Call Centers with a Postponed Callback Offer

Transcription

1 Call Center with a Potponed Callback Offer Benjamin Legro, Sihan Ding, Rob Van Der Mei, Oualid Jouini To cite thi verion: Benjamin Legro, Sihan Ding, Rob Van Der Mei, Oualid Jouini. Call Center with a Potponed Callback Offer. OR Spectrum, Springer Verlag, <hal > HAL Id: hal Submitted on 15 Nov 2017 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 Noname manucript No. will be inerted by the editor) Call Center with a Potponed Callback Offer Benjamin Legro Sihan Ding Rob van der Mei Oualid Jouini the date of receipt and acceptance hould be inerted later Abtract We tudy a call center model with a potponed callback option. A cutomer at the head of the queue whoe elaped waiting time achieve a given threhold receive a voice meage mentioning the option to be called back later. Thi callback option differ from the traditional one found in the literature where the callback offer i given at cutomer arrival. We approximate thi ytem by a two-dimenional Markov chain, with one dimenion being a unit of a dicretization of the waiting time. We next how that thi approximation model converge to the exact one. Thi allow u to obtain explicitly the performance meaure without abandonment and to compute them numerically otherwie. From the performance analyi, we derive a erie of practical inight and recommendation for a clever ue of the callback offer. In particular, we how that thi time-baed offer outperform traditional one when conidering the waiting time of inbound call. Keyword Queueing Sytem Markov Chain Performance Evaluation Waiting Time Callback B. Legro EM Normandie, Laboratoire Méti, 64 Rue du Ranelagh, Pari, France benjamin.legro@centralien.net S. Ding Center for Mathematic and Computer Science CWI), Stochatic group, Science Park 123, 1098 XG, Amterdam, The Netherland dingihan@hotmail.com R. van der Mei Center for Mathematic and Computer Science CWI), Stochatic group, Science Park 123, 1098 XG, Amterdam, The Netherland R.D.van.der.Mei@cwi.nl O. Jouini CentraleSupélec, Univerité Pari-Saclay, Laboratoire Genie Indutriel, Grande Voie de Vigne, Chatenay-Malabry, France oualid.jouini@centraleupelec.fr

3 2 Benjamin Legro et al. 1 Introduction Call center erve a the public face in variou area and indutrie: inurance companie, emergency center, bank, information center, help-dek, tele-marketing, jut to name a few. The ucce of call center i due to the technological advance in information and communication ytem. The mot ued form of communication i the direct telephone contact. However in the context of highly congeted call center, the ue of alternative option can be propoed to cutomer o a to better match demand and capacity. Alternative option could be , chat, blog, callback ervice, etc. The callback offer allow the call center to change the nature of the channel from an inbound call to an outbound one. For the call center manager, thi change i valuable becaue it reduce the congetion in the inbound queue. Another important apect in call center i cutomer abandonment e.g., ee [26], [9]). While waiting in the inbound queue, a cutomer may decide to leave the ytem without being erved. Thi cutomer i then lot for the call center without poibilitie to be recontacted. Intead, an outbound cutomer can be reached later. Even with a long delay before being called back, thi cutomer i potentially not lot. From cutomer perpective, the willingne to accept future proceing depend on the urge to get an anwer and the waiting cot. If waiting i painful and getting an anwer i not urgent, then a cutomer may accept the callback offer. In practice everal type of callback offer are developed with the ame purpoe of changing inbound call into outbound one. A large number of patent reflect thi wide variety and the technological challenge to implement thi option in the Automatic Call Ditributor ACD) [24, 27, 29, 7]. Neverthele, from our dicuion with our partner INTERACTIV GROUP, the effect of the callback option are not well undertood by manager and the implementation till need to be improved to achieve ome ervice level objective. In call center, a percentile of the waiting time i the uually choen a a ervice level objective. Thi metric i often preferred to the average peed of anwer becaue the former wa perceived to be more informative; ee [4]. It i therefore important for manager to develop a callback offer which can be adjuted to thi type of ervice level agreement. At the ame time, the callback offer hould be carefully ued. Even when the callback offer i accepted by a cutomer, mot cutomer would prefer being erved directly. So, the callback offer hould not be automatically propoed but hould be propoed in a way which allow the call center to control the proportion of outbound call. A mentioned above, the other apect i abandonment. In cae of a too important ue of the callback offer the proportion of non-abandoning cutomer may get too important which in turn may lead to the impoibility to enure a ufficiently hort delay for callback cutomer. In ummary, an efficient callback offer hould: Help the manager to achieve a ervice level objective for inbound call; Control the proportion of outbound call; Be eay to implement in the ACD;

4 Call Center with a Potponed Callback Offer 3 Be ufficiently imple to develop taffing olution and predict performance. In the literature on operation reearch, different callback option have already been tudied and optimized [2, 3, 15, 11, 20]. Thee callback model will be dicued in detail below. A common element in thee model i that the deciion to propoe a callback offer i baed on the ytem ize. For intance, above a threhold on the queue length, a callback option i propoed to all arriving cutomer. Unlike thee model, we propoe a new callback option given to the firt cutomer in line when it experienced waiting time reache a given waiting time threhold; the ervice level objective. We call thi callback option the potponed call back offer. Thi make ene both from a theoretical and a practical point of view, epecially for objective that are function of the waiting time uch a the percentage of call that have waited horter than a pecific threhold. One can imagine, and it i indeed hown in thi paper, that a policy that ue actual waiting time information perform well for thi type of objective. The motivation to let cutomer wait before the callback offer in our model i to avoid giving a callback offer to a cutomer who could have been erved in a reaonable time. If a callback offer i given at arrival baed eventually on the queue ize, it may be poible due to the variability in the ervice time to encounter a erie of mall ervice time which would have enable to erve thi cutomer in a reaonable time. By letting the cutomer wait before the callback offer, the call center give a chance to erve the cutomer without uing the callback option. Recall that mot cutomer prefer being directly erved than being called back later. In addition, we aume that cutomer have a probabilitic reaction to the callback offer and that a non-preemptive priority i given to inbound call ince thee one are more urgent. A precie definition of the queueing model in given in Section 2. Another value of thi callback model i that it i completely tractable. Without abandonment, cloed-form expreion of the performance meaure can be obtained. Thi allow for workforce management olution and a imple implementation of the callback offer. In Section 3, we determine the proportion of cutomer who have waited le than the waiting time objective and the proportion of callback cutomer. In order to differentiate between inbound and outbound cutomer, we are alo intereted in their repected expected waiting time. Cloed-form expreion of thee performance meaure are derived without abandonment and a numerical method i developed with abandonment. The difficulty to compute thee metric i that the deciion to change a high priority cutomer into a low priority one doe not depend on a claical tate definition like the number of high priority cutomer but on the experienced waiting time of a given cutomer. To overcome thi difficulty, we propoe the following approach: 1. We develop an approximating model, in which the waiting time of the firt cutomer in line i modeled by a ucceion of exponential phae. The number of waiting phae and the elaping of time rate per phae are the control parameter of the approximation.

