J. Service Science & Management, 2010, 3, 186-197
doi:10.4236/jssm.2010.32023 Published Online June 2010 (http://www.SciRP.org/journal/jssm)
Modeling Customer Reactions to Congestion in
Competitive Service Facilities
Mohammad Saidi-Mehrabad, Ebrahim Teimory, Ali Pahlavani*
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran.
Received February 16th, 2010; revised March 21st, 2010; accepted April 25th, 2010.
ABSTRACT
This paper reviews classic approaches for modeling customers’ choice behavior in competitive facility planning prob-
lems. They are either deterministic or probabilistic and work by a utility function based on some factors whether cus-
tomer-independent or dependent. This paper focuses especially on congestion, the most important factor in customer to
service or fixed-server systems. Various behaviors which customers may divulge when they face with a congested facil-
ity are extensively studied. We also define a new congestion-sensitivity reaction which has not been considered in the
literature. Relevant modeling approaches are proposed to formulate customers-sensitivity to congestion. An illustrative
example is also given to analyze and compare the proposed approaches.
Keywords: Competitive Planning Models, Customers’ Choice Behavior, Congestion, Waiting Time
1. Introduction
A large part of planning problems which ask for firms’
location and pricing decisions occurs in a situation in
which there exist other facilities providing the same or
homogonous service or product. In the relevant models,
planner aims to devise the better alternatives for a firm
competing for customers’ purchasing power with other
firms. For example, the problem of locating shop centers,
banks, ATMs, super-markets and restaurants could be
modeled using this paradigm [1]. This problem known as
competitive facility location model maximizes market
share, revenue or profit.
Prior to coming to a decision using the model, it is re-
quired to determine how customers behave or how they
decide to choose a facility and furthermore what is their
expected expenditure.
There are two main categories on retail facilities choi-
ce models: descriptive-determinist approach and explica-
tive-stochastic approach.
Descriptive approaches are based on observation. They
rely on unreal assumptions such as customers choose the
closest facility. Most classic location problems such as
p-median [2] and MCLP [3] are often formulated based
on this assumption. Hotelling [4] was the first on study-
ing a competitive location planning model using a de-
scriptive approach. MAXCAP [5] is a well known com-
petitive location problem based on this approach. Cus-
tomers’ purchasing power is distributed among different
facilities according to a deterministic or zero-one ap-
proach which is called also full capture [6]. In this case,
the whole demand of a customer is captured by a facility
which is the best for him/her according to a utility func-
tion. Conventionally, the utility function is defined based
on only distance or travelling time. This is true when
differences between facilities are negligible, or in areas
where shopping opportunities are few and transportation
is difficult [7]. In many cases however, facilities are mul-
tiform, i.e., they do differ in other aspects than the mere
site where they are located, and customers will take these
differences into account in the way they feel attracted to
them [8].
In the explicative approach for formulating customers’
behavior, historical information is implemented to com-
prehend dynamics of retail selling competition and how
customers choose purchasing opportunities. Spatial in-
teraction model as the most important branch of the ex-
plicative approach is first developed by Huff [9]. Spatial
interaction is the process whereby entities at different
points in physical space make contacts, demand/supply
decisions or locational choices [10].
Spatial interaction models postulate that customers
compare alternatives based on their evaluation of the
total utility of the facility and not merely on its location.
Huff argued that when customers have several alterna-
tives, they may consider visiting different facilities rather
than restricting their patronage to only one facility. Based

Modeling Customer Reactions to Congestion in Competitive Service Facilities187
on this claim, Huff coined his idea that assumes the cus-
tomers’ behavior to be probabilistic rather than determi-
nistic. He defined a utility function as 2
.
jij
A
d

 where A
is the facility’s attraction measure and d is the distance to
the facility and
β
2 is the sensitivity of customers to dis-
tance. In his model, the probability of patronizing facility
j by customer i (xij) is determined as
2
2
.,,
.
ijj ij
ij
ik kik
kE kE
uAd
x
iNjE
uAd





