Performance Evaluation 70 (2013) 440–453
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Performance Evaluation journal homepage: www.elsevier.com/locate/peva
Tandem queueing system with impatient customers as a model of call center with Interactive Voice Response Chesoong Kim a,∗ , Alexander Dudin b , Sergey Dudin b , Olga Dudina b a
Sangji University, Wonju, Kangwon, 220-702, Republic of Korea
b
Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus
article
info
Article history: Received 14 March 2012 Received in revised form 24 January 2013 Accepted 13 February 2013 Available online 24 February 2013 Keywords: Call center Interactive Voice Response Markovian arrival process Phase type service time distribution Impatient customers
abstract A tandem queueing system with a Markovian Arrival Process (MAP) useful in modeling a call center with Interactive Voice Response (IVR) is investigated. The first stage has a finite number of servers without buffer while the second stage of the tandem has a finite buffer and a finite number of servers. The service time at the first (second) stage has an exponential (phase type) distribution. A special approach for reducing the number of states of the stochastic process that describes the behavior of the system is used. The main performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution is derived. The numerical results are presented. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Call center is a centralized office used by companies for contact with its clients. To decrease load of operators in some call centers IVR (Interactive Voice Response) technology is used. IVR is a call center option that allows customers to interact with a company and serve their own inquiries via a telephone keypad without the involvement of an agent. Typically IVR is used in call centers of banks, insurance companies, travel agencies, mobile operators, etc. For example, if the subscriber of mobile network wants to know the balance of his(her) account, enable or disable the service, to find out some information about tariffs, he(she) can easily do it himself(herself) using IVR. If the customer cannot solve his(her) problem by using IVR and requires some assistance, he(she) can request to connect with an agent. In call centers with IVR the load of operator is lower, so the performance of call centers is higher. For modeling call centers, queueing theory is used. Adequate mathematical modeling call centers leads to a substantial increase of their economic efficiency, because exact prediction at the design stage can significantly reduce maintenance costs. Mathematical models allow to solve a problem of call center optimal design. For the references and the present state of art in investigation of call centers the reader is referred to survey [1], papers [2–5] and references therein. In [6], a call center with IVR is investigated. In this paper authors consider a two-stage queueing model with Poisson arrival process and exponential service time distribution at both stages with impatient customers and without balk of customers. Performance measures of this system are calculated approximately in an asymptotic Quality and Efficiency Driven (QED) regime. In our paper, we present an exact analytical analysis of the model without the restriction that the number of operators is large enough. We consider a tandem multi-server queueing system with a finite intermediate buffer and impatient
∗
Corresponding author. Tel.: +82 33 730 0464. E-mail addresses:
[email protected] (C. Kim),
[email protected] (A. Dudin),
[email protected] (S. Dudin),
[email protected] (O. Dudina).
0166-5316/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.peva.2013.02.001
C. Kim et al. / Performance Evaluation 70 (2013) 440–453
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Fig. 1. Structure of the system.
customers which can be used for modeling and optimizing call centers with IVR. The first stage consists of a finite number of servers (IVR). After the service at the first stage customers leave the system or move to the second stage. If after completing the service at the first stage the customer moves to the second stage and all servers (agents) are busy, the customer is reported about the current queue length (so called ‘‘visible’’ queue) and its estimated waiting time, and based on the provided information decides whether to balk (leave the system permanently without the service) or wait for the service. Statistics show that customers, who receive information about their place in a buffer or waiting time, are 1.5–2 times more patient, than customers who do not have such an information. As a result, the number of unserved customers is greatly reduced, therefore, consideration of the ‘‘visible’’ queue is an important point in modeling modern call centers. In many papers, see, e.g., [4,6], authors mention that the distribution of the service time of a customer by an agent is not exponential and the arrival flow of customers is not Poisson. Our statistical analysis of real arrival flows and service processes of the call center of one of the largest banks in Belarus has shown the same behavior. In our paper we consider a Markovian arrival process which is more general than Poisson arrival process and allows to take into account bursty nature of flows in modern call centers. Also, we consider a phase type (PH) service time distribution at the second stage of the system instead of an exponentially distributed service time considered in [6]. This allows to take into account variation of the service time carefully. It is known that the subset of phase-type distributions is dense in the set of all positive-valued distributions, that is, it can be used to approximate any nonnegative valued