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assumed to be constant and equal to
ma
j
. As a shorthand, we will use
j
D
.
u
j
/
to represent the demand rate of customer node j
2
J. The inter-arrival times of
the demand processes generated by different customer locations are assumed to be
independent.
We should also note that while it is tempting to relax the Poisson assumption
for the demand process, this must be done with care as the facilities see aggregate
demands from different customer locations, i.e., a superposition of the demand
processes. In order to apply standard queueing results to the facilities, the demand
process seen by each facility must be a renewal process. While the superposition
of Poisson processes is Poisson, which is obviously a renewal process, in general,
the superposition of renewal processes is not a renewal process. This quickly leads
to a loss of tractability for the models. Thus, except for some trivial extensions,
the Poisson assumption for demand streams appears unavoidable (one interesting
exception occurs when customer demand space is continuous, rather than finite, in
which case facilities see Poisson arrivals under much loser conditions—see Baron
et al. (
2008
) for the development and required assumptions). However, there is no
problem, at least from the analytical point of view, in assuming that the demand
process at each node j
2
J is not time-homogenous, i.e., that the demand rate
is a function of time. To simplify the presentation, we will stick with the time-
homogenous assumption.
An important implicit assumption in all SLCIS models we are aware of is that
all customers generate “identical” demands (in terms of service requirements), i.e.,
that the streams of demand are indistinguishable once they reach the facility.
17.2.2
Facilities
Customer demands are serviced by the
facilities
that contain
service resources
(or
“servers”). All aspects related to the facilities, including their number, locations,
and the amount/types of resources allocated to them can, potentially, be treated as
decision variables in the model. In describing the system dynamics below we will
initially treat the values of these variables as having already been determined, but
will relax this assumption when describing model formulations later.
We will assume that facility locations must belong to some set I and that at
most m
0 facilities can be located; we will use i
2
I; to represent the location
(site) of facility i. By far, the most common assumption in SLCIS literature is that
set I is discrete, i.e., that all potential locations for the facilities have already been
enumerated. In this case, we can assume without loss of generality that I
J
(since any point in I not containing customers can be treated as a customer demand
point with the maximum demand rate equal to 0). Other options, include I
R
2
,
leading to
continuous
SLCIS models (see, for example, Brimberg and Mehrez
1997
;Brimbergetal.
1997
), or I
J
[
A for a network G, leading to
network