 
 (1)
where the denominator of Equation (1) sums up the utili-
ties of customer i from all facilities (E).
As a result, if there are Di customers resided at demand
point i, the expected number of customers visiting facil-
ity j will be
., ,
iji ij
EDxiNjE (2)
Later, the inclusion of other characteristics in Huff's
model originated other models. According to [11], char-
acteristics of a retail facility could be categorized into
two groups. The characteristics included in the first
group are independent of customer’s origin (e.g. product
quality, price, facility’s convenience level and its size).
The other group includes characteristics that are depend-
ent on the customer’s origin such as distance or travelling
time.
By a Multinomial Logit model [12], the above prob-
ability is given as the following.
exp() ,,
exp( )
ij
ij
ik
kE
V
x
iNjE
V



y a central planner.
(3)
where Vij is the utility perceived by customer i from fa-
cility j. Conventionally, this utility is expressed as a lin-
ear additive function of facilities’ characteristics.
As pointed before, in addition to distance there are
other criteria affecting customer’s choice behavior. Au-
thors in [13] developed a competitive location and design
model in which customers decide based on distance and
some other design variables. These may be quality, con-
gestion level or offered price. Among them, congestion is
very important especially in a competitive service market.
In the service sector, customers’ impatience to being
served has been considered as a main issue of competi-
tive advantage. Convenience in terms of service speed is
usually accounted for premier on price. In this atmos-
phere, a competitor will succeed if it responses fairly to
this requirement.
Customers divulge their impatience by reacting to the
level of congestion at facilities. For service-to-customer
systems, the congestion is reflected to customers by wai-
ting time or response time and for customer-to-service
systems it is measured by waiting time or system occu-
pancy level.
For a competing firm that plans to maximize its market
share, the congestion should be taken into account as a
main customer’s choice criterion. A considerable part of
the literature is devoted to an approach by which cus-
tomers consider the congestion of facilities at their ori-
gins. In this approach, it is assumed that customers know
facilities’ congestion level at the beginning and decide a
facility or a set of facilities based on a measure such as
mean waiting time or mean occupancy level and/or total
admissions. It has been extensively studied by various
researchers.
For instance, Lee and Cohen [14] studied the existence
and uniqueness of equilibrium demand for service facili-
ties serving congestion-sensitive customers. Congestion
is considered by customers in their initial decisions. In
[15] MAXCAP model is improved to include waiting
time as a customers’ choice criterion along with travel-
ling time. However it utilizes a deterministic choice ap-
proach and assumes all facilities to be single server. A
multi-server facility location problem was developed
[16] in which customers’ demand is distributed according
to a Multinomial Logit model based on travelling and
waiting time. There have been also some simultaneous
optimization models. A simultaneous location and capac-
ity optimization model is presented for a competitor in a
market with customers considering the mean waiting
time in their initial choices [17]. Aboolian et al. [18]
presented a competitive web server location and design
problem in which customers make choice based on the
difference in expected response times between new and
old facilities. Demand elasticity to congestion has been
also studied in facilities planning issues [14,19]. In a dif-
ferent approach [20], the author formulated a model for
locating multiple-server, congestible facilities. He de-
fined demand to be elastic to travelling time and also
system occupancy level. However the customers' alloca-
tion is deterministic b
This paper criticizes the common approach for model-
ing congestion-sensitivity of customers involved in a
competitive service market. We study the obvious reac-
tions of customers that face by congested facilities. We
also develop five different approaches for formulating
such reactions in competitive planning models. Except
one approach that presents a learning process on conges-
tion level, the other approaches follow a two steps frame-
work. In the first step customers decide probabilistically
based on a utility function depending on distance and
offered price. In the second step they take congestion
into account and determine whether to patronize a facil-
ity or not. The manner how they react to the congestion
defines the behavior.
The rest of the paper is organized as follows. Section 2
describes our proposed frameworks for modeling cus-
tomers’ reactions to congestion. Section 3 gives some

Modeling Customer Reactions to Congestion in Competitive Service Facilities
188
experimental results on the models and finally Section 4
concludes the paper and proposes future research issues.
2. Our Modeling Frameworks
Suppose that the market is a network that includes some
nodes (N = {1, 2, ..., n}) as demand origins and also as
potential facility sites. Let E ⊂ N (|E| = q) be the set of
our firm's facilities and E' ⊂ N (|E'| = q') be the set of
other competitors’ facilities. There are also some edges
(G) each of them indicates the availability of a direct
path between two nodes. The network is in a metric
space equipped with distance d being the shortest path
distance.
Without loss of generality, it is assumed that custom-
ers arrive from multiple infinite sources according to a
Poisson process with mean demand generating rate
,
iiN

 . They are served with FIFO discipline in fa-
cilities which utilize m servers all with exponentially
distributed service time with mean 1/μ. Buffer volume of
each facility is also limited to K. We have the following
performance indicators:
(Utilization factor) /

