Ann Oper Res (2016) 239:401–428 DOI 10.1007/s10479-014-1626-2
Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option Alexander Dudin · Chesoong Kim · Olga Dudina · Sergey Dudin
Published online: 28 May 2014 © Springer Science+Business Media New York 2014
Abstract A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented. Keywords Call center · Call-back · Generalized phase-type distribution · Markovian arrival process · Multi-server queueing system 1 Introduction Call centers are used by companies for receiving and servicing their clients’ requests by telephone. Call centers are a very important part of companies that activities are related A. Dudin · O. Dudina · S. Dudin Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus e-mail:
[email protected] O. Dudina e-mail:
[email protected] S. Dudin e-mail:
[email protected] C. S. Kim (B) Sangji University, Wonju, Kangwon 220-702, Korea e-mail:
[email protected]
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to the communication with their customers. Customer’s service is becoming increasingly important due to high competition among companies, so image and profit of the company depend on the effective operation of its call centers. The problem of effective service of a large number of calls with minimal losses is of primary importance. This problem can be successfully solved by providing a so called call-back option and informing customers about anticipated delays. The call-back option means the following. A customer who calls to a call center during the epoch when all operators are busy and does not want to wait for service in line has an opportunity to ask for call-back and an operator will contact him (her) for service later on. This option allows to essentially decrease the loss probability of calls, to avoid frustration of customers, to make more smooth load of operators and increase the effectiveness of their work. In call centers that do not provide the call-back option some part of the customers’ service time may be spent on listening customers’ complaints about the long waiting time. At the same time, in call centers with the call-back option the customer has a choice—to wait for the response of the operator or to ask for call-back and due to this customers rarely complain. So, using of the call-back option can reduce the average customer’s service and waiting time. The call-back option also can solve another important problem—the problem of call center staff attrition. It is not a secret, that the job of call center operators is stressful. And high nonsmooth load, customers’ complaints, etc., can lead to operators attrition and weariness, which in turn can lead to high staff turnover. The high staff turnover can lead to not only negatively impact on call center performance but also to heavy expenses associated with recruiting and training new staff. Adequate mathematical modeling call centers leads to substantial increase of their economic efficiency, reduces the maintenance costs and improves the quality of customers’ service. For modeling call centers the queueing theory is often used. For background information and an overview of the present state of art in the study of call centers, the reader is referred to the survey Aksin et al. (2007) and the papers Koole and Mandelbaum (2002), Kim and Park (2010), Dudin and Dudina (2011), as well as references therein. In this paper, we consider a multi-server queueing system with an infinite buffer and impatient customers which can be used for modeling and optimizing call centers with the callback option. If all operators are busy during the customer arrival epoch, the customer becomes aware of the current queue length (visible queue) and his (her) estimated waiting time, and, based on the information provided, decides whether to balk (leave the system permanently without service), wait in line, or leave his (her) phone number. In the latter case, an operator of the call center will call back to the customer later when the operator will be free. Statistics shows that customers who receive information about their place in the buffer or waiting time, are 1.5–2 times more patient, than the customers who do not have such information. As a result, the number of unserviced customers is greatly reduced, therefore the consideration of visible queue and call-back option is an important point in the modeling modern call centers. In the papers Armony and Maglaras (2004a, b), Iravani and Balcioglu (2008), Kim et al. (2012), models of call centers with the call-back option are considered. In Armony and Maglaras (2004a) and Armony and Maglaras (2004b), an asymptotic analysis of the model of the call center with visible queue and call-back option in the case of heavy load for a large number of operators is presented. In the paper (Iravani and Balcioglu 2008), an M/M/N type queue with the call-back option (but without visibility of a queue) is analyzed. The advantage of work Kim et al. (2012) comparing to Armony and Maglaras (2004a) and Armony and Maglaras (2004b) consists of presenting an exact analytical analysis of the model with visible queue and call-back option without the restriction that the number of operators is large enough. The advantage of work Kim et al. (2012) comparing to Iravani
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and Balcioglu (2008) consists of the following: (i) more general customers’ arrival process is considered; (ii) the possibility of leaving the system without service depending on information about the current queue length is allowed; (iii) presence of an additional time necessary to provide service to customers who choose the call-back option (comparing to the customers who do not use this option) is taken into account. The disadvantage of work Kim et al. (2012) consists of the assumption that the service and dial times have an exponential distribution which rarely fits real data in call centers. In the present paper, we consider more general model where the service and dial times have a phasetype distribution (PH) what allows us to take into account variance of the service and dial processes. Since the class of phase-type distributions is dense in the set of all positive-valued distributions, PH distribution can be used to approximate any positive-valued distribution. So, from the practical point of view the consideration of PH service and dial time distributions is very important for adequate modelling and correct performance evaluating call centers. However, the consideration of a phase-type distribution instead of an exponential implies essential increase of complexity of the generator of a Markov chain describing behavior of the system. To significantly simplify the study of the system and to avoid cumbersome mathematical expressions, in this paper, we introduce the notion of a generalized phase-type distribution instead of the separate consideration of phase-type service and dial time distributions. It allows us to derive the ergodicity condition, calculate the stationary distribution of the system states, and obtain formulas for performance measures. It is worth noting that, comparing to the paper Kim et al. (2012), we derive the Laplace–Stieltjes transform of the sojourn and waiting time distributions of virtual customers in significantly simpler form, and we also derive the Laplace–Stieltjes transform of the sojourn and waiting time distributions of real customers. Moreover, numerical results illustrating the behavior of the system characteristics depending on its parameters are presented. The paper is organized as follows. In Sect. 2, the mathematical model is described. The notion of a generalized phase-type distribution is presented in Sect. 3. The process of the system states, the ergodicity condition and the stationary distribution of the system states is analyzed in Sect. 4. The expressions for the main performance measures of the system are given in Sect. 5. In Sects. 6 and 7, we focus on the analysis of the sojourn and waiting time distributions of real and virtual customers. Section 8 contains some numerical illustrations. Section 9 concludes the paper.
2 Mathematical model The system under study consists of N identical operators (servers), buffer of capacity R for real customers and infinite buffer for virtual customers. The structure of the system is presented in Fig. 1. Customers (calls) arrive to the system according to the Markovian arrival process (MAP). Arrivals can occur at the epochs of jumps in the underlying process νt , t ≥ 0, which is an irreducible continuous time Markov chain with the state space {0, 1, . . . , W }. The sojourn time of this chain in the state ν is exponentially distributed with the positive finite parameter (0) λν . When the sojourn time in the state ν expires, with probability pν,ν the process νt jumps (1)
to the state ν without a generation of a customer, and with probability pν,ν the process νt jumps to the state ν with the generation of a customer, ν, ν = 0, W . The notation ν = 0, W means that the parameter ν takes the values in the set {0, 1, . . . , W }.
