Industrial Management & Data SystemsSolving two-dimensional Markov chain model for call centersChih-Chin Liang Hsing Luh

Article information:To cite this document:Chih-Chin Liang Hsing Luh , (2015),"Solving two-dimensional Markov chain model for call centers",Industrial Management & Data Systems, Vol. 115 Iss 5 pp. 901 - 922Permanent link to this document:http://dx.doi.org/10.1108/IMDS-12-2014-0363

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Solving two-dimensional Markovchain model for call centers

Chih-Chin LiangGraduate Institute of Business and Management,

National Formosa University, Hu wei, Taiwan, ROC, andHsing Luh

Department of Applied Mathematics, National Chenchi University,Taipei, Taiwan, ROC

AbstractPurpose – The purpose of this paper is to develop a novel model of a call center that must treat callswith distinctly different service depending on whether they orginate from VIP or regular customers.VIP calls must be responded to immediately but regular calls can be routed to a retrial queue if theoperators are busy.Design/methodology/approach – This study’s proposed model can easily reveal the optimalarrangement of operators while minimizing computational time and without losing any precision of theperformance measure when dealing with a call center with more operators.Findings – Based on the results of the comparison between the exact method and the proposedapproximation method, the approach shows that the larger the number of operators or inbound calls,the smaller the error between the two methods.Originality/value – This investigation presents a computational method andmanagement cost functionintended to identify the optimal number of operators for a call center. Because of computationallimitations, many operators could not be easily analyzed using the exact method. For the manager of a callcenter, the sooner the optimal solution is found, the faster business strategies are deployed. This studydevelops an approximation method and compares it with the exact method.Keywords Call center, Management cost, Operators management, Two-dimensional Markov chainPaper type Research paper

1. IntroductionMotivated by a research project involving call centers at a selected company in Taiwan(Kim et al., 2012; Liang et al., 2005, 2009), this study discusses an approximation methodsuitable for use when an exact solution is not attainable to calculation of the managementcost of a call center that involves blocking probability and waiting time (Huang, 2010;Liang and Luh, 2013; Melikov and Babayev, 2006; Xu et al., 2002; Bright and Taylor,1995).A large-scale service sector regards uninterrupted customer service as a key operationaltarget (Kim et al., 2012; Liang et al., 2005, 2009). To meet customer demands, a servicecompany must manage human resources in a call center to answer inbound calls using acomputer system. Human resource management is a strategic approach to managingemployees who individually and collectively contribute to the achievement of businessobjectives (Armstrong and Taylor, 2014; Yang et al., 2007). Moreover, a call centercharged with satisfying customer inquiries assigns staff members as phone operators(Kim et al., 2012). Additionally, operators should make all customer arrangements during

Industrial Management & DataSystems

Vol. 115 No. 5, 2015pp. 901-922

©Emerald Group Publishing Limited0263-5577

DOI 10.1108/IMDS-12-2014-0363

Received 4 December 2014Revised 1 February 2015

18 March 2015Accepted 23 March 2015

The current issue and full text archive of this journal is available on Emerald Insight at:www.emeraldinsight.com/0263-5577.htm

The authors would like to thank the Ministry of Science and Technology of the Republic of China,Taiwan, for financially supporting this research under Contract No. 103-2410-H-150-004.Additionally, the authors many thanks to Mr Han-Sheng Huang for his work about literaturereview and data collection.

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customer phone calls (Dudina et al., 2013; Gans et al., 2003). The number of operatorsrepresent the capacity of handing inbound calls. Managers can implement actions toreduce management expenses (Armstrong and Taylor, 2014, Benaroch et al., 2010;Li and Chandra, 2007; Lee and Park, 2005). That is, through analyzing the managementcosts associated with specific solutions, managers can determine the optimal operatorassignment strategy (Benaroch et al., 2010; Dubelaar et al., 2005; Lee and Park, 2005).Analysis of the management costs associated with daily call center operations canassist managers in making operator allocation decisions. Operators thus can beassigned efficiently based on analysis of the management costs associated with callcenter operations.

Obviously, the main daily job of the operator is to answer customer calls. However, notall customers benefit the company equally. On average, the revenues contributed bycustomers of telecommunication companies in Taiwan range from $7.9 US dollars/month(regular users) to $60 US dollars/month (important users) (National CommunicationCommission, 2014). This statistic implies that VIP customers can benefit a company farmore than regular customers do (Kim et al., 2012; Liang et al., 2005, 2009). Therefore basedon financial considerations companies should adopt business strategies by which theyrespond to VIP customers faster than regular ones. Obviously, the best solution for acompany is a system that prioritizes VIP calls (denoted as v-calls) over regular calls(denoted as r-calls). However, given limitations of human resources, how to assignoperators to efficiently serve customers of different levels of importance becomes crucialto developing a successful call-center service.

To optimize the assignment of operators in a call center, the guard channel scheme,which suggests the reservation of partial channels for packets with high privilege,could be adopted to help assign operators for v-calls (Choi and Chang, 1999; Do, 2010;Dudina et al., 2013; Gans et al., 2003; Servi, 2002; Winkler, 2013). This investigationconsiders a novel model for call center operations that is based on a guard channelscheme. Based on the proposed design, v-call are assigned high privilege, while stillproviding regular customers with satisfactory service. Restated, some operators shouldbe dedicated exclusively to answering v-calls, while regular customers are guaranteedto receive services from the call center. The risk of this approach is that if the regularcustomer is angry because of the long waiting time, they may spread negative feedback(Kim et al., 2012; Liang et al., 2005, 2009). Therefore, the proposed method should considerwaiting time. This study models two types of calls as a two-dimensional Markov chain(Phung-Duc et al., 2010; Tran-Gia and Mandjes, 1997). This investigation applies theV-model to call center behavior (Dudina et al., 2013; Gans et al., 2003). Specifically, twotypes of customers can enter the call center and be served. Our objective is to deriveperformance measures of the investigated model, as well as upper and lower bounds ofthe waiting time of regular customers.

