Doeksen, Gerald A.

Dunn, James W.

English

Research Papers in Economics

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  (1)  0 = vPk-1 + uPk+l - (u+v)Pk  k=1,2,...,n-1
the provision of emergency services.  Their con-
stituency wants more and better services,  but  (2)  0  = uP, - vPo
financial considerations  limit the quantity and
quality of services  provided.  This  classic eco-  (3)  0 = vPn-1 - uPn
nomic confrontation  requires a decision based
on the trade-offs between  the benefits  of pro-  where Pk is the probability that there will be k
tection provided by additional services and the  persons in the queue. From equation 2,
cost  of providing these  services.  Such a  deci-
sion is needed  for ambulance  service,  fire pro-  P  = v 
tection, and law enforcement.  u
Ideally,  decision  makers  would like  to pro-  whichwhen substitutedintoequation 1 yields
vide emergency  system capacity adequate  for 
the worst catastrophe.  However,  as no budget  P2 = (u)  Po
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-  Pk = (uV)kP
mate the  number  of times  per  year  an  emer- 
gency service  will  be unable  to respond  when  At this point  define p =  which  isgenerally
needed because its units are employed on other  called  the  traffic intensity ratio.  Because the
emergencies.  The  technique  is  illustrated  by  probabilities must sum to one, for a queue with
application to an ambulance  service in a rural  no maximum length the probability  of having
Oklahoma  county.  Information  obtained  by  k or more persons in the system when only k-1
use of this technique,  combined with a budget  can be served is
analysis,  will enable  local  decision  makers  to
estimate  the  need  for  and  cost  of  additional  P(>k) = 1 - P-  P1 - ...-  Pk-i
ambulance capacity.
(4)  = 1-  (l-p)-  (1-p)p -...- (1-p)p k-
METHOD  = pk
The decision maker supposedly  is willing to The general model of queueing theory can be  accept  a  probability,  a,  of a person requiring
used  to  determine  the  probability  that  the  emergency  service when  all servers  ar  occu-
number of demanders for an emergency service  pied. For an emergency  service with k-l ser-
will exceed the capacity to provide this service.  v
The  derivation  of  this  general  model  can  be
found in [2,  p. 38-40] and in other introductory  a=  P(  k).
queueing theory texts.
Then
The basic model assumes an average  arrival
rate at the service  point of v,  and an average  (5)  p  = al/k
service rate of u. If the maximum queue length
is  n,  then the  following  equations,  known  as  where  p*  is  the  traffic  intensity  ratio  asso-
the Erlang equations, can be derived.  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  Economics,
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.
121AN APPLICATION  Table 1 shows budgets for the main ambulance,
a new ambulance as a back-up unit, and a used
The  community  leaders  of  Alfalfa  County,  ambulance  as a back-up  unit.  This budget  in-
Oklahoma,  were  forced  to provide  ambulance  formation  is taken from  [1]  and is updated to
service  when the private  suppliers  refused  to  present prices. It is based on the assumption of
continue the service. The leaders were not only  a hospital-based  system with four Emergency
confronted with the expensive problem of pro-  Medical  Technicians  (EMT)  and  a  Licensed
viding one complete  ambulance,  but also  had  Practical  Nurse  or  a  Registered  Nurse
to  decide  whether  a  back-up  ambulance  was  accompanying the ambulance on calls. It  is as-
needed for a multiple injury  accident or over-  sumed that if a back-up is provided, one of the
lapping  requests  for  service.  Alfalfa  County  EMTs will be on call during each eight hours of
has a population of 7,224; the town of Cherokee  the day and will be paid  $5  for being on call.
with  2,119  residents  is  in  the  center  of the  Pocket pagers are provided to allow the EMT
county.'  A  small  hospital  is  located  in  freedom of movement within town.
Cherokee.  Estimated yearly costs for providing a one-
A  calculation  procedure  developed  by  ambulance  system  with  the  ambulance
Doeksen,  Frye,  and  Green  [1]  indicated  that  replaced after 75,000 miles or every three years
335  calls per  year  could  be expected  for  the  would  be  $39,053  (Table  1).  If  the  decision
county system.  In addition,  an analysis of the  makers  provide  a  new  back-up  unit,  mileage
previous year's calls revealed an average round  could be alternated  between  the vehicles  and
trip service time of 77.9 minutes.  An estimate  each  would  last  six  years.  Thus,  additional
of the service rate, u, can be calculated  as (60  yearly capital depreciation because of the back-
min/hr)/(77.9  min/call)  =  0.7702  calls  handled  up  would  be  much  less  than  for  the  original
per hour.  ambulance.  In  either  case  many  of  the