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Fig. 1 Structure of the system
The behavior of the MAP is completely characterized by the matrices D0 and D1 defined (0) , ν, ν = 0, W , ν = ν , and by the entries (D0 )ν,ν = −λν , ν = 0, W , (D0 )ν,ν = λν pν,ν (1)
(D1 )ν,ν = λν pν,ν , ν, ν = 0, W . The matrix D(1) = D0 + D1 represents the generator of the process νt , t ≥ 0. The average arrival rate is given as λ = θ D1 e, where θ is the unique solution to the system θ D(1) = 0, θe = 1. Here e is a column vector of appropriate size consisting of 1’s and 0 is a row vector of appropriate size consisting of zeroes. The squared coefficient 2 of variation of intervals between successive arrivals is given as cvar = 2λθ (−D0 )−1 e − 1. The coefficient of correlation of two successive intervals between arrivals is given as ccor = 2 . (λθ (−D0 )−1 (D(1) − D0 )(−D0 )−1 e − 1)/cvar For more information about the MAP see (Lucantoni 1991). If there is an available server during an arbitrary customer arrival epoch, the customer is admitted to the system and occupies a free server. Otherwise, the customer has the following three possible options: (i) to leave the system without service, (ii) to become a real customer, i.e., enter the buffer and wait in the system until some server will become available for him (her), or (iii) to become a virtual customer, i.e., to join an infinite size virtual pool of customers and wait until he (she) will be picked up for service. We assume that both real and virtual customers are served according to the rule First In - First Out. The difference between two types of customers is that the real customer physically presents in the system (holds a line or occupies a place in the buffer) while the virtual customer just leaves his (her) phone number and an operator will call him (her) and offer service later on when the system will be not congested. We suggest that service to a virtual customer is offered only when the queue of real customers is empty and there is a free server. We assume the following discipline for making a choice among options (i)–(iii). The number of real customers in the buffer can not exceed a certain threshold R. So, if all servers are busy and all R positions in the buffer for real customers are occupied during an arbitrary customer arrival epoch, then this customer has only options (i) or (iii). He (she) leaves the system with probability δ or becomes a virtual customer with the complementary probability. If all servers are busy and there are r, r = 0, R − 1, real customers in the buffer during an arbitrary customer arrival epoch, the arriving customer leaves the system with probability qr , with the complementary probability the customer decides to wait for service. In the latter case, he (she) becomes a real customer with probability 1 − pr , and with probability pr he (she) becomes a virtual customer. The dependence of the probability of joining the buffer on the current number of customers in the buffer realizes the conception of the visible queue, usefulness of which was mentioned in introduction.
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The service time of a customer for each server has a PH distribution with an irreducible representation (β (s) , S (s) ). This service time can be interpreted as the time until the underlying (1) Markov process ηt , t ≥ 0, with the finite state space {1, . . . , M (s) , M (s) + 1} reaches the single absorbing state M (s) + 1 conditioned on the fact that the initial state of this process is selected among the states {1, . . . , M (s) } according to the probabilistic row vector (s) (s) β (s) = (β1 , . . . , β M (s) ). The transition rates of the process ηt within the set {1, . . . , M (s) } are defined by the sub-generator S (s) and the transition rates into the absorbing state (which (s) lead to the service completion) are given by the entries of the column vector S0 = −S (s) e. The mean service time is calculated by b1 = β (s) (−S (s) )−1 e. The squared coefficient of variation is given by cvar = b2 /b12 − 1 where b2 = 2β (s) (−S (s) )−2 e. For more information about PH distribution and its usefulness see, e.g., Neuts (1981). The methods of modelling PH process using a set of service times obtained at the real system, and in particular call center, can be founded in Panchenko and Thummler (2007). If there is no real customer in the buffer during the service completion epoch, a free server starts to dial to the virtual customer placed first into the virtual pool. During a dial time the server is blocked. The dial time has a phase-type distribution with M (d) + 1 states, (2) an irreducible representation (β (d) , S (d) ) and the underlying Markov process ηt , t ≥ 0. We assume that the server does not succeed to connect to the customer (the customer’s phone is busy or does not answer) with probability h. In this case, the virtual customer is lost and the blocked server becomes free. With probability 1 − h the server connects to the customer and starts service of the virtual customer. The mean dial time is calculated by b1dial = β (d) (−S (d) )−1 e. Real customers are impatient, i.e., a customer leaves the buffer after an exponentially distributed with the parameter α, 0 < α < ∞, time, due to a lack of service. In the case of leaving the buffer due to impatience, a real customer leaves the system forever with probability γ or becomes a virtual customer with the complementary probability. 3 Generalized phase type distribution To investigate the formulated queueing system, it is necessary to construct the multidimensional continuous time Markov chain describing behavior of this system. There is a lot of different ways for constructing the Markov chain describing behavior of a multiserver queueing system with a phase-type distribution of the service time. So, the problem of choosing the best way arises. Speaking about the best way, we suggest that this way allows to reach the following two goals: (i) to write down the generator of the chain without essential analytical and mental difficulties and (ii) to provide minimally possible dimension of the blocks of the constructed generator. Unfortunately, it is quite rare case when these goals do not interfere with each other and, so, it is desirable to find some trade-off between them. Essential difficulty in writing down the generator of the chain, which describes behavior of the system under study in this paper, stems from the fact that each busy server may either to provide service to a customer or to dial to a virtual customer and phase-type distributions of the service and dial times have different parameters. So, during each time moment it is necessary to count separately the busy servers providing service and dialing to customers. This complicates the problem of writing explicit expressions for the blocks of the generator. Aiming to decrease the dimension of the stochastic process that describes the behavior of a multi-server system with a phase-type distribution, it is necessary to optimize description of the joint behavior of phases of service in all busy servers. In the overwhelming majority of the existing papers, see, e.g., (Dudin and Dudina 2011; He et al. 2000; Breuer et al. 2005; Dudin
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et al. 2013), the behavior of the queue with a phase-type service time distribution is described (1) (N ) (m) by the stochastic process including the components ζt , . . . , ζt , t ≥ 0, where ζt is the state of the PH underlying process on the mth busy server, m = 1, N . The dimension of the (1) (N ) process ζt = {ζt , . . . , ζt }, t ≥ 0, is equal to K¯ = M N where M is the dimension of the PH process state space including the absorbing state. If we use this way of description, the dimension K¯ may be too high for easy computing the system performance measures even for small number of servers N . For example, if we fix N = 10 and M = 4, the process ζt , t ≥ 0, has huge dimension K¯ = 410 = 1048576. Thus, to investigate the system with a phase-type distribution, we consider another description of the joint behavior of phases of service in all busy servers, i.e., we consider the stochastic (1) (M) (m) process including the components ηt , . . . , ηt , where ηt is the number of servers at phase (m) (m) M ηt = N , during the epoch t, t ≥ 0. Note m of service, m = 1, M, ηt = 0, N , m=1 (m) that the meaning of the components ηt , m = 1, M, is chosen according to the approach by Ramaswami and Lucantoni, see Ramaswami (1985), Ramaswami and Lucantoni (1985). (1) (M) . The dimension of the process ηt = {ηt , . . . , ηt }, t ≥ 0, is equal to K˜ = N +M−1 M−1 For example, if we fix, as above, N = 10 and M = 4, the process ηt , t ≥ 0, has dimension K˜ = 286 that is significantly less than dimension K¯ = 410 . In order to greatly simplify the problem of writing the blocks of the generator of queue under study, instead of the separate consideration of service and dial times we propose to consider a generalized processing time having distribution which we call as a generalized phase-type distribution with an irreducible representation (β (1) , β (2) , S). Time having such a distribution can be interpreted as the time until the underlying Markov process ηt , t ≥ 0, with the finite state space {1, . . . , M, M + 1}, where M = M (d) + M (s) , reaches the single absorbing state M + 1. The initial state of this process is selected among the states {1, . . . , M} depending on the type of a customer who is chosen for service. If a virtual customer is chosen for service, the initial state of this process is selected according to the probabilistic row vector β (1) = (β (d) , 0 M (s) ) and, if a real customer is chosen for service, the initial state is selected according to the probabilistic row vector β (2) = (0 M (d) , β (s) ) . The transition rates of the process ηt within the set {1, . . . , M} are defined by the sub-generator (d) −(1 − h)S (d) eβ (s) S and the transition rates into the absorbing state (which S = O S (s) lead to the service completion) are given by the entries of the column vector S0 = −Se. The main idea of a generalized phase-type distribution is to treat processing of all customers in unified way independent of their type, but not to consider the service process of each type of customers separately. A generalized phase-type distribution can be successfully used in an investigation of a variety of systems with heterogeneous customers and different service time distributions for each type of customers (see examples in Appendix 1). For each specific multi-server queue with heterogeneous customers, the generator S of a generalized phase-type distribution may be constructed differently. What is worth to note, the use of a generalized phase-type with the state space (1, . . . , M1 + · · · + M K + 1) instead of the consideration of K phase-type service processes with dimensions Mk + 1, k = 1, K , of the state space does not change (increase or decrease) the dimension of the stochastic process that describes the behavior of system under consideration (with the use of V. Ramaswami and D.M. Lucantoni approach). The proof of this assertion is presented in Appendix 2. Summarizing this section, we can say that the use of a generalized phase-type distribution of the customer processing time in combination with the use of V. Ramaswami and D.M. Lucantoni approach: (i) significantly simplifies the problem of writing the explicit expressions
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for the blocks of the generator; (ii) does not lead to the increase of the dimension of the blocks of the generator what is important in further computer computation of system performance measures.
