Performance Evaluation

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1 Performance Evaluaton 70 (2013) Contents lsts avalable at ScVerse ScenceDrect Performance Evaluaton journal homepage: wwwelsevercom/locate/peva Tandem queueng system wth mpatent customers as a model of call center wth Interactve Voce Response Chesoong Km a,, Alexander Dudn b, Sergey Dudn b, Olga Dudna b a Sangj Unversty, Wonju, Kangwon, , Republc of Korea b Belarusan State Unversty, 4, Nezavsmost Ave, Mnsk, , Belarus a r t c l e n f o a b s t r a c t Artcle hstory: Receved 14 March 2012 Receved n revsed form 24 January 2013 Accepted 13 February 2013 Avalable onlne 24 February 2013 Keywords: Call center Interactve Voce Response Markovan arrval process Phase type servce tme dstrbuton Impatent customers A tandem queueng system wth a Markovan Arrval Process (MAP) useful n modelng a call center wth Interactve Voce Response (IVR) s nvestgated The frst stage has a fnte number of servers wthout buffer whle the second stage of the tandem has a fnte buffer and a fnte number of servers The servce tme at the frst (second) stage has an exponental (phase type) dstrbuton A specal approach for reducng the number of states of the stochastc process that descrbes the behavor of the system s used The man performance measures are calculated The Laplace Steltjes transform of the sojourn tme dstrbuton s derved The numercal results are presented 2013 Elsever BV All rghts reserved 1 Introducton Call center s a centralzed offce used by companes for contact wth ts clents To decrease load of operators n some call centers IVR (Interactve Voce Response) technology s used IVR s a call center opton that allows customers to nteract wth a company and serve ther own nqures va a telephone keypad wthout the nvolvement of an agent Typcally IVR s used n call centers of banks, nsurance companes, travel agences, moble operators, etc For example, f the subscrber of moble network wants to know the balance of hs(her) account, enable or dsable the servce, to fnd out some nformaton about tarffs, he(she) can easly do t hmself(herself) usng IVR If the customer cannot solve hs(her) problem by usng IVR and requres some assstance, he(she) can request to connect wth an agent In call centers wth IVR the load of operator s lower, so the performance of call centers s hgher For modelng call centers, queueng theory s used Adequate mathematcal modelng call centers leads to a substantal ncrease of ther economc effcency, because exact predcton at the desgn stage can sgnfcantly reduce mantenance costs Mathematcal models allow to solve a problem of call center optmal desgn For the references and the present state of art n nvestgaton of call centers the reader s referred to survey [1], papers [2 5] and references theren In [6], a call center wth IVR s nvestgated In ths paper authors consder a two-stage queueng model wth Posson arrval process and exponental servce tme dstrbuton at both stages wth mpatent customers and wthout balk of customers Performance measures of ths system are calculated approxmately n an asymptotc Qualty and Effcency Drven (QED) regme In our paper, we present an exact analytcal analyss of the model wthout the restrcton that the number of operators s large enough We consder a tandem mult-server queueng system wth a fnte ntermedate buffer and mpatent Correspondng author Tel: E-mal addresses: dowoo@sangjackr (C Km), dudn@bsuby (A Dudn), dudn@madrdcom (S Dudn), dudna_olga@emalcom (O Dudna) /$ see front matter 2013 Elsever BV All rghts reserved

2 C Km et al / Performance Evaluaton 70 (2013) Fg 1 Structure of the system customers whch can be used for modelng and optmzng call centers wth IVR The frst stage conssts of a fnte number of servers (IVR) After the servce at the frst stage customers leave the system or move to the second stage If after completng the servce at the frst stage the customer moves to the second stage and all servers (agents) are busy, the customer s reported about the current queue length (so called vsble queue) and ts estmated watng tme, and based on the provded nformaton decdes whether to balk (leave the system permanently wthout the servce) or wat for the servce Statstcs show that customers, who receve nformaton about ther place n a buffer or watng tme, are 15 2 tmes more patent, than customers who do not have such an nformaton As a result, the number of unserved customers s greatly reduced, therefore, consderaton of the vsble queue s an mportant pont n modelng modern call centers In many papers, see, eg, [4,6], authors menton that the dstrbuton of the servce tme of a customer by an agent s not exponental and the arrval flow of customers s not Posson Our statstcal analyss of real arrval flows and servce processes of the call center of one of the largest banks n Belarus has shown the same behavor In our paper we consder a Markovan arrval process whch s more general than Posson arrval process and allows to take nto account bursty nature of flows n modern call centers Also, we consder a phase type (PH) servce tme dstrbuton at the second stage of the system nstead of an exponentally dstrbuted servce tme consdered n [6] Ths allows to take nto account varaton of the servce tme carefully It s known that the subset of phase-type dstrbutons s dense n the set of all postve-valued dstrbutons, that s, t can be used