Performance Evaluation

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1 Performance Evaluation 95 (206) 40 Content lit available at ScienceDirect Performance Evaluation journal homepage: Optimal cheduling in call center with a callback option Benjamin Legro a, Oualid Jouini a,, Ger Koole b a Laboratoire Genie Indutriel, CentraleSupélec, Univerité Pari-Saclay, Grande Voie de Vigne, Chatenay-Malabry, France b VU Univerity Amterdam, Department of Mathematic, De Boelelaan 08a, 08 HV Amterdam, The Netherland a r t i c l e i n f o a b t r a c t Article hitory: Received Augut 204 Received in revied form 29 June 205 Accepted 23 September 205 Available online November 205 Keyword: Call center Callback option Routing optimization Queueing ytem Markov chain Markov deciion procee We conider a call center model with a callback option, which allow to tranform an inbound call into an outbound one. A delayed call, with a long anticipated waiting time, receive the option to be called back. We aume a probabilitic cutomer reaction to the callback offer (option). The objective of the ytem manager i to characterize the optimal call cheduling that minimize the expected waiting and abandonment cot. For the ingle-erver cae, we prove that non-idling i optimal. Uing a Markov deciion proce approach, we prove for the two-erver cae that a threhold policy on the number of queued outbound call i optimal. For the multi-erver cae, we numerically characterize a witching curve of the number of agent reerved for inbound call. It i a function of the number of queued outbound call, the number of buy agent and the identity of job in ervice. We alo develop a Markov chain method to evaluate the ytem performance meaure under the optimal policy. We next conduct a numerical tudy to examine the impact of the policy parameter on the ytem performance. We oberve that the value of the callback offer i epecially important for congeted ituation. It alo appear 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. We moreover oberve in mot cae that the callback offer hould be given upon arrival to any delayed call. However, if balking and abandonment are very high (which help to reduce the workload) or if the overall treatment time pent to erve an outbound call i too large compared to that of an inbound one, there i a value in delaying the propoition of the callback offer. 205 Elevier B.V. All right reerved.. Introduction Context and motivation. 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 telephone. However, in the context of highly congeted call center, the ue of alternative ervice channel can be propoed to cutomer o a to better match demand and capacity. Alternative channel could be , chat, blog, or potponed callback ervice. We focu on thi lat alternative. The idea i that cutomer, who are expected to experience long waiting time, receive the option to be called back later. Thi lead to a contact center with two channel, one for inbound call (inbound), and another for outbound call (outbound). The recent tudy of ICMI [], baed on the analyi of 36 large contact center, report that 76% of them ue the outbound channel. Correponding author. addree: benjamin.legro@centralien.net (B. Legro), oualid.jouini@centraleupelec.fr (O. Jouini), ger.koole@vu.nl (G. Koole) / 205 Elevier B.V. All right reerved.

