  This paper studies a stochastic congested location problem in the network of
a service system that consists of facilities to be established in a finite
number of candidate locations. Population zones allocated to each open service
facility together creates a stream of demand that follows a Poisson process and
may cause congestion at the facility. The service time at each facility is
stochastic and depends on the service capacity and follows a general
distribution that can differ for each facility. The service capacity is
selected from a given (bounded or unbounded) interval. The objective of our
problem is to optimize a balanced performance measure that compromises between
facility-induced and customer-related costs. Service times are represented by a
flexible location-scale stochastic model. The problem is formulated using
quadratic conic optimization. Valid inequalities and a cut-generation procedure
are developed to increase computational efficiency. A comprehensive numerical
study is carried out to show the efficiency and effectiveness of the solution
procedure. Moreover, our numerical experiments using real data of a preventive
healthcare system in Toronto show that the optimal service network
configuration is highly sensitive to the service-time distribution. Our method
for convexifying the waiting-time formulas of M/G/1 queues is general and
extends the existing convexity results in queueing theory such that they can be
used in optimization problems where the service rates are continuous.