This study presents a model of two call types, v-calls and r-calls, as a two-dimensionalMarkov chain. Based on analysis of the two-dimensional Markov chain, we can obtainthree measures of model performance: the probability that the system fails to serve aregular customer on the first attempt (Pr), the probability that the system fails to servea VIP on first attempt (Pv), and the average number of regular customers in the retrialgroup (Lq). Additionally, the upper and lower bounds of the waiting time for regularcustomers could be found by analysis of the two-dinensional Markov chain.

To identify the optimal assignment among K operators, this investigation constructs amanagement cost function per unit of time using a queueing model and sets the optimalthresholdN. For example, if the number of occupied operators who are too busy to answer

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customer inquiries is N or larger, the remaining K-N operators are reserved for servingVIPs. On the other hand, if the incoming r-call which finds no operators available on itsarrival, the call sent back to the retrial group to wait for the next available operators.Based on the above descriptions, the management cost (MC (N,K)) is expressed as follows,where Cw denotes the holding cost of calls in the retrial group, while Cr and Cv representthe opportunity costs of blocking calls of regular customers and VIPs:

MC N ;Kð Þ ¼ CwLqþCrPrþCvPv (1)

.From Equation (1), the results could be obtained by computing a two-dimensional Markovchain as explained below. However, applying the proposed model to a real case, it wouldbe difficult to identify the optimal N because of the long computing time. Because thenumber of operators is generally large, with more than 100 operators in a large-scaled callcenter, the above complex computation necessitates the expenditure of considerable effortto solve stationary probabilities (Choi and Chang, 1999; Do, 2010; Dudina et al., 2013;Gans et al., 2003; Servi, 2002; Winkler, 2013). Since computers necessarily have limitedcomputational ability, processes might be delayed if performed using a low-end computer.Additionally, such limited computational ability wastes excessive energy together withthe excessive waiting time. On the other hand, a high-end computational machine is neededto solve this problem for the large number of operators, if no appropriate solution can beprovided. In order to resolve this problem, the investigation provides an approximationmethod known as the phase merging algorithm, introduced by Melikov and Babayev(2006). Instead of the exact method, the algorithm is rephrased in the form applicable toa two-dimensional Markov chain and applied to the model to approximated stationaryprobability distributions and measures of their performance.

This study also examines estimated errors between the approximate and precisemethods. The numerical results demonstrate that the approximation method enablesthe manager to save computational time without sacrificing accuracy given numerousoperators. Using the approximation method, it is easy to accurately estimate managementcosts and effectively allocate operators in a large call center.

The remainder of this paper is structured as follows. In Section 2, a quasi-birth-and-death (QBD) process model is designed based on a two-dimensional Markovchain and the estimation of waiting time is further calculated. Section 3 illustratesthe computation of stationary probability distribution. Section 4 then describes theapproximation method, which is applied to a case to determine the optimum allocationof operators that minimizes management cost. Section 5 discusses the error estimationbetween the exact and approximation methods. Finally, Section 6 illustrates theconclusions and suggests directions for future research.

2. A queueing modelTo estimate the management cost, the queueing model, which describes a call center thatdeals with calls from two categories of users, is derived to compute the opportunity ofincoming calls are the r-calls from regular customers, and the v-calls from VIPs. A totalof K operators is considered in a call center, where K is a large finite number. Namely,a maximum of K operators can serve customers. Let λγ and λυ denote the Poisson arrivalrates of r-calls and v-calls, respectively. The service rates for both call types by eachoperator are μ, where the service time is exponentially distributed. If at least one freeoperator is available on the arrival of a v-call, that call is admitted for immediate service;otherwise, v-call are immediately terminated. Let Pv represent the probability that v-callsare not served immediately.

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Figure 1 shows that r-calls can legitimately connect with the call center when thenumber of busy operators is less than N, N is a nonnegative integer and has a valueof K or less. Otherwise, the incoming r-calls are blocked or routed to the retrial group.The r-calls in the retrial group try to re-call following a period that follows anexponential distribution with rate α. Since all waiting calls in the retrial group arestored on a computer system with limited storage capacity, this study assumes that theretrial group has finite capacity, say R. Let Pr represent the probability that r-calls donot receive immediate service. Thus, Pr denotes the probability that r-calls do notreceive service when the retrial group is full. Generally, inbound r-calls may leave thesystem immediately after it learns that no available operators exist for service even ifthe retrial group is not full. This study thus assumes that r-calls either enter the retrialgroup with probability H0 or leave the system forever with probability 1-H0. The re-callrate of r-calls in the retrial group is α. Regarding the r-calls in the retrial group,following an unsuccessful re-call they either return to the retrial group with probabilityH1 or leave the system with probability 1-H1.

