Modeling Customer Reactions to Congestion in Competitive Service Facilities

doi:10.4236/jssm.2010.32024

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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.

Email: [email protected]

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



 where A

is the facility’s attraction measure and d is the distance to

the facility and

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

.,,

ijj ij

ik kik

kE kE

uAd

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( )

iNjE







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

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 c

kE E

iNjEE















(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.

jj iij

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

fork m

orm kK

fork K



















(12)

Pr 1!!

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,

wjE

2) Compute ()

.(), ,

ij ijij j

cpftwiIjEE



,

3) Compute ,,

iIjEE



   using Equation

(7),

4) Compute the arrival rates as (1)

1.,

iij









jEE

 and the effective arrival rates using Equa-

tion (11),



5) Compute E using Equation (6),

(1)

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

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; )

km if kK

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

iNjEEk K





 (16)

As a result, the effective arrival rate of facility j would

be as the following:

110

.Pr ()

.Pr()..,

nnK

jijijkk j

iik

kkjiij

jEE

 













 

(17)

where λj is defined as

jiij

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)

5 6 7

8 9

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Figure 1. A typical balking function

Modeling Customer Reactions to Congestion in Competitive Service Facilities

190

Pr ,

...

Pr Pr,

nn m

nnm

mnK































(19)

...

Pr !!

nnn m

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:

..Pr()..

kkkj iij









(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

Rkm



  (22)

where .m







.

The state probabilities are derived according to death

and birth flow as the following,

Pr ,

...

Pr Pr,

!(1)

nm nn m

mn K



 































(23)

...

Pr !! (1)

nm nn m

nnm 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



is the balking prob-

ability for the customers interested initially in facility l

and

0(1).Pr(),

kkl

klEE









Pr.(1).Pr (

lkk







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/).,, ,

ljl mc

kE E

zQd lEEkj









 

l

(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











(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

..Pr.,

jiijilllj

ilEEiN

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

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

..Pr.

.Pr.,

jiijilllj

ilEEiN

illlj

lE EiN

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 ,,

iNjEE





 using Equations

(7) and (14);

2) Set t = 0 and () 0







. Compute

()

Pr (),







0,1, 2,...,

using Equations (19)-(20) or Equations

(23)-(24);

3) Compute using Equation (28) or (29);

)( )(t



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

jE i







(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.

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