5 4 Benjamin Legro et al. 2. Since thi new model i a Markov chain, the tranition rate can be obtained and the tationary probabilitie can be derived. 3. Finally, a the control parameter of the approximating model tend to infinity, we how that thi model converge to the exact one which in turn lead to the exact performance meaure. The key operational finding derived in Section 4 are that 1) the callback offer can be ued a a tool to reduce a waiting time percentile, 2) the value of a callback option i more apparent under intermediate loaded ituation, with abandonment, for mall call center, or when cutomer react motly poitively to the callback option, 3) two rational trategie are poible for cutomer; either they all accept or they all reject the callback offer, 4) the time at which the callback offer i propoed hould be ufficiently potponed, epecially when the abandonment i ignificant or when cutomer do not have a rational reaction to the callback offer, and 5) compared to a non-potponed callback option, a potponed offer improve the waiting time of inbound call and the proportion of abandonment, epecially in highly loaded ituation. In what follow we dicu the related literature. Literature review. There i an extenive and growing literature on call center. We refer the reader to [12] and [1] for an overview. The main topic encountered in call center tudie are routing deciion e.g., ee [14], [30], [22]), taffing e.g., ee [8], [23]), or performance evaluation e.g., ee [17], [32], [31]). Our article focue on performance evaluation baed on a particular routing mechanim defined through a callback offer. There are a few paper on different callback option in call center. [2] conider a model in which cutomer are given a choice of whether to wait online for their call to be anwered or to leave a number and be called back within a pecified time or to immediately balk. Upon arrival, cutomer are informed or know from prior experience) of the expected waiting time if they chooe to wait and the delay guarantee for the callback option. Their deciion i probabilitic and baed on thi information. Under the heavy-traffic regime, [2] develop an etimation cheme for the anticipated real-time delay that i aymptotically correct. They alo propoe an aymptotically optimal routing policy that minimize real-time delay ubject to a deadline on the potponed ervice mode. In [3], the author develop an aymptotically optimal routing rule, characterize the unique equilibrium regime of the ytem, and propoe a taffing rule that pick the minimum number of agent that atifie a et of operational contraint on the performance of the ytem. There are two recent paper by [15] and [11]. [15] conider a call center model with a callback option where the capacity of the queue for the inbound call i finite. Cutomer balking and abandonment are allowed. They provide an efficient algorithm for calculating the tationary probabilitie of the ytem. Moreover, they derive the Laplace-Stieltje tranform of the ojourn time ditribution of virtual cutomer. [11] conider a lightly different model, where agent make outbound call to thoe lot cutomer. There are two agent team,

6 Call Center with a Potponed Callback Offer 5 one that handle in priority inbound call, and another that handle in priority outbound call. They compute the tationary probabilitie, and deduce from that ome performance meaure. They alo numerically addre the taffing iue of the two team. Finally, [20] conider in their callback model, a probabilitic cutomer reaction to the callback offer. They how uing a Markov deciion proce approach, that the optimal reervation policy for inbound call i of witch type. Thereafter, the ytem performance meaure are computed under the optimal policy. It appear from thi tudy that the value of the callback offer i apparent for congeted ituation and that the benefit of a reervation policy are more apparent in large call center, while they almot diappear in the extreme ituation of light or heavy workload. Moreover, if balking and abandonment are very high or if the overall treatment time pent to erve an outbound call i very large compared to that of an inbound one, there i a value in delaying the propoition of the callback offer. Another tream of literature le cloely related to our article deal with the analyi of queueing multi-channel call center model with blending. Thi can be related to callback model by auming an infinite amount of cutomer to callback at the next working period. Some paper focu on performance evaluation, and other addre the analyi of blending policie or taffing deciion. [10] develop variou continuou Markov chain model for a call center with inbound and outbound call. The author conider a threhold policy and characterize the rate of outbound and the waiting time ditribution of inbound. Other call center paper addre the analyi of blending policie. [13] and [6] prove that a threhold policy on the number of idle agent i optimal to maximize the outbound throughput under a ervice level contraint on the inbound waiting time. Similar reult are alo found in [19], for a nontationary model where inbound call arrive according to a non-homogeneou Poion proce. [28] conider a large call blending model and propoe a logarithmic afety taffing rule, combined with a threhold control policy to enure that agent utilization i alway cloe to one with alway idle agent preent. 2 Setting In thi ection we define the queueing model and preent an approximation model which can be tudied through a Markov chain analyi. 2.1 Queueing model We conider a multi-erver ingle queue with identical, parallel erver. The arrival proce of cutomer i Poion with rate λ. Service time are independent and exponentially ditributed with rate µ. When a cutomer call, if at leat one agent i available then thi cutomer i directly erved, otherwie he/he i routed to a firt-come-firt-erved queue called Queue 1. After having

7 6 Benjamin Legro et al. waited K time unit, the firt cutomer in line waiting in Queue 1 hear a voice meage, propoing to be called back later. We aume that a proportion r of cutomer accept the callback offer and become then outbound call. Thee call are routed to another queue called Queue 2. Since inbound call are more urgent, a non preemptive priority i given to Queue 1. Another reaon for the priority of inbound call i the cot of waiting. In many call center inbound cutomer pay per waiting time unit wherea an outbound cutomer would not pay. A priority for inbound call would then help to reduce their waiting cot. Moreover, cutomer patience i limited. We aume that the patience of a cutomer in Queue 1 i exponentially ditributed with rate β. Cutomer in Queue 2 are infinitely patient ince they are outbound call. Our queueing model i equivalent to a particular V-queueing model with two queue; Queue 1 and Queue 2, where cutomer in Queue 1 have a non-preemptive priority over cutomer in Queue 2. The arrival proce in Queue 1 i Poion with parameter λ and the arrival proce in Queue 2 i generated by cutomer in Queue 1 who have waited exactly K time unit without being erved and accept the callback offer. Thi equivalent queueing model i depicted in Figure 1. For thi Y d / &/> < W n Y d Fig. 1 Queueing Model queueing model, we are intereted in the proportion of callback cutomer, P c, the proportion of abandonment, P a, the expected waiting time of cutomer erved from Queue 1, EW 1 ), the expected waiting time of callback cutomer, EW 2 ) it include the time alo pent in Queue 1) and the probability of waiting le than the intant at which the callback option i propoed, P W < K), where W i the waiting time of an arbitrary cutomer. Note that without abandonment, thi queueing model can be een a an M/M/ queue where the queue dicipline ha been modified. 2.2 An approximating model In order to have a Markov chain, one may only have exponential duration between two ucceive event. Yet, the time at which the callback offer i given