distribution, so, consideration of PH service process is very important for adequate modeling call centers. It is well known that if we investigate a multi-server queueing system with PH service process by standard methods, stochastic process that describes the behavior of the system can have a huge state space even for a small number of servers. This is due to the fact that at a given moment we must know the state of PH underlying process for each busy server. So, the standard technique is not appropriate for modeling call centers. In this paper we use the special method proposed by Ramaswami and Lucantoni, see [7,8], for reducing the dimension of systems with a phase type service time distribution. The paper is organized as follows. In Section 2, the mathematical model is described. The stationary distribution of system states is analyzed in Section 3. The expressions for the main performance measures of the system are given in Section 4. In Section 5, we focus on the analysis of the sojourn time distribution. Section 6 contains some numerical illustrations. Section 7 concludes the paper. 2. Mathematical model The structure of the system under consideration is presented on Fig. 1. The queueing system consists of two sequential stages. The first stage of tandem is R server queueing system without a buffer, the second stage is N server queueing system with a buffer of capacity K . Customers arrive at the system according to the MAP. The MAP is defined by the underlying process νt , t ≥ 0, which is an irreducible continuous time Markov chain with finite state space {0, 1, . . . , W }. Arrivals occur only at epochs of jumps in the underlying process νt , t ≥ 0. The intensities of transitions of the process νt , t ≥ 0, which are accompanied (are not accompanied) by an arrival of a customer, are defined by the square matrix D1 (D0 ) of size W + 1. The matrix generating function of these matrices is D(z ) = D0 + D1 z , |z | ≤ 1. The matrix D(1) is an infinitesimal generator of the process νt , t ≥ 0. The stationary distribution vector θ of this process satisfies the system of equations θ D(1) = 0, θ e = 1. Here and in the sequel 0 is a zero row vector and e denotes unit column vector. If the dimension of a vector is not clear from the context, it ¯ = W + 1. is indicated as a subscript, e.g., eW¯ denotes the unit column vector of dimension W The average intensity λ (fundamental rate) of the MAP is defined by λ = θ D1 e. The coefficient cvar of variation of intervals between customer arrivals is calculated by cvar = 2λθ(−D0 )−1 e − 1, while the coefficient ccor of correlation of intervals 2 between successive arrivals is given as ccor = (λθ(−D0 )−1 D1 (−D0 )−1 e − 1)/cvar . Different methods for the estimation of MAP parameters using a finite set of observed data such as a set of customer arrival times recorded at the real call center are presented, e.g., in [9]. More information about the MAP and related research is given, e.g., in [10–13]. The service time at the first stage server is assumed to be exponentially distributed with the rate µ, 0 < µ < ∞. After completing the service at the first stage, the customer leaves the system forever with probability q, 0 ≤ q ≤ 1. With supplementary probability 1 − q the customer moves to the second stage of the tandem. Note that in contrast to paper [6],
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C. Kim et al. / Performance Evaluation 70 (2013) 440–453
where it is assumed that the trunk will not be released when a customer leaves the IVR queue and switches to the service of agents, we assume that the first stage server is released after a customer leaves the first stage. If there is a free server at the second stage at a customer arrival epoch from the first stage, the customer occupies this server. If all servers at the second stage are busy at a customer arrival epoch and there are l, l ∈ {0, 1, . . . , K − 1}, customers in the buffer, then this customer leaves the system with probability pl or moves to the buffer with supplementary probability. It means that the arriving customer, who does not succeed to enter the service immediately, is reported about the queue length. Some customers are patient enough to wait for the service, while others abandon the system immediately if they consider this length as inappropriate. If the buffer is full at a customer arrival epoch at the second stage, he(she) leaves the system forever. Customers can be impatient, i.e., the customer leaves the intermediate buffer after an exponentially distributed with the parameter α, 0 < α < ∞, time after arrival, conditioned on the fact that this customer is not servicing. The service time of a customer by a second stage server has PH distribution with an irreducible representation (β, S ). This service time can be interpreted as a time until the underlying Markov process ηt , t ≥ 0, with finite state space {1, . . . , M , M + 1} reaches the single absorbing state M + 1 conditioned on the fact that the initial state of this process is selected among the states {1, . . . , M } according to the probabilistic row vector β = (β1 , . . . , βM ). Transition rates of the process ηt within the set {1, . . . , M } are defined by the sub-generator S and transition rates into the absorbing state (what leads to service completion) are given by entries of the column vector S0 = −Se. Note that representation (β, S ) is irreducible if the matrix S + S0 β is irreducible. ∞ The service time distribution function has the form A(x) = 1 − βeSx e, Laplace–Stieltjes transform (LST ) 0 e−sx dA(x) −1 −1 of this distribution is β(sI − S ) S0 , Re s > 0. The mean service time is calculated by b1 = β(−S ) e. The coefficient of variation is given by cvar =
b2 /b21 − 1 where b2 = 2β(−S )−2 e.
