R would improve outcomes for everyone. Higher rewards would attract more participants, which could spread the load and improve satisfaction. But the researchers found something unexpected.
When R increased, more people started to join blindly. They were motivated by the higher reward and were willing to wait. But this led to longer queues. As a result, waiting costs rose and the average experience worsened. Social welfare, instead of improving, started to decline.
In contrast, reducing the inspection cost CI had a positive effect. When people could easily and cheaply access queue information, they made better decisions. Those who found the queue too long chose to leave. This filtered out excess demand and kept the system more stable. Shorter queues followed, and overall efficiency improved.
To visualize this, the researchers generated a welfare contour map across a grid of different R and CI values. The results were clear. Welfare improved more steeply when the inspection cost decreased than when the reward increased. In simple terms, it was more beneficial to make inspection easier than to make waiting more attractive.
This insight goes against common thinking in service design. Many organizations try to reduce customer frustration by improving the service itself. They add staff, offer bonuses, or enhance amenities. But this research showed that a smaller investment in making queue data more accessible can lead to bigger system-wide benefits.
Examples include installing visible, real-time queue displays in public spaces. Optimizing apps so they load faster also helps. Even simple tools like QR codes that lead directly to live wait times can shift behavior in productive ways.
The deeper lesson is this: systems don’t only depend on how well they function. They also depend on what people believe about how they function. Making it easier for people to form accurate beliefs about the queue can be just as powerful as increasing the actual service speed.
This realization reframes how we think about user experience. Efficiency is not only about performance metrics. It is also about the decisions people make in response to the signals we give them.
The Real-World Impacts: From Hospitals to Airports
The implications of this research extend far beyond hospitals. It offers insights into how we design smart services in various settings.
In healthcare, the findings suggest that hospitals and public health systems like the NHS should focus more on making queue information accessible and less on increasing service rewards. This may include public signage, simplified interfaces, and real-time updates that reduce inspection friction.
Airports and amusement parks, which already deal with large crowds and fluctuating demand, could also benefit. If too many people rely on queue-tracking tools and make the same decision, they can overload one area while others remain underused. This creates a synchronization problem that amplifies delays instead of reducing them.
This scenario resembles Braess’s Paradox in traffic planning. In that paradox, adding a new road can slow down traffic because everyone chooses the same new route. The same thing can happen with queue inspection tools.
The research encourages us to rethink what optimization really means. More control and information do not always result in better outcomes.
What This Says About Optimization in the Digital Age
We live in a world where data is always at our fingertips. The more we know, the more we expect to gain control over our lives and our time.
But this research challenges that assumption. It shows that the pursuit of perfect knowledge and complete control can backfire. Systems may become less efficient, not more.
This study is not only about queueing theory. It reflects a larger issue in the digital age. We are flooded with signals, metrics, and dashboards designed to help us make better choices.
So the next time you reach for your phone to check the line at a hospital, an airport, or a ride, consider this. Is that information really making your experience better, or is it feeding a cycle that makes the system worse?
In an age obsessed with knowing everything, perhaps the most radical decision is to stop checking and simply trust the line.
Reference: “Queues with Inspection Cost: To See or Not to See?” by Jake Clarkson, Konstantin Avrachenkov, and Eitan Altman, 25 March 2024, arXiv preprint.
DOI: arXiv:2503.13232
TL;DR
Checking queue times might make waits worse. A new study shows too much info causes crowding. Easier inspection helps more than faster service.