8 Call Center with a Potponed Callback Offer 7 i determinitic. To overcome thi difficulty, we develop here an approximating model in which all duration are exponential. The reulting Markov chain will be tudied in Section 3 to obtain the performance meaure of the exact model. The approximation i baed on a Markov chain where the tate contitute a dicrete repreentation of the waiting time of the firt cutomer in line FIL) in Queue 1 when one or more cutomer are waiting. The waiting time of the FIL in Queue 1 i modeled by a ucceion of exponential phae with rate per phae a propoed in [18]. Intead, Queue 2 i modeled a in mot queueing model by it number of cutomer. The number of waiting phae in Queue 1 after which the callback offer i propoed to the FIL i denoted by n. After leaving thi waiting phae a cutomer -if not erved- i routed to Queue 2 with probability r or tay in Queue 1 with probability 1 r. The queue dicipline in both queue i till FCFS. After giving a tate definition and the tranition rate we will explain how thi approximation converge to the real model. State definition. The ytem i modeled uing a two dimenional continuoutime Markov chain. We denote by x, y) a tate of the ytem for x and y 0, where x repreent the erver tate or the waiting time in Queue 1 and y repreent the number of cutomer in Queue 2. More preciely, tate with x 0 correpond to an empty Queue 1 and + x buy agent. State with x > 0 correpond to the phae at which the FIL in Queue 1 i waiting and all agent are buy. Tranition. We next decribe the 7 poible tranition in the Markov chain. When the FIL change, becaue of a ervice completion or an abandonment ee tranition Type 5), or becaue of the current FIL moving to Queue 2 ee tranition Type 8), the waiting time phae change from x > 0 to x h with probability q x,x h. Thi mean that either the new firt in line i in waiting phae x h > 0 or that Queue 1 i empty if x h = 0, for 0 h < x. The probabilitie q x,x h are given in Theorem 2 of [21] by q x,x h = 1 [ 1 + λ ) ] x h 1 + β x k=x h+1 [ 1 + λ ) ] k 1 + β for 0 h < x and q x,0 = [ x 1 + λ k=1 ) ] k 1. + β Moreover, the probability of abandonment after a given waiting phae i ee Table 1, Line 3 in [21]) β +β 1. An arrival with rate λ while Queue 1 i empty x 0, y = 0), which change the tate to x + 1, 0). If x < 0, then he number of buy erver i increaed by 1. Otherwie if x = 0, then the FIL entity i created.

9 8 Benjamin Legro et al. 2. A ervice completion with rate + x)µ while Queue 1 and 2 are empty < x 0, y = 0), which change the tate to x 1, y). The number of buy erver i reduced by A ervice completion with rate µ while Queue 1 i empty, Queue 2 i not empty and all erver are buy x = 0, y 1), which change the tate to 0, y 1). The number of cutomer in Queue 2 i reduced by A ervice completion with rate µq x,x h or an abandonment with rate while Queue 1 i not empty x > 0, y 0), which change the tate to x h, y), that i, the new FIL i in waiting phae x h. 5. A phae increae without abandonment with rate +β while Queue 1 i β +β not empty and the FIL i not in waiting phae n 0 < x < n, y 0), which change the tate to x + 1, y). The waiting phae of the FIL i increaed by A phae increae with rate 1 r) while the FIL i in waiting phae n y 0), which change the tate to n + 1, y). The waiting phae of the FIL i increaed by A phae increae with rate rq x,x h while the FIL in Queue 1 i in waiting phae n x = n, y 0), which change the tate to x h, y + 1), that i, the new FIL i in waiting phae x h and the number of cutomer in Queue 2 i increaed by 1. Convergence to the real ytem. We approximate the determinitic duration before giving the callback offer by an Erlang random variable with n phae and rate per phae. We chooe n and uch that n = K. The Laplace tranform of the Erlang ditribution with parameter n and i n. +) We have = e n ln1+/) 1 ) + ln1 /) en e n/ = e K, where we write fa) ga) to expre that lim a a0 ga) = 1, for a 0 R. Applying the Levy continuity theorem for Laplace tranform, thi reult enure that a n and go to infinity, the conidered Erlang random variable converge in ditribution to the determinitic duration K. The other approximation i the tranition from Queue 1 to Queue 2. It i aumed in our modeling that after one -tranition from tate x = n only one cutomer i routed to Queue 2. However, more than one cutomer could be in phae n a in any other phae). More preciely with no abandonment), given that one cutomer i in phae n, thi cutomer i the only one with probability λ+, or two cutomer or more are in phae n with probability λ λ+. Again, a tend to infinity, the probability that only one cutomer i in one phae i equal to one. a a 0 fa)

10 Call Center with a Potponed Callback Offer 9 3 Performance Analyi In Section 3.1, we derive explicitly the performance meaure without abandonment. The method developed here i adapted numerically in Section to include abandonment. 3.1 Explicit performance meaure without abandonment In Section 3.1, we give the tationary probabilitie of the dicretized ytem. Next, in Section 3.1.2, we let the elaping of time rate tend to infinity in order to obtain the exact performance meaure Stationary probabilitie Recall that in the cae with no abandonment β = 0), we imply have q x,x h = λ ) ) h for 0 h < x and q x,0 = ) x a in Theorem 2.1 of [18]. Let u introduce the notation a = λ µ and a = a+/µ +/µ. The ratio a repreent the traffic intenity of the ytem and a i a modified verion of the traffic intenity. The parameter a i an increaing function of which i equal to a for = 0 and equal to for =. Propoition 1 give the tationary probability p x,y to be in tate x, y) for x and y 0.