to approxmate any nonnegatve valued dstrbuton, so, consderaton of PH servce process s very mportant for adequate modelng call centers It s well known that f we nvestgate a mult-server queueng system wth PH servce process by standard methods, stochastc process that descrbes the behavor of the system can have a huge state space even for a small number of servers Ths s due to the fact that at a gven moment we must know the state of PH underlyng process for each busy server So, the standard technque s not approprate for modelng call centers In ths paper we use the specal method proposed by Ramaswam and Lucanton, see [7,8], for reducng the dmenson of systems wth a phase type servce tme dstrbuton The paper s organzed as follows In Secton 2, the mathematcal model s descrbed The statonary dstrbuton of system states s analyzed n Secton 3 The expressons for the man performance measures of the system are gven n Secton 4 In Secton 5, we focus on the analyss of the sojourn tme dstrbuton Secton 6 contans some numercal llustratons Secton 7 concludes the paper 2 Mathematcal model The structure of the system under consderaton s presented on Fg 1 The queueng system conssts of two sequental stages The frst stage of tandem s R server queueng system wthout a buffer, the second stage s N server queueng system wth a buffer of capacty K Customers arrve at the system accordng to the MAP The MAP s defned by the underlyng process ν t, t 0, whch s an rreducble contnuous tme Markov chan wth fnte state space {0, 1,, W} Arrvals occur only at epochs of jumps n the underlyng process ν t, t 0 The ntenstes of transtons of the process ν t, t 0, whch are accompaned (are not accompaned) by an arrval of a customer, are defned by the square matrx D 1 (D 0 ) of sze W + 1 The matrx generatng functon of these matrces s D(z) = D 0 + D 1 z, z 1 The matrx D(1) s an nfntesmal generator of the process ν t, t 0 The statonary dstrbuton vector θ of ths process satsfes the system of equatons θd(1) = 0, θe = 1 Here and n the sequel 0 s a zero row vector and e denotes unt column vector If the dmenson of a vector s not clear from the context, t s ndcated as a subscrpt, eg, e W denotes the unt column vector of dmenson W = W + 1 The average ntensty λ (fundamental rate) of the MAP s defned by λ = θd 1 e The coeffcent c var of varaton of ntervals between customer arrvals s calculated by c var = 2λθ( D 0 ) 1 e 1, whle the coeffcent c cor of correlaton of ntervals between successve arrvals s gven as c cor = (λθ( D 0 ) 1 D 1 ( D 0 ) 1 e 1)/c 2 var Dfferent methods for the estmaton of MAP parameters usng a fnte set of observed data such as a set of customer arrval tmes recorded at the real call center are presented, eg, n [9] More nformaton about the MAP and related research s gven, eg, n [10 13] The servce tme at the frst stage server s assumed to be exponentally dstrbuted wth the rate µ, 0 < µ < After completng the servce at the frst stage, the customer leaves the system forever wth probablty q, 0 q 1 Wth supplementary probablty 1 q the customer moves to the second stage of the tandem Note that n contrast to paper [6],

3 442 C Km et al / Performance Evaluaton 70 (2013) where t s assumed that the trunk wll not be released when a customer leaves the IVR queue and swtches to the servce of agents, we assume that the frst stage server s released after a customer leaves the frst stage If there s a free server at the second stage at a customer arrval epoch from the frst stage, the customer occupes ths server If all servers at the second stage are busy at a customer arrval epoch and there are l, l {0, 1,, K 1}, customers n the buffer, then ths customer leaves the system wth probablty p l or moves to the buffer wth supplementary probablty It means that the arrvng customer, who does not succeed to enter the servce mmedately, s reported about the queue length Some customers are patent enough to wat for the servce, whle others abandon the system mmedately f they consder ths length as napproprate If the buffer s full at a customer arrval epoch at the second stage, he(she) leaves the system forever Customers can be mpatent, e, the customer leaves the ntermedate buffer after an exponentally dstrbuted wth the parameter α, 0 < α <, tme after arrval, condtoned on the fact that ths customer s not servcng The servce tme of a customer by a second stage server has PH dstrbuton wth an rreducble representaton (β, S) Ths servce tme can be nterpreted as a tme untl the underlyng Markov process η t, t 0, wth fnte state space {1,, M, M + 1} reaches the sngle absorbng state M + 1 condtoned on the fact that the ntal state of ths process s selected among the states {1,, M} accordng to the probablstc row vector β = (β 1,, β M ) Transton rates of the process η t wthn the set {1,, M} are defned by the sub-generator S and transton rates nto the absorbng state (what leads to servce completon) are gven by entres of the column vector S 0 = Se Note that representaton (β, S) s rreducble f the matrx S + S 0 β s rreducble The servce tme dstrbuton functon has the form A(x) = 1 βe Sx e, Laplace Steltjes transform (LST ) e sx da(x) 0 of ths dstrbuton s β(si S) 1 S 0, Re s > 0 The mean servce tme s calculated by b 1 = β( S) 1 e The coeffcent of varaton s gven by c var = b 2 /b where b 2 = 2β( S) 2 e For more nformaton about PH dstrbuton and ts usefulness