For more information about PH distribution and its usefulness see, e.g., [14]. Methods of modeling the PH process using a set of service times obtained at the real system, and in particular call center, can be found in paper [15]. Note that the main goal of the analysis presented in this paper is to elaborate the algorithms for computing the key performance measures of the system under consideration. Once these measures are calculated, we can solve various optimization problems of real call center management. 3. The process of system states Let it be the number of customers at the first stage, it ∈ {0, 1, . . . , R}, nt be the number of customers at the second stage, (m) nt ∈ {0, 1, . . . , K + N }, νt be the state of the underlying process of the MAP , νt ∈ {0, 1, . . . , W }, ηt be the number of M (m) (m) servers at the phase m of service, m ∈ {1, 2, . . . , M }, ηt ∈ {0, 1, . . . , min{nt , N }}, = min{nt , N }, at the epoch m=1 ηt (m)
t , t ≥ 0. Note that the meaning of components ηt is chosen according to the approach by Ramaswami and Lucantoni, see [7,8]. The behavior of the system under study can be described in terms of the regular irreducible continuous-time Markov chain
ξt = {it , nt , νt , ηt(1) , . . . , ηt(M ) },
t ≥ 0,
with state space
{i, n, ν, η(1) , . . . , η(M ) }, i ∈ {0, 1, . . . , R}, n ∈ {0, 1, . . . , K + N }, ν ∈ {0, 1, . . . , W }, η
(m)
∈ {0, 1, . . . , min{n, N }}, m ∈ {1, 2, . . . , M },
M
η
(m)
= min{n, N } .
m=1
(1)
(M )
The number of states of the process ηt = {ηt , . . . , ηt
}, t ≥ 0, when n servers are busy is equal to K¯ n =
n+M −1 M −1
. In
the overwhelming majority of existing papers, the behavior of queues with PH service time distribution when n servers are (1) (n) (m) busy is described by stochastic process including the components ζt , . . . , ζt , t ≥ 0, where ζt , m ∈ {1, 2, . . . , n}, is ¯ the state of the PH underlying process on the mth busy server. Note that the number Kn is significantly less than the number (1) (n) of states of the process {ζt , . . . , ζt }, t ≥ 0, which is equal to K˜ n = M n . For example, if we fix n = 25 and M = 2, the 25 ˜ number Kn = 2 while the number K¯ n is only 26. Since the Markov chain ξt is regular irreducible and has finite state space, then for any choice of system parameters there exist the stationary probabilities of system states which are defined as follows:
π (i, n, ν, η(1) , . . . , η(M ) ) = lim P {it = i, nt = n, νt = ν, ηt(1) = η(1) , . . . , ηt(M ) = η(M ) }, t →∞
i ∈ {0, 1, . . . , R},
n ∈ {0, 1, . . . , K + N },
ν ∈ {0, 1, . . . , W },
C. Kim et al. / Performance Evaluation 70 (2013) 440–453 M
η(m) ∈ {0, 1, . . . , min{n, N }}, m ∈ {1, 2, . . . , M },
443
η(m) = min{n, N }.
m=1
Let us form the row vectors π(i, n, ν) of probabilities π (i, n, ν, η(1) , . . . , η(M ) ), i ∈ {0, 1, . . . , R}, n ∈ {0, 1, . . . , K + N }, ν ∈ {0, 1, . . . , W }, enumerated in the reverse lexicographic order of components η(1) , . . . , η(M ) . Then let us form the row vectors
π(i, n) = (π(i, n, 0), π(i, n, 1), . . . , π(i, n, W )), n ∈ {0, 1, . . . , K + N }, πi = (π(i, 0), π(i, 1), . . . , π(i, K + N )), i ∈ {0, 1, . . . , R}. It is well known that the probability vectors π i , i ∈ {0, 1, . . . , R}, satisfy the following system of linear algebraic equations:
(π0 , π1 , . . . , πR )Q = 0,
(π0 , π1 , . . . , πR )e = 1
(1)
where Q is the infinitesimal generator of the Markov chain ξt , t ≥ 0. Lemma 1. The infinitesimal generator Q of the Markov chain ξt , t ≥ 0, has the block tridiagonal structure: Q0,0 Q1,0 O
Q = ..