11 10 Benjamin Legro et al. Propoition 1 Under the tability condition λ < µ, we have 1 p,0 = a x x! + a! x=0 ) 1 + a λ r a 1 + λ a ) 1 a/) 1 r a a ) p x,0 = ax x! p,0, for 0 x, a ) x a λ µ λ1 r)) rλ p x,0 = p 0,0 µ λ1 r) rλ a, for 1 x n, ) x n p x,0 = p 0,0 1 r) λ µ λ) a µ λ1 r) rλ a, for x > n, p x,y = λ a ) x a p µ λ1 r)) rλ 0,0 µ λ µ λ1 r) a ) x a rλ a µ λ1 r) rλ a rλ µ λ a ) y µ µ λ1 r) rλ a, for 1 x n, y 1, a ) x n p x,y = 1 r) pn,y, for x > n, y 1. 1, Proof. We adopt the following approach to derive the tationary probabilitie. Firt, we determine a et of equilibrium equation. Next, uing thee equilibrium equation we derive a imple explicit expreion of the probability that the FIL in Queue 1 i in waiting phae x; p x = p x,y for x 0. Conidering thi probability lead to a one-dimenional problem which in turn allow u to compute the probability of an empty ytem uing the normalizing condition. Finally, we derive the other tationary probabilitie. y=0 Equilibrium equation. Let S be the tate pace. Conider the cut between ) A 1 = {, 0),, x, 0)} and S\A 1, where x. Oberving that x+ λ+ x 1 ) ) l ) λ λ+ λ+ = h, λ+ we deduce that the cumulative tranition rate l=h ) h from tate x, y) to tate 0, y), 1, y) x h, y) i µ, for 0 h < x < n and y 0. Therefore, by equating flow acro the cut, one may

12 Call Center with a Potponed Callback Offer 11 write λp x,0 = + x + 1)µp x+1,0, for x < 0, 1) ) i λp 0,0 = µp 0,1 + µ p i,0, 2) i=1 ) i x p x,0 = µp 0,1 + µ p i,0, for 0 < x n, 3) i=x+1 ) i x p x,0 + rp n,0 = µp 0,1 + µ p i,0, for x > n. 4) i=x+1 Conider now the cut between A 2 = {x, y ) : y y} and S\A 2, where y 0. Thi lead to rp n,y = µp 0,y+1, for y 0. 5) Finally, from the cut between A 3 = {0, y), 1, y), x, y)} and S\A 3, where x 0 and y 1, we get ) i µ + λ)p 0,y = µp 0,y+1 + µ p i,y 6) i=1 + r p n,y 1, for y 1, ) i x p x,y + µp 0,y = µp 0,y+1 + µ p i,y 7) i=x+1 x + r p n,y 1, for 0 < x n and y 1, ) i x p x,y + µp 0,y = µp 0,y+1 + µ p i,y + rp n,y 1, 8) for x > n and y 1. i=x+1 Probability of an empty ytem. Summing up Equation 4) and 8) for y 1, yield ) k p x = µ p x+k, k=1 for x > n. Let u denote by z, a root of the related homogeneou equation. We then have ) k = µ z k, k=1

13 12 Benjamin Legro et al. which lead to 1 z)) = µz. Thi equation ha a unique olution; z = λ+ µ+ = a. Therefore we have p x+n+1 = a ) x pn+1, for x 0. Summing up now Equation 3) and 7) for y 1 and x = n, yield ) k 1 r)p n = µ p n+k, k=1 o we deduce that p x+n = 1 r) a ) x pn for x 0. We now prove by ) x induction on x that p n x = a pn, for 0 x < n. Thi relation i clearly true for x = 0. Aume now that thi relation hold for p n, p n 1,, p n x. Summing up now Equation 3) and 7) for y 1, yield ) ) x ) 2 ) x 1 p n x+1) = µ p n + µ p n + a a ) x ) ) x+1 + µ p n + r + µ) p n a ) x+1+k a ) k + µ1 r) pn k=1 x+1 ) i ) x+1 i ) x+1 = µ p n + r p n a i=1 ) x r) p n. ) ) 1 Uing λ+ a = µ+, we may write p n x+1) = µ a = µ a = a ) x+1 x+1 µ + i=1 ) x+1 µ + 1 ) x+1 p n, ) i p n + µ+ 1 µ+ ) x+1 ) x+1 p n ) x+1 p n + p n which prove the induction tep. ) Uing Equation 6), with the ame approach n we alo obtain p 0 = λ a pn, therefore p x = λ a ) x p0 for 1 x n and p x = 1 r) λ a ) x p0 for x > n. From the lat expreion, the tability condition i a < 1. Thi i equivalent to λ < µ a for a imple M/M/ queue. Moreover, umming up Equation 5) for y 0, lead to µp 0 p 0,0 ) = rp n. p So, p 0 = 0,0 1 r a a ). Uing now Equation 1), we finally deduce that p n 0 = a! p,0 1 r a a n. Uing the fact that the overall um of the tationary probabilitie )

14 Call Center with a Potponed Callback Offer 13 i equal to one, we obtain the probability of an empty ytem a in Propoition 1. ) x pn,0 Other tationary probabilitie. We can how that p n+x,0 = 1 r) α for x > 0. The proof i identical to the proof for p n+x above. We now how by induction on x that { ) x p n x,0 = p n,0 + rλ ) x 1)}, 9) a µ λ a for 0 x < n. Thi relation i clearly true for x = 0. Aume now that thi relation hold for p n,0, p n 1,0, p n x,0. One may write uing Equation 3) that x ) x+1 k p n x+1),0 = µp 0,1 + µ p n k,0 k=0 ) x+1+k α ) k + µ1 r) pn,0. k=0 k=1 We now replace p n,0, p n 1,0,, p n x,0 by their expreion a a function of p n,0 and µp 0,1 by rp n,0 Equation 5)). We obtain ) x+1+k α ) k p n x+1),0 = rp n,0 + µ1 r) pn,0 k=1 x ) { x+1 k ) k + µp n,0 + rλ ) k 1)}. a µ λ a Uing now µ x k=0 λ+ a ) x+1 λ+ ) x+1 k = λ ) x+1 ), and ) ) x+1 1 λ+, k=1 λ+ x k=0 ) x+1+k α λ+ ) x+1 k ) k = λ+ µ ) k a = ) x+2, λ+ we prove the induction tep. Oberve that Equation 2) i almot identical to Equation 3) in which we would replace x by 0. The only difference i the multiplicative coefficient on the left hand ide of Equation 2). Thi one i λ intead of. Therefore, uing the corrective coefficient λ, we deduce the explicit expreion of p 0,0 ; p 0,0 = { λ p n,0 a + rλ 1)}. µ λ a Thi lat equation relate p 0,0 and p n,0. By ubtituting the expreion of p n,0 a a function of p 0,0 into Equation 9), we get ) n x ) ) n x λ a + rλ µ λ a 1 p x,0 = p 0,0 ) n ) a + rλ µ λ a 1 a ) x a λ µ λ1 r)) rλ = p 0,0 µ λ1 r) rλ a,

