A Continuous-Class Queueing Model With Proportional Hazards-Based Routing

Abstract

Motivated by jail overcrowding and the U.S. correctional system's widespread use of risk models to aid in inmate release decisions both prior to trial (i.e., pretrial release) and near the end of their jail sentence (i.e., split sentencing), we formulate and analyze a queueing network model with two novel features: there is a continuum of customer classes corresponding to an inmate's continuous risk level p, and routing in the network is dictated by a Cox proportional hazards model, where the hazard rate associated with recidivism (i.e., committing another crime) during release is proportional to eγp for some parameter γ. We perform an exact analysis of a continuous-class M/M/c/c (i.e., Erlang B) model where preemptive priority is awarded according to the risk level p, and use it to develop approximate performance measures for a queueing network of a jail with a two-threshold policy, which dictates who is granted pretrial release and who receives a split sentence. For a slightly simplified version of the model, we derive sufficient conditions under which no inmates are offered pretrial release unless all inmates are given a split sentence.