  (4)
(Mean queue length) (5) ()P
K
n
nm
Lnm


r
(Mean waiting time) /wL

 (6)
where λ is the arrival rate of a facility and defined as the
sum of demand generating rates of demand nodes pa-
tronizing the facility,  is the effective arrival rate and
Prn is the probability that there are n customers at a facil-
ity. This probability is a function of the arrival rate, λ and
is defined according to the structure of queuing system.
With the conventional approach for formulating cong-
estion-sensitivity as explained in Section 1, customers
consider all criteria simultaneously and they have to
make a definite decision on destination facilities when
they are at their origins. The planner assumes that they
cannot deviate from their initial decisions. Obviously this
is not a real adaptation from human decision making.
Since customers usually follow a changing mood and
moreover don’t know all criteria simultaneously, they
follow a sequential decision making process.
Whether they employ a simultaneous or sequential de-
cision making, the probability distribution of their de-
mand should be determined.
We define this probability according to Multinomial
Logit model [12] as
,,
ij
ik
c
ij c
kE E
e
x
iNjEE
e









(7)
where υ is a parameter defined as /
 6, and σ is
the standard deviation in taste of the customers [16]. The
dispersion in facility choice increases with smaller values
for υ resulted from higher values of σ.
The main indicator of the probability is the cost in-
curred by customers to being served, cij. The determining
factors of this cost may differ for different customers.
The manner through which the congestion is included
in customers’ choices is the main issue considered by this
paper. We describe different reactions of customers to
congested competitive facilities and present appropriate
approaches to determine the effective arrival rates of
facilities and the firms’ market share.
2.1 Customers are Insensitive to Congestion
For the case which the arriving customers are not con-
gestion-sensitive, cij in Equation (7) is defined as the sum
of offered price and cost of travelling time to the facility.
., ,
ij ijij
cpft iNjEE

  (8)
where tij is the travelling time between nodes i and j, pa-
rameter f is the cost of unit time and pij is the service
price offered by the facility located at j to customer i.
We have the following term for the effective arrival
rate of facility j.
1
.,
n
jj iij
i
x
jEE  

  
 (9)
2.2 Customers Revise their Decisions According
to their New Observations on Waiting Times
In this case the congestion level is stated by mean wait-
ing time. It is assumed that customers initially don’t
know anything about waiting time levels in a new estab-
lished facility and they cannot foreknow congestion level.
Therefore, at their first trip they choose facilities based
on factors other than congestion. Their experienced wait-
ing times are included in their second trip. This process
will be continued until an equilibrium demand distribu-
tion is found.
Therefore cij is defined as the sum of offered price,
cost of travelling time to the facility and cost of waiting
time at the facility.
.( ),,
ij ijij j
cpftw iNjEE



(10)
where wj is the mean waiting time of facility j according
to Equation (6). The effective arrival rate in this case will
be as the following
(1Pr ),
jj K
jEE 
  
(11)
where K is the maximum capacity of the system and
probability PrK denotes the probability that there are K
customers at the facility. The state probabilities of the
considered queuing system are computed according to
[21] as

Modeling Customer Reactions to Congestion in Competitive Service Facilities189
Prk
0
0
Pr
!
Pr
!
0
k
k
km
fork m
k
f
orm kK
mm
fork K












(12)
1
0
11
Pr 1!!
nm
nm
mK
nnm
nm m
 





 










(13)
By this approach, it is assumed that at the first usage
of network after a new facility’s establishment or net-
work redesign, the mean waiting times of facilities are
not known to the customers and their renewed knowl-
edge about waiting time levels affects their next choices.
To formulate this framework, we present a procedure
with the following steps:
1) Set t = 0 and E
,
() 0, t
j
wjE
2) Compute () .(), ,
tt
ij ijij j
cpftwiIjEE

,
3) Compute ,,
t
ij
x
iIjEE

   using Equation
(7),
4) Compute the arrival rates as (1)
1.,
n
tt
j
iij
i
x




jEE
 and the effective arrival rates using Equa-
tion (11),

5) Compute E using Equation (6),
(1)
,
t
j
wjE


6) Check convergence condition. If it holds, stop with
the current arrival rates else set t = t + 1 and go to step 2.
Convergence is reached when the value of two succes-
sive results for ,
jjEE


  become close together,
i.e., (1)()tt
jj
jE E





, where ε is a nonnegative
small real number.
Since the approach assumes that decisions made by
competing firms change the congestion level and the cus-
tomers need to learn how to apply it in their choices, it
better suits with decision making situations which highly
affect the congestion level. Therefore it is efficient for
competitive location and design planning models rather
than pricing models.
2.3 Customers Balks from Entering the Facility
When they Arrive
Similar to Subsection 2.2, it is assumed that customers
initially don’t know anything about the congestion level
of facilities. However they never consider congestion at
their origins but behave in a sequential manner. At the
first step they decide based on the sum of offered price
and cost of travelling time i.e.
., ,
ij ijij
cp ft iNjEE