see, eg, [14] Methods of modelng the PH process usng a set of servce tmes obtaned at the real system, and n partcular call center, can be found n paper [15] Note that the man goal of the analyss presented n ths paper s to elaborate the algorthms for computng the key performance measures of the system under consderaton Once these measures are calculated, we can solve varous optmzaton problems of real call center management 3 The process of system states Let t be the number of customers at the frst stage, t {0, 1,, R}, n t be the number of customers at the second stage, n t {0, 1,, K + N}, ν t be the state of the underlyng process of the MAP, ν t {0, 1,, W}, η (m) t be the number of servers at the phase m of servce, m {1, 2,, M}, η (m) t {0, 1,, mn{n t, N}}, M m=1 η(m) t = mn{n t, N}, at the epoch t, t 0 Note that the meanng of components η (m) t s chosen accordng to the approach by Ramaswam and Lucanton, see [7,8] The behavor of the system under study can be descrbed n terms of the regular rreducble contnuous-tme Markov chan ξ t = { t, n t, ν t, η (1) t,, η (M) t }, t 0, wth state space {, n, ν, η (1),, η (M) }, {0, 1,, R}, n {0, 1,, K + N}, ν {0, 1,, W}, η (m) {0, 1,, mn{n, N}}, m {1, 2,, M}, M η (m) = mn{n, N} m=1 The number of states of the process η t = {η (1) t,, η (M) n+m 1 t }, t 0, when n servers are busy s equal to Kn = In M 1 the overwhelmng majorty of exstng papers, the behavor of queues wth PH servce tme dstrbuton when n servers are busy s descrbed by stochastc process ncludng the components ζ (1) t,, ζ (n) t, t 0, where ζ (m) t, m {1, 2,, n}, s the state of the PH underlyng process on the mth busy server Note that the number Kn s sgnfcantly less than the number of states of the process {ζ (1) t,, ζ (n) t }, t 0, whch s equal to Kn = M n For example, f we fx n = 25 and M = 2, the number Kn = 2 25 whle the number Kn s only 26 Snce the Markov chan ξ t s regular rreducble and has fnte state space, then for any choce of system parameters there exst the statonary probabltes of system states whch are defned as follows: π(, n, ν, η (1),, η (M) ) = lm t P{ t =, n t = n, ν t = ν, η (1) t = η (1),, η (M) t = η (M) }, {0, 1,, R}, n {0, 1,, K + N}, ν {0, 1,, W},

4 η (m) {0, 1,, mn{n, N}}, m {1, 2,, M}, C Km et al / Performance Evaluaton 70 (2013) M η (m) = mn{n, N} Let us form the row vectors π(, n, ν) of probabltes π(, n, ν, η (1),, η (M) ), {0, 1,, R}, n {0, 1,, K + N}, ν {0, 1,, W}, enumerated n the reverse lexcographc order of components η (1),, η (M) Then let us form the row vectors π(, n) = (π(, n, 0), π(, n, 1),, π(, n, W)), n {0, 1,, K + N}, π = (π(, 0), π(, 1),, π(, K + N)), {0, 1,, R} It s well known that the probablty vectors π, {0, 1,, R}, satsfy the followng system of lnear algebrac equatons: m=1 (π 0, π 1,, π R )Q = 0, (π 0, π 1,, π R )e = 1 (1) where Q s the nfntesmal generator of the Markov chan ξ t, t 0 Lemma 1 The nfntesmal generator Q of the Markov chan ξ t, t 0, has the block trdagonal structure: Q 0,0 Q + O O O Q 1,0 Q 1,1 Q + O O O Q 2,1 Q 2,2 O O Q = O O O Q R 1,R 1 Q + O O O Q R,R 1 Q R,R The non-zero blocks Q,j,, j 0, have the followng form: C (0) O O O O C (1) C (1) O O O (2) O C C (2) O O Q, =, 0 R, O O O C (K+N 1) O (K+N) O O O C C (K+N) B (0) B (0) O O O O B (1) B (1) O O O O B (2) O O Q, 1 =, 1 R, O O O B (K+N 1) B (K+N 1) O O O O B (K+N) Q + = D 1 I N n=0 K n, +K KN where I s the dentty matrx, O s a zero matrx of approprate dmenson; and are symbols of Kronecker sum and product respectvely, see, eg, [16]; C (n) = (D 0 µi W ) (A n (N, S) + (n) ), {0,, R 1}, n {0,, N}; C (n) = (D 0 (µ + (n N)α)I W ) (A N (N, S) + (N) ), {0,, R 1}, n {N + 1,, K + N}; C (n) R = (D(1) RµI W ) (A n (N, S) + (n) ), n {0,, N}; C (n) R = (D(1) (Rµ + (n N)α)I W ) (A N (N, S) + (N) ), n {N + 1,, K + N}; C (n) = I W L N n (N, S), n {1,, N}; C (n) = (n N)αI W L 0 (N, S)PN 1 (β), n {N + 1,, K + N}; B (n) = qµi W Kn, {0,, R}, n {0,, N 1}; B (n) = (q + (1 q)p n N )µi W KN, {0,, R}, n {N,, K + N 1}; B (K+N) = µi W KN, {0,, R}; B(n) = (1 q)µi W P n (β), {0,, R}, n {0,, N 1}; B(n) = (1 q)(1 p n N )µi W KN, {0,, R}, n {N, K + N 1};

5 444 C Km et al / Performance Evaluaton 70 (2013) S = 0 0 ; S 0 S (n) = dag{a n (N, S)e + L N n (N, S)e}, n {1, 2,, N}, (0) = O 1 1 The matrx P (β) defnes the transton probabltes of the process η t, t 0, at the epoch of startng the new servce gven that servers are busy at ths epoch The matrx L N (N, S) defnes the ntenstes of transtons of ths process at the servce completon epoch gven that servers are busy at ths epoch The matrx A (N, S) defnes the ntenstes of transtons of the process η t, t 0, whch do not lead to the servce completon gven that servers are busy Modules of dagonal entres of the matrx () defne the total ntensty of leavng the correspondng states of the process η t, t 0, gven that servers are busy The detaled descrpton of matrces P n (β), L n (N, S), n {0,, N 1}, and An (N, S), n {0,, N}, and the algorthms for ther calculaton can be found n [17] In case the servce tme at the second stage has an exponental dstrbuton wth the parameter µ herenafter we have to put P n (β) = 1, L n (N, S) = (N n) µ, n {0,, N 1}, Kn = 1, A n (N, S) = 0, (n) = n µ, n {0,, N} Proof The proof of the lemma s mplemented by means of the analyss of all transtons of the Markov chan ξ t, t 0, durng the nterval of an nfntesmal length and rewrtng ntenstes of