Let X1(t) represent the random variable representing the number of r-calls in theretrial group and let X2(t) denote the random variable representing the number of callsin service at time t. Taking a long-term view, let S represent the state space, whereS¼ {(X1, X2), 0⩽X1⩽R, 0⩽X2⩽K}. Then (X1, X2) denotes a two-dimensional Markovchain whose elements in Q-matrix are represented by:

q i; jð Þ; i; j0� �� �

¼

lrþlv; if i0 ¼ i; j

0 ¼ jþ1; 0p ipR; 0p jp N�1

lv; if i0 ¼ i; j

0 ¼ jþ1; 0p ipR; Np jpK

ia; if i0 ¼ i�1; j

0 ¼ jþ1; 1p ipR; 0p jpN�1

ia 1�H 1ð Þ; if i0 ¼ i�1; j

0 ¼ j; 1p ipR; Np jp K

iaH 1; if i0 ¼ i; j

0 ¼ j; 1p ipR; Np jpK

lrH 0; if i0 ¼ iþ1; j

0 ¼ jþ1; 0p ipR; Np jpK

jm; if i0 ¼ i; j

0 ¼ j�1; 0p ipR; 1p jpK

0; otherwise

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

RetrialGroup

K

K-1

N

N-1

2

1

Blocked r-calls

r-calls

v-calls

Operators

······

······

Figure 1.A queueing system

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Figure 2 illustrates the diagram of the state transition rate for call center operations.The parameters of the proposed model are: K, the numbers of operators in thecall center; R, the capacity of the retrial group; N, the decision variable of numberof operators; λr, the arrival rate of r-calls following a Poisson distribution;λv: the arrival rate of v-calls following a Poisson distribution; H0, the probabilitythat a r-call enters the retrial group; α, the rate of associated with a re-call time inthe retrial group following an exponential distribution; H1, the probability of r-callsthat re-enter the retrial group after an unsuccessful retry; μ, the service rate ofan operator; π(i, j), stationary probability at state (i, j), where i is the random variablerepresenting the number of r-calls in the retrial group, j is the random variablerepresenting the number of calls in service from VIP and regular customers; Lq,the average number of calls in retrial group; and λ, the total arrival rate of VIPand regular customers.

The two-dimensional Markov chain resembles the birth and death process model(QBD) (Artalejo and Gómez-Corral, 2008; Artalejo et al., 2005; Gómez-Corral, 2006;Do et al., 2014; Phung-Duc, 2014; Phung-Duc et al., 2013). This investigationestablishes the infinitesimal generator (Q-matrix) as follows. Let dik represent thediagonal element on the kth row in the Bi, where i is larger than or equal to zero andsmaller than or equal to R, while k is larger than or equal to one and smaller thanor equal to K:

Q ¼

B0 A0 0 0 � � � 0

A2 B1 A0 0 � � � 0

0 A2 B2 A0 & � � �0 0 & & & 0

^ ^ 0 A2 BR�1 A0

0 0 0 0 A2 BR

0BBBBBBBBB@

1CCCCCCCCCA

Let Bi, A0 and A2 defined as follows:

Bi ¼

0

1

^

N�1

N

Nþ1

^

K�1

K

�dðiÞ1 lv 0 0 0 0 � � � 0 0

m �dðiÞ2 lv 0 0 0 � � � 0 0

0 & & & & ^ ^ ^ ^

^ & N�1ð Þm �dðiÞN�1 ia 0 � � � 0 0

0 0 0 Nm �dðiÞN lv � � � 0 0

0 0 0 0 Nþ1ð Þm �dðiÞN þ 1 lv 0 0

^ ^ ^ ^ 0 & & lv 0

0 0 0 0 0 0 K�1ð Þm �dðiÞK�1 lv

0 0 0 0 0 0 0 Km �dðiÞK

266666666666666666664

377777777777777777775

;

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0, K

1, K

2, K

0, N

+1

0, N

0, 0

1, 0

2, 0

K�

(N+

2)�

(N+

1)�

N�

K�

(N+

2)�

(N+

2)�

N�

N�

K�

K�

(N+

2)�

R, K

R�

(1-H

1)

R�H

1

N�

R�

R�

(1-H

1)R

�H1

K�

N�

R-1

, K

R-1

, N+

1

R-1

, NR

, N

(R-1

)�H

1

R�

(1-H

1)

R�H

1

R, N

+1

(R-1

)�H

1

(N+

1)�

� rH0

� v� v � v

� v � v

� v � v

� v � v� v � v

� v� v

� v� v

� ��

��

�

��

��

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

� rH0

�(1

-H1)

�H1

2�H

1

�(1-

H1) �

2�

2�

R �

2�(1

-H1)

2�(1

-H1)

2�H

1

3�(1

-H1)

(R

-1)�

(1-H

1)

(R-1

)�H

1(R

-1)�

3�(1

-H1)

(R

-1)�

(1-H

1)

�H1

2�H

1

�H1

�(1

-H1)

�

2�(1

-H1)

3�(1

-H1)

(R

-1)�

(1-H

1)

• • •• • •

• • •• • •

• • •

• • •• • •

•••

• • •• • •

• • •

•••

R-1

, 0

(R-1

) �

R, 0

��

��

�

1, N

+1

2, N

+1

1, N

2, N

••• •

•••••

Figure 2.State transitiondiagram

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A0 ¼

0

1

^

N�1

N

Nþ1

^

K�1

K

0 0 0 0 0 0 � � � 0 0

0 & 0 0 0 0 � � � 0 0

0 0 & 0 0 0 � � � 0 0

^ ^ 0 0 0 0 � � � 0 0

^ ^ ^ 0 lrH 0 0 � � � 0 0

^ ^ ^ ^ 0 & 0 0 0

^ ^ ^ ^ ^ & & 0 0

^ ^ ^ ^ ^ 0 0 lrH 0 ^

0 0 0 0 0 0 0 0 lrH 0

266666666666666664

377777777777777775

;

and:

A2 ¼

0

1

^

N�1

N

Nþ1

^

K�1

K

0 ia 0 0 0 0 � � � 0 0

0 & ia 0 0 0 � � � 0 0

0 0 & & 0 0 � � � 0 0

^ ^ 0 0 ia 0 � � � 0 0

^ ^ ^ 0 ia 1�H 0ð Þ 0 � � � 0 0

^ ^ ^ ^ 0 ia 1�H 0ð Þ 0 0 0

^ ^ ^ ^ 0 & & 0 0

^ ^ ^ ^ 0 0 0 ia 1�H 0ð Þ ^

0 0 0 0 0 0 0 0 ia 1�H 0ð Þ

266666666666666664

377777777777777775

:

Additionally, let π¼ [π0, π1, π2,…., πR−1, πR] represent the exact solution, where πi¼(π(i, 0), π(i, 1),…, π(i,K)) for i¼ 0, 1, 2,…, R, and ∀πi∈M1×(K+1), where the above resultssatisfy equations:

pQ ¼ 0; (2)

and:

XRi¼0

XKj¼0

p i; jð Þ ¼ 1 : (3)

Hence, this investigation adopts the following balance equations:

p0B0þp1A2 ¼ 0

p0A0þp1B1þp2A2 ¼ 0

p1A0þp2B2þp3A2 ¼ 0

^

pR�2A0þpR�1BR�1þpRA2 ¼ 0

pR�1A0þpRBR ¼ 0

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Applying Equations (2) and (3), we could obtain πi¼ (π(i, 0), π(i, 1),…, π(i, K)) fori¼ 0, 1, 2,…, R, by using the Matlab software given relatively small K. The threeperformance measures derived from the above equations are as follows:

(1) The probability that the system fails to serve a regular customer on the firstattempt is Pr ¼

PRi¼0

PKj¼N p i; jð Þ, because calls from regular users will be

redirected to the retrial group when N or more operators are busy;

(2) The probability that the system fails to serve a VIP caller is Pv ¼PR

i¼0 p i;Kð Þ,because all operators are too busy to answer v-call; and

(3) The average number of regular customers in the retrial group isLq ¼

PRi¼1

PKj¼0 ip i; jð Þ.

3. Computational timeBecause of a large KW100, the abovementioned measures require computationthrough experiments. To determine the computational complexity of the threeperformance measures, this investigation establishes parameters for an experimentbased on a real case. Although the number of operators is 71 (K¼ 71), to clarify theproblem of long computational time, this investigation also extended K from 71 to 451.Table II shows the time spent to calculate the blocking probability of calls from regularcustomers based on the data listed in Table I for K with different capacity throughEquations (2) and (3). The experimental results demonstrate that because this queueingmodel is a two-dimensional Markov chain, the balance equations are so complex thattheir algorithmic computing time grows exponentially (Figure 3). The experiment isimplemented using a mathematical tool (Matlab® 7.0) with a high-end workstation(Windows® 7 Business version, Intel® Core™ i7 CPU at 2.69 GHZ, and three gigabytesof random access memory (RAM)) (Tables I and II).

Obviously, such an exact method becomes highly impractical given large K.Over 20 hours (73,072 seconds) are needed to calculate the results when K is126. Therefore, Section 3 of this study introduces an approximation known as thephase merging algorithm to calculate Equations (2) and (3) to solve the computationalproblem.

4. The approximation methodNumerous approximation methods have been proposed to solve the computationalproblem associated with a large number of channels (exceeding 1,000) with heavytraffic in the telecommunication research literature (Abdrabou and Zhuang, 2011;Allon et al., 2013; Bandi and Bertsimas, 2012; Bicen et al., 2012; Halfin and Whitt,1981; Nelson, 2012; Sriram and Whitt, 1986; Whitt, 1983). However, call centers havetheir own limitations and unique features in handling calls from real people, and do

Parameter Value Parameter Value

N K H1 0.999R 15 λr 7.78μ 1/3 λv 3.89α 1.8 λ 11.67H0 0.15 K 71

Table I.Parameter set

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not merely transmit data. It is necessary to consider simple approximation methods.One such method is known as phase merging algorithm and was introduced byMelikov and Babayev (2006) and adopted by Choi, Melikov and Velibekov (Choi et al.,2008; Liang and Luh, 2013; Ponomarenko et al., 2010) to compute stationaryprobability. This method can be implemented in call centers. The approximationmethod is shown as follows.

Application of the phase merging algorithm to this model requires making thefollowing assumptions (based on Figure 2):

lv * lrH 0 (4)

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

200,000

1 11 21 31 41 51 61 71 81 91 101 111 121 151 251 351 451

seconds

KFigure 3.

Computational time

K Seconds K Seconds

1 0.02 86 237.566 0.17 91 305.5711 0.56 96 362.8016 1.31 101 428.3221 2.76 106 476.8826 6.30 111 558.3631 10.25 116 638.9536 16.44 121 731.5441 24.79 126 761.4146 35.30 151 1,325.8051 49.16 201 3,484.2056 65.94 251 8,271.8061 87.36 301 14,773.0066 111.10 351 54,720.0071 139.09 401 99,006.7076 168.04 451 180,890.0081 208.39

Table II.Computational time

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Km * Ra (5)

lv * Rað1�H 1Þ (6)

The above assumptions are rational in a call center. When the arrival rate of v-callssignificantly exceeds that of r-calls multiplied by the probability of r-calls entering theretrial group being blocked, a high chance of more than N customers in service impliesprotection for VIPs. In this regard, this study assumes the up vertical flow mustsignificantly exceed the left and right horizontal flow (see Figure 2). Additionally, the downvertical flow significantly exceeds the up slanting flow shown in Figure 2. If the modelsatisfies the above conditions, then it is possible to adopt the phase merging algorithm.