  (14)
At the second step, they react to the congestion when
they arrive at a facility. As stated in [20], in the case of
non-essential services, some of the arriving customers
will choose not to wait if they see a long queue, i.e., they
balk from waiting in the queue. We define a parameter β
 [0,1] which accounts for the decrease of the demand
with respect to the system’s occupancy level faced by the
customer.
Parameter βk is the percentage of the customers willing
to wait in the queue given that k other customers are at-
tending in the facility. It is defined as
max(0; )
1
0
k
km if kK
Km
otherwise










(15)
where m is the number of servers and K is the maximum
possible capacity of the facility. A typical instance for
balking function for m = 2 and K = 10 is given in Figure 1.
With xij defined by Equations (7) and (14), the per-
centage of customers i that patronize facility j and joins
the queue, given that there are k other customers in the
facility, is
..,,,1, 2,...,
ijkki ij
x
iNjEEk K 

 (16)
As a result, the effective arrival rate of facility j would
be as the following:
110
10
.Pr ()
.Pr()..,
nnK
jijijkk j
iik
nK
kkjiij
ik
x
jEE
 







 
(17)
where λj is defined as
1
.,
n
jiij
i
x
jEE 

 
 (18)
The state probabilities (Prk) of the system are derived
according to death and birth flow diagram [21] as the
following,
Number of customers
Beta (Captured Percentage)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 1. A typical balking function

Modeling Customer Reactions to Congestion in Competitive Service Facilities
190
0
12
0
0
Pr ,
!
...
Pr Pr,
!
0
n
n
nn m
nnm
nm
n
mnK
mm
nK

 













(19)
1
12
0
01
... 1
Pr !!
n
mK
nnn m
nm
nnm
nm m
 














 
(20)
2.4 Joined Customers Renege from Waiting
In the previous case, it is assumed that all the customers
that join the queue stay until served by a server. However,
it is also possible for an impatient customer to depart the
queue, i.e., he/she may renege from waiting. In this case,
after joining the queue each customer will wait a certain
length of time to being served. If the service has not be-
gun by then, he/she departs. This time is a random vari-
able whose density function is () t
rt e



. Consequ-
ently, the effective arrival rate will be as the following:
10
..Pr()..
nK
j
kkkj iij
ik
Rx 

 
(21)
where λj, βk, γi and xij are the same as Subsection 2.3 and
Rk is the probability that a new arrived customer will
survive to be serviced given that there are k customers in
the facility on arrival and given that it joins. In [22] it is
proved that
1
k
Rkm


  (22)
where .m



.
The state probabilities are derived according to death
and birth flow as the following,
0
12
0
0
Pr ,
!
...
Pr Pr,
!(1)
0
n
m
nm nn m
n
nm
nm
n
mn K
m
nK

 

















(23)
1
12
0
01
...
Pr !! (1)
nm
mK
nm nn m
nnm nm
nm
  


 









  


(24)
where /

 .
2.5 Balked or Reneged Customers may Veer
from their Initial Destinations
In the case of essential services or a fierce competitive
market, the balked or reneged customers may go directly to
another facility rather than coming back to their origins, i.e.,
they veer and deviate from their initial decisions. In this
case the second facility would indirectly capture their de-
mand. We assume that customers do such upturns only for
one cycle due to travelling time and cost issues.
Figure 2 illustrates the two situations in a simple net-
work where there is a single demand node with four cust-
omers and three facilities are serving them. Initially three
customers choose to patronize facility F1 and one of th-
em chooses facility F2 (Figure 2(a)). Because of unbea-
rable congestion at facility F1, the customers balk or re-
nege from waiting and return to their origins (Figure 2(b)).
But this is not the case for all situations. They may go di-
rectly to another facility close to facility F1 with the aim of
being served in a less congested facility (Figure 2(c)).
We need to determine the percentage of customers
which may divulge such behavior. Assume that PrB
l

is the balking prob-
ability for the customers interested initially in facility l
and
0(1).Pr(),
K
kkl
klEE




0
Pr.(1).Pr (
K
R
lkk
kR



),
kl
lEE



is the
reneging probability for them. To reflect the veering be-
havior in the model, we also define a variable zlj which
stands for the probability that a customer balked or re-
neged from facility l, will choose facility j. This proba-
Figure 2. An example for comparison of two congestion-
sensitivity manners: (a) Customers dispatching from the
demand node; (b) customers balk or renege and return to
their origins; (c) Some of balked or reneged customers go to
another facility (veer)