15 14 Benjamin Legro et al. for 1 x n, and ) x n p x,0 = p 0,0 1 r) λ µ λ) a µ λ1 r) rλ a, for x > n. With the ame approach one can how by induction that p n+x,y = p n,y 1 r) a ) x, for x > 0 and { ) x p n x,y = p n,y + rλ ) x 1)} 10) a µ λ a + rλ [ ) x ] µ λ p n,y 1 1, for 0 x < n. Combining now Equation 6) with Equation 10), we get { p 0,y = p n,y + rλ 1)} λ µ λ a + rλ µ λ p n,y 1 λ [ 1 a a a ]. Thi lat equation relate p 0,y, p n,y and p n,y 1. Since µp 0,y = rp n,y 1 for y 1 Equation 5)), we obtain a relation between p 0,y and p n,y ; { p 0,y = p n,y + rλ 1)} λ µ λ a + λ µ λ p µ 0,y λ [ 1 Thi lat equation can be finally implified into a ]. a p n,y = λ p µ λ a 0,y µ λ1 r) rλ a, for y 1. Equation 5) give an expreion of p n,y 1 a a function of p 0,y. Inerting thee two reult into Equation 10) lead to an expreion of p x,y a a function of p 0,y ; p x,y = λ p µ λ1 r) a ) x a rλ 0,y µ λ1 r) rλ a, for 0 < x n and y 1. Finally, from Equation 5) we get rλ µ λ a ) y p n,y = µ µ λ1 r) rλ a p n,0. Thi finihe the proof of the Propoition.

16 Call Center with a Potponed Callback Offer Performance meaure In Theorem 1, we derive the performance meaure. In order to relate the performance meaure to thoe of an M/M/ queue, we introduce the notation C, a) = P W > 0) i.e., probability of queueing in an M/M/ queue). Recall from [16] page 103 that C, a) = Theorem 1 We have P c = r C, a) 1 a! a x x! + a! x=0 1 a/)e µ1 a/) K 1 r a e µ1 a/) K, P W > K) = C, a) 1 r a )e µ1 a/) K 1 r a e µ1 a/) K, EW 1 ) = a! µ EW 2 ) = 1 + µ K µ1 a/). 1 1 a/ 1 1 a/. 1 re µ1 a/) K 1 + µ1 a/) K) 1 1 a/) 2 r a a x e µ1 a/) K) 1 x! + a! x=0 1 re µ1 a/) K 1 a/ Proof. The approach to derive the performance meaure firt conit of defining the embedded Markov chain at pecific intant choen in order to reach the performance meaure at arbitrary intant. Next, by letting and n tend to infinity we obtain the reult. ), The embedded Markov chain. Arriving cutomer either enter ervice upon arrival, enter ervice from Queue 1 after ome wait, or are routed to Queue 2. Call the intant when one of thee three event occur Q-intant. Since the event at Q-intant all occur one at a time, in the long-run the ytem i identical at arrival intant and Q-intant. Since the Poion arrival proce of cutomer i independent of the ytem tate, the ytem i identical at arrival intant and arbitrary intant. So, the ytem i alo identical at arbitrary intant and Q-intant. We therefore chooe to conider the ytem at Q- intant to obtain the performance meaure the arrival intant cannot be een in our Markov chain). The Q-intant are determined by λ-tranition from tate with a vacant erver, µ-tranition from the other tate except in tate 0, y) and -tranition from tate n, y), for y 0. The overall cutomer flow at Q- intant i identical to the cutomer flow at arrival intant and ha a rate λ. Therefore, the probability at Q-intant that x erver are buy for 0 x < i λ λ p +x,0 = p +x,0. The probability that the FIL i in waiting phae x and y cutomer are in Queue 2 i µ λ p x,y for 0 < x < n or x > n, 0 for x = 0 and µ+r λ p n,y for x = n. The tationary probabilitie at Q-intant are then completely known. Thi allow u to derive the performance meaure.

17 16 Benjamin Legro et al. Performance meaure. The approach to obtain the performance meaure i to let and n tend to infinity with repect to n = K. Firt, we have a lim = e µ1 a/) K. n, We now derive the proportion of cutomer who are routed to Queue 2, P c. A cutomer move from Queue 1 to Queue 2 due to a -tranition from tate n, y), y 0. The proportion of cutomer which are moved from Queue 1 to Queue 2 i therefore P c = lim r n, λ p n. Recall from the proof of Propoition 1 that p n = λ a p0 and p 0 = a! p,0 1 r a a ). n Therefore, r λ p a a! n = r p,0 1 r a a. 11) From the expreion of p,0 in Propoition 1, we get the probability of an empty ytem in an M/M/ queue: lim p,0 = n, [ 1 a x x=0 x! + a! ] ) 1 a/ By applying the lat reult in Equation 11), we obtain the explicit expreion of P c. We now derive the proportion of cutomer who wait le than K, P W < K). A cutomer i erved from Queue 1 due to a µ-tranition from tate x, y), y 0. Therefore P W < K) = Therefore we get P W < K) = lim p,0 +p +1,0 + +p 1,0 + µ n, λ p 1 + p p n ). lim n, p,0 1 a x a x! +! x=0 1 a/ 1 λ+ 1 r a µ+ λ+ µ+, thi in turn lead to the reult of the Theorem. Conider now the erved cutomer from Queue 1. A erved cutomer from Queue 1 wait x -phae with probability µ λ p x for x > 0 and each phae ha an expected duration of 1/. Therefore, µ x 1 P c )EW 1 ) = lim n, λ p x x=1 µ a = lim p 0 n, 2 rn + 1) 1 a ) a a + 1 r a 1 ) 2.