4 The process of system states The behavior of the system under consideration can be described in terms of the regular irreducible continuous-time Markov chain (1)
(M)
ξt = {i t , rt , νt , ηt , . . . , ηt
}, t ≥ 0,
where i t is the total number of customers in the system, i t ≥ 0, rt is the number of customers in the buffer, rt = 0, max{0, min{i t − N , R}}, νt is the state of the underlying process of (m) the M A P, νt = 0, W , ηt is the number of servers at phase m of generalized service, M (m) (m) ηt = min{i t , N }, during the epoch t, t ≥ 0. m = 1, M, ηt = 0, min{i t , N }, m=1 For further use thought this paper, we introduce the following notation: • I is an identity matrix; ⊕ and ⊗ indicate the Kronecker sum and product, respectively (see, e.g., Graham (1981)); • W¯ = W + 1; • Cl = diag{0, 1, . . . , l}, l = 1, max{N , R}, P¯l = diag{ p0 , p1 , . . . , pl }, l = 0, R − 1; • Δr = diag{q0 , q1 , . . . , qr }, r = 0, R − 1, Δ = diag{q0 , q1 , . . . , q R−1 , δ}, Δ¯ = diag{(1 − q0 )(1 − p0 ), (1 − q1 )(1 − p1 ), . . . , (1 − q R−1 )(1 − p R−1 ), 0}, Δ˜ = diag{(1 − q0 ) p0 , (1 − q1 ) p1 , . . . , (1 − q R−1 ) p R−1 , 1 − δ}; • El− , l = 1, max{N , R}, are the matrices of size (l + 1) × l with all zero entries except the entries (El− )0,0 , (El− )i,i−1 , i = 1, l, which are equal to 1; • El+ , E˜ l+ , l = 0, max{N , R} − 1, are the matrices of size (l + 1) × (l + 2) with nonzero entries (El+ )i,i+1 , ( E˜ l+ )i,i , i = 0, l, which are equal to 1; ˆ i,i+1 , i = • Eˆ is the square matrix of size R + 1 with all zero entries except the entries ( E) 0, R − 1, which are equal to 1; • El , l = 1, max{N , R}, are the square matrices of size l + 1 with all zero entries except the entries (El )i,i−1 , i = 1, l, which are equal to 1; • I¯l , l = 1, R, are the matrices of size (l + 1) × l with all zero entries except the entry ( I¯l )0,0 which is equal to 1; • Iˆ is the square matrix of size R + 1 withnonzero entry ( Iˆ)0,0 which equals 1; M−1 0 0 , i = 0, N ; S˜ = • K i = i+M−1 . S0 S Let us enumerate the states of this Markov chain in the reverse lexicographic order of the (1) (M) components ηt , . . . , ηt and in the direct lexicographic order of the component νt , and refer to (i, r ) as a macro-state consisting of (W + 1)K min{i,N } states (i, r, ν, η(1) , . . . , η(M) ). Let Q be the generator of the Markov chain ξt , t ≥ 0, consisting of the blocks Q i, j , which, in turn, consist of the matrices (Q i, j )r,r of the transition rates of this chain from the macro-state (i, r ) to the macro-state ( j, r ), r, r = 0, max{0, min{i − N , R}}. The diagonal entries of matrices Q i,i are negative, and the modulus of the diagonal entries of the blocks (Q i,i )r,r define the total intensity of leaving the corresponding state of the Markov chain ξt , t ≥ 0. Analysing all transitions of the Markov chain ξt , t ≥ 0, during an interval of infinitesimal length and rewriting the intensities of these transitions in block matrix form we obtained the following result.
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Lemma 1 The infinitesimal generator Q = (Q i, j )i, j≥0 of the Markov chain ξt , t ≥ 0, has a block-tridiagonal structure. The non-zero blocks Q i, j , i, j ≥ 0, have the following form: Q 0,0 = D0 , Q i,i = D0 ⊕ Ai (N , S) + IW¯ ⊗ Δ(i) , 1 ≤ i < N , Q N ,N = (D0 + q0 D1 ) ⊕ A N (N , S) + IW¯ ⊗ Δ(N ) , Q i,i = Ii−N +1 ⊗ (D0 ⊕ A N (N , S) + IW¯ ⊗ Δ(N ) ) + (−αCi−N + (1 − γ )αCi−N E i−N ) ⊗ IW¯ K N + Δi−N ⊗ D1 ⊗ I K N , N < i < N + R, Q i,i = Q 1 = I R+1 ⊗ (D0 ⊕ A N (N , S) + IW¯ ⊗ Δ(N ) ) + Δ ⊗ D1 ⊗ I K N + (−αC R + (1 − γ )αC R E R ) ⊗ IW¯ K N , i ≥ N + R, ˜ Q i,i−1 = IW¯ ⊗ L N −i (N , S), 1 ≤ i ≤ N, − ¯ Q i,i−1 = Ii−N ⊗ IW¯ ⊗ L 1 + (E i−N − I¯i−N ) ⊗ IW¯ ⊗ L 2 − + γ αCi−N E i−N ⊗ IW¯ K N , N < i ≤ N + R,
Q i,i−1 = Q 0 = Iˆ ⊗ IW¯ ⊗ L 1 + E R ⊗ IW¯ ⊗ L 2 + γ αC R E R ⊗ IW¯ K N , i > N + R, Q i,i+1 = D1 ⊗ Pi (β (2) ), 0 ≤ i < N , + Q i,i+1 = (Ii−N +1 − Δi−N )(Ii−N +1 − P¯i−N )E i−N ⊗ D1 ⊗ I K N + + (Ii−N +1 − Δi−N ) P¯i−N E˜ i−N ⊗ D1 ⊗ I K N , N ≤ i < N + R,
˜ ⊗ D1 ⊗ I K N , i ≥ N + R, Q i,i+1 = Q 2 = (Δ¯ Eˆ + Δ) where ˜ N −1 (β (1) ), L 2 = L 0 (N , S)P ˜ N −1 (β (2) ), L 1 = L 0 (N , S)P ˜ Δ(0) = 0, Δ(i) = −diag{Ai (N , S)e + L N −i (N , S)e}, i = 1, N . A detailed description of the matrices Pi (β (r ) ), r = 1, 2, i = 0, N − 1, Ai (N , S) and ˜ i = 0, N , and the algorithms for their calculation can be found in Kim et al. L i (N , S), (2013, 2012). Note that the matrix Pi (β (r ) ) defines the transition probabilities of the process t ≥ 0, during the epoch of starting new service given that i servers are ˜ defines the intensities of the transitions busy during this epoch. The matrix L N −i (N , S) of this process during the service completion epoch given that i servers are busy during this epoch. The matrix Ai (N , S) defines the intensities of the transitions of the process (1) (M) {ηt , . . . , ηt }, t ≥ 0, which do not lead to the service completion given that i servers are busy. The modules of the diagonal entries of the matrix Δ(i) define the total intensity of (1) (M) leaving the corresponding states of the process {ηt , . . . , ηt }, t ≥ 0, given that i servers are busy. Since the Markov chain ξt , t ≥ 0, belongs to the class of continuous-time quasi-birthand-death processes, then, see, e.g., Neuts (1981), the ergodicity condition of this process is the fulfillment of the inequality
(1) (M) {ηt , . . . , ηt },
yQ 0 e > yQ 2 e,
(1)
where the row vector y = ( y0 , y1 , . . . , y R ) is the unique solution to the following system of linear algebraic equations y(Q 0 + Q 1 + Q 2 ) = 0,
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ye = 1.