these transtons n the block matrx form If the dmenson of system (1) s small, t can be easly solved on a computer by standard methods Otherwse, to solve ths system the algorthm that was elaborated n [18] can be appled 4 Performance measures As soon as the vectors π, {0, 1,, R}, have been calculated, we are able to fnd varous performance measures of the call center: The probablty that an arbtrary customer wll be lost at the frst stage P (1) = 1 λ π RQ + e The average number of busy servers at the frst stage R N (1) = π e =1 The average ntensty of flow of customers who get servce at the frst stage λ (1) out = N (1) µ The average number of busy servers at the second stage R K+N N (2) = mn{n, N}π(, n)e =0 n=1 The average number of customers n the buffer R K+N N buffer = (n N)π(, n)e =0 n=n+1 The average number of customers n the system R K+N L system = ( + n)π(, n)e =0 n=0 The average ntensty of flow of customers who get servce at the second stage R K+N λ out = π(, n)(i W L max{0,n n} (N, S))e =0 n=1 The probablty that an arbtrary customer wll be lost at the second stage λ out P (2) = 1 (1 q)λ (1) out The probablty that an arbtrary customer wll be lost at the entrance to the second stage P (2-ent) = (λ (1) out) 1 R µπ(, K + N)e =1

6 C Km et al / Performance Evaluaton 70 (2013) The probablty that an arbtrary customer arrves to the second stage when all servers at ths stage are busy, the buffer s not full, and the customer does not jon the buffer and leaves the system P (2-esc) = (λ (1) out) 1 R =1 K+N 1 n=n p n N µπ(, n)e The probablty that an arbtrary customer after arrval to the second stage wll go to the buffer and leave t due to mpatence P (2-mp) = P (2) P(2-ent) P (2-esc) 5 Dstrbuton of the sojourn and watng tmes of an arbtrary customer Let V(x) be the dstrbuton functon of the sojourn tme of an arbtrary customer n the system under study and v(s) = e sx dv(x), Re s > 0, be ts LST 0 We wll derve an expresson for the LST v(s) by means of the method of collectve marks So, v(s) has the meanng of probablty that catastrophe does not arrve durng the sojourn tme of an arbtrary customer We wll tag an arbtrary customer and wll keep track of hs(her) stayng n the system Let v(s,, n, ν, η (1),, η (M) ) be the probablty that a catastrophe wll not arrve durng the rest of the tagged customer sojourn tme n the system condtoned on the fact that, at the gven moment, the number of customers at the frst stage ncludng the tagged customer s equal to, {1, 2,, R}, the number of customers at the second stage s n, n {0, 1,, K + N}, and the states of processes ν t, η (1) t,, η (M) t, t 0, are ν, η (1),, η (M), ν {0, 1,, W}, η (m) {0, 1,, mn{n, N}}, m {1, 2,, M}, M m=1 η(m) = mn{n, N} Let us enumerate the LST s v(s,, n, ν, η (1),, η (M) ) n the reverse lexcographc order of components η (1),, η (M) and drect lexcographc order of the component ν and form from these LST s the column vectors v(s,, n) If we have the functons v(s,, n) calculated, by condtonng on the states of the system at the moment of customer arrval, the LSTv(s) wll be computed as v(s) = P (1) + R 1 K+N λ 1 π(, n)(d 1 I Kmn{n,N} )v(s, + 1, n) =0 n=0 The system of lnear algebrac equatons for vectors v(s,, n) s derved usng the law of total probablty: Here v(s,, n) = (s + max{0, n N}α + µ)i D 0 A mn{n,n} (N, S) I (mn{n,n}) 1 ( 1)µ ( δ 0 n<n (1 q)i W P n (β) + δ N n<k+n (1 q)(1 p n N )I W KN )v(s, 1, n + 1) + (δ n,k+n + δ N n<k+n (1 q)p n N + (1 δ n,n+k )q)v(s, 1, n) + (1 δ,r )D 1 I Kmn{n,N} v(s, + 1, n) + δ,r D 1 I Kmn{n,N} v(s, R, n) + ( δ N<n K+N (I W L 0 (N, S)PN 1 (β) + (n N)αI) + δ 0<n N I W L N n (N, S))v(s,, n 1) + µ (1 δ n,k+n )qe W Kmn{N,n} + (1 q) δ 0 n<n β(si S) 1 S 0 e W Kn + (δ n,k+n + δ N n<k+n (1 q)p n N )e W KN + δ N n<k+n (1 q)(1 p n N )e W y(s, n N + 1), {1, 2,, R}, n {0, 1,, N + K} (2) δ n Z = 1, f n Z, 0, otherwse, the vectors y(s, l), l {1,, K}, consst of probabltes y(s, l, η (1),, η (M) ) enumerated n the reverse lexcographc order of components η (1),, η (M) The probabltes y(s, l, η (1),, η (M) ) are the LST s of dstrbuton of the tagged customer s sojourn tme condtoned on the fact that, at the gven moment, the tagged customer s stayng n the buffer and ts poston s equal to l, l {1, 2,, K}, and the states of processes η (1) t,, η (M) t, t 0, are η (1),, η (M) respectvely Theorem 2 The vectors y(s, l), l {1,, K}, are calculated as follows: y(s, 1) = ((s + α)i A) 1 [Le KN β(si S) 1 S 0 + αe KN ], (3) y(s, l) = ((s + lα)i A) 1 [(L + (l 1)αI)y(s, l 1) + αe KN ], l {2, 3,, K}, (4)

7 446 C Km et al / Performance Evaluaton 70 (2013) where A = A N (N, S) + (N), L = L 0 (N, S)PN 1 (β) Proof Based on a probablstc sense of the LST, we obtan the system for calculaton of vectors y(s, l): ( (s + lα)i + A)y(s, l) + (1 δ l,1 )(L + (l 1)αI)y(s, l 1) + αe KN + δ l,1 Le KN β(si S) 1 S 0 = 0 T, l {1, 2,, K} (5) It s easy to verfy that system (5) can be wrtten as (3) (4) Havng the vectors y(s, l) been computed, we can calculate the vectors v(s,, n) To ths end let us ntroduce the column vectors Here v(s, ) = (v(s,, 0),, v(s,, K + N)) T, v(s) = (v(s, 1),, v(s, R)) T System (2) of lnear algebrac equatons can be rewrtten nto the matrx form as (si Q, )v(s, ) + (1 δ,r )Q + v(s, + 1) + (1 δ,1 )Q 1, 2 v(s, 1) + a(s) = 0 T, {1, 2,, R} (6) a(s) = (a T (s), 0 at (s),, 1 at K+N (s))t, a n (s) = µ (1 δ n,k+n )qe W Kmn{N,n} + (1 q) δ 0 n<n