18 Call Center with a Potponed Callback Offer 17 In order to compute thi limit, we eparate the lat expreion in three part. Firt, we may write lim p 0 = n, lim n, a! p,0 a 1 r a = [ 1 a a x! x! + a! x=0 ] a/ 1 r a. 13) e µ1 a/) K Second, we have µ a lim n, 2 Finally, one may write rn+1) 1 a 1 1 a ) a ) 2 = lim n, = +1 r a Applying the aumption n = K yield lim rn + 1) n, 1 µ1 a/) 2. µ a) 2 ) ) a + µ + µ 2 14) a r n + 1) a) = 1 r + /µ 1 a ) a a + 1 r a = 1 re µ1 a/) K 1 + µ1 a/) K). 15) Combining Equation 13), 14) and 15) lead to the expreion of EW 1 ). We now conider the expected waiting time of cutomer who are routed to Queue 2. The probability of having y cutomer in Queue 2 at Q-intant y 0) i x=1 µ λ p x,y + r λ p n,y. Uing the reult of Propoition 1, we can compute explicitly thi expreion by letting n and tend to infinity. 3.2 Numerical analyi with abandonment The complexity of the tranition tructure doe not allow u to obtain explicit expreion for the performance meaure with abandonment. However, ince the tranition tructure i completely known, uing pace tate truncation with a bound, D 1, for the number of waiting phae in Queue 1 and a bound, D 2, for the number of cutomer in Queue 2, we can derive the performance meaure including the proportion of abandonment.

19 18 Benjamin Legro et al. Let S be the tate pace. Conider the cut between A 1 = {, 0),, x, 0)} and S\A 1, where x D 1. By equating flow acro the cut, one may write λp x,0 = + x + 1)µp x+1,0, for x < 0, 16) ) β D1 λp 0,0 = µp 0,1 + µ + p i,0 q i,0, + β 17) p x,0 = µp 0,1 + µ + β + β p x,0 + rp n,0 = µp 0,1 + µ + i=1 ) D1 i=x+1 p i,0 β + β x q i,k, for 0 < x n, 18) k=0 ) D1 i=x+1 p i,0 x q i,k, for n < x < D 1. k=0 19) Conider now the cut between A 2 = {x, y ) : y y} and S\A 2, where 0 y D 2. Thi lead to rp n,y = µp 0,y+1, for 0 y < D 2. 20) Finally, from the cut between A 3 = {0, y), 1, y), x, y)} and S\A 3, where x D 1 and 1 y D 2, we get β µ + λ)p 0,y = µp 0,y+1 + µ + + β for 1 y < D 2, ) D1 p i,y q i,0 + rq n,0 p n,y 1, 21) p x,y + µp 0,y = 22) ) β D1 x x µp 0,y+1 + µ + p i,y q i,k + r q n,k p n,y 1, + β i=x+1 k=0 i=1 for 0 < x n, and 1 y < D 2, ) β D1 p x,y + µp 0,y = µp 0,y+1 + µ + + β for n < x < D 1 and 1 y < D 2. i=x+1 k=0 p i,y x q i,k + rp n,y 1 k=0 23) We then get a finite number of equation due to the tate pace truncation. In addition with the normalizing condition i.e., the um of the overall probabilitie i equal to one), on may obtain numerically all tationary probabilitie. Arriving cutomer either enter ervice upon arrival, enter ervice from Queue 1 or Queue 2 after ome wait, abandon from Queue 1 after experiencing ome wait, or move from Queue 1 to Queue 2 after waiting n phae. The

20 Call Center with a Potponed Callback Offer 19 proportion of cutomer which accept the callback offer, P c i then given by P c = r D 2 p n,y. λ The proportion of cutomer who have waited le than K time unit, P W < K), i 1 D 2 n µ + β +β P W < K) = p x,0 + p x,y. λ x= The proportion of abandonment, P a, i D 2 D 1 P a = y=0x=1 y=0 y=0x=1 β +β λ p x,y. The expected waiting time in Queue 1, EW 1 ), i given by D 2 D 1 1 P c )EW 1 ) = y=0x=1 µ + β +β x λ p x,y. We now conider the expected waiting time of cutomer who are routed to Queue 2. The probability of having y cutomer in Queue 2 y 0) i D 1 µ+ β +β λ p x,y + r λ p n,y. Thi lead to the expected number in Queue 2. x=1 Next, applying Little Law lead to EW 2 ). One difficulty in the computation i the choice for the two parameter and D 1. The truncation parameter D 1 introduce the rik of having a large probability ma in the truncated tate, particularly for large value of. The value of ha an important influence on the approximation. Increaing mean that more tate are required for the truncation. At the ame time, hould be ufficiently large to repreent the continuou elaping of time. 4 Operational Finding, Dicuion and Inight We invetigate the iue related to a potponed callback offer. We derive a erie of inight which can be proven in the cae without abandonment. The proven reult are next dicued with abandonment. More preciely, in Section 4.1, we how how a potponed callback offer can improve a waiting time percentile. In Section 4.2, we analyi how the cutomer behavior may impact the ytem performance and what may be a cutomer rational trategy. In Section 4.3, we invetigate the impact of the control parameter K on the performance meaure to obtain recommendation to better control the ytem performance. Finally, in Section 4.4, we conduct a comparion between our potponed callback option and a callback option given at cutomer arrival a developed in the literature e.g., ee [2], [20]).

21 20 Benjamin Legro et al. 4.1 The callback offer, a tool to improve a waiting time percentile We evaluate the impact of the callback offer on P W < K). Analyi without abandonment. We denote by R the ratio between P W > K) with the callback offer and P W > K) without the callback offer. Without the callback offer, we have P W > K) = C, a) e µ1 a/) K. Therefore, uing the expreion of P W > K) in Theorem 1, we get 1 r a R = 1 r a 1. e µ1 a/) K So, a a firt inight, we obtain Inight 1 The callback offer allow the manager to reduce a waiting time percentile. In Figure 2, we repreent P W > K) and R a a function of the workload for three different value for the callback acceptance parameter r. We oberve that the higher i r, the maller are P W > K) and R. Thi can be proven by P W > K) r R r = a = C, a)e µ1 a/) K 1 e µ1 a/) K ) 1 r a e µ1 a/) K) 2 < 0, and a 1 e µ1 a/) K ) 1 r a e µ1 a/) K) 2 < 0. One would expect that the impact of accepting the callback offer i tronger under high workload ituation. Yet, the highet improvement i for intermediate workload ituation a hown in Figure 2b) and 2d). Thi can be explained a follow. For low workload ituation, the probability of waiting le that the threhold K i high. Therefore, mot cutomer do not hear the callback offer. Under high workload ituation, mot cutomer hear the callback offer but whether they accept it or not, they will wait more than K. The comparion between Figure 2a) and 2c) illutrate that the abolute improvement i tronger in mall call center. The reaon i related to the pooling effect. It i well etablihed that the pooling effect in large call center reduce the improvement that a good routing trategy could bring [5, 19]. In ummary, our obervation lead to a econd inight: Inight 2 The more cutomer are likely to accept the callback offer, the more trongly P W > K) can be improved. The maximal improvement i for intermediate workload ituation and for mall call center ize.

