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 unknown ex-ante and modelled as a decision variable. The model is a bilocation-allocation optimization-based queuing problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions. The model takes the form of a nonlinear integer problem which is intractable in its original form. We provide a reformulation and algorithmic framework to allow for the exact solution of large instances. 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 most efficient and scales well. The data-driven analysis based on real-life overdose data reveals that the use of drones could: 1) decrease the response time by 85%, 2) increase the survival chance by 466%, 3) save up to 36 additional lives per year, and 4) provide an additional quality-adjusted life year (QALY) of up to 305 years in Virginia Beach.