Community Bail Fund Systems: Fluid Limits and Approximations

Paper: arXiv 2507.05490

Authors: Yidan Zhang, Jamol Pender

Abstract

Community bail funds (CBFs) assist individuals who have been arrested and cannot afford bail, preventing unnecessary pretrial incarceration along with its harmful or sometimes fatal consequences. By posting bail, CBFs allow defendants to stay at home and maintain their livelihoods until trial. This paper introduces new stochastic models that combine queueing theory with classic insurance risk models to capture the dynamics of the remaining funds in a CBF. We first analyze a model where all bail requests are accepted. Although the remaining fund balance can go negative, this model provides insight for CBFs that are not financially constrained. We then apply the Skorokhod map to make sure the CBF balance does not go negative and show that the Skorokhod map produces a model where requests are partially fulfilled. Finally, we analyze a model where bail requests can be blocked if there is not enough money to satisfy the request upon arrival. Although the blocking model prevents the CBF from being negative, the blocking feature gives rise to new analytical challenges for a direct stochastic analysis. Thus, we prove a functional law of large numbers or a fluid limit for the blocking model and show that the fluid limit is a distributed delay equation. We assess the quality of our fluid limit via simulation and show that the fluid limit accurately describes the large-scale stochastic dynamics of the CBF. Finally, we prove stochastic ordering results for the CBF processes we analyze.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical formulation combining queueing theory and insurance risk models to solve a novel application in social operations research. While the theoretical derivations (fluid limits, Skorokhod maps, stochastic ordering) are advanced and well-explained, the empirical rigor is primarily limited to simulation rather than backtesting on real-world bail fund datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Stochastic Dynamics<br/>of Community Bail Funds] --> B[Inputs: Bail Request Arrivals,<br/>Amounts & Trial Durations]
    
    B --> C[Develop Stochastic Models:<br/>Queueing Theory + Insurance Risk]
    
    C --> D[Model 1: Unconstrained Fund<br/>Allows Negative Balance]
    C --> E[Model 2: Skorokhod Map<br/>Partial Request Fulfillment]
    C --> F[Model 3: Blocking System<br/>Rejects Unaffordable Requests]
    
    F --> G[Derive Fluid Limit:<br/>Functional Law of Large Numbers]
    
    D & E & G --> H[Outcomes: Validated Fluid Limit<br/>& Stochastic Ordering Results]

Cited References (48)

1. The Hitchhiker’s Guide to the Galaxy (1995) — link
2. The Physics of the $ M _t/ G /\infty$ Queue (1993)
3. The Price of Jails: Measuring the Taxpayer Cost of Local Incarceration (2015)
4. Combining Machine Learning and Queueing Theory for Data-driven Incarceration-Diversion Program Management (2024)
5. Systemic failure to appear in court (2023)
6. Service operations for justice-on-time: A data-driven queueing approach (2024)
7. Bail Fund Co-Optation and the Purpose of Cash Bail (2021)
8. Celebrity Activism on Racial Justice During COVID-19: The Death of George Floyd, the Rise of Naomi Osaka, and the Celebritization of Race in Pandemic Times (2022)

0 of 48 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.