22 Call Center with a Potponed Callback Offer 21 d d l l l dl d d e d e d a) = 1 l l l d d e d e d b) = 1 d d l l l dl d e d ed d e d ed d e d ed d c) = 50 l l l dl d e d ed d e d ed d e d ed d d) = 50 Fig. 2 P W > K) µ = 1, K = 0.5, β = 0) ld ld ldd dl d e d e d e d e d Fig. 3 P W > K) µ = 1, = 10, K = 0.5, r = 90%) Impact of the abandonment. The callback offer can be ued to prevent ome cutomer with too long waiting time to leave the ytem. It i then intereting to oberve how abandonment may impact P W > K). In Figure 3, we give P W > K) a a function of the ratio a/ for different value of the abandonment rate. An intereting obervation i that the abandonment feature trongly help to reduce P W > K). Thi i particularly apparent in high workload ituation. Callback cutomer then benefit from the abandonment of cutomer in Queue 1 becaue the abandonment participate in the departure flow from Queue 1.

23 22 Benjamin Legro et al. 4.2 Cutomer behavior We invetigate here the cutomer reaction to the callback offer. Impact of r on EW 2 ). The parameter r i aumed to capture the cutomer behavior. An intereting obervation i that thi parameter r i not part of the expreion of EW 2 ) without abandonment. Thi mean that the delay for callback cutomer i inenitive to the willingne of cutomer to accept the callback offer. Hence, we get the following inight: Inight 3 Without abandonment, the delay for callback cutomer i inenitive to the parameter r. However, Figure 4 reveal that with abandonment, the parameter r influence the delay for callback cutomer. More preciely, a r increae, EW 2 ) increae. Thi obervation i intuitive. A r increae, the proportion of callback cutomer alo increae. Thee cutomer do not abandon which in turn lead to a higher congetion of the ytem. ed ed dd dd dd ld ld ldd dd dd d dl d Fig. 4 EW 2 ) µ = 1, = 10, K = 0.5, λ = 9.9) Rational cutomer. We tudy here cutomer rational behavior. Firt, with rational cutomer, one may neglect the exponential patience. A hown in [25], rational abandonment can occur only upon arrival zero or infinite patience for each cutomer). We then invetigate the willingne to accept the callback offer without abandonment. The choice for a cutomer to accept the callback offer or not can be een a the reult of a rational deciion. When hearing the callback offer at time K, a cutomer ha the choice to tay in Queue 1 with a remaining 1 expected waiting time of µ becaue the callback offer i given to the firt cutomer in line) or can chooe to be called back later with an expected delay of EW 2 ) K. Of coure, accepting the callback offer lead to higher waiting time but waiting to be called back i le cotly/annoying than continuing to wait for an agent to be available. We capture by c 1 and c 2 the cot per time

24 Call Center with a Potponed Callback Offer 23 unit of waiting in the initial queue Queue 1) or in the callback queue Queue 2), repectively. The parameter r hould therefore be ) 1 r = arg min 1 r)c 1 µ + rc 2EW 2 ) K), with c 1 c 2. Since EW 2 ) i inenitive to r, the optimal value for r i either 0 or 1. More preciely, we get: Inight 4 Only two rational cutomer trategie are poible. Either all cutomer who hear the callback offer accept thi offer if c 2 1 a/ < c 1, otherwie 1+λ K they all reject the offer. 1+λ K The condition c 2 1 a/ < c 1 induce that the higher the workload i, the more likely cutomer will refue the callback offer. Intuitively, thi can be explained by the long delay for callback cutomer in cae of high workload ituation due to their low priority. The econd conequence i that the maller i K, the more likely a cutomer will accept the callback offer. The reaon i related to the proportion of callback cutomer. When K i mall, a high proportion of cutomer will hear the offer. Therefore if they all accept the offer, the proportion of thoe who are in Queue 1 i mall and the effect of the low prioritization i reduced which in turn make the callback offer attractive. 4.3 The control parameter K The control parameter for the call center i the time at which the callback offer i propoed, K. With rational cutomer. A mentioned in Section 4.2, by chooing a too high value for K, a call center with rational cutomer will induce a rejection of the callback offer r = 0). In thi cae, the value of K i irrelevant. Under a waiting time threhold for the callback offer, all cutomer accept the offer r = 1). In the cae r = 1, both EW 1 ) and EW 2 ), are trictly increaing in K. Thi argue for a value of K = 0. However, in that cae with r = 1 and K = 0, the call center manager may looe the control of the proportion of callback cutomer and the inbound queue will alway be empty. Thi might be unwanted becaue inbound call can be a ource of revenue for the call center; contrary to outbound call they may pay a waiting cot per waiting time unit. So, the choice of K alo depend on the wanted proportion of callback cutomer. Thi proportion, P c, i trictly decreaing in K. Thi can be een by P c K = µr C, a) 1 a/)2 e µ1 a/) K 1 r a e µ1 a/) K ) 2 < 0.

25 24 Benjamin Legro et al. ld e ld e ld ed d d e d e d d d d Fig. 5 EW 1 ) = 10, µ = 1, r = 0.8) With irrational cutomer. In the cae r < 1, the element mentioned above till hold except the monotonicity of EW 1 ). In Figure 5, we preent EW 1 ) a a function of K for different workload ituation. Propoition 2 If 0 < r < 1, there exit a unique value for K which minimize EW 1 ). It i the unique olution in K of with x = µk1 a/) and A = xa + re x = 1, 24) 1 a x x! + a!1 a/) x=0 1 a a x x! + a!1 a/) x=0 Note that in the cae r = 0, EW 1 ) i inenitive to K. Proof. We obtain Equation 24) from EW1) K = 0. Conider the function fx) = xa + re x 1. We want to how that fx) = 0 ha a unique olution. We have f x) = A re x. Since x > 0, r < 1 and A > 1, we have f x) > 0 for x 0. So, the function f i increaing in x for x 0. Moreover, f0) = r 1 < 0 and fx) = +. Thi prove that there exit a unique olution of lim x + Equation 24). One way to obtain the unique olution of Equation 24) i to apply the Newton algorithm by defining recurively x k by x 0 = 0 and x k+1 = x k fx k) f x k ) for k 0 and f defined a in the proof of Propoition 2. Note that ince f x) > 0 for x 0, the recurion i well defined. The reaon which explain why EW 1 ) i not increaing in K i the definition of the callback offer. Increaing K doe not necearily mean that le cutomer have the callback propoition. Recall that only the firt cutomer in line can hear the callback offer. In cae of high workload ituation and low value for K, the probability to be the FIL at waiting time K i low except if r = 1). Mot likely, at waiting time K a cutomer will have other cutomer in front of him and will not have the callback offer. Increaing K in thi cae, lead to a higher chance to be the FIL at waiting time K. Therefore, increaing.

