SOUTHERN JOURNAL OF AGRICULTURAL ECONOMICS
DECEMBER, 1978
CAPACITY DECISIONS FOR AN EMERGENCY SERVICE
James W. Dunn and Gerald A. Doeksen
Decision makers face two opposing forces in
the provision of emergency services. Their con-
stituency wants more and better services, but
financial considerations limit the quantity and
quality of services provided. This classic eco-
nomic confrontation requires a decision based
on the trade-offs between the benefits of pro-
tection provided by additional services and the
cost of providing these services. Such a deci-
sion is needed for ambulance service, fire pro-
tection, and law enforcement.
Ideally, decision makers would like to pro-
vide emergency system capacity adequate for
the worst catastrophe. However, as no budget
will permit such a system, they must deter-
mine how much service capacity to provide.
The objective of this article is to derive and
illustrate a technique which can be used to esti-
mate the number of times per year an emer-
gency service will be unable to respond when
needed because its units are employed on other
emergencies. The technique is illustrated by
application to an ambulance service in a rural
Oklahoma county. Information obtained by
use of this technique, combined with a budget
analysis, will enable local decision makers to
estimate the need for and cost of additional
ambulance capacity.
METHOD
The general model of queueing theory can be
used to determine the probability that the
number of demanders for an emergency service
will exceed the capacity to provide this service.
The derivation of this general model can be
found in [2, p. 38-40] and in other introductory
queueing theory texts.
The basic model assumes an average arrival
rate at the service point of v, and an average
service rate of u. If the maximum queue length
is n, then the following equations, known as
the Erlang equations, can be derived.
(1) O = vPk-ι + uPk+1 - (u+v)Pk k=l,2,...,n-l
(2) O = uP1-vPo
(3) 0 = vPn~1-uPn
where Pk is the probability that there will be к
persons in the queue. From equation 2,
P=vP
ɪɪ uio
which when substituted into equation 1 yields
P2=Q2P0
pk=Qkpo
At this point define p ≡ ɪ which is generally
called the traffic intensity ratio. Because the
probabilities must sum to one, for a queue with
no maximum length the probability of having
к or more persons in the system when only k-1
can be served is
P(>k) = 1 - P0 - P1 - ... - Pk_j
(4) = 1 - (l-p) - (l-p)p - ... - (l-p)pk-1
= pk.
The decision maker supposedly is willing to
accept a probability, a, of a person requiring
emergency service when all servers are occu-
pied. For an emergency service with k-1 ser-
vers
α= P(>k).
Then
(5) p* = α1'k,
where p* is the traffic intensity ratio asso-
ciated with this a.
James W. Dunn is a former research assistant at Oklahoma State University and is now Assistant Professor of Agricultural Economics, Pennsylvania State Univer-
sity. Gerald A. Doeksen is a former Economist with the Economic Development Division, ERS, USDA, and is now Associate Professor of Agricultural Eeonnmiea,
Oklahoma State University. The authors acknowledge the helpful suggestions of Walter W. Haessel, Robert D. Weaver, Milton C. Hallberg, Sam M. Cordes, and the
anonymous reviewers.
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