  We propose a new stochastic emergency network design model that uses a fleet
of drones to quickly deliver naxolone in response to opioid overdoses. The
network is represented as a collection of M/G/K queuing systems in which the
capacity K of each system is a decision variable and the service time is
modelled as a decision-dependent random variable. The model is an
optimization-based queuing problem which locates fixed (drone bases) and mobile
(drones) servers and determines the drone dispatching decisions, and takes the
form of a nonlinear integer problem, which is intractable in its original form.
We develop an efficient reformulation and algorithmic framework. Our approach
reformulates the multiple nonlinearities (fractional, polynomial, exponential,
factorial terms) to give a mixed-integer linear programming (MILP) formulation.
We demonstrate its generalizablity and show that the problem of minimizing the
average response time of a network of M/G/K queuing systems with unknown
capacity K is always MILP-representable. We design two algorithms and
demonstrate that the outer approximation branch-and-cut method is the most
efficient and scales well. The analysis based on real-life overdose data
reveals that drones can in Virginia Beach: 1) decrease the response time by
78%, 2) increase the survival chance by 432%, 3) save up to 34 additional lives
per year, and 4) provide annually up to 287 additional quality-adjusted life
years.