26 Call Center with a Potponed Callback Offer 25 K lead to a higher chance to leave Queue 1. Thi explain how EW 1 ) can be decreaing in K. In cae of low workload ituation, increaing K reduce the proportion of callback cutomer and therefore increae EW 1 ). With abandonment. Figure 6a) and 6b) illutrate the impact of K on EW 1 ) and EW 2 ), for different value of the abandonment rate β. We oberve that with abandonment, the value of K which minimize EW 1 ) i higher than the one obtained without abandonment. With abandonment, the increaing of the number of cutomer in Queue 1 increae alo the departure rate after abandonment or ervice) of inbound from the ytem, which make the ytem more efficient and may decreae EW 1 ). Therefore higher value for K may lead to a better performance for inbound call. We oberve that EW 2 ) i till increaing in K Figure 6b)) although the abandonment in Queue 1 alo reduce the waiting time in Queue 2. ed ld ld l ed l l dd l d d e dd d e dd dd d dd d d d d d d d d a) EW 1 ) b) EW 2 ) Fig. 6 Impact of the abandonment = 10, r = 0.8, a/ = 0.95, µ = 1) The abandonment play a important role in the choice of K. Since by definition outbound call do not abandon, reducing K reduce abandonment, which i poitive. Yet, thi may alo increae the workload and lead to higher waiting time. Thi lead to another inight. Inight 5 The callback offer may help to reduce the proportion of abandonment. However, the time at which the callback offer i propoed hould be carefully choen in order to avoid congetion. 4.4 Comparion with a non-potponed callback offer The callback offer tudied in thi article differ from the one in the literature by the intant at which it i propoed. In mot callback model, the callback offer i given at arrival of a new call if the expected waiting time i too high e.g., ee [2], [20]). Intead, we conider in thi article a callback offer given after experimenting ome wait. We propoe to conduct a comparion between thee two trategie.

27 26 Benjamin Legro et al. We call Model A our potponed callback offer and by Model B a callback offer propoed at arrival of a new call. For Model B, we aume that at and above a given number of cutomer in Queue 1 or equivalently at and above a given expected waiting time for an arriving cutomer) the callback offer i propoed to all arriving cutomer. Hence, in Model B, Queue 1 ha a limited capacity n. All arriving cutomer are routed to Queue 2 if Queue 1 ize i equal to n. Therefore n i the control parameter of Model B. The performance meaure in Model B can be obtained through a Markov chain analyi or can be deduced from Propoition 3 of [20]. We obtain the following performance meaure for Model B: P c = C, a) 1 a/) a 1 ) a n+1, a! EW 1 ) = µ 1 ) a n 1 + n1 a/)) ) +1 1 EW 2 ) = 1 a/) 2 1 a 1 + n µ1 a/). a x x! + a! x=0 ), 1 a 1 a/ The difficulty in the comparion i the cutomer reaction to the offer. It may differ whether the callback offer i given at arrival or later. To avoid thi complexity, we aume that all cutomer accept the callback offer in both Model. Thi correpond to a rational behavior in Model A. Comparion without abandonment. In Theorem 2, we conider a context for which the call center manager want to maintain the proportion of callback cutomer at a given level. Under thi contraint which force the two model to have the ame proportion of callback cutomer we prove that our potponed callback offer lead to a better expected waiting time for inbound call and a wore expected waiting time for outbound one. Theorem 2 Given that the control parameter K Model A) and n Model B) are choen uch that the proportion of callback cutomer in identical in both model, EW 1 ) i lower in Model A and EW 2 ) i lower in Model B. Proof. To obtain the ame proportion of callback cutomer in both model, the control parameter n and K hould be related by ) a n = e µ1 a/)k. Thi equation i equivalent to n lna/) = µ1 a/)k. Let u denote by EW 1 ) A and EW 1 ) B, the expected waiting time of inbound call in Model A and B. We have EW 1 ) A EW 1 ) B = a! µ e µ1 a/) K n µk) 1 1 a/) a a x e µ1 a/) K) 1 x! + a! x=0 ). 1 e µ1 a/) K 1 a/

28 Call Center with a Potponed Callback Offer 27 Thu, the ign of thi difference depend on the igh of n µk. One may write, n µk = µk lna/) + 1 a/). lna/) Since a/ < 1, µk lna/) > 0. Thu, the ign of the expreion depend on the ign of lna/) + 1 a/. Conider the function in x, fx) = lnx) + 1 x for x > 0. We have f x) = 1 x 1. So f x) > 0 for 0 < x 1. Since f1) = 0, lna/)+1 a/ < 0. Thi prove that EW 1 ) A EW 1 ) B < 0. With the ame approach we can prove that the expected waiting time for outbound call i higher with Model A. In Figure 7a) and 7b), we repreent EW 1 ) and EW 2 ) a a function of the workload for Model A and Model B auming a fixed value of K = 0.5 for Model A and n i adjuted in Model B with the relation n lna/) = µ1 a/)k uch that the two model achieve the ame proportion of callback cutomer. An intereting obervation i that the improvement for EW 1 ) with Model A i higher under high workload ituation wherea the improvement for EW 2 ) with Model B i higher under low workload ituation. Thi lead to a lat inight. e d d D d D d D d D d d d d d e d e d a) EW 1 ) d d d d e d e d b) EW 2 ) Fig. 7 Comparion between the callback offer = 10, r = 1, µ = 1, β = 0, K = 0.5, n lna/) = µ1 a/)k) Inight 6 A potponed callback offer i preferred under high workload ituation. Comparion with abandonment. In Figure 8a), 8b), and 8c), we repreent P a, EW 1 ) and EW 2 ) a a function of the arrival rate for Model A and Model B auming a fixed value of n = 5 for Model B and K i adjuted in Model A uch that the two model achieve the ame proportion of callback cutomer. We obtain the ame qualitative obervation a in Figure 7. A mentioned in Inight 6, with abandonment the potponed callback offer i preferred under high workload ituation. In Addition, Figure 8a) reveal that for a given proportion of callback cutomer, the potponed callback offer