β(si S) 1 S 0 e W Kn + (δ n,k+n + δ N n<k+n (1 q)p n N )e W KN + δ N n<k+n (1 q)(1 p n N )e W y(s, n N + 1), n {0,, K + N} Let us ntroduce the followng notaton Q 1,1 Q + O O O Q 1,0 Q 2,2 Q + O O O Q 2,1 Q 3,3 O O Q = O O O Q R 1,R 1 Q + O O O Q R 1,R 2 Q R,R Usng ths notaton we can rewrte system (6) nto the form ( Q si)v(s) + er a(s) = 0 T (7) It can be verfed that dagonal entres of the matrx Q si domnate n all rows of ths matrx So, the nverse matrx exsts Thus we have proved the followng asserton Theorem 3 The vector v(s) consstng of condtonal LST s v(s,, n, ν, η (1),, η (M) ), {1, 2,, R}, n {0, 1,, K + N}, ν {0, 1,, W}, η (m) {0, 1,, mn{n, N}}, m {1, 2,, M}, s calculated by v(s) = ( Q si) 1 (e R a(s)) (8) Formula (8) gves the explct form of the vector v(s), but n practce the matrx Q si usually has a bg dmenson Usng the fact that ths matrx has block trdagonal form the vector v(s) can be easly found from system (7) usng the trdagonal matrx algorthm, see [19], that s modfed for block trdagonal matrces Ths algorthm s presented n the followng theorem Theorem 4 The components v(s, ), {1,, R}, of the vector v(s) are calculated as v(s, R) = γ R+1 (s), v(s, ) = Z +1 (s)v(s, + 1) + γ +1 (s), = R 1, R 2,, 1, where the matrx functons Z (s), {2,, R}, are calculated usng the recurson Z (s) = (Q 1, 1 si + Q 2, 3 Z 1 (s)) 1 Q +, {3,, R},

8 C Km et al / Performance Evaluaton 70 (2013) under the ntal condton Z 2 (s) = (Q 1,1 si) 1 Q +, and the vectors γ (s), {2,, R + 1}, are gven as follows γ 2 (s) = (Q 1,1 si) 1 a(s), γ (s) = (Q 1, 1 si + Q 2, 3 Z 1 (s)) 1 (a(s) Q 2, 3 γ 1 (s)), {3,, R + 1} Havng the LST s v(s,, n, ν, η (1),, η (M) ) been computed we can fnd LST v(s) Corollary 1 The average sojourn tme V soj of an arbtrary customer n the system s calculated by R 1 K+N v(s, V soj = λ 1 + 1, n) π(, n)(d 1 I Kmn{n,N} ) s =0 n=0 Here the column vectors v(s,+1,n) s s=0 s=0 are calculated as blocks of the vector dv(s) 1 ds s=0 = Q [e R a (0) v(0)] Remark 1 Based on the presented above results t s easy to obtan, eg, the formulas for the LST of the dstrbuton of the sojourn tme of an arbtrary customer, who gets servce at both stages, the LST of the dstrbuton of watng tme of an arbtrary customer n the ntermedate buffer, and other mportant performance characterstcs of the system Remark 2 The moment V (m) = ( 1) m dm v(s) ds m R 1 K+N V (m) = ( 1) m λ 1 =0 n=0 Here the column vectors m v(s,+1,n) s m 1, where d0 v(s) = v(s) ds 0 s=0, m 1, of order m of the sojourn tme dstrbuton can be found as follows: π(, n)(d 1 I Kmn{n,N} ) m v(s, + 1, n) s m s=0 are calculated as blocks of the vector dm v(s) ds m = ( Q si) 1 (m dm 1 v(s) ds m 1 e R dm a(s) ), ds m m Remark 3 By analogy wth the column vectors y(s, l) of condtonal LST s y(s, l, η (1),, η (M) ) of the sojourn tme dstrbuton, t s possble to ntroduce the vectors w(s, l) of condtonal LST s of the watng tme dstrbuton of the tagged customer condtoned on the fact that, at the gven moment, the tagged customer s stayng n the buffer and ts poston s equal to l, l {1, 2,, K}, and the states of processes η (1) t,, η (M) t, t 0, are η (1),, η (M) respectvely and the tagged customer wll not leave the buffer due to mpatence It can be shown that these vectors can be computed from equatons w(s, 1) = ((s + α)i A) 1 Le KN, w(s, l) = ((s + lα)i A) 1 (L + (l 1)αI)w(s, l 1), l {2, 3,, K} By numercally nvertng the components of the vector w(s, l), see, eg, [20], t s possble to compute the condtonal watng tme dstrbuton of the customer who jons the second stage when all agents are busy and at the moment when he(she) gets poston number l, l {1,, K}, n the buffer, the states of processes η (1) t,, η (M) t, t 0, are η (1),, η (M) respectvely and ths customer wll not leave the buffer due to mpatence Ths dstrbuton can be used for nformng the customer about the expected tme tll the begnnng of the servce 6 Numercal examples To demonstrate feasblty of developed algorthms and show numercally some nterestng features of the system under consderaton we present the results of three experments In Experment 1, we nvestgate the mpact of correlaton and varaton n nput flow on system performance measures For ths purpose let us ntroduce four MAPs defned by the matrces D 0 and D 1 All these MAPs have the same average total arrval rate λ = 20, but dfferent coeffcents of correlaton and varaton The frst process MAP 1 s the statonary Posson one It s defned by the matrces D 0 = 20 and D 1 = 20 It has the coeffcent of correlaton c cor = 0 and the coeffcent of varaton c var = 1 The second process MAP 2 defned by the matrces D 0 = , D 1 = has the coeffcent of correlaton c cor = 025 and the coeffcent of varaton c var = 54 The thrd process MAP 3 defned by the matrces D 0 =, D = has the coeffcent of correlaton c cor = 025 and the coeffcent of varaton c var = 145