Modeling Customer Reactions to Congestion in Competitive Service Facilities191
bility depends on the difference of prices offered by two
facilities and travelling time between them as follows:
(1/).,, ,
lj
lk
c
ljl mc
kE E
e
zQd lEEkj
e





 
l
j
(25)
.,, ,
ljijillj
cppftjlEEl


(26)
where dm is the maximum distance between two nodes in
the network and Ql is the centrality index for facility l. It
is defined as
,
'1
lj
jl
l
d
Ql
qq




EE

(27)
We implement the centrality index to determine how
far the facility is from other facilities, i.e., the density of
network’s areas. The density measure is utilized to define
what part of balked or reneged customers from a facility
will go to other facilities. For a facility established in a
dense area, the probability that a balked or reneged cus-
tomer will go directly to another facility will be higher
than that for sparse areas. Figure 3 illustrates the effect
of centrality index for the example depicted by Figure 2.
2.5.1 Only Balked Customers Choose to Go to
Another Facility
In this case, the arrival rate could be partitioned into two
parts, one part for directly captured demand and another
one for indirectly captured demand of balked customers.
Therefore, Equation (18) becomes
1
..Pr.,
lj
nB
jiijilllj
ilEEiN
x
zjEE  



 
 (28)
Figure 3. An example illustrating the effect of centrality
index: (a) Facility F3 is closer to F1 and larger part of
balked or reneged customers decide to go to F3; (b) Facility
F3 is farther to F1 and smaller part of them decide to go to
F3
The effective arrival rate in this case is the same as
Equation (17) where probabilities Prk are computed using
Equations (19)-(20).
2.5.2 Both Balked and Reneged Customers Choose to
Go to Another Facility
In this case, the arrival rate could be partitioned into
three parts, the first part for directly captured demand,
the second part for indirectly captured demand of balked
customers and the third part for indirectly captured de-
mand of reneged customers.
Therefore, Equation (18) becomes
1
..Pr.
.Pr.,
lj
lj
nB
jiijilllj
ilEEiN
R
illlj
lE EiN
xz
zjEE
 



 

 




 
(29)
The effective arrival rate in this case is the same as
Equation (21) where probabilities Prk are computed from
Equations (23)-(24).
Obviously, the right-hand side of Equations (28) and
(29) is a function of λjs (jEE


). Therefore, it can be
written as a system of equations,
(),
jjEE




(30)
where


is the vector of facilities’ arrival rates. Since
(.)

is a non-linear function and there are (q + q') facili-
ties, Equation (30) indicates a non-linear system of (q +
q') equations and (q + q') variables.
For solving this system of equations we employ a pro-
cedure similar to fixed point iteration approach [23]. This
procedure has the following steps:
1) Compute ,,
ij
x
iNjEE


 using Equations
(7) and (14);
2) Set t = 0 and () 0
t



. Compute
()
Pr (),
t
kj

k


0,1, 2,...,
K
using Equations (19)-(20) or Equations
(23)-(24);
3) Compute using Equation (28) or (29);
)( )(t
j

4) Compute a new value
)().1(. )()()1(ttt 




, 10 

(31)
where Γ is the vector of right hand side functions of
Equation (28) or (29),(.)

and θ is a problem-depen-
dant factor.
5) Check the convergence condition. If it holds, stop
with the current solution else set t = t + 1 and go to step
3. Convergence is reached when the value of two su-
ccessive results for 

become close together i.e.
 
)()1( tt


, where ε is a nonnegative small real
number.

Modeling Customer Reactions to Congestion in Competitive Service Facilities
192
Having defined the possible reactions of impatient
customers in congestible facilities, we can analyze them
in a competitive planning model. The utilized measure
for this purpose is the market share of firms or their fa-
cilities. However, other measures could also be derived.
The market share of our firm is defined as
1
/
n
j
i
jE i
MS


