Stochastic Location Models with Congestion - Location Science - page 432

by different facilities are independent of each other. Thus, each facility i 2 I acts as
a stand-alone queueing system with Poisson arrivals and general service times, i.e.,
an M=G=1 (or M=G= i ) queue with service rate i .
System stability (i.e., ensuring that queue lengths are finite) requires that

i i ;i 2 I; (17.7)

which acts as a constraint on capacity assignment decisions. In addition, the
framework defined above allows us to express the key performance characteristics
of the facilities, such as the steady-state system waiting time W i D W. i ; i /
(this includes both queueing and service times), and the steady-state number of
customers in the system L i D L i . i ; i /, both of which are random variables
whose distributions can, in principle, be obtained. We will come back to these
quantities when we discuss system costs and service-level constraints in the next
section.
It may also be useful to require that each facility face some minimum demand
rate min in order to ensure that it can be operated economically; sometimes these
minimum demand rates are imposed by regulators for public service facilities (see,
e.g., Zhang et al. 2010 ). These constraints take the form

i min y i ;i 2 I:

(17.8)

We note that many models make additional assumptions regarding the operations
of facilities. For example, the assumption that the distribution of service times is
exponential is quite common (though likely not very realistic in many real-life
systems; e.g., see the discussion in Boffey et al. 2006 ). Some authors (e.g., Boffey
et al. 2010 ) assume limited buffer space at the facilities. We will delay the discussion
of these additional aspects until Sect. 17.5 . For the moment we regard each facility
as an infinite-buffer M=G=1 or M=G= queue.

Remark The fact that each facility (once location, capacity and customer allocation
decisions are made) can be viewed as an independent queueing system is the
main characteristic distinguishing immobile from mobile server models; in mobile
server models the systems operated by different facilities cannot be decoupled.
This is because in these models the typical assumption is that server assignments
are dynamic, i.e., depend on the state of the system. Thus a server from a given
facility may service demands from customers at point j under some conditions,
but not under others. This leads to a system which is not, in general, separable,
and where servers located at different facilities must be treated as distinguishable.
Such queueing networks are analytically intractable even when all location, capacity
and allocation decisions are made. Thus, all modeling approaches involve strong
approximations and/or descriptive/simulation components (e.g., the Hypercube
model proposed by Larson ( 1974 ) is frequently used as the modeling foundation).
In contrast, SLCIS models decompose into a set of queues with Poisson
arrivals—systems for which strong analytical results (both exact and approximate)