The steps of the phase merging algorithm are illustrated based on the aboveassumptions, as follows:

(1) Split state space of the original two-dimensional Markov chain by (Choi et al., 2008):

S ¼ [Ri¼0

Si; Si \ Si0 ¼ |; ia i0 (7)

where Si¼ {(i, j)∈S:0⩽ j⩽K}.

(2) The conditional stationary distribution p j9i� � ¼ p i;jð ÞPK

j¼1p i;jð Þ

within each class Si has

been defined as an M/M/K/K queueing system given the following arrival rates:

lrþlv; if joN

lv; if j⩾N:

(

Therefore:

p j9i� � ¼

rj

j!p 09i� �

; if 1p jpN

rN

j! rj�Nv p 09i

� �; if Np jpK

:

8<:

where r ¼ lr þlvm , rv ¼ lv

m , 0p jpK and:

p 09i� � ¼ XN

j¼0

rj

j!þrN

XKj¼N þ 1

rj�NN

j!

!�1

(8)

From Equation (8) (Choi et al., 2008), we find the probability π(j∣i) which does notdepend on i. Therefore, p jð ÞD¼ p j9i

� �is omitted.

(3) All states within subset Si could be merged into a single merged state o iW .The merged model is a birth and death process and the transition rates aremanipulated with little algebra, which yields:

q o i4 ; o i04

� � lrH 0

XKJ¼N

p jð Þ; if i ⩾ 0; i0 ¼ iþ1;

iaXN�1

j¼0

p jð Þþ 1�H 1ð ÞXKJ¼N

p jð Þ !

;

0; otherwise

if i ⩾ 1; i0 ¼ i�1:

8>>>>>>>><>>>>>>>>:

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The stationary distribution of the merged model ψ(o iW ), o iW , i¼ 0, 1,…, Ris thus obtained. Based on the stationary probability of M/M/R/K, this studyprovides the following equations (Choi et al., 2008):

c o i4ð Þ ¼ ti

i!c o04ð Þ; (9)

where:

c o04ð Þ ¼XRi¼0

ti

i!

!�1

; (10)

t ¼ lrH 0PK

j¼N p jð ÞaPN�1

j¼0 p jð Þþ 1�H 1ð ÞPKj¼N p jð Þ

� �; (11)

Define:

p i; jð Þ ¼ p jð Þc o i4ð Þ: (12)

Based on the above descriptions, we obtain the following lemmas.

Based on the assumptions of Equations (4)-(6), the phase merging algorithm canapproximate π using Equation (12). It is explained in the following.

Lemma 1. Because π(i, j) is the stationary probability of state (i, j) and the expectedrates of flow in and out of state (i, j) must be equal, this study considersthree cases with given state (i, j) and Si.

• Case 1. jWN.

Flow in state (i, j), it gives:

p iþ1; jð Þ iþ1ð Þa 1�H 1ð Þ½ �þp i; j�1ð ÞlvþlrH 0þp i; jþ1ð Þ jþ1ð Þm (13)

Flow out state (i, j), it gives:

p i; jð Þþ lvþlrH 0þ jmþ ia 1�H 1ð Þð (14)

Because Equation (13)¼Equation (14), by little algebra, we have:

p i; jð Þ lvþ jmð Þ ¼ p i; j�1ð Þlvþp i; jþ1ð Þ jþ1ð Þmþej (15)

where ej ¼ p iþ1; jð Þ iþ1ð Þa 1�H 1ð Þ½ �þp i�1; jð ÞlrH 0 � p i; jð ÞlrH 0 � p i; jð Þia 1�H 1ð Þ:Let Equation(15) be divided by λv and it produces:

p i; jð Þ lvþ jmð Þlv

¼ p i; j�1ð Þlvlv

þ p i; jþ1ð Þ jþ1ð Þmlv

þ p iþ1; jð Þ iþ1ð Þa 1�H 1ð Þlv

þp i�1; jð ÞlrH 0

lv� p i; jð ÞlrH 0

lv� p i; jð Þia 1�H 1ð Þ

lv

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The εj in Equation (15) might be ignored, because (4) and (6) imply εj is approaching tozero. The Equation (16) represents the balance equation within Si.

• Case 2. j¼N:

p i; jð Þ lvþ jmð Þ ¼ p i; j�1ð Þ lvþlrð Þþp i; jþ1ð Þ jþ1ð Þmþej (16)

where ej ¼ p iþ1; jð Þ iþ1ð Þa 1�H 1ð Þ½ �þp i�1; jð ÞlrH 0�p i; jð ÞlrH 0�p i; jð Þia 1�H 1ð Þ:Let (16) be divided by λvKμ and it produces:

p i; jð Þ lvþ jmð ÞlvKm

¼ p i; j�1ð ÞðlvþlrÞlvKm

þ p i; jþ1ð Þ jþ1ð ÞmlvKm

þp iþ1; jð Þ iþ1ð Þa 1�H 1ð ÞlvKm

þp i�1; jð ÞlrH 0

lvKm� p iþ1; j� 1ð Þ iþ1ð Þa

lvKm

� p i; jð ÞlrH 0

lvKm� p i; jð Þia 1�H 1ð Þ

lvKm

The εj in Equation (16) might be ignored, because (4)-(6) imply εj is approaching to zero.• Case 3. joN:

p i; jð Þ lvþlrþ jmð Þ ¼ p i; j� 1ð Þ lvþlrð Þþp i; jþ1ð Þ jþ1ð Þmþej (17)

where ej ¼ p iþ1; j� 1ð Þa iþ1ð Þ � p i; jð Þia:Let (17) be divided by Kμ which leads to:

p i; jð Þ lvþluþ jmð ÞKm

¼ p i; j� 1ð Þ lvþlrð ÞKm

þp i; jþ1ð Þ jþ1ð ÞmKm

þp iþ1; j� 1ð Þ iþ1ð ÞaKm

� p i; jð Þ iaKm

The εj in Equation (17) might be ignored, because (5) implies εj is approaching to zero.Using Equations (15)-(17) and given Si, we can ignore the horizontal and slanting flowsin the system (see Figure 2) except for vertical flows. Therefore, based on Equations(15)-(17), we conclude the following: If εj is small and can be ignored, p (i, j) is anε-approximation of π(i, j). Based on four approximations, (18)-(21) could be derived forthe measures of performance and the average waiting time (Choi et al., 2008):

Pr � 1� p 0ð ÞXN � 1

i¼0

ri

i!; (18)

Pv � p 0ð ÞrNrK � N

v

K!; (19)

Lq � c o04ð ÞXRJ¼1

t j

j� 1ð Þ!; (20)

Now we compute an effective arrival rate of r-calls in the retrial group denoted by lnr .Because the total calls in the retrial group is completely owing to regular customers,

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it is calculated by the average number of r-calls admitted in the retrial group minussome of them leave the system with probability (1-H1) or find no servers availablesubsequently. Let P(R) denote the probability that the retrial group is full, and Pn

rrepresents the probability that r-calls may be routed to the retrial group where P Rð Þ ¼PK

J¼0 p R; jð Þ and Pn

r ¼PR � 1

i¼0

PKj¼N p i; jð Þ.

Lemma 2. The effective arrival rate of r-calls is given by:

lnr ¼ lrPn

r H 0 1� P Rð Þð Þ 1 � Pn

r1 � H 1P

n

r ð1 � P Rð ÞÞ

� ��and 0p 1 � Pn

r1 � H 1P

n

r 1 � P Rð Þð Þp1

Proof: Let D ¼ lrPn

r H 0 1� P Rð Þð Þ. We have:

lnr ¼ D� DPn

r 1�H 1ð ÞþDPn

r H 1P Rð Þ� �� DPn

r 1�H 1ð ÞH 1Pn

r 1� P Rð Þð Þ�þDPn

r H 1P Rð ÞH 1Pn

r 1� P Rð Þð Þ � UUU

¼ D 1� Pn

r ð1�H 1Þ1�H 1P

n

r 1� P Rð Þð Þ �Pn

r H 1P Rð Þ1�H 1P

n

r 1� P Rð Þð Þ

�

¼ D1� Pn

r

� �1�H 1P

n

r 1� P Rð Þð Þ

�

¼ lrPn

r H 0 1� P Rð Þð Þ 1� Pn

r

� �1�H 1P

n

r 1� P Rð Þð Þ

�

_0pH 1p1 and 0pP Rð Þp1

‘0p 1� Pn

r

1�H 1Pn

r 1� P Rð Þð Þp1

Finally, from the abovementioned derivations, we obtain the mean waiting time in theretrial group based on the formula of Little is:

Wr ¼Lq

lnr(21)

The call center must determine the number of servers required to ensure that bothtypes of customers (regular and VIP customers) receive satisfactory service.Satisfactory service (Chen and Henderson, 2001) means that at least qr percent ofr-calls are served withinm seconds. This definition of satisfactory service is common inthe call center industry. Restated, partial r-calls are answered immediately (set m¼ 0).This investigation considers two waiting time with r-calls: N equals zero and N doesnot equal zero (0oN⩽K).

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• Case 1. N¼ 0.Clearly, any regular customer cannot enter the system to receive service. On the otherhand, theM/M/K/K queueing system only serves VIP customers. Since VIP customersdo not have waiting time in the model, the waiting times for v-calls and r-calls bothequal zero. Thus we can conclude that for any time when w is zero or greater, it has:

P waiting time of both calls4wð Þ ¼ 0: (22)

• Case 2. 0oN⩽K.

Because VIP customers are served immediately, this investigation only needs toconsider the waiting time for regular customers in the retrial group. Abate and Whitthave proposed algorithms for numerically calculating tail probabilities. However,the proposed algorithms require complex arithmetic, which is inconvenient for themanager of a call center manager to apply to calculate waiting time. Let Wr denotea random variable of the waiting time of r-calls. Given Markov’s inequality for anonnegative random variable X and constants x, β larger than zero indicates:

P X ⩾ xð ÞpE Xb� �xb

;

where P(.) denotes a probability function (Ross, 2002). Hence, we conclude that:

P Wr4wð ÞpWr

w: (23)

Equation (5) provides an upper bound on the tail probabilities ofWr, and hence a lowerbound of P(WrWw), namely, P(WrWw∣λv¼ 0) for any w that exceeds zero.