Q+ Q1,1 Q2,1
O Q+ Q2,2
.. .
··· ··· ··· .. .
O O O
O O
O O
··· ···
QR−1,R−1 QR,R−1
.. .
. O O
O O O
.. .
.. . . Q+ QR,R
The non-zero blocks Qi,j , i, j ≥ 0, have the following form:
(0)
Ci C¯ (1) O
Qi,i = . .
O (1) Ci C¯ (2)
O O (2) Ci
.. .
··· ··· ··· .. .
O O
O O
··· ···
.. .
. O O
(0)
(0)
Bi O O
Qi,i−1 = . .
B¯ i (1) Bi O
O (1) B¯ i (2) Bi
.. .
··· ··· ··· .. .
O O
O O
··· ···
.. .
. O O
Q + = D1 ⊗ IN
¯ ¯ n=0 Kn +K KN
O O O
O O O
.. .
.. .
(K +N −1)
Ci C¯ (K +N ) O O O
.. .
(K +N −1)
Bi
O
O
(K +N )
,
0 ≤ i ≤ R,
Ci
O O O
, .. . ¯Bi(K +N −1)
1 ≤ i ≤ R,
(K +N )
Bi
,
where
• • • • • • • • • • • • •
I is the identity matrix, O is a zero matrix of appropriate dimension; ⊕ and ⊗ are symbols of Kronecker sum and product respectively, see, e.g., [16]; (n) Ci = (D0 − iµIW¯ ) ⊕ (An (N , S ) + ∆(n) ), i ∈ {0, . . . , R − 1}, n ∈ {0, . . . , N }; (n)
= (D0 − (iµ + (n − N )α)IW¯ ) ⊕ (AN (N , S ) + ∆(N ) ), i ∈ {0, . . . , R − 1}, n ∈ {N + 1, . . . , K + N }; CR = (D(1) − RµIW¯ ) ⊕ (An (N , S ) + ∆(n) ), n ∈ {0, . . . , N }; (n) CR = (D(1) − (Rµ + (n − N )α)IW¯ ) ⊕ (AN (N , S ) + ∆(N ) ), n ∈ {N + 1, . . . , K + N }; C¯ (n) = IW¯ ⊗ LN −n (N , S˜ ), n ∈ {1, . . . , N }; C¯ (n) = (n − N )α IW¯ ⊕ L0 (N , S˜ )PN −1 (β), n ∈ {N + 1, . . . , K + N }; (n) Bi = qiµIW¯ K¯ n , i ∈ {0, . . . , R}, n ∈ {0, . . . , N − 1}; (n) Bi = (q + (1 − q)pn−N )iµIW¯ K¯ N , i ∈ {0, . . . , R}, n ∈ {N , . . . , K + N − 1}; (K +N ) Bi = iµIW¯ K¯ N , i ∈ {0, . . . , R}; ¯B(i n) = (1 − q)iµIW¯ ⊗ Pn (β), i ∈ {0, . . . , R}, n ∈ {0, . . . , N − 1}; (n) B¯ i = (1 − q)(1 − pn−N )iµIW¯ K¯ N , i ∈ {0, . . . , R}, n ∈ {N , K + N − 1}; Ci
(n)
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C. Kim et al. / Performance Evaluation 70 (2013) 440–453
• S˜ = (n)
• ∆
0 S
0 S0
;
= −diag{An (N , S )e + LN −n (N , S˜ )e}, n ∈ {1, 2, . . . , N }, ∆(0) = O1×1 .
The matrix Pi (β) defines the transition probabilities of the process ηt , t ≥ 0, at the epoch of starting the new service given that i servers are busy at this epoch. The matrix LN −i (N , S˜ ) defines the intensities of transitions of this process at the service completion epoch given that i servers are busy at this epoch. The matrix Ai (N , S ) defines the intensities of transitions of the process ηt , t ≥ 0, which do not lead to the service completion given that i servers are busy. Modules of diagonal entries of the matrix ∆(i) define the total intensity of leaving the corresponding states of the process ηt , t ≥ 0, given that i servers are busy. The detailed description of matrices Pn (β), Ln (N , S˜ ), n ∈ {0, . . . , N − 1}, and An (N , S ), n ∈ {0, . . . , N }, and the algorithms for their calculation can be found in [17]. In case the service time at the second stage has an exponential distribution with the parameter µ ˜ hereinafter we have to put Pn (β) = 1, Ln (N , S˜ ) = (N − n)µ, ˜ n ∈ {0, . . . , N − 1}, K¯ n = 1, An (N , S ) = 0, ∆(n) = −nµ, ˜ n ∈ {0, . . . , N }. Proof. The proof of the lemma is implemented by means of the analysis of all transitions of the Markov chain ξt , t ≥ 0, during the interval of an infinitesimal length and rewriting intensities of these transitions in the block matrix form. If the dimension of system (1) is small, it can be easily solved on a computer by standard methods. Otherwise, to solve this system the algorithm that was elaborated in [18] can be applied. 4. Performance measures As soon as the vectors π i , i ∈ {0, 1, . . . , R}, have been calculated, we are able to find various performance measures of the call center:
• The probability that an arbitrary customer will be lost at the first stage (1)
Ploss =
1
λ
π R Q + e.
