Stochastic Location Models with Congestion - Location Science - page 430

SLCIS models (see, e.g., Berman et al. 2014 ). Unless stated otherwise, we will
generally assume I to be discrete.
To take advantage of the discreteness of I we will follow the typical convention
in location modeling and define y i 2f 0;1 g to be a binary indicator variable with
the value 1 if a facility is open at site i 2 I, and 0 otherwise. To ensure that the total
number of open facilities does not exceed m we require:
X

y i m:

(17.1)

i2I

If a facility is opened at i 2 I (i.e. y i D 1), it must be allocated some service
capacity i >0, which can be thought of as the average processing rate. We will
assume that i D 0 whenever y i D 0, which can be assured by using the constraints

i My i ;i 2 I; (17.2)

where M is the maximum possible processing capacity that can be assigned to a
facility.
As noted in Baron et al. ( 2008 ), there are two standard approaches to represent
facility capacity in queuing environment: as a “single-server” facility where the
capacity level can take on any value in some interval i 2 Œ0; max ,where max is
the maximum practical capacity level, or as a “multi-server” facility housing i 0
parallel servers each with fixed capacity 0 ,where i 2f 0;:::;k g is an integer,
i D i 0 is the processing capacity of facility i,andk is the maximum number of
servers that can be stationed at a facility (with max D k 0 ).
While there are some important differences between the single-server and multi-
server models (these will be touched on later) our bias is to favor the single-server
representation. It is more transparent, typically leads to cleaner analytical results,
and seems more practical as well: a typical facility will house a variety of processing
resources and discrete “servers” may be hard to identify. For example, a medical
clinic will often house doctors, nurses, examination rooms, X-ray machines, etc.
While it is sensible for a planner to think of processing capacity of a clinic in
terms of patients per hour (and how this processing capacity changes when certain
resources are added or removed), it is harder to think of the clinic containing
distinct servers (are these doctors? nurses? rooms?). Thus, unless stated otherwise,
each facility will be assumed to house a single “server” with capacity .
The service times at each facility are assumed to be stochastic. More specifically,
following Baron et al. ( 2008 ), we assume First Come First Serve (FCFS) service
discipline and that service requirements (which can be thought of as the amount
of work required to process one customer request) are independent and identically
distributed random variables with a cumulative distribution function (CDF)

F S . w /,
and a well-defined moment generating function (MGF) G S ./. We also assume that
the mean service time EŒS D 1—this assumption is made with no loss of generality
as it simply rescales service times. Note that in this framework, since i represents
the service rate of facility i, the mean service time is 1= i and it is not hard to show
that the distribution of service times is given by F S . i w / with MGF G S .= i /.