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Dynamic Control of Service Systems with Returns: Application to Design of Post-Discharge Hospital Readmission Prevention Programs

Timothy C. Y. Chan, Simon Y. Huang, Vahid Sarhangian

TL;DR

The paper develops a fluid-control framework for Erlang-R queues with controllable return probabilities to balance post-discharge intervention costs against holding and return costs. It derives the structure of both long-run equilibrium policies and bias-optimal transient policies using Pontryagin’s minimum principle, revealing a switching, line-based policy geometry in the congested region and a piecewise-linear segmentation under cost piecewise-forms. Through extensive simulations and a hospital readmission case study, the authors show substantial cost savings (up to 25.4% long-run and 33.7% finite-horizon) when congestion-aware interventions are dynamically adjusted, compared to congestion-agnostic or aggressive baseline policies. The work provides actionable surge-protocol designs and extends to time-varying arrivals, with robustness analyses and discussion of practical extensions, including partial observability and return-time effects.

Abstract

We study a control problem for queueing systems where customers may return for additional episodes of service after their initial service completion. At each service completion epoch, the decision maker can choose to reduce the probability of return for the departing customer but at a cost that is convex increasing in the amount of reduction in the return probability. Other costs are incurred as customers wait in the queue and every time they return for service. Our primary motivation comes from post-discharge Quality Improvement (QI) interventions (e.g., follow up phone-calls, appointments) frequently used in a variety of healthcare settings to reduce unplanned hospital readmissions. Our objective is to understand how the cost of interventions should be balanced with the reductions in congestion and service costs. To this end, we consider a fluid approximation of the queueing system and characterize the structure of optimal long-run average and bias-optimal transient control policies for the fluid model. Our structural results motivate the design of intuitive surge protocols whereby different intensities of interventions (corresponding to different levels of reduction in the return probability) are provided based on the congestion in the system. Through extensive simulation experiments, we study the performance of the fluid policy for the stochastic system and identify parameter regimes where it leads to significant cost savings compared to a fixed long-run average optimal policy that ignores holding costs and a simple policy that uses the highest level of intervention whenever the queue is non-empty. In particular, we find that in a parameter regime relevant to our motivating application, dynamically adjusting the intensity of interventions could result in up to 25.4% reduction in long-run average cost and 33.7% in finite-horizon costs compared to the simple aggressive policy.

Dynamic Control of Service Systems with Returns: Application to Design of Post-Discharge Hospital Readmission Prevention Programs

TL;DR

The paper develops a fluid-control framework for Erlang-R queues with controllable return probabilities to balance post-discharge intervention costs against holding and return costs. It derives the structure of both long-run equilibrium policies and bias-optimal transient policies using Pontryagin’s minimum principle, revealing a switching, line-based policy geometry in the congested region and a piecewise-linear segmentation under cost piecewise-forms. Through extensive simulations and a hospital readmission case study, the authors show substantial cost savings (up to 25.4% long-run and 33.7% finite-horizon) when congestion-aware interventions are dynamically adjusted, compared to congestion-agnostic or aggressive baseline policies. The work provides actionable surge-protocol designs and extends to time-varying arrivals, with robustness analyses and discussion of practical extensions, including partial observability and return-time effects.

