  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.