That is, given ∀wW0 then:

P Wr4w9ln ¼ 0� �

pP Wr4wð ÞpWr

w: (24)

Apparently, Equation (24) provides lower and upper bounds for regular customer waitingtimes. The upper bound is calculated by Equation (21). The manager can use these resultsto determine the optimum N that satisfies regular customers the service requirements canbe met. The procedure used to calculate N for satisfactory service is shown in AlgorithmWP. The manager could simply increase N until Equation (23) with w¼m falls below(100-qr) percent. Therefore, we can be reasonably confident of approaching these goalsusing formulae (21) and (24). WP algorithm is written as follows:

Step 1. Given N, w, qr.Step 2. Compute x ¼ E Wr9Nð Þ

wStep 3. If xo (100− qr) percent, then STOP.Step 4. Increase N and go to Step 2.

Based on satisfactory service requirements, we compute the probability of waiting timewhich is more than w seconds for regular customers when the traffic intensity of regularcustomers is lr=Nm. From (24), we can determine the lower bound and upper bound ofthe probability. The effective arrival rate is lnr . Given w, we have Wr=w ¼ Lq=wl

n

r .Furthermore, the lower bound can be found when lv ¼ 0: In this case, we consider it as a

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traditional M/M/N/K queueing model where K¼N+R. To illustrate satisfactory servicerequirements, we have set up the parameters in Table III.

The experiment results show that the upper bound of probability of waiting time goesup along with the increasing arrival rate of regular customers. Additionally, under thestudy it expects that a satisfactory service represents at least 90 percent of r-calls areserved within 50 seconds. We adopt WP algorithm to compute Nwithw¼ 50 and qr¼ 90.The experiment results show that the probability of waiting time more than 50 seconds isbelow 10 percent when lr=Nm ¼ 1:556 and N≧ 48 (Figure 4). Hence, the call center maycontrol the probability of waiting time of regular customers less than 50 seconds is morethan 90 percent by assigning more than 48 servers.

Finally, from Equations (1) and (2), we obtain πi¼ (π(i, 0), π(i, 1),…, π(i, K)) fori¼ 0, 1, 2,…, R, by using a mathematical tool (Matlab® 7.0) with a high-endworkstation, we can calculate the results using the above method (the exact method).However, it is easy to identify the optimal N when K is small. When K is large, suchcomplex computations might slow the computer system to a standstill. For example,when the number of operators is ⩾70 (the number of the operators of the case

0.8 1 1.2 1.4 1.6 1.8 20.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

X : 1.556Y : 0.1069

�r /N�

Pro

babi

lity

of w

aitin

g m

ore

then

50

seco

nd fo

r re

gula

r cu

stom

ers

X : 1.556Y : 0.1011 X : 1.556

Y : 0.0958

(1)Upper bound of waiting time with N=46

(2)Upper bound of waiting time with N=47

(3)Upper bound of waiting time with N=48

Figure 4.Probability of

waiting time morethan 50 seconds

Parameter Upper bound Lower bound

K 70 70R 15 15N 35 351/μ 3 3α 1.8 1,000H0 0.15 1H1 0.999 1λr 7.78; 10.37; 12.96; 15.56; 18.15; 20.74; 23.34 7.78; 10.37; 12.96; 15.56; 18.15; 20.74; 23.34λv 3.89; 5.19; 6.48; 7.78; 9.08; 10.37; 11.67 0; 0; 0; 0; 0; 0; 0w 50 (sec.) 50 (sec.)

Table III.Parameters for upper

bound and lowerbound tests

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company), almost six hours are needed to complete the computation. Therefore, theapproximation method must be discussed.

Finally, to clarify the errors between the exact and approximation methods, thisinvestigation obtained the results listed in Table III. In the experiments, we assumeR¼ 15, α¼ 1.8, μ¼ 0.33, λ¼ 11.67 and N¼K. This investigation determined that theerrors between the exact and approximation methods decrease with increasingK. FromTable III, we found almost no error between the exact method and the approximationmethod once KW40. Restated, if K exceeds 51, Pr decreases to zero based on theoriginal assumptions (Table IV). To clarify the comparison of the exact andapproximation methods, λ must exceed the original assumptions. This study thusincreased λ from 11.67 to 211.67. The results are listed in Table V. Additionally, the

KComputational time by

approximation method (sec.)Computational time byexact method (sec.)

Speedratio

Pr byapproximation

methodPr by exactmethod

1 0.0312 0.0624 2.00 0.972 0.9916 0.0468 0.39 8.33 0.834 0.93911 0.0468 0.5928 12.67 0.697 0.81316 0.078 1.2792 16.40 0.563 0.65521 0.0936 2.652 28.33 0.433 0.50226 0.1092 5.85 53.57 0.310 0.35731 0.156 9.4537 60.60 0.198 0.22636 0.1716 14.3053 83.36 0.107 0.12041 0.2028 21.7621 107.31 0.044 0.04846 0.2184 31.8086 145.64 0.012 0.01351 0.2964 44.9283 151.58 0.002 0.00256 0.3432 59.14 172.32 0.000 0.00061 0.3744 76.7993 205.13 0.000 0.00066 0.4368 98.9358 226.5 0.000 0.00071 0.4836 123.194 254.74 0.000 0.000

Table IV.Exact vsapproximation(λ¼ 11.67)

KComputational time by

approximation method (sec.)Computational time byexact method (sec.)