The same method can be used to derive  the  operating  expenses  can  be allocated  between
annual number of calls consistent with a given  vehicles.  The  main  additional  charge  for  the
probability of the queue length exceeding  the  back-up unit is labor costs. Total yearly costs
number of service facilities.  A similar method  under these conditions for a new back-up unit
determines  the  expected  probability  of  are $8,760. If a used ambulance is purchased as
capacity  being  exceeded  associated  with  a  a  back-up  unit for  $3,000  and  is  depreciated
given number of annual calls.  over  three  years,  then  total  yearly  costs  are
Suppose the community  leaders  are willing  $7,930.
to accept at most  a probability  of one occur-
rence per year that more than one patient will
require  the ambulance  at  the same  time,  i.e.,  TABLE 1.  ANNUAL  BUDGET  FOR
for  the  335  calls  expected  a  =  1/335.  For  a  FIRST-RUN  AMBULANCE
single ambulance system, k = 2, and from equa-  AND  FOR  BACK-UP
tion 5,  this probability has an associated traf-  AMBULANCE
fic intensity ratio,  p*, of 0.0546. Solving for v  Back-Up  Ambulance
from the traffic intensity ratio formula, one ob- 
tains v = p*u =  (0.0546) (.7702) = 0.0421.  The  First-Run  Ambue  Used ·v  pu  =  ~  UAPZ~L.  LJ~iV  Ambulance  Ambulance  Ambulance
mean arrival rate for service is 0.0421 calls per
hour,  which translates  into  1.01  calls per day  Equipment  Expenses
or 369  calls per  year,  slightly  more  than  the  Depreciation-ambulance  $  5,272  $  500  $1,000
335 expected calls estimated previously.  Depreciation-communication  165  165  165
By  a  reverse  method,  given  that  the  Depreciation-pagers  0  200  200
estimated number of calls is 335 per year,  the  Iterest  1,440  ,520  240
average  arrival  rate v=  (335)  (1/365)  (1/24)  =
0.03824 calls per hour.  As before  u is 0.7702.  Insurance  50  50 
Substituting into equation 4, one obtains  Subtotal  $7,377  $2,885  $1,855
Operating  Expenses
P(>2)  p  =  p  )  =  0.00247.  Vehicle  2,401  300  500
Communication  50  100  100
This probability  is  approximately  one  occur-  Medical  521  0  0
rence in every 400 calls.  Subtotal  $ 2,972  $  400  $  600
To weigh the trade-offs associated with over-  Labor  $284  5 
lapping  demand,  the  decision  makers  must
know  the  costs  of  providing  a  back-up  unit.  Total  $39,053  $8,760  $7,930
'1970 Census population estimates.  Population has changed very little since  1970.
122The extra costs for a back-up unit do not ap-  and would not basically change the fiscal situa-
pear to  be large until they are compared  with  tion.
revenue for the system. For Alfalfa County,  as-
suming a charge of $25 per call plus $1  per mile
one  way  with 80 percent  of  the users  paying  SUMMARY  AND CONCLUSIONS
their  bill,  approximately  $18,000  is  received
each year. Thus, the one-unit system incurs an  The method illustrated can be used to derive
approximately yearly loss of $21,000 without a  the annual  number  of calls consistent  with a
back-up  unit.  The  cost  for  the  back-up  unit  given  probability  that the  number  of  service
would increase the yearly loss about $8,000.  demanders  will  exceed  the number  of  service
The  back-up  ambulance  can  reduce  the  facilities.  A  reverse  method  can  be  used  to
probability  of overlapping  need  for the avail-  determine the expected probability of capacity
able  ambulance  from  approximately  once  being exceeded associated with a given number
every  400  calls  to  approximately  once  every  of annual calls.  In either case,  beginning  with
8000  calls,  i.e., P(>3).  Whether this protection  an estimate of average  service time,  an objec-
is worth $8,000 is the decision the local people  tive procedure can be applied to a decision that
must make.  The  back-up  unit,  in  addition  to  otherwise  would  be very  subjective. As  local
providing  reserve  coverage  for  overlapping  decision  makers  continue  to  be  urged  to
calls, provides  coverage during down  time for  provide additional emergency services in rural
the  primary  ambulance  and  may  generate  areas, information from this technique,  in con-
some  revenue  from  interhospital  transfers  or  junction  with budgets  and  models  to  predict
other  nonemergency  uses  which would  other-  usage, will be extremely useful in rural decision
wise be foregone.  Such uses are rare,  however,  making.
REFERENCES
[1]  Doeksen,  Gerald A.,  Jack Frye,  and Bernal L.  Green. Economics of Rural Ambulance Service
in the Great  Plains, USDA, ERS, Agricultural Economics Report No. 308, November 1975.
[2]  Saaty, Thomas L. Elements of Queueing Theory with Applications. New York:  McGraw-Hill
Book Company,  Inc., 1961.
123

CAPACITY DECISIONS FOR AN EMERGENCY SERVICE

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