 (32)
In the next section we test the approaches through an
illustrative example.
3. An Illustrative Example
Suppose that an area is formed as a network that includes
50 demand nodes. Three firms are competing with each
other for customers’ purchasing power. They have just
established some facilities. The deployment outline of
the demand nodes and also the firms’ facilities is exhib-
ited by Figure 4.
Note that all nodes in the network indicate a demand
node. An oval node indicates that a facility of firm 1 has
been located in that node. A square node indicates that a
facility belonging to firm 2 has been established in the
node and a diamond node shows a facility of firm 3. The
circles show the demand nodes with no established facility.
The length of available direct paths in the network is
known and the shortest distance between each pair of
nodes could be determined.
Table 1 gives the demand generating rates for demand
nodes. Table 2 gives the queuing parameters for the
competing firms. The price charged for customers is as-
sumed to be p = 12, the same for all three firms. The val-
ues for other parameters are given in Table 3.
Now we apply different congestion-sensitivity reac-
tions and their relevant modeling approaches on the de-
fined problem. The obtained results are given in Table 4.
The table gives the market share of competing firms and
their facilities from demand nodes. The last column gives
the percentage of total captured demand of the market.
Table 4 illustrates also the results for the case which
disregards congestion effects (Subsection 2.1). It is given
only for comparison purposes.
Figure 4. An outline of the market area in the example
Table 1. The nodes’ demand generating rates
Node Rate Node Rate Node Rate Node Rate Node Rate
1 1.2 11 3.9 21 1.9 31 1.1 41 3.8
2 1.7 12 1.8 22 2.7 32 4 42 3.7
3 4.1 13 4 23 1.2 33 2.3 43 0.1
4 1.5 14 0.8 24 1.1 34 2.1 44 4
5 3.3 15 0.7 25 1 35 2.4 45 1
6 2 16 1.5 26 2.2 36 1.5 46 2.6
7 1.5 17 1.7 27 0.9 37 2.9 47 2.2
8 0.8 18 1.3 28 1.7 38 1.7 48 1.3
9 3.1 19 0.7 29 0.1 39 0.4 49 3.3
10 2.8 20 1 30 2.3 40 1.2 50 0.2
Table 2. The queuing specifications of the firms
Firm Number of FacilitiesFacility Nodes (No. of Servers) System Capacity (K) Service Rate (μ)
Firm 1 2 37 (2), 39 (3) 10 5
Firm 2 7 2 (2), 11 (1), 14 (2), 21 (2), 34 (1), 38 (1), 45 (3)8 4
Firm 3 2 4 (2), 29 (2) 15 5

Modeling Customer Reactions to Congestion in Competitive Service Facilities193
Table 3. Other parameters of the network
Measure Value
Customers’ behavior uniformity (υ) 0.1
Cost of travelling and waiting time (f) 1
Reneging rate (α) 5
As it can be seen from Table 4, in the cases which
customers are congestion-insensitive or decide at origin
based on their knowledge on waiting time levels, the
whole available demand of the market is captured. This
is because that; in these two cases the customers don’t
escape from congestion but accept it as a usual phe-
nomenon.
The case of “Balking and reneging” results in the least
market capture because the congestion-sensitive or impa-
tient customers leave highly congested facilities and re-
turn to their origins. When a part of those leaving cus-
tomers doesn’t return and decides to being served by
other facilities, the overall capture increases. This is re-
flected by “Balking, reneging and veering” case. A simi-
lar analysis could be stated for “Balking” and “Balking
and veering” cases.
It is interesting to note that facility F2 of firm 1 cap-
tures maximum share of the market except for conges-
tion-insensitivity case. This is because of its better loca-
tion and also its larger number of servers. In the contrast,
facility F2 of firm 2 captures the minimum share of the
market except for congestion-insensitivity case. This is
because that it has only one server and its system capac-
ity and service rate are smaller than other facilities. In the
congestion-insensitivity case, since the congestion effect
is disregarded, the only parameters affecting customers’
behavior are price and facilities’ location. Since price is
assumed to be the same for all facilities, their locations
play the main role in determining market share. There-
fore it is expected that a facility located at a dense area
would capture a larger share of the market.
In the second set of experiments we analyze the effect
of different parameters such as the default number of
servers, mean service rate and system capacity on the
firms’ market shares. The results are given by Figures
5-8.
Figure 5 presents the analysis with respect to the firm
3’s mean service rate which changes by −80% to +80%
(in steps of 40%) around its base value (µ0 = 5).
Table 4. The market share of firms and facilities (percentage)
Firm 1 Firm 2 Firm 3
Customer behavior
F1 F2 Total F1 F2F3F4 F5F6F7TotalF1 F2 Total
Overall
Capture
Congestion-insensitive 9.3 9.9 19.1 9.1 8.7 7.910.09.89.78.863.98.0 8.9 17.0100.0
Learning to revise 9.6 10.4 20.0 9.3 8.08.110.18.98.99.362.68.3 9.1 17.4100.0
Balking 8.4 9.4 17.9 8.0 6.47.08.7 7.17.18.352.47.6 8.4 16.086.3
Balking and veering 9.4 10.6 20.0 8.7 7.07.79.5 8.18.09.158.18.2 9.2 17.495.4
Balking and reneging 7.4 9.0 16.3 6.7 4.36.07.2 4.74.67.841.26.6 7.2 13.971.4
Balking, reneging and
veering 9.3 11.1 20.4 8.2 5.8 7.59.0 6.96.99.453.77.8 8.8 16.690.8
(a)
(b)

