Community Bail Fund Systems: Fluid Limits and Approximations

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.