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For calculation of the sub-vectors yr , r = 0, R, we propose the following numerically stable algorithm. Step 1. Calculate the matrices Q¯ r,r , Q¯ r,r +1 , Q¯ r,r −1 as follows: Q¯ r,r = D0 ⊕ A(N , S) + IW¯ ⊗ Δ(N ) + δr,0 IW¯ ⊗ L 1 +((1 − δr,R )((1 − qr ) pr + qr ) + δr,R )D1 ⊗ I K N − r α IW¯ K N , r = 0, R, Q¯ r,r +1 = (1 − qr )(1 − pr )D1 ⊗ I K N , r = 0, R − 1, Q¯ r,r −1 = IW¯ ⊗ L 2 + r α IW¯ K N , r = 1, R, where δi, j is a symbol of Kronecker delta. Step 2. Calculate the matrices Tr , r = 0, R − 1, using the backward recursion Tr = − Q¯ r,r +1 ( Q¯ r +1,r +1 + Tr +1 Q¯ r +2,r +1 )−1 , r = R − 2, R − 3, . . . , 0, under the initial condition TR−1 = − Q¯ R−1,R ( Q¯ R,R )−1 . Step 3. Calculate the matrices Fr using the recurrent formulas F0 = I,
Fr = Fr −1 Tr −1 , r = 1, R.
Step 4. Calculate the vector y0 as the unique solution to the following system y0 ( Q¯ 0,0 + T0 Q¯ 1,0 ) = 0,
y0
R
Fr e = 1.
r =0
Step 5. Calculate the vectors yr = y0 Fr , r = 1, R. If the ergodicity condition (1) of the Markov chain ξt is fulfilled, then the stationary probabilities of the system states π(i, r, ν, η(1) , . . . , η(M) ), i ≥ 0, r = 0, max{0, min{i − N , R}}, ν = 0, W , η(m) = 0, min{i, N }, m = 1, M, exist. Let us form the row vectors π (i, r, ν) of these probabilities enumerated in the reverse lexicographic order of the components η(1) , . . . , η(M) . Then let us form the row vectors π(i, r ) = (π (i, r, 0), π (i, r, 1), . . . , π (i, r, W )), and π i = (π (i, 0), π (i, 1), . . . , π (i, max{0, min{i − N , R}})). To calculate the probability vectors π i , i ≥ 0, the numerically stable algorithm presented in the paper Kim et al. (2012) (Theorem 2) can be used.
5 Performance measures As soon as the vectors π i , i ≥ 0, have been calculated, we are able to find various performance measures of the system under consideration. The average number of customers in the system is calculated as L=
∞
iπ i e.
i=1
The average number of (real and virtual) customers, who wait for service, is calculated as N bu f f er =
∞
(i − N )π i e.
i=N +1
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The average number of real customers, who wait for service, is defined as bu f f er
Nr eal
∞
min{i−N ,R}
i=N +1
r =1
=
r π (i, r )e.
The average number of virtual customers, who wait for service, is defined as bu f f er
Nvir t
∞
min{i−N ,R}
i=N +1
r =0
=
(i − N − r )π (i, r )e.
The average number of busy and blocked servers is calculated as N ser ver =
∞
min{i, N }π i e.
i=1
-loss that an arbitrary real customer arrives when all servers are busy, The probability Presc eal r, r < R, real customers are present in the buffer, and this customer does not join the system is defined as -loss Presc eal
−1
=λ
∞ min{i−N ,R−1}
qr π (i, r )(D1 ⊗ I K N )e.
r =0
i=N
The probability that an arbitrary real customer leaves the system forever or becomes a virtual customer at the entrance to the system due to the presence of R real customers in the buffer is calculated as -loss = λ−1 Prent eal
∞
π (i, R)(D1 ⊗ I K N )e.
i=N +R
-vir t that an arbitrary customer arrives when all servers are busy, The probability Prtoeal r, r < R, real customers are present in the buffer, and this customer becomes a virtual customer is defined as -vir t = λ−1 Prtoeal
∞ min{i−N ,R−1} i=N
(1 − qr ) pr π (i, r )(D1 ⊗ I K N )e.
r =0
Let us introduce the probabilities z(r, η(1) , . . . , η(M) ), r = 1, R, that during the waiting time of a real customer in the buffer this customer does not leave the system due to impatience conditioned on the fact that during his (her) arrival epoch there are r − 1 real customers in (1) (M) the buffer and the states of the processes ηt , . . . , ηt are η(1) , . . . , η(M) , respectively. The column vectors z(r ) consisting of the probabilities z(r, η(1) , . . . , η(M) ) enumerated in the reverse lexicographic order of the components η(1) , . . . , η(M) are defined as follows: z(1) = (α I − A)−1 L 2 e K N ,
z(r ) = (r α I − A)−1 (L 2 + (r − 1)α I )z(r − 1), r = 2, R,
where A = A N (N , S) + Δ(N ) . imp -loss The probability Pr eal that an arbitrary real customer arrives when all servers are busy, r, r < R, real customers are present in the buffer, and this customer will go to the buffer and leave it due to impatience is defined as
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imp -loss
Pr eal
=
411
∞ min{i−N ,R−1} 1 (1 − qr )(1 − pr )π(i, r )(D1 e ⊗ I K N )(e K N − z(r + 1)). λ r =0
i=N
The probability that an arbitrary real customer leaves the system forever or becomes a virtual customer is calculated as imp -loss P loss = P esc-loss + P ent -loss + P to-vir t + P . r eal
r eal
r eal
r eal
r eal
The intensity of output flow of customers from the system is calculated as λout =
∞
˜ π i (I(max{0,min{R,i−N }}+1) ⊗ IW¯ ⊗ L max{0,N −i} (N , S))e.
i=1
The intensity of output flow of virtual customers is calculated as λvir t =
∞
π i (I(max{0,min{R,i−N }}+1) ⊗ IW¯ ⊗ L max{0,N −i} (N , S˜ (1) ))e
i=1
0 0 . (d) T T (−(S e) , 0 M (s) ) S The intensity of output flow of customers, to whom an operator was unable to dial, from the system is calculated as λnot -dial = hλ .