9 448 C Km et al / Performance Evaluaton 70 (2013) Fg 2 The average number L system of customers n the system and the average number N buffer of customers n the buffer as functons of the number of servers at the second stage for dfferent MAPs Fg 3 The average number of busy servers N (1) at the frst stage and N (2) at the second stage as functons of the number of servers at the second stage for dfferent MAPs Fg 4 The average ntensty λ (1) out of flow of customers, who get servce at the frst stage of the system, and the average ntensty λ out of flow of customers, who get servce at the second stage of the system, as functons of the number of servers at the second stage for dfferent MAPs The fourth process MAP 4 s defned by the matrces D 0 =, D = It has the coeffcent of correlaton c cor = 04 and the coeffcent of varaton c var = 124 We assume that the number of servers at the frst stage R = 80, the servce rate at the frst stage µ = 087, the probablty q = 06, buffer capacty K = 30, the ntensty of mpatence α = 05, the probabltes p l = 002(l + 1), l {0, 1,, 29} PH servce process at the second stage s characterzed by the vector β = (02, 08) and the matrx S = The mean servce tme b 1 at the second stage s equal to 2027, squared coeffcent of varaton s equal to 198 Let us vary the number of servers N at the second stage n the nterval [1, 25] Fgs 2 4 llustrate the dependence of the average number L system of customers n the system and the average number N buffer of customers n the buffer, the average numbers of busy servers N (1) at the frst stage and N (2) at the second stage, the average ntensty λ (1) out of flow of customers, who get servce at the frst stage, and the average ntensty λ out of flow of customers, who get servce at the second stage, on the number of servers at the second stage for MAPs presented above From these fgures, one can conclude that coeffcents of correlaton and varaton n the arrval flow have a profound effect on system performance measures The dependence of the average number of customers n the buffer N buffer on coeffcents of correlaton and varaton n the arrval flow can be explaned as follows For small N, the average number of customers n the buffer decreases wth ncreasng coeffcents of correlaton and varaton due to hgher probablty for flows wth large coeffcents of correlaton and varaton When the number of servers N ncreases, the probablty to meet a free server at arrval epoch of the customer from arrval process wth low coeffcents of correlaton and varaton s larger than for the customer from arrval process wth large coeffcents of correlaton and varaton, so the average number of customers n the buffer s larger for arrval processes wth large coeffcents of correlaton and varaton

10 C Km et al / Performance Evaluaton 70 (2013) Fg 5 Loss probabltes P (1) and P(2-ent) as functons of the number of servers at the second stage for dfferent MAPs Fg 6 Loss probabltes P (2-esc) and P (2-mp) as functons of the number of servers at the second stage for dfferent MAPs Fg 7 The average sojourn tme V soj as a functon of the number of servers at the second stage for dfferent MAPs The average number of busy servers at the frst stage and the average ntensty of flow of customers, who get servce at the frst stage, do not depend on the number of servers at the second stage, but depend on coeffcents of correlaton and varaton n the arrval flow The average number of busy servers at the second stage, customers n the system and the ntensty of flow of customers, who get servce at the second stage, ncrease wth ncreasng the number N, and decrease when coeffcents of correlaton and varaton n arrval flow ncrease It can be explaned as follows The probabltes of the customer at the second stage essentally decrease wth ncreasng N and ncrease wth ncreasng coeffcents of correlaton and varaton n the arrval flow Fgs 5 7 show the dependence of probabltes P (1), P(2-ent), P (2-esc), P (2-mp) and the average sojourn tme V soj on the number of servers for dfferent MAPs The value of P (1) s very close to zero for all arrval processes under consderaton except MAP 4 For ths arrval process the probablty P (1) s equal to 00129, so for ths flow 80 servers at the frst stage are not enough to provde satsfactory value of probablty at the frst stage One can see that dependences of the probablty P (2-mp) and the average sojourn tme V soj on the number of servers are very smlar to the same dependences of N buffer and L respectvely It can be explaned by the formula λ (1) out P(2-mp) = αn buffer Ths formula holds true because the rght and left sdes of ths equaton determne the ntensty of customers leavng the system due to mpatence Also results of ths experment show that the Lttle s formulas holds true for the system under study, e, V soj = Lsystem λ Experment 2 The problem of optmal choce of the number of agents has great practcal mportance In ths experment we solve numercally the problem of optmal choce of the number of agents whch conssts n determnng the number of agents for whch the average proft of the system per unt tme s maxmal Note that the proft can be consdered by dfferent ways We assume that the average proft J(N) of the system under the fxed number of agents N can be found as follows: J(N) = aλ out (1 q)λ (1) out(d 1 P (2-ent) + d 2 P (2-esc) + d 3 P (2-mp) ) cn