Speedratio

Pr byapproximation

methodPr by exactmethod

1 0.016 0.031 2.00 0.998 0.9996 0.016 0.031 2.00 0.991 0.99211 0.078 0.593 7.60 0.983 0.98516 0.094 1.217 13.00 0.975 0.97821 0.094 2.184 23.33 0.967 0.97126 0.140 4.976 35.44 0.959 0.96431 0.125 8.518 68.25 0.951 0.95736 0.125 13.869 111.13 0.943 0.95041 0.187 20.904 111.67 0.936 0.94346 0.250 30.358 121.63 0.928 0.93651 0.296 41.434 139.79 0.920 0.92956 0.296 56.940 192.11 0.912 0.92261 0.343 76.425 222.68 0.904 0.91566 0.406 98.187 242.08 0.896 0.90871 0.484 123.184 254.72 0.888 0.901

Table V.Exact vsapproximation(λ¼ 211.67)

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speed ratio Computational timeapproximation=Computational timeexact� �

is used to clarifythe outstanding performance of the approximation method.

5. DiscussionHowever, to merely identify the errors between the exact and approximation methodsis insufficient, because the manager needs to know whether their company needs toadopt the approximation method to determine the management costs. The manager’squestion can be answered by discussing the difference in errors between theapproximation and exact methods.

Restated, this study demonstrates the errors of the approximation and exactmethods of different performance measures along with the increasing number of totaloperators. Table VI lists the parameters for finding the errors of Pr, Pv, Lq and MCbetween the approximation and exact methods. Figures 5 and 6 lists the experimentalresults. Moreover, to identify the errors between the absolute and exact methods, thisstudy assumes a large λ (23.34, which is twice 11.67).

The numerical results are illustrated in Figures 5 and 6 when λ¼ 11.67 and 23.34,respectively. The error between approximation and exact methods are increasing atfirst (the top error values when λ¼ 11.67 and 23.34 are K¼ 11 and K¼ 6, respectively)and decreasing gradually. The error between the approximation and exact methodswill approach to zero when λ¼ 11.67 and 23.34 are K¼ 86 and K¼ 51, respectively.The MC could be found quickly with small error (if the acceptable error rate is smallerthan 5 percent) when λ¼ 11.67 and 23.34 are K¼ 36 and K¼ 16, respectively. Theaforementioned numerical results imply that using approximation method is feasiblefor the manager of a call center to calculateMC value in a busy call center. If 5 percentof error rate can be accepted by a manager, the manager can decide the optimal K.Thus, the manager of a call center can hire operators explicitly based on the estimatedMC. Finally, this study shows the computational results of the MC based on differentretrial rates (α) that represent the frequency of sending the blocking calls in the waitingqueue to operators. The computational results show that MC will be 13.5, 61.62 and70.26 when retrial rates are 1, 30 and 120, respectively.

6. ConclusionThis investigation presents a computational method and management cost functionintended to identify the optimal number of operators for a call center. Because ofcomputational limitations, many operators could not be easily analyzed using the exact

Parameter Value (λ¼ 11.67) Value (λ¼ 23.34)

N 25 25R 15 15μ 1/3 1/3α 1.8 1.8H0 0.15 0.15H1 0.999 0.999λr 7.78 15.56λv 3.89 7.78Cr 1 1Cv 2 2Cw 20 20

Table VI.Experimentalparameters

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16.0

0

14.0

0

12.0

0

10.0

0

8.00

6.00

4.00

2.00

0.00

Pr(

%)

16

1116

2126

3136

4146

5156

6166

7176

8186 0.

0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

9196

101

106

111

116

121

126

131

136

141

146

151K 15

616

116

6

0.2

0.2

0.0

0.1

0.5

0.5

0.1

0.2

1.0

1.0

0.3

0.5

0.6

0.6

0.4

0.5

2.6

2.6

0.7

1.3

3.7

3.7

0.9

1.7

4.8

4.8

1.2

2.3

5.9

5.9

1.6

2.9

7.1

7.1

2.1

3.6

8.4

8.3

2.7

4.5

9.6

9.6

3.5

5.6

10.

10.

4.9

7.2

11.

11.

6.8

9.3

10.

10.

8.4

10.

14.

14.

10.

13.

0.7

11.

11.

8.8

11.

0.7

5.0

5.2

Pv(%

)

L q MC

16.0

0

14.0

0

12.0

0

10.0

0

8.00

6.00

4.00

2.00

0.00

%

Figure 5.Absolute errors ofapproximation minusexact methods withK (λ¼ 11.67)

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16.0

0

14.0

0

12.0

0

10.0

0

8.00

6.00

4.00

2.00

0.00

25.0

0

20.0

0

15.0

0

10.0

0

5.00

0.00

Pr(

%)

16

1116

2126

3136

4146

5156

6166

7176

8186

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.2

0.4

1.1

2.7

4.7

4.7

2.7

1.8

0.8

1.3

2.2

4.5

9.9

12.

13.

10.

2.2

2.6

1.9

1.9

7.0

11.

11.

4.3

9.2

9.2

2.8

6.9

6.9

0.5

1.3

1.3

0.0

0.0

0.1

0.1

0.4

0.4

0.0

0.0

9196

101

106

111

116

121

126

131

136

141

146

151

156

161

166

K

Pv(%

)

L qMC

%

Figure 6.Absolute errors of

approximation minusexact methods with

K (λ¼ 23.34)

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method. For the manager of a call center, the sooner the optimal solution is found, thefaster business strategies are deployed. This study develops an approximation methodand compares it with the exact method. The approximation method demonstratesoutstanding computational performances when the state variables are too big to behandled in matrix.

However, a good call center management scheme must consider not only operationalcosts, but also traffic conditions and other managerial variables. Future researchshould include more variables to identify nonlinear optimization with stochasticparameters.

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Corresponding authorDr Chih-Chin Liang can be contacted at: lgcwow@gmail.com

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