Matt Chase
What Queueing Theory Reveals About Criminal Justice
- By
- November 17, 2025
- CBR - Operations Management
Queueing theory—a mathematical framework that’s typically used to study, understand, and manage wait times and lines—is usually applied to improve efficiency in settings such as grocery stores, hospitals, and manufacturing facilities. But it could also be used to help improve the overall functioning of the criminal-justice system.
For example, Chicago Booth PhD student Zhiqiang Zhang, Purdue’s Pengyi Shi, and Booth’s Amy Ward sought to quantify the trade-off between fairness and efficiency when allocating resources. To do this, they developed a simplified mathematical model in which the criminal-justice system operates like a service system, whereby offenders are routed between two different stages. After an initial processing step (sentencing and jail), a routing decision assigns people to monitoring (standard probation or intensive supervision, which can involve connecting people with community-support services such as job training or housing assistance). The model assumes that the justice system has unlimited capacity—everyone gets processed immediately without delays.
Things became interesting after the initial processing. The researchers modeled how a person’s individual risk score and supervision type would affect their probability of reoffending, with the assumption that higher-risk individuals getting standard monitoring would be much more likely to cycle back through the system than those receiving more-expensive intensive services. Quantifying the fairness-efficiency trade-off, the researchers compared fair policies with optimized routing and established that the additional cycling back—in the form of resentencing and returning to prison—is the cost of fairness. And even with unlimited capacity, the theoretical system can become overwhelmed by poor routing decisions.
This mathematical approach revealed how routing decisions ripple through the entire network. When you assign someone to intensive monitoring instead of standard probation, you’re not just helping that individual; you’re reducing future load on the whole system, not to mention avoiding the theoretical crimes that bring them back in. Make the wrong choice, and today’s efficiency shortcut becomes tomorrow’s overcrowded docket.
A balancing act
Expanding a diversion program can improve social outcomes by reducing recidivism. But it involves accepting more higher-risk applicants, who tend to get kicked out.
Queueing theory can also help answer other practical questions: How well does a prison-diversion program work? And if a state wants to expand such a program, which seeks to rehabilitate rather than lock up people who commit low-level offenses, how many participants could it take in and serve without becoming overwhelmed?
Using the approach, University of Illinois PhD student Bingxuan Li, Hebrew University of Jerusalem’s Antonio Castellanos, Shi, and Ward find that expanding eligibility criteria would lead to not just more participants but also new staffing requirements.
The researchers built a simulation that treated diversion programs like a typical system with limited capacity. They tracked thousands of individual journeys as people flowed in, received services (treatment, counseling, case management) for varying lengths of time, and eventually flowed out with different outcomes. Their model revealed that optimal program size isn’t about minimizing immediate problems such as people dropping out or being asked to leave. Instead, it’s about balancing short-term disruptions against long-term crime reduction, accepting that larger programs will have more immediate issues but deliver greater societal benefits over time.
The queueing approach reveals a chain of consequences that policymakers typically miss. These mathematical models transform policy debates from guesswork into concrete operational planning.
- Bingxuan Li, Antonio Castellanos, Pengyi Shi, and Amy Ward, “Combining Machine Learning and Queueing Theory for Data-Driven Incarceration-Diversion Program Management,” Proceedings of the AAAI Conference on Artificial Intelligence, March 2024.
- Zhiqiang Zhang, Pengyi Shi, and Amy Ward, “Routing for Fairness and Efficiency in a Queueing Model with Reentry and Continuous Customer Classes,” 2022 American Control Conference, June 2022.
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