In this note we give a short and easy proof of the equivalence of Hakimi's one-median problem and the k-server-facility-loss median problem as discussed by Chiu and Larson in Computer and Operation Research. The proof makes only use of a stochastic monotonicity result for birth and death processes and the insensitivity of the M/G/k/k loss model.Hakimi median;stochastic location;stochastic monotonicity

Frenk, J.B.G.

LabbÃ©, M.

Zhang, S.

English

Research Papers in Economics

Operations Research Letters 13 (1993) 213-214  May 1993 
North-Holland 
A  note  on  a  stochastic  location  problem 
J.B.G. Frenk 
Econometric Institute, Erasmus University, Rotterdam,  The Netherlands 
M. Labb6 
NFSR Research Associate, C.E.M.E., Unit~ersitd Libre de Bruxelles, Belgium 
S.  Zhang 
Department of Econometrics, Unicersity of Groningen, Groningen, The Netherlands 
In  this note we  give a  short and  easy proof of the  equivalence of Hakimi's one-median problem  and  the  k-server-facility-loss 
median problem as discussed by Chiu and Larson in Computer and Operation Research. The proof makes only use of a stochastic 
monotonicity result for birth and death processes and the insensitivity of the M/G/k/k  loss model. 
stochastic location; Hakimi median; stochastic monotonicity 
1.  Model formulation and results  loss  model.  For  this  queueing  process  it  is  well- 
known  (cf.  [2])  that  for fixed  x ~  9-  the  random 
In [1]  Chiu  and  Larson  consider  the  so-called  variable  Xt(x)  converges  in  distribution  to  the 
k-server-single-facility-loss  median  model  (k-  random variable  J(x)  representing  the  number 
SFLM).  In  this  model  customers  at  fixed  loca-  of customers  being  served  in  the  steady  state  if 
tions  generate  calls  according  to  a  Poisson  pro-  x c  9- denotes the location.  Moreover, the distri- 
cess  with  rate  A >  0.  If  upon  arrival  of  a  cus-  bution  of X(x)  depends  only on the  arrival  rate 
tomer's call at the service facility all its  k  identi-  and  the  first  moment  re(x)  of the  service  time 
cal  servers  are  busy this  customer  is  lost  at  cost  distribution  (Erlang's Loss Formula).  If the facil- 
Q  >  0.  Moreover,  if  upon  arrival  some  of  the  ity  is  located  at  x ~  ~-,  the  cost  function  z(x) 
servers  are  idle  one  of  them  is  assigned  to  this  takes the following form (cf. [1]). 
customer  and  travels to  the  location of that  cus- 
tomer at a fixed cost rate. The objective is now to  z(x) =Pk(x)Q + (1 -Pk(x))m(x)  (1) 
determine  among a  set  ~___ ~2  of feasible  loca- 
tions  that  location  for  the  service  facility which  with 
will  minimize  the  average cost  of the  system.  In  •  Pk(x):= Pr{customer  finds  upon  arrival  in 
order to analyze this model define  steady state  all  k  servers busy) =  Pr{customer 
•  J't(x) := number of customers being served  at  arriving in steady state is lost), 
time  t  by one of the  k  servers if the facility is  •  m(x) := expected total travel time of server to 
located at  x ~  ~-.  arbitrary customer when  the  facility is located 
As argued in [1] the queueing process underlying  at  x, and 
the  k-SFLM  location  problem  is  a  M/G/k/k  *  Q := cost per lost customer,  Q >  0. 
The  main  result  proved  in  [1]  using  lengthy 
calculations  states  that  the  cost  function  z(x)  is 
increasing in m(x). This implies that the k-SFLM 
Correspondence to: J.B.G. Frenk, Econometric Institute, Eras-  location  problem  is  solved  by  determining  the 