• The average number of busy servers at the first stage N (1) =
R
iπ i e.
i=1
• The average intensity of flow of customers who get service at the first stage 1) λ(out = N (1) µ.
• The average number of busy servers at the second stage N (2) =
R K +N
min{n, N }π(i, n)e.
i=0 n=1
• The average number of customers in the buffer N buffer =
R K +N
(n − N )π(i, n)e.
i=0 n=N +1
• The average number of customers in the system Lsystem =
R K +N
(i + n)π(i, n)e.
i=0 n=0
• The average intensity of flow of customers who get service at the second stage λout =
R K +N
π(i, n)(IW¯ ⊗ Lmax{0,N −n} (N , S˜ ))e.
i=0 n=1
• The probability that an arbitrary customer will be lost at the second stage λout (2) . Ploss = 1 − 1) (1 − q)λ(out • The probability that an arbitrary customer will be lost at the entrance to the second stage (2-ent)
Ploss
1) −1 = (λ(out )
R i =1
iµπ(i, K + N )e.
C. Kim et al. / Performance Evaluation 70 (2013) 440–453
445
• The probability that an arbitrary customer arrives to the second stage when all servers at this stage are busy, the buffer is not full, and the customer does not join the buffer and leaves the system (2-esc)
Ploss
1) −1 = (λ(out )
R K+ N −1 i =1
pn−N iµπ(i, n)e.
n =N
• The probability that an arbitrary customer after arrival to the second stage will go to the buffer and leave it due to impatience (2-imp)
Ploss
(2) (2-ent) (2-esc) = Ploss − Ploss − Ploss .
5. Distribution of the sojourn and waiting times of an arbitrary customer Let V (x) be the distribution function of the sojourn time of an arbitrary customer in the system under study and ∞ −sx e dV (x), Re s > 0, be its LST . 0 We will derive an expression for the LST v(s) by means of the method of collective marks. So, v(s) has the meaning of probability that catastrophe does not arrive during the sojourn time of an arbitrary customer. We will tag an arbitrary customer and will keep track of his(her) staying in the system. Let v(s, i, n, ν, η(1) , . . . , η(M ) ) be the probability that a catastrophe will not arrive during the rest of the tagged customer sojourn time in the system conditioned on the fact that, at the given moment, the number of customers at the first stage including the tagged customer is equal to i, i ∈ {1, 2, . . . , R}, the number of customers at the second stage is n, n ∈ {0, 1, . . . , K + N }, and the states (M ) (1) of processes νt , ηt , . . . , ηt , t ≥ 0, are ν, η(1) , . . . , η(M ) , ν ∈ {0, 1, . . . , W }, η(m) ∈ {0, 1, . . . , min{n, N }}, m ∈ M {1, 2, . . . , M }, m=1 η(m) = min{n, N }. Let us enumerate the LST s v(s, i, n, ν, η(1) , . . . , η(M ) ) in the reverse lexicographic order of components η(1) , . . . , η(M ) and direct lexicographic order of the component ν and form from these LST s the column vectors v(s, i, n). If we have the functions v(s, i, n) calculated, by conditioning on the states of the system at the moment of customer arrival, the LST v(s) will be computed as
v(s) =
(1) v(s) = Ploss + λ−1
R −1 K +N
π(i, n)(D1 ⊗ IK¯ min{n,N } )v(s, i + 1, n).
i =0 n =0
The system of linear algebraic equations for vectors v(s, i, n) is derived using the law of total probability: v(s, i, n) = (s + max{0, n − N }α + iµ)I − D0 ⊕ Amin{n,N } (N , S ) − I ⊗ ∆(min{n,N })
−1
× (i − 1)µ (δ˜0≤n