Modeling Customer Reactions to Congestion in Competitive Service Facilities
194
(c)
(d)
(e)
Figure 5. Sensitivity of firms’ market share to the change in
mean service rate
Similarly, Figure 6 presents the analysis with respect
to the default number of servers for firm 3’s facilities and
Figure 7 presents the analysis with respect to the default
capacity of firm 3’s facilities.
Figure 8 shows the results of analyzing the effect of
reneging rate of customers on the firms’ market share.
We change its base value (α = 5) by −80% to +80% (in
steps of 40%). This test could be applied only on two
cases which deal with reneging customers.
We summarize our observations of the sensitivity ana-
lyses as the following:
 From Figures 5 and 6, we conclude that more mar-
ket demand will be captured by firm 3 when the servers’
number assigned to its facilities or the mean service rate
of its servers is high. The market shares of other firms
decrease except for “Balking” and “Balking and reneg-
ing” cases because in these two cases, the parameters and
also the arrival rates of other firms are not changed.
(a)
(b)
(c)
(d)
(e)
Figure 6. Sensitivity of firms’ market share to the servers’
number

Modeling Customer Reactions to Congestion in Competitive Service Facilities195
The result achieved from analyzing the effect of sys-
tem capacity (Figure 7) is similar to the mean service
rate and servers’ number except for the case of “Learning
to revise” in which a larger system capacity has a nega-
tive effect on firm 3’s market share. This can be reasoned
regarding the fact that a larger system capacity will cause
longer waiting time.
 From Figure 8, we conclude that high reneging
rates lower the market share of all firms. The decrement
slope in the case of “Balking and reneging” is high be-
cause all reneged customers return to their homes.
 Variation of other parameters such as price, demand
rates and time to cost parameter has not considerable
impacts on the final results.
(a)
(b)
(c)
(d)
(e)
Figure 7. Sensitivity of firms’ market share to the system
capacity
(a)
(b)
Figure 8. Sensitivity of firms’ market share to reneging rate