mus University, Rotterdam, The Netherlands.  location  x ~  9-  which  minimizes  the  expected 
0167-6377/93/$06.00 © 1993 -  Elsevier Science Publishers B.V. All rights reserved  213 Volume 13, Number 4  OPERATIONS RESEARCH LETTERS  May 1993 
total travel time m(x).  Hence, in the special case  particular for the Markovian loss model with ar- 
where  ,qr denotes some network  X  this reduces  rival  rate  I  and  service rate  1/m(x)  d and  so we 
to finding the so-called Hakimi median (cf. [3]) at  are  finished  by  showing  that  A/"(1)  < X  (z)  with 
one of the nodes of X.  The above result can be  JV (i) the number of customers in the steady state 
verified easily without any calculations by using a  in  a  Markovian  M/M/k/k  loss  system  i  with 
well-known  stochastic  monotonicity  result  for  arrival  rate  h  and  service  rate  1/m(xi).  If  Xt  (° 
birth and death processes. Before proving this we  denotes the number of customers at time t  in the 
need  the  following observations.  By Little's  for-  same Markovian loss system i  then the stochastic 
mula (cf. [5]) the quantity (1 -  P~(x))m(x)  equals  processes  {Art  ~i),  t >_ 0},  i =  1,  2,  are  birth  and 
(1/h)L(x)where  L(x) is the expected number of  death  processes  on  the  finite  state  space 
customers in the system and  h  is the arrival rate  {0, 1  ..... k} with nonzero transition rates qs,s+~ = 
of the Poisson process. Moreover, by the PASTA  A, 0 <j < k -  1 and qj,j-1 =j/m(xg) for system i, 
property, i.e. Poisson Arrivals See Time Averages  j  =  1  ..... k.  By assumption  we  know  that  m(x t) 
(cf. [5]), we obtain  <  m(x 2)  and  hence by a  well-known monotonic- 
P~(x) =  Pr{X(x)  = k}  ity result for birth and death processes (cf. Prop  d. 
4.2.10 of [4]) it follows that for every t >  0 APt  (1) < 
and hence by (1)  Xt  (2). This implies A/°) d AP(2) and so the result is 
1  proved.  [] 
z(x)  = Q  Pr{X(x)  =k}  +  -~L(x).  (2) 
Using (2) one can now prove the following result.  Acknowledgement 
Lemma 1.1.  Let x t, x 2 ~ 3 r. Then m(x 0  < m(x2)  The authors like to thank the anonymous ref- 
implies z(x l) < z(x2),  eree for his useful comments. 
Proof.  Since  the  expected  number  L(x)  of cus- 
tomers  in the  steady state  equals  ~k  Pr{A;'(x)  References  j=l 
>j},  it is sufficient by (2) to show that  At(x1) is 
stochastically  smaller  than  ,/]/(X2)(,/~(X1)  ~  [1]  S.S.  Chiu and R.C.  Larson, "Locating an n-server facility 
X(x2)),  i.e.  Pr{Ar(xl) > j} < Pr{X(x  2) > j}  for  in a stochastic environment", Comp. Oper. Res. 12, 509-516 
--  --  --  (1985). 
every 0 < j  < k. In order to prove this we observe  [2] J.W. Cohen,  On Regenerative Processes in Queueing The- 
that for the M/G/k/k  loss model corresponding  ory, Lecture notes in  economics and  mathematical sys- 
to a facility location in  x  (cf. [2])  terns, volume 121, Springer-Verlag,  Berlin, 1976. 
p(X) J//J!  [3]  S.L.  Hakimi, "Optimal locations of switching centers in a 
communication  network  and  median  of  a  graph",  Oper. 
Pr{X(x)  =j}  =  g  Res. 12, 450-459 (1964). 
p( x)i/i!  [41 D.  Stoyan,  Comparison Methods for  Queues and  other 
Stochastic Models, Wiley, New York, 1983. 
i=0  [5]  H.C. Tijms, Stochastic modelling and analysis (A computa- 
for  0 <j < k  with  p(x) = Am(x).  This  holds  in  tional approach), Wiley, New York, 1986. 
214 

A note on a stochastic location problem

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