11 450 C Km et al / Performance Evaluaton 70 (2013) Fg 8 The cost crteron J(N) as a functon of the number of servers at the second stage for dfferent MAPs Table 1 The optmal values of the cost crteron J(N ) and the number N of servers for dfferent MAPs (a = 20, d 1 = 4, d 2 = 2, d 3 = 5, c = 4) MAP 1 MAP 2 MAP 3 MAP 4 J(N ) N Table 2 The optmal values of the cost crteron J(N ) and the number N of servers for dfferent MAPs (a = 15, d 1 = 4, d 2 = 7, d 3 = 3, c = 25) MAP 1 MAP 2 MAP 3 MAP 4 J(N ) N Here a s the average proft obtaned by the call center from servcng one customer, λ out s the ntensty of the flow of customers who get successful servce n the system, (1 q)λ (1) out s the average ntensty of customers arrval at the second stage, d 1, d 2, d 3 are the charges of the call center when an arrvng customer s lost due to absence of the place n the buffer, unwllngness of a customer to wat and mpatence, respectvely, c s the charge pad for mantenance of one agent per unt tme Our goal s to fnd the optmal value N whch provdes the maxmal value of ths cost crteron J(N) It s worth to note, that the problem of the sutable choce of cost coeffcents (n our case coeffcents a, d r, r = 1, 3, c) n the cost crteron always play a crucal role n successful mplementaton of optmzaton We assume here that n our model the cost coeffcents can come from the experts n the real call center to whch the model wll be appled So, at frst we fxed the followng cost coeffcents n the cost crteron: a = 20, d 1 = 4, d 2 = 2, d 3 = 5, c = 4 The rest of parameters are the same as ones presented n the frst experment Fg 8 presents the dependence of the cost crteron J(N) for dfferent MAPs The optmal values of the cost crteron J(N ) and the number N of servers are gven n Table 1 Based on Table 1, one can conclude that values of the cost crteron J(N ) are very senstve to the coeffcent of correlaton n the arrval process In order to show that the optmal number of servers N s also senstve to the coeffcent of correlaton n the arrval process, let us change the coeffcents n the cost crteron as follows: a = 15, d 1 = 4, d 2 = 7, d 3 = 3, c = 25 The optmal values of the cost crteron J(N ) and the number N of servers are presented n Table 2 So, from Table 2 one can conclude that the optmal number of servers N and values of the cost crteron J(N ) are essentally depend on the coeffcent of correlaton n the arrval process The results confrm the mportance of takng nto account coeffcents of correlaton and varaton n the arrval process for performance evaluaton and correct predcton of the system operaton For example, as t s seen from Table 1, f we assume that the arrval flow of customers to the call center s descrbed by a statonary Posson arrval process, whle actually the correlated flow MAP 4 arrves to the call center, we wll hre N = 18 agents and expect the proft J = , but actually n ths case we wll get a proft of only that s less than half of what s expected In Experment 3, we show the mpact of the coeffcent of varaton n the servce process at the second stage on the man performance measures of the system To ths end, we consder four PH servce processes wth the same mean servce tme b 1 = 2027, but dfferent values of the coeffcent of varaton c var The frst process corresponds to the exponental dstrbuton and s defned by the vector β = (1) and the matrx S = ( ), M = 1 It has the coeffcent of varaton c var = 1 The second process concdes wth the PH servce process defned n the prevous experment As was mentoned t has the coeffcent of varaton c var = 14 The thrd process has the coeffcent of varaton c var = 5 and corresponds to hyper-exponental dstrbuton of order 2 It s characterzed by the vector β = (005, 095) and the matrx S =

12 C Km et al / Performance Evaluaton 70 (2013) Fg 9 The average ntensty λ out of flow of customers, who get servce at the second stage of the system, and the probablty P (2-ent) as functons of the number of servers at the second stage for dfferent PHs Fg 10 Loss probabltes P (2-esc) and P (2-mp) as functons of the number of servers at the second stage for dfferent PHs Fg 11 The average value V soj and the varance D of the sojourn tme as functons of the number of servers at the second stage for dfferent PHs The fourth process has the coeffcent of varaton c var = 10 and also corresponds to hyper-exponental dstrbuton of order 2 It s characterzed by the vector β = (098, 002) and the matrx S = We assume that arrvals are defned by the thrd MAP process presented n the frst experment (the arrval rate λ = 20, the coeffcent of correlaton c cor = 025 and the coeffcent of varaton c var = 145) The rest of parameters are the same as ones presented n the prevous experment Fgs 9 11 show the dependence of the average ntensty λ out, the probabltes P (2-ent), P (2-esc), P (2-mp), the average sojourn tme V soj and the varance D = V (2) (V soj ) 2 of the sojourn tme dstrbuton on the number of servers for PHs wth dfferent coeffcents of varaton Lookng at Fgs 9 11, one can observe that system performance measures are senstve wth respect to the servce tme varaton at the second stage Impact s a especally strong on the sojourn tme of an arbtrary customer, n partcular, on the varance of the sojourn tme 7 Concluson In ths paper we consder a novel tandem queueng system wth mpatent customers whch can be suted for modelng a real-lfe call center wth IVR The specal approach for reducng the system state space s used The man performance measures and the Laplace Steltjes transform of the sojourn tme dstrbuton are derved Numercal results confrm the mportance of the analyss of queueng models wth the MAP arrval process because correlaton and varaton n the arrval process essentally mpact on the system performance Importance of takng nto account the coeffcent of varaton of