where S˜ (1) =
out
vir t
The intensity of output flow of customers, who receive service in system, is calculated as v λser out =
∞
π i (I(max{0,min{R,i−N }}+1) ⊗ IW¯ ⊗ L max{0,N −i} (N , S˜ (2) ))e
i=1
0 0 . (0 M (d) , −(S (s) e)T )T S The probability that an arbitrary customer leaves the system forever without receiving service is calculated as λout . P loss = 1 − λ where S˜ (2) =
6 Distribution of the sojourn and waiting times of a real customer Let Vr eal (x) be the distribution function
∞ of the sojourn time of an arbitrary real customer in the system under study and vr eal (s) = 0 e−sx d Vr eal (x), Re s > 0, be its Laplace–Stieltjes transform (L ST ). Theorem 1 The L ST vr eal (s) of the distribution of the sojourn time of an arbitrary real customer in the system is calculated as v (s) = P ent -loss + P esc-loss + P to-vir t r eal
r eal
+ λ−1
N −1
r eal
r eal
(s)
π i (D1 ⊗ I K i )eβ (s) (s I − S (s) )−1 S0
i=0
∞ min{i−N ,R−1} + (1 − qr )(1 − pr )π(i, r )(D1 e ⊗ I K N )vr eal (s, r + 1) i=N
(2)
r =0
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where the vectors vr eal (s, r ), r = 1, R, are computed recursively by formulas (s)
vr eal (s, 1) = ((s + α)I − A)−1 [L 2 e K N β (s) (s I − S (s) )−1 S0 + αe K N ], −1
vr eal (s, r ) = ((s + r α)I − A)
(3)
[(L 2 + (r − 1)α I )vr eal (s, r − 1) + αe K N ], r = 2, R. (4)
Proof Let us tag an arbitrary real customer and keep track of his (her) processing in the system. We will derive the expression for the L ST vr eal (s) using the method of collective marks (method of additional event, method of catastrophes) for references, see, e.g., Kesten and Runnenburg (1956); Danzig (1955). To this end, we interpret the variable s as the intensity of some virtual stationary Poisson flow of catastrophes. Thus, vr eal (s) has the meaning of the probability that no catastrophe arrives during the sojourn time of the tagged customer. The following situations are possible during the arrival epoch of the tagged customer: (a) The customer leaves the system immediately upon arrival. The probability of this event -loss + P esc-loss . In this case, the probability that no catastrophe arrives during is Prent eal r eal the sojourn time is equal to 1. (b) The customer arrives to the system, when all servers are busy, and becomes virtual. -vir t . In this case, the probability that no catastrophe The probability of this event is Prtoeal arrives during the sojourn time of this customer as real one is also equal to 1. (c) There is an idle server during the arrival epoch of the tagged customer and this customer N −1 immediately goes to service. The probability of this event is λ−1 i=0 π i (D1 ⊗ I K i )e. In this case, the probability that no catastrophe arrives during the sojourn time is equal to the probability that no catastrophe arrives during the service time and is defined as (s) β (s) (s I − S (s) )−1 S0 . (d) All servers are busy during the arrival epoch and the tagged customer joins the buffer. Let vr eal (s, r, η(1) , . . . , η(M) ) be the L ST of the distribution of the tagged real customer’s sojourn time conditioned on the fact that, during the given moment, the position of the tagged customer in the buffer is equal to r, r = 1, R and the states of the processes M (1) (M) ηt , . . . , ηt , t ≥ 0, are η(1) , . . . , η(M) , respectively, η(m) = 0, N , m=1 η(m) = N , m = 1, M. Let us enumerate probabilities vr eal (s, r, η(1) , . . . , η(M) ) in the reverse lexicographic order of the components η(1) , . . . , η(M) and form from these probabilities the column vectors vr eal (s, r ). If we find vr eal (s, r ), then the probability that the tagged customer will be admitted to the system during the epoch when all servers are busy and no catastrophe arrives during its sojourn time will be written as: λ−1
∞ min{i−N ,R−1} i=N
(1 − qr )(1 − pr )
r =0
N
π(i, r )(D1 e ⊗ I K N )vr eal (s, r + 1).
n=0
Based on a probabilistic sense of the L ST , we obtain the system of equations for calculation of vr eal (s, r ), r = 1, R : (−(s + r α)I + A)vr eal (s, r ) + (1 − δr,1 )(L 2 + (r − 1)α I )vr eal (s, r − 1) + αe K N (s)
+ δr,1 L 2 e K N β (s) (s I − S (s) )−1 S0 = 0T , r = 1, R,
(5)
It is easy to verify that the solution of (5) can be written as (3), (4). Using the formula of total probability we prove the theorem.
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Corollary 1 The average sojourn time of an arbitrary real customer in the system is defined by: N −1 soj Vr eal = λ−1 b1 π i (D1 ⊗ I K i )e i=0
−
(1 − qr )(1 − pr )π (i, r )(D1 e ⊗ I K N )vr eal (0, r + 1) (6)
∞ min{i−N ,R−1} i=N
r =0
where vr eal (0, 1) = −(α I − A)−1 (I + b1 L 2 )e K N ,
vr eal (0, r ) = −(r α I − A)−1 [e K N − (L 2 + (r − 1)α I )vr eal (0, r − 1)], r = 2, R.
Proof Formula (6) for calculation of the average sojourn time of an arbitrary real cussoj tomer in the system is based on definition Vr eal = −vr eal (s)|s=0 e, taking into account
that vr eal (s, r )|s=0 = e K N , r = 1, R. Remark 1 Based on Theorem 1, the LST of the distribution of an arbitrary real customer’s (s) waiting time can be found by deleting the multiplier β (s) (s I − S (s) )−1 S0 from formulas (2) and (3). Corollary 2 The average waiting time of an arbitrary real customer in the system is calculated as follows: ∞ min{i−N ,R−1} (1 − qr )(1 − pr )π (i, r )(D1 e ⊗ I K N )yr eal (0, r + 1)
−1 Vrwait eal = −λ
r =0
i=N
where yr eal (0, 1) = −(α I − A)−1 e K N ,
yr eal (0, r ) = −(r α I − A)−1 [e K N − (L 2 + (r − 1)α I )yr eal (0, r − 1)], r = 2, R.
Theorem 2 The L ST wr eal (s) of the distribution of the sojourn time of an arbitrary serviced real customer in the system is calculated as wr eal (s) =
N −1
1 λ(1 − Prloss eal ) +
(s)
π i (D1 ⊗ I K i )eβ (s) (s I − S (s) )−1 S0
i=0
⎤
∞ min{i−N ,R−1}
(1 − qr )(1 − pr )π(i, r )(D1 e ⊗ I K N )wr eal (s, r + 1)⎦
i=N
(7)
r =0
where wr eal (s, r ), r = 1, R, are computed by formulas (s)
wr eal (s, 1) = ((s + α)I − A)−1 L 2 e K N β (s) (s I − S (s) )−1 S0 , wr eal (s, r ) = ((s + r α)I − A)−1 (L 2 + (r − 1)α I )wr eal (s, r − 1), r = 2, R.
(8)
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Corollary 3 The average sojourn time of an arbitrary serviced real customer in the system is defined as follows: N −1 1 soj -ser v Vr eal = π i (D1 ⊗ I K i )e b1 loss λ(1 − Pr eal ) i=0 ⎤ ∞ min{i−N ,R−1} − (1 − qr )(1 − pr )π (i, r )(D1 e ⊗ I K N )wr eal (0, r + 1)⎦ r =0
i=N
where wr eal (0, 1) = −(α I − A)−1 [(α I − A)−1 + b1 I ]L 2 e K N ,
wr eal (0, r ) = −(r α I − A)−1 [(r α I − A)−1 (L 2 + (r − 1)α I )wr eal (0, r − 1) −(L 2 + (r − 1)α I )wr eal (0, r − 1)], r = 2, R.
