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Third, there is an important class of models where servers are assumed to
be “mobile”, i.e., servers travel to customers rather than customers traveling to
facilities. Examples of the underlying systems include emergency services (fire,
ambulance, police) as well as repairmen making house calls. These models are close
“cousins” of the fixed-server models as they do include most of the same com-
ponents: stochastic demand streams, stochastic service times, congestion/queuing
behavior. However, these models also include additional significant levels of
complexity, such as dynamic dispatching and routing of servers, where servers can
be repositioned between facilities, re-routed before completion of the call, etc. The
underlying queuing models are analytically intractable, even if the facility locations
are assumed fixed, leading to various approximation-based approaches. In contrast,
the queuing systems underlying models with fixed servers are often (though not
always) analytically tractable, allowing for more (theoretically) precise solutions
in many cases. We refer the reader to a survey by Berman and Krass (
2002
)and
to a more recent survey on emergency systems planning by Ignolfsson (
2013
)for
more details on models with mobile servers. We note that the technical distinction
between models with fixed and mobile servers does not lie in the server mobility per
se, but rather in how the underlying queuing network is modeled (in fact, some of the
models described in this chapter have been applied in mobile server contexts). We
will provide more precision for this distinction below, once the underlying technical
framework is properly introduced.
The field of
Stochastic Location models with Congestion and Immobile Servers
(SLCIS), the main focus of this chapter, has seen a rather explosive growth over a
relatively recent time period. As noted in Berman and Krass (
2002
), by the early
2000s, only a handful of papers on SLCIS could be found. However, by 2006 over
20 contributions were listed in the comprehensive review by Boffey et al. (
2006
)(we
are only counting the papers that meet the assumptions for SLCIS models discussed
earlier). In the last eight years, this number has roughly doubled. It is our intent to
review the current state of the field, as well as to systematize the many variants of
SLCIS models that have been proposed.
We note that much of the recent work has been on models with elastic demand—
i.e., where the intensity of customer demands depends on the quality of the service
provided by the facilities. In this regard it is important to mention a review by
Brandeau et al. (
1995
) that describes early foundation for much of this work.
As with most other location models, one could focus on cost minimization or on
net revenue (profit) maximization. Cost minimization is more appropriate when the
revenues are either not well-defined (e.g., in the case of public health facilities), or
are assumed to be exogenous to the model (e.g., when customer demand levels and
prices are fixed). While most SLCIS models in the literature are formulated with the
cost minimization objective, profit optimization is more general and is much more
natural when demand is elastic. Therefore, we will assume this objective type in our
general formulation in the following section.
The remainder of this chapter is organized as follows. We start by describing
the main model components in Sect.
17.2
. These components include customers,
facilities, and the objective function of the model. A crucial part of any SLCIS