13 452 C Km et al / Performance Evaluaton 70 (2013) the servce tme dstrbuton s also llustrated numercally Possblty of effectve use of presented results for solvng the problems of optmal desgn of modern call centers s demonstrated Acknowledgments Ths research was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton, Scence and Technology (Grant No ) References [1] OZ Aksn, M Armony, V Mehrotra, The modern call centers: a mult-dscplnary perspectve on operatons management research, Producton and Operaton Management 16 (2007) [2] O Joun, Y Dallery, Z Aksn, Queung models for flexble mult-class call centers wth real-tme antcpated delays, Internatonal Journal of Producton Economcs 120 (2009) [3] JW Km, SC Park, Outsourcng strategy n two-stage call centers, Computers & Operatons Research 37 (2010) [4] T Aktekn, R Soyer, Call center arrval modellng: a Bayesan state-space approach, Naval Research Logstcs 58 (2011) [5] S 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SA Dudn, VI Klmenok, The MAP/PH/1/N queue wth tme phased arrvals as model for traffc control n telecommuncaton networks, Performance Evaluaton 66 (2009) [12] SR Chakravarty, n: JJ Cochran (Ed), Markovan Arrval Processes, n: Wley Encyclopeda of Operatons Research and Management Scence, John Wley and Sons, 2010, Publshed Onlne: 15 June [13] JR Artalejo, A Gomez-Corral, QM He, Markovan arrvals n stochastc modelng: a survey and some new results, Statstcs and Operatons Research Transactons (SORT) 34 (2010) [14] M Neuts, Matrx-Geometrc Solutons n Stochastc Models An Algorthmc Approach, John Hopkns Unversty Press, 1981 [15] A Panchenko, A Thummler, Effcent phase-type fttng wth aggregated traffc traces, Performance Evaluaton 64 (2007) [16] A Graham, Kronecker Products and Matrx Calculus wth Applcatons, Ells Horwood, Cchester, 1981 [17] Ch Km, S Dudn, O Taramn, J Baek, Queueng system MAP/PH/N/N + R wth mpatent heterogeneous customers as a model of call center, Appled Mathematcal Modellng [18] V Klmenok, CS Km, D Orlovsky, A Dudn, Lack of nvarant property of Erlang model n case of the MAP nput, Queueng Systems 49 (2005) [19] SD Conte, C deboor, Elementary Numercal Analyss An Algorthmc Approach, McGraw-Hll, New York, 1972 [20] J Abate, W Whtt, Numercal nverson of the Laplace transforms of probablty dstrbutons, ORSA Journal on Computng 7 (1995) Chesoong Km took hs Master degree and PhD n Engneerng from the Department of Industral Engneerng at Seoul Natonal Unversty n 1989 and 1993, respectvely He was a Vstng Scholar n the Department of Mechancal Engneerng at the Unversty of Queensland, Australa from September 1998 to August 1999 He was foregn scentst at the School of Mathematcs and Statstcs at Carleton Unversty, Canada from July 2003 to August 2004 He was also a Vstng Professor n the Department of Industral Engneerng at the Unversty of Washngton, USA from August 2004 to August 2005 He had scentfc vsts to Belarusan State Unversty n Belarus and Unversty of Debrecen n Hungary, respectvely He was selected for Who s Who n Asa and Who s Who n the World n 2007 He s currently Full Professor and Head of Department of Industral Engneerng at Sangj Unversty Hs current research nterests are n stochastc process, queueng theory wth partcular emphass on computer and wreless communcaton network, queueng network modelng and ther applcatons He has publshed around 60 papers n nternatonally refereed journals He has been the recpent of a number of grants from the Korean Scence and Engneerng Foundaton (KOSEF) and Korea Research Foundaton (KRF) of Korea Alexander Dudn has got PhD degree n Probablty Theory and Mathematcal Statstcs n 1982 from Vlnus Unversty and Doctor of Scence degree n 1992 from Tomsk Unversty He s Head of Laboratory of Appled Probablstc Analyss n Belarusan State Unversty, Professor of the Probablty Theory and Mathematcal Statstcs Department He s author of 280 publcatons ncludng more than 80 papers n top level journals (Journal of Appled Probablty, Queueng Systems, Performance Evaluaton, Operatons Research Letters, Annals of Operatons Research, Computers and Operatons Research, Computer Networks, IEEE Communcatons Letters, etc) He s the Charman of IPC of Belarusan Wnter Workshops n Queueng Theory whch are held snce 1985 He serves as the member of IPC of several nternatonal conferences Felds of scentfc nterests are: Random Processes n Queueng Systems, Controllable Queueng Systems and ther Optmzaton, Queueng Systems n Random Envronment, Retral Queueng Systems, Applcatons of Queueng Theory to Telecommuncaton He was nvted for lecturng and research to USA, UK, Germany, France, Holland, Japan, South Korea, Inda

14 C Km et al / Performance Evaluaton 70 (2013) Sergey Dudn was graduated from Belarusan State Unversty n 2007 In 2010, he got PhD degree from Belarusan State Unversty n System Analyss, Control and Informaton Processng and works currently as senor scentfc researcher of Research Laboratory of Appled Probablstc Analyss n Belarusan State Unversty Hs man felds of nterests are queueng systems wth sesson arrvals and controlled tandem models Olga Dudna was graduated from Belarusan State Unversty n 2007 In 2010, she got PhD degree from Belarusan State Unversty n Probablty Theory and Mathematcal Statstcs and works currently as senor scentfc researcher of Research Laboratory of Appled Probablstc Analyss n Belarusan State Unversty Her man felds of nterests are queueng tandem queueng models wth correlated arrval flows, non-markovan queueng systems