3.8.3 Application to Facility Location and Districting One could apply the ideas of spatial Poisson
processes to a problem
of facility location and districting of a city. Suppose that demands
are distributed
uniformly throughout the plane and suppose that travel distance is
rightangle. We can
consider two applications: (1) for each service request, a response
unit is dispatched
from the nearest facility (the service system need not be an
emergency service; for
instance, it could be a social service agency whose personnel make
home visits); and (2)
the individual requiring service travels to the nearest facility
(e.g., hospital, library,
"little city hall," police district station house). Each
district about a
facility would consist of all points closer to that facility than to
any other. Use of Poisson model to generate "upper
bound." At one
extreme, you could ask: What are the response distance
characteristics of the system if
facilities are distributed at random? We can answer this question by
assuming that
facilities are distributed as a homogeneous spatial Poisson process.
This corresponds to a
totally unplanned system (in terms of districting) in which the
facility locations could
be viewed as occurring from "throwing darts blindfolded"
at a map of the city.
That is, given n facilities in any particular region, their
locations would be
independently, uniformly distributed over the region (following the
"unordered
arrival times" argument of Chapter 2 for a time Poisson
process). where y is the average density of facilities. Lower bound. To achieve minimal mean
travel distance, the
facilities should be positioned in a regular lattice, as shown in
Figure 3.34. This makes
intuitive sense since a diamond gives the set of points within a
given distance of its
center, when rightangle distance is used (analogous to a circle for
Euclidean distance),
so diamonds can be used to partition a city into districts of equal
coverage, where
coverage of a district is measured by maximum possible distance from
its facility.