Modeling Customer Reactions to Congestion in Competitive Service Facilities
196
4. Conclusions and Future Research
In this paper we have considered customers' patronizing
behavior in a competitive market. It has been concluded
that the better approach for formulating customers’ cho-
ice behavior in spatial competitive modeling is a prob-
abilistic model based on three variables, distance, waiting
time and price. With emphasis on congestion effects, we
have also studied customers’ reactions to congested fac-
ilities. These are especially balking, reneging and veering.
This is the first paper considering congestion-sensitivity
reactions in competitive congested systems and the first
work studying veering as a usual event in congested sys-
tems. By veering we mean the case in which after a cus-
tomer balked or reneged from a facility, he/she may de-
cide to patronize another facility rather than coming back
to his/her origin.
Although the prevailing approach in the literature as-
sumes that customers take congestion into account at
their origins, it has been claimed that they initially don’t
know a lot about facilities’ congestion level. Our pro-
posed approaches retain customers unaware until they
reach at the facilities. The first approach assumes that
customers amend their future decisions according to the
waiting time faced by them at the previous experiences.
The four other approaches assume that customers react to
the congestion when they reach at the facilities. They
may balk, renege, veer or divulge a combination of them.
An illustrative example has also given to demonstrate
differences between the outcomes of proposed ap-
proaches. We have seen that congestion-sensitivity of
customers has a considerable effect on the firms’ market
share. Therefore, a much attention must be paid for for-
mulating the congestion-sensitivity of customers in spa-
tial planning models.
We have tried to study all possible reactions to the
congestion. However, a special type of queues has been
considered. It will be interesting to study other types of
queuing systems.
REFERENCES
[1] T. Drezner and Z. Drezner, “Finding the Optimal Solution
to the Huff Based Competitive Location Model,” Com-
putational Management Science, Vol. 1, No. 2, 2004, pp.
193-208.
[2] S. L. Hakimi, “Optimum Locations of Switching Centres
and the Absolute Centres and Medians of a Graph,” Op-
erations Research, Vol. 12, No. 3, 1964, pp. 450-459.
[3] R. L. Church and C. ReVelle, “The Maximal Covering
Location Problem,” Papers of Regional Science Associa-
tion, Vol. 32, No. 1, 1974, pp. 101-118,
[4] H. Hotelling, “Stability in Competition,” Economic Jour-
nal, Vol. 39, No. 153, 1929, pp. 41-57.
[5] C. ReVelle, “The Maximum Capture or Sphere of Influ-
ence Location Problem: Hotelling Revisited on a Net-
work,” Journal of Regional Science, Vol. 26, No. 2, 1986,
pp. 343-358.
[6] R. Aboolian, O. Berman and D. Krass, “Competitive
Facility Location Model with Concave Demand,” Euro-
pean Journal of Operational Research, Vol. 181, No. 1,
2007, pp. 598-619.
[7] M. D. Garcia Perez, P. F. Hernandez and B. P. Pelegrin,
“On Price Competition in Location-Price Models with
Spatially Separated Markets,” TOP, Vol. 12, No. 2, 2004,
pp. 351-374.
[8] F. Plastria, “Static Competitive Facility Location: An
Overview of Optimisation Approaches,” European Jour-
nal of Operational Research, Vol. 129, No. 3, 2001, pp.
461-470.
[9] D. Huff, “Defining and Estimating a Trading Area,”
Journal of Marketing, Vol. 28, No. 3, 1964, pp. 34-38.
[10] J. R. Roy and J. C. Thill, “Spatial Interaction Modeling,”
Papers in Regional Science, Vol. 83, No. 4, 2004, 339-
361.
[11] A. Jain and V. Mahajan, “Evaluating the Competitive
Environment in Retailing Using the Multiplicative Com-
petitive Interactive Model,” In: J. Sheth, Ed., Research in
Marketing, JAI Press, Greenwich, Vol. 2, 1979, pp. 217-
235.
[12] D. McFadden, “Conditional Logit Analysis of Qualitative
Choice Behaviour,” In: P. Zarembka, Ed., Frontiers in
Econometrics, Academic Press, New York, 1974.
[13] R. Aboolian, O. Berman and D. Krass, “Competitive Fac-
ility Location and Design Problem,” European Journal of
Operational Research, Vol. 182, No. 1, 2007, pp. 40-62.
[14] H. Lee and M. Cohen, “Equilibrium Analysis of Disagg-
Regate Facility Choice System Subject to Congestion-
Elastic Demand,” Operations Research, Vol. 33, No. 2,
1985, pp. 293-311,
[15] F. Silva and D. Serra, “Incorporating Waiting Time in
Competitive Location Models,” Networks and Spatial
Economics, Vol. 7, No. 1, 2007, pp. 63-76.
[16] V. Marianov, M. Rios and M. J. Icaza, “Facility Location
for Market Capture When Users Rank Facilities by
Shorter Travel and Waiting Times,” European Journal of
Operational Research, Vol. 191, No. 1, 2008, pp. 32-44.
[17] A. M. Kwasnica and E. Stavrulaki, “Competitive Loca-
tion and Capacity Decisions for Firms Serving Time-
Sensitive Customers,” Naval Research Logistics, Vol. 55,
No. 7, 2008, pp. 704-721.
[18] R. Aboolian, Y. Sun and G. J. Koehler, “A Location–
Allocation Problem for a Web Services Provider in a
Competitive Market,” European Journal of Operational
Research, Vol. 194, No. 1, 2009, pp. 64-77.
[19] M. L. Brandeau, S. S. Chiu, S. Kumar and T. A. Gro-
ssman, “Location with Market Externalities,” In: Z. Dre-
zner, Ed., Facility Location, Springer-Verlag, 1995, pp.
121-150.
[20] V. Marianov, “Location of Multiple-Server Congestible
Facilities for Maximizing Expected Demand, When Serv-
Ices are Non-Essential,” Annals of Operations Research,

Modeling Customer Reactions to Congestion in Competitive Service Facilities
197
Vol. 123, No. 4, 2003, pp. 125-141.
[21] D. Gross and C. M. Harris, “Fundamentals of Queuing
Theory,” 3rd Edition, John Wiley and Sons, New York,
2002.
[22] D. Q. Yue and Y. P. Sun, “Waiting Time of M/M/C/N
Queuing System with Balking, Reneging, and Multiple
Synchronous Vacations of Partial Servers,” Systems En-
gineering-Theory & Practice, Vol. 28, No. 2, 2008, pp.
89-97.
[23] C. T. Kelley, “Iterative Methods for Linear and Nonlinear
Equations,” No. 16 in Frontiers in Applied Mathematics,
SIAM, Philadelphia, 1995.