First Time’s the Charm: A Queueing Theory Approach to Reducing Hospital Readmission Rates

Abstract

This thesis develops three new queueing models to represent the hospital care process, entitled “Star 1,” “Star 2,” and “Star 3.” We are motivated by the Hospital Readmissions Reduction Program (HRRP), created as part of the 2010 Patient Protection and A↵ordable Care Act to incentivize hospitals to reduce their 30-day readmission rates. We model the probability of readmission as a function of two variables: the hospital’s treatment rate (or average patient length of stay) and the level of congestion in the hospital. We introduce several families of functions to model the probability of readmission. Each of our three queueing models represents the hospital care process in a di↵erent manner. In Star 1, we model the hospital as an infinite-server queue and the probability of readmission as a function of only the hospital treatment rate, μ. In Star 2, we represent the hospital as a c-server delay model. In Star 3, we model the probability of readmission as dependent on both μ and Q1, the congestion in the hospital. We develop equations to characterize both the time-varying and steady-state behaviors of these three queueing models. We also approximate the transient systems through the creation of fluid and di↵usion models. Having developed these models, we analyze their sensitivities to changes in their arrival and service rates, as well as their dependence on the function used to model the probability of readmission. We find that our queueing systems are sensitive to changes in their parameters, with heightened sensitivity to the hospital treatment rate, μ. This follows because μ a↵ects not only patients’ service at the hospital, but also the probability of readmission. We find that the Star 2 queueing system is particularly sensitive to its parameter values, because its finite number of servers can cause the system not to reach steady state when its arrival and service rates are marginally adjusted. Furthermore, certain parameter settings cause steady state to be unachievable in the Star 3 queueing system, because in that model the length of the hospital queue implicitly relies on its own value. We posit that, under certain conditions, increasing average patient length of stay by one day can be an e↵ective means to reducing the rate of readmissions, while under other, more extreme conditions, such a change can cause the length of the hospital queue to grow unmanageably in the long run.