Remark 2 Based on Theorem 2, the LST of the distribution of an arbitrary serviced real (s) customer’s waiting time can be found by deleting the multiplier β (s) (s I − S (s) )−1 S0 from formulas (7), (8) of this theorem. Corollary 4 The average waiting time of an arbitrary serviced real customer is calculated -ser v = V soj -ser v − b . as Vrwait 1 r eal eal 7 Distribution of the sojourn and waiting times of a virtual customer Let Vvir t (x) be the distribution
∞function of the sojourn time of an arbitrary virtual customer in the system and vvir t (s) = 0 e−sx d Vvir t (x), Re s > 0, be its L ST . Let us tag an arbitrary virtual customer and keep track of his (her) staying in the system. Let vvir t (s, l, r, ν, η(1) , . . . , η(M) ) be the probability that the catastrophe will not arrive during the rest of the tagged customer sojourn time in the system conditioned on the fact that, during the given moment, the tagged customer has the position number l, l ≥ 1, in the queue of virtual customers (i.e., there are l − 1 virtual customers in the buffer that arrived earlier than the tagged customer), the number of real customers in the buffer is equal to r, r = (1) (M) 0, R, the states of the processes νt , ηt , . . . , ηt , t ≥ 0, are ν, η(1) , . . . , η(M) , ν = 0, W , M respectively, η(m) = 0, N , m=1 η(m) = N , m = 1, M. Let us enumerate probabilities vvir t (s, l, r, ν, η(1) , . . . , η(M) ) in the reverse lexicographic order of the components η(1) , . . . , η(M) and direct lexicographic order of the component ν and form from these probabilities the column vectors vvir t (s, l, r ). Theorem 3 The LST vvir t (s) of the distribution of an arbitrary virtual customer’s sojourn time in the system is computed by ⎡ ∞ min{i−N ,R−1} ⎣ vvir t (s) = λ−1 (1 − qr ) pr π (i, r )(D1 ⊗ I K N )vvir t (s, i − N − r + 1, r ) vir t r =0
i=N ∞
+ (1 − δ)
π (i, R)(D1 ⊗ I K N )vvir t (s, i − N − R + 1, R)
i=N +R
+ (1 − γ )
123
∞
min{i−N ,R}
i=N +1
r =1
⎤ r απ (i, r )vvir t (s, i − N − r + 1, r − 1)⎦
Ann Oper Res (2016) 239:401–428
415
where the vectors vvir t (s, l, r ) can be found from the following system of linear algebraic equations: vvir t (s, l, r ) = (((s + r α)I − D0 ⊕ A)−1 × δr,0 [(1 − δl,1 )(IW¯ ⊗ L 1 )vvir t (s, l − 1, 0) + δl,1 IW¯ ⊗ L 1 eβ (1) (s I − S)−1 S0 ] + (1 − δr,0 )(IW¯ ⊗ L 2 + r α I )vvir t (s, l, r − 1) + (1 − δr,R )[(1 − qr )(1 − pr )(D1 ⊗ I K N )vvir t (s, l, r + 1) + ((1 − qr ) pr + qr )(D1 ⊗ I K N )vvir t (s, l, r )] + δr,R (D1 ⊗ I K N )vvir t (s, l, r ) , l > 0, r = 0, R.
(9)
Proof The tagged customer can arrive to the system by the following ways: (a) The customer arrives to the system when all servers are busy, there are r, 0 ≤ r < R real customers in the buffer and an arriving customer decides to become a virtual customer. ∞ min{i−N ,R−1} (1−qr ) pr π(i, r )(D1 ⊗ I K N ). The probability of this event is λ−1 i=N r =0 vir t In this case, the probability that no catastrophe arrives during the sojourn time, conditioned on the fact, that the number of customers in the system is i and the number of the real customers in the buffer is r, is equal to vvir t (s, i − N − r + 1, r ). (b) The customer arrives to the system when all servers are busy, there are R real customers are present in the buffer and an arriving customer ∞ decides to become a virtual customer. The probability of this event is λ−1 (1 − δ) i=N +R π(i, R)(D1 ⊗ I K N ). In this case, vir t the probability that no catastrophe arrives during the sojourn time of this customer, conditioned on the fact that the number of customers in the system is i, is equal to vvir t (s, i − N − R + 1, R). (c) The customer leaves the system due to impatience and decides to become a virtual cus∞ min{i−N ,R} tomer. The probability of this event is λ−1 r απ (i, r ). In i=N +1 r =1 vir t (1 − γ ) this case, the probability that no catastrophe arrives during the sojourn time, conditioned on the fact that the number of customers in the system is i and the number of the real customers in the buffer is r, is defined as vvir t (s, i − N − r + 1, r − 1)]. System (9) for calculation of the vectors vvir t (s, l, r ), l ≥ 1, r = 0, R, can be obtained based on a probabilistic sense of the L ST .
To solve system (9), let us introduce the column vectors T T T vvir t (s, l) = (vvir t (s, l, 0), . . . , vvir t (s, l, R)) ,
and introduce • the vector a(s) = e˜ ⊗ (IW¯ ⊗ L 1 e)β (1) (s I − S)−1 S0 where e˜ is a column vector of size R + 1 with all zero entries, except the entry e˜ 0 = 1; ¯ Δ˜ + Δ) ⊗ D1 ⊗ I K N + (I R+1 − E) ¯ ⊗ D1 ⊗ • Ω1 = I R+1 ⊗ (D0 ⊕ A) − αC R ⊗ IW¯ K N + E( ¯ ˆ ¯ I K N + E Δ E ⊗ D1 ⊗ I K N + E R ⊗ (IW¯ ⊗ L 2 ) + αC R E R ⊗ IW¯ K N , • Ω0 = Iˆ ⊗ (IW¯ ⊗ L 1 ) where E¯ is the square matrix of size R + 1 with all zero entries ¯ i,i , i = 0, R − 1, which are equal to 1. except the entries ( E) Using this notation, we can rewrite system (9) into the form (Ω1 − s I )vvir t (s, l) + δl,1 a(s) + (1 − δl,1 )Ω0 vvir t (s, l − 1) = 0T , l ≥ 1.
(10)
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System (10) can be rewritten as vvir t (s, 1) = −(Ω1 − s I )−1 a(s),
vvir t (s, l) = −(Ω1 − s I )−1 Ω0 vvir t (s, l − 1), l > 1. soj
Corollary 5 The average sojourn time Vvir t of an arbitrary virtual customer is calculated by Vvir t = −vvir t (s)|s=0 ⎡ ∞ min{i−N ,R−1} ⎣ = −λ−1 (1 − qr ) pr π(i, r )(D1 ⊗ I K N )vvir t (s, i − N −r +1, r )|s=0 vir t soj
i=N ∞
+ (1 − δ)
i=N +R
− (1 − γ )
r =0
π (i, R)(D1 ⊗ I K N )vvir t (s, i − N − R + 1, R)|s=0
∞ min{i−N ,R} i=N
r =1
⎤ ⎦ r απ (i, r )vvir t (s, i − N − r + 1, r − 1)|s=0 .
Here the column vectors vvir t (s, l, r )|s=0 are calculated as the blocks of the vector vvir t (s, l)|s=0 which can be calculated as follows −1 vvir t (s, 1)|s=0 = Ω1 [e − a (s)|s=0 ],
−1 vvir t (s, n)|s=0 = Ω1 [e − Ω0 vvir t (s, l − 1)|s=0 ], l > 1. wait of an arbitrary virtual customer is calculated Corollary 6 The average waiting time Vvir t soj wait dial by Vvir t = Vvir t − b1 − (1 − h)b1 .