Abstract

We study a control problem for queueing systems where customers may return for additional episodes of service after their initial service completion. At each service completion epoch, the decision maker can choose to reduce the probability of return for the departing customer but at a cost that is convex increasing in the amount of reduction in the return probability. Other costs are incurred as customers wait in the queue and every time they return for service. Our primary motivation comes from post-discharge Quality Improvement (QI) interventions (e.g., follow up phone-calls, appointments) frequently used in a variety of healthcare settings to reduce unplanned hospital readmissions. Our objective is to understand how the cost of interventions should be balanced with the reductions in congestion and service costs. To this end, we consider a fluid approximation of the queueing system and characterize the structure of optimal long-run average and bias-optimal transient control policies for the fluid model. Our structural results motivate the design of intuitive surge protocols whereby different intensities of interventions (corresponding to different levels of reduction in the return probability) are provided based on the congestion in the system. Through extensive simulation experiments, we study the performance of the fluid policy for the stochastic system and identify parameter regimes where it leads to significant cost savings compared to a fixed long-run average optimal policy that ignores holding costs and a simple policy that uses the highest level of intervention whenever the queue is non-empty. In particular, we find that in a parameter regime relevant to our motivating application, dynamically adjusting the intensity of interventions could result in up to 25.4% reduction in long-run average cost and 33.7% in finite-horizon costs compared to the simple aggressive policy.
Paper Structure (35 sections, 17 theorems, 60 equations, 20 figures)

This paper contains 35 sections, 17 theorems, 60 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model Description
  4. The Fluid Model
  5. Long-Run Average Control
  6. Optimizing Amongst Equilibria
  7. Transient Control
  8. Formulation of the Associated Fluid Control Problem and Preliminaries
  9. Pontryagin's Minimum Principle.
  10. Optimal Transient Policy Starting with a Non-Empty Queue
  11. Optimal Policy as a Function of Time.
  12. Optimal Policy as a Function of State.
  13. Piecewise Linear Intervention Cost and Switching Control.
  14. Optimal Transient Policy Starting with an Empty Queue
  15. Illustrative Examples
  16. ...and 20 more sections

Key Result

Proposition 1

$(i)$ For any given initial state $(x^0, y^0) \geq 0$, there exists a time $t_{{\mathcal{A}}} < \infty$ such that $(x(t_{\mathcal{A}}), y(t_{\mathcal{A}})) \in {\mathcal{A}}$ under any admissible policy $p(\cdot)\in{\mathcal{P}}$. $(ii)$ If $( x^0, y^0 ) \in {\mathcal{A}}$, then $(x(t), y(t)) \in {\

Figures (20)

  • Figure 1: An illustration of the three partitions of the state-space.
  • Figure 2: Optimal transient policies under three intervention cost structures: linear $C(p) = 5(0.2-p)$ (left), piecewise linear $C(p) = 2(0.2-p) + 6(0.15-p)^+$ (middle), and quadratic $C(p) = 50(0.2-p)^2$ (right). Other system parameters are $N=50, \lambda=9.5, \mu=1/4, \nu=1/15, p_l=0.1, p_u=0.2, r=1.0, h=0.25$. Two (isochromatic) contour lines in the ${\mathcal{C}}$ region for the quadratic intervention cost are highlighted in red.
  • Figure 3: Optimal trajectories starting from three initial conditions in $\mathcal{N}$ (left) and stream plot under the optimal policy (right). System and cost parameters are $N=50,\bar{\lambda}=0.95,\mu=1/4, \nu=1/15$; and $r=1, M=0.5, h=1/4$ with $C(p) = 5(0.2-p)$. The dashed line corresponds to the intervention boundary.
  • Figure 4: Long-run average probability of return under the fluid policy for three periods and varying amplitudes of the arrival rate. Other system and cost parameters are $N=50,\bar{\lambda}=0.95,\mu=1/4, \nu=1/15$; and $r=1, M=0.5, h=1/4$ with $C(p) = 5(0.2-p)$ (left) and $C(p) = 50(0.2-p)^2$ (right).
  • Figure 5: Relative reduction in expected cost for both finite-horizon and long-run experiments under the fluid policy with respect to the equilibrium policy (top row) and simple policy (bottom row) under the quadratic intervention cost and for different values of cost parameters. System parameters are fixed at $N=50,\lambda=0.95,\mu=1/4,\nu=1/15$ and and the return cost is normalized at $r=1$.
  • ...and 15 more figures

Theorems & Definitions (17)

  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Lemma 1
  • Proposition 7
  • Theorem 1
  • ...and 7 more