Remark 3 By analogy with the vectors vvir t (s, l), let us introduce the column vectors wvir t (s, l), l ≥ 1, of conditional L ST s of the waiting time distribution of an arbitrary virtual customer. It can be shown that these vectors can be computed from the equations wvir t (s, 1) = −(Ω1 − s I )−1 (IW¯ ⊗ L 1 e), wvir t (s, l) = −(Ω1 − s I )−1 Ω0 wvir t (s, l − 1), l > 1. It is possible to compute the average waiting time of the virtual customer who joins the system during the moment when he (she) gets the position number l, l ≥ 1, in queue of virtual customers. This average waiting time can be used for informing the arriving customer about the expected time till the moment when he (she) will be called for service.
8 Numerical examples In Experiment 1, we show the impact of the coefficient of variation in the service and dial processes on the system operation for different values of the intensity of impatience α. To this end, we consider five cases of service and dial time distributions: 1. The service and dial times have an exponential distribution. Let β (s) = β (d) = (1), S (s) = (−0.72), S (d) = (−2). The coefficients of variation cvar are equal to one.
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2. The service time has an exponential distribution (β (s) = (1), S (s) = (−0.72)). The (d) dial time has a phase = distribution with cvar = 5 and is characterized by β −0.12404 0 (d) . (0.05; 0.95), S = 0 −9.80326 3. The service time has a phase distribution with cvar =0.5, the dial time has an exponential −1.44 1.44 (s) (s) , β (d) = (1), S (d) = (−2). distribution: β = (1; 0), S = 0 −1.44 4. The service time has a phase distribution with cvar = 5, the dial time has an expo−0.04465 0 nential distribution: β (s) = (0.05; 0.95), S (s) = , β (d) = 0 −3.52917 (1), S (d) = (−2). 5. The service and dial times have a phase distribution with the same coefficient of −0.04465 0 (s) (d) (s) variation cvar = 5: β = (0.05; 0.95), S = = ,β 0 −3.52917 −0.12404 0 . (0.05; 0.95), S (d) = 0 −9.80326 The curves on the figures corresponding to five cases described above are marked as (M, M), (M, PH5 ), (PH0.5 , M), (PH5 , M), (PH5 , PH5 ) respectively. Note, that in all cases 7 the mean service time b1 is equal to 1 18 , the mean dial time b1dial is equal to 0.5. The MAP arrival process with the average arrival rate λ = 4 is defined by the matrices D0 =
−5.406562 0 , 0 −0.17548
D1 =
5.370616 0.035946 . 0.097739 0.077741
We assume that the number of servers N = 5, the buffer capacity R = 10, the probabilities are defined as: γ = 0.4, δ = 0.5, h = 0.15, qr = 0.1, pr = 0.05(r + 1), r = 0, 9. Let us vary the intensity of impatience α in the interval [0.1, 10]. v not -dial and λ The dependencies of the intensities of output flow of customers λser out , out , λout imp -loss esc loss ent loss loss the loss probabilities Pr eal , Pr eal , Pr eal , Pr eal , and the average sojourn times soj -ser v soj , Vvir t on the intensity of impatience α for service and dial processes with different Vr eal coefficient of variation are presented in Figs. 2, 3 and 4. It is evident that increase of the intensity of impatience α leads to increase of the number of real customers who leave the buffer due to impatience (they are lost forever or become imp -loss virtual customer), therefore the loss probabilities Pr eal and Prloss eal increase. As a result, the v number of successfully served customers decreases, thus the intensities λser out and λout , the soj -ser v average sojourn time Vr eal also decrease. The growth of the number of virtual customers not -dial of output flow of customers to whom the operator causes the increase of the intensity λout was unable to dial from the system. With increasing the intensity of impatience α the system -loss and P esc-loss , as well as the average gets less overloaded, so the loss probabilities Prent eal r eal soj sojourn time of virtual customers Vvir t decrease. It is interesting that for small values of α in the cases (PH5 , M) and (PH5 , PH5 ) the -loss increases. This growth is caused the sharp decrease of the loss probability probability Presc eal ent loss Pr eal of real customers at the entrance to the system. It can be explained as follows. Under the small value of α the buffer is often full, therefore arriving customers often are lost due to -loss becomes close to zero for buffer overflow. Note that with increase of α the value of Prent eal all cases, so the buffer size is enough to service customers without their loss at the entrance to the system.
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v not -dial and λ Fig. 2 Intensities of output flow of customers λser out as functions of the intensity of out , λout impatience α
Based on the presented figures, one can conclude that the system performance measures essentially depend on the coefficient of variation in the service process. Note, that the ergodicity condition for the cases (M, M), (M, PH5 ) and (PH0.5 , M) holds true only if α ≥ 0.3, while for other cases if α ≥ 0.1. It is also clear that key performance measures of the sys-
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-loss , P imp-loss and P ent -loss as functions of the intensity of impatience α Fig. 3 Loss probabilities Presc eal r eal r eal
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soj -ser v soj Fig. 4 Loss probability Prloss , Vvir t as functions of the intensity eal , and the average sojourn times Vr eal of impatience α
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tem are sensitive with respect to the coefficient of variation in the dial process, but not as significantly as to the service time variance. Furthermore, the dependence of performance measures on the variation in the service and dial processes and their behavior can not be predicted correctly without computer realization of results obtained in the presented paper. In Experiment 2, we intend to show that the correlation in the input process has a great impact on performance measures of the system. Below we introduce three MAPs defined by the matrices D0 and D1 . All these MAPs have the same average arrival rate λ = 6, but different coefficients of correlation and variation. The first process is defined by the matrices D0 = −6 and D1 = 6. It has the coefficient of correlation ccor = 0 and the coefficient of variation cvar = 1. This is the stationary Poisson process. The second process has ccor = 0.2 and cvar = 12.4, and is defined by the matrices −8.110725 0 8.0568 0.053925 D0 = , D1 = . 0 −0.26325 0.146625 0.116625 The third process has ccor = 0.4 and cvar = 12.4, and is defined by the matrices −20.38935 0 20.17698 0.21237 D0 = , D1 = . 0.00609 −0.66144 0.07281 0.58254 The service and dial times have an exponential distribution and are the same as ones presented in the first experiment (case 1). We assume that R = 20, α = 0.2, γ = 0.4, δ = 0.5, h = 0.15, qr = 0.1, pr = 0.02(r + 1), r = 0, R − 1. Figures 5, 6 and 7 illustrate the dependence of the average number of customers in the system L, the average number of real and virtual customers in the buffer, the probabilities imp -loss -loss , P to-vir t , P loss , the intensity of output flow of customers λ , and the Pr eal , Prent out eal r eal average waiting time Vrwait eal of an arbitrary real customer in the system on the number of servers N for different MAPs presented above. The ergodicity condition holds true for all arrival processes if the number of servers N is greater or equal to 8. It is evident that the coefficient of correlation in the arrival process strongly affects the system performance measures. The increase of the coefficient of correlation in the arrival process leads to the degradation of quality of service in the system: the average number of customers L bu f f er in the system, the average number of real customers Nr eal and the average number of virtual bu f f er imp -loss -loss , P to-vir t , P loss in the buffer, the loss probabilities Pr eal , Prent customers Nvir t eal r eal and the average waiting time Vrwait eal increase, while the intensity of output flow λout decreases. The difference in the values is essential for small values of the number of servers N . So, traditional in the literature assumption that the arrival flow is described by the stationary Poisson process can lead to the pure prediction of the system performance measures. It is worth noting that the numerical experiments also show that the famous Little’s formula holds true for real and virtual customers, i.e., Vrwait eal =
bu f f er
Nr eal λ
=
imp -loss Pr eal α
and wait Vvir t =
bu f f er
Nvir t . λvir t
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bu f f er
Fig. 5 Average number of customers in the system L, the average number of real customers Nr eal
and the
bu f f er average number of virtual customers Nvir t in the buffer as functions of the number of servers for different
MAPs
Experiment 3 Due to the fact that most of costs of call center involves wages of operators, the problem of the optimal choice of the number of call center operators is very important for effective call center performance. In this experiment, we solve numerically the problem of
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imp -loss
-loss and P to-vir t as functions of the number of servers for different , Prent eal r eal
Fig. 6 Loss probabilities Pr eal MAPs
the choice of the minimal number of operators N ∗ for which the requirements of the quality of service are fulfilled. As the requirements of the quality of service the following constraints are considered:
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Fig. 7 Probability P loss , the intensity of output flow of customers λout , the average waiting time Vrwait eal as functions of the number of servers for different MAPs wait Vvir t wait -ser v Vr eal loss
P
123
< V1 , < V2 , < .
Ann Oper Res (2016) 239:401–428 Table 1 Value of optimal number of operators N ∗ and the wait , values of P loss , Vvir t wait ser v Vr eal under the fixed N ∗ for different arrival processes
425 N∗
P loss
wait Vvir t
-ser v Vrwait eal
ccor = 0
11
0.0427
1.3
0.057
ccor = 0.2
13
0.0425
1.7
0.052
ccor = 0.4
27
0.0315
2.3
0.076
It is worth noting, that the problem of the suitable choice of the parameters V1 , V2 and plays crucial role in successful implementation of the optimal choice of the number of operators. We assume here that in our model the values of parameters are obtained from experts in the real call center to which the model will be applied. So, we fixed the following values of the parameters: V1 = 3, V2 = 0.1, = 0.05. The rest of parameters are the same as ones presented in the first experiment. wait , V wait -ser v The value of optimal number of operators N ∗ and the values of P loss , Vvir t r eal ∗ under the fixed N for different arrival processes are presented in Table 1. Based on Table 1, one can conclude that the optimal number of servers N ∗ is very sensitive to the coefficient of correlation in the arrival process. For example, if we assume that the arrival flow of customers to the call center is described by a stationary Poisson arrival process, while actually the Markovian arrival flow with coefficient of correlation ccor = 0.4 arrives to the call center, we will hire N ∗ = 11 operators and expect that the requirements of the quality of service are fulfilled, but actually, in this case, the loss probability is P loss = 0.206, the average wait = 32, 90, and the average waiting time waiting time of an arbitrary virtual customer is Vvir t wait ser v of an arbitrary serviced real customer Vr eal = 0.981, i.e., the call center will operate much worse than it is expected. So, taking into account the coefficient of correlation in the arrival process is very important for optimal choice of the number of call center operators.
9 Conclusion The queueing system with a generalized phase-type service time distribution as a model of call center with the call-back option is investigated. Formulas for main performance measures and the Laplace–Stieltjes transforms of the sojourn and waiting time distributions of real and virtual customers are obtained. Numerical examples are presented. The obtained results can be used for performance evaluation and optimization of call centers of banks, emergency and information services, mobile operators. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2011-0015214).
Appendix 1 Let us present some examples of using the generalized phase-type distribution in an investigation of queues with heterogeneous customers and different service time distributions for each type of customers. (i) If we consider the system with K types of customers and the service time of type k customer, k = 1, K , has an exponential distribution with the parameter μk , it
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is reasonable to consider a generalized PH distribution with an irreducible representation (β (1) , . . . , β (K ) , S) where β (k) , k = 1, K , is the vector of size K with all zero entries except the kth entry, which is equal to 1, and the matrix S of the form S = −diag{μ1 , . . . , μ K }. (ii) If we consider the system with K types of customers and the service time of type k customer, k = 1, K , has a phase-type distribution with an irreducible representation (k) (βˆ , S (k) ) and the finite state space {1, . . . , Mk , Mk + 1}, it is reasonable to consider a generalized PH distribution with an irreducible representation (β (1) , . . . , β (K ) , S) where (k) β (k) = (0k−1 , βˆ , 0 K ), k = 1, K , and the matrix S of the form S = l=1
Ml
l=k+1
diag{S1 , . . . , S K }.
Ml
Appendix 2 In Appendix 2, we prove the assertion that the use of the generalized phase-type with the state space (1, . . . , M1 + · · · + Ml + 1) instead of the consideration of l phase type service processes with dimensions Mn +1, n = 1, l of the state space does not change the dimension of the stochastic process that describes the behavior of the system under consideration. Let us assume that l different types of customers are serviced in any of N servers. The service time of type n customer has a phase-type distribution with the dimension of the space of the non-absorbing states Mn , n = 1, l. We use approach by V. Ramaswami and D.M. Lucantoni, so, the dimension of components of the Markov chain that describes the service process of all types of customers K 1 is equal to K1 =
−kl N N
N −(kl +kl−1 +···+k3 )
···
kl =0 kl−1 =0
×
k 2 + M2 − 1 M2 − 1
k2 =0
kl + Ml − 1 Ml − 1
× ···
N − (kl + kl−1 + · · · + k2 ) + M1 − 1 , M1 − 1
where the summation index kn has meaning of the number of type n customers presenting in the system. If we describe processing of customers by a generalized phase-type service process, the N + M1 + · · · + Ml − 1 dimension of its state space is K 2 = . M1 + · · · + Ml − 1 So, we assert that K 1 = K 2 for any N and l. Let us first consider the case l = 2, i.e., there are two types of customers and the service time of type n customer has a phase-type distribution with dimension of the space of the non-absorbingstates Mn , n = 1, 2. In this case the dimension K 1 is given by formulae N N − k 2 + M1 − 1 k 2 + M2 − 1 . Using the following formulae K 1 = k2 =0 M2 − 1 M1 − 1 N k+a b−k k=0
a
b−n
=
a+b+1 n
(11)
and setting a = M 2 − 1, b = N + M1 − 1 it is easy to verify that K 1 = N + M1 + M2 − 1 N + M1 + M2 − 1 = = K 2 . So, for l = 2 the dimensions N M1 + M2 − 1 K 1 and K 2 are equal.
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427
Let us assume that K 1 = K 2 for the case l, l ≥ 2, types of customers, i.e., −kl N N
N −(kl +kl−1 +···+k3 )
···
kl =0 kl−1 =0
k2 =0
kl + Ml − 1 Ml − 1
× ··· ×
N − (kl + kl−1 + · · · + k2 ) + M1 − 1 k + M2 − 1 × 2 M2 − 1 M1 − 1 N + M1 + · · · + Ml − 1 = , M1 + · · · + Ml − 1
(12)
and prove that K 1 = K 2 for the case l + 1 types of customers as well. K1 =
N
N −(kl+1 +kl +···+k3 )
N −kl+1
kl+1 =0 kl =0
···
k2 =0
kl+1 + Ml+1 − 1 Ml+1 − 1
× ··· ×
N − (kl+1 + kl + · · · + k2 ) + M1 − 1 k 2 + M2 − 1 = [(A2)] M2 − 1 M1 − 1 N N − kl+1 + M1 + · · · + Ml − 1 kl+1 + Ml+1 − 1 = [(A1)] = Ml+1 − 1 M1 + · · · + Ml − 1 kl+1 =0 N + M1 + · · · + Ml+1 − 1 = = K2. M1 + · · · + Ml+1 − 1
×
So, by induction the assertion is proved.
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