2 2 B. Legro et al. / Performance Evaluation 95 (206) 40 (a) Inbound. (b) Outbound. Fig.. Effect of the callback option on performance (arrival rate = 5.5, ervice rate = 0.2, number of agent = 28). The flexibility of the callback option come from the willingne of ome cutomer to accept future proceing. The call center can then make ue of thi opportunity to better manage arrival uncertainty, which in turn would improve the ytem performance. An illutration of callback option benefit i provided in Fig.. The figure give imulated performance meaure of a call center example with variou level for the ue of the callback option. We conider a non-idling ytem where inbound have a non-preemptive higher priority over outbound. We oberve that the expected waiting time of inbound and outbound call are coniderably improved by uing the callback option. For intance, the expected waiting time of inbound could be divided by around 20 (it decreae from 8 min and 55 to 23 ) while only 0% of arriving call chooe to be called back. The unpredicted and flexible call center environment offer the potential for a routing optimization that would lead to a ignificant operational improvement. It i a non-expenive approach compared to taffing optimization [2,3]. One important quetion for manager in our context i how hould be the routing rule of job that would enure non-exceive waiting time for both job type, i.e., upon a ervice completion, hould the agent handle an inbound or an outbound call? when hould be propoed the callback offer? We addre thee quetion under a queueing modeling framework and a probabilitic cutomer reaction to the callback option. A call center where agent imultaneouly handle inbound and outbound call i commonly referred to a call blending. The key ditinction of call center problem with blending come from the fact that outbound tak have le urgency relative to inbound call. Blended operation problem have led to reearch on performance evaluation [4 6], taffing [7] and analyi of blending policie [8 3]. Becaue of the lack of ervice level requirement on outbound, it i bet to give higher priority to inbound. Moreover, to reduce the number of inbound who may experience long waiting before ervice, one ha to guarantee that there i ufficient idlene in the ytem. In the patent of Duma et al. [4], baed on extenive imulation experiment, it i hown that blending inbound and outbound call and employing a threhold policy, enure that the outbound throughput rate i met while waiting time of inbound are very hort. It i alo hown that blending the two type of call in one pool require le agent than employing two ditinct pool. Bhulai and Koole [9] and Gan and Zhou [2], prove thi optimal control, which i of threhold type, when the ervice rate of the two type of job are equal. More preciely, they how that it i optimal to chedule outbound tak only when no outbound are in the queue and the number of idle agent exceed a certain threhold. In the cae of a callback option, thi policy cannot be directly applied. The reaon i that the above literature conider an infinite amount of non-priority job. In a call center with a callback option, the number of cutomer waiting to be called back ha to be finite in order to avoid infinite waiting. The routing policy hould then account for the length of the callback queue. Another difference, compared to cae with claical infinite amount of outbound tak, i that inbound and outbound arrival are negatively correlated. Thi require further analyi, and may lead to different managerial recommendation. Contribution. We conider a call center with a ingle cutomer type. A delayed call, with a long anticipated waiting time, receive the option to be called back. We develop a modeling that account for balking, abandonment, probabilitic cutomer reaction to a tate-dependent delay information, unequal ervice requirement for job type, and the eventual non-availability of a called back cutomer. The objective of the ytem manager i to find the optimal call cheduling policy that minimize the expected operating cot of inbound and outbound. The control action concern the number of agent reerved for inbound and the ytem tate ituation at which the callback offer hould be propoed. We ditinguih three main contribution. The firt contribution i related to the agent reervation policy. We prove for the ingle-erver cae that non-idling i optimal. Uing a Markov deciion proce (MDP) approach, we prove for the twoerver cae with equal ervice requirement that a threhold policy on the number of queued outbound i optimal. Baed on the two-erver reult, we conjecture for the multi-erver cae that the optimal policy i of witch type. The number of agent to reerve for inbound depend on the number of queued outbound, the number of buy agent and the identity of job in ervice. Moreover, we examine the impact of the ytem exogenou parameter on the agent reervation policy. We oberve, for example, that a reervation policy i not likely to be ued under light or heavily loaded ituation.

3 B. Legro et al. / Performance Evaluation 95 (206) 40 3 The econd contribution i the performance analyi under the optimal reervation policy. The performance meaure of interet are related to the job type waiting time and abandonment. We develop a controlled numerical approximation to obtain thee performance meaure for the general modeling. For variou particular cae, uing a Markov chain method, we go further by providing either exact numerical algorithm, or cloed-form expreion for the performance analyi. The third contribution i the analyi of the impact of the policy parameter on performance. We derive the firt and econd monotonicity reult in the number of agent for the performance meaure in the non-idling cae. Thee reult upport that the benefit of a reervation policy i more apparent in large call center. Moreover, in mot cae, the callback offer hould be given upon arrival to any delayed call. We prove thi reult in the non-idling cae uing firt order monotonicity reult. However, if balking and abandonment are very high (which help to reduce the workload) or if the overall treatment time pent to erve an outbound call i too large compared to that of an inbound one, there i a value in delaying the callback offer to all cutomer. Literature review. There i a rich literature on the operation management in call center. We refer the reader to the two urvey by Gan et al. [8] and Akşin et al. [3]. For a background on the pecific context of multi-channel call center, we refer the reader to Chapter 7 in [5]. A mentioned above, there are only few paper dealing with routing trategie in the context of a finite amount of callback. The firt two paper directly addreing the problem of the callback option are by Armony and Maglara [0,6]. The author 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, Armony and Maglara [0] develop an etimation cheme for the anticipated real-time delay. They alo propoe an aymptotically optimal routing policy that minimize real-time delay ubject to a deadline on the potponed ervice mode. In [6], 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. To the contrary to Armony and Maglara [0,6], we account here for the feature of abandonment, unequal ervice requirement and the poible non-availability of an outbound call. Yet, our modeling i retricted to policie with trict non-preemptive priority for inbound. Armony and Maglara [0,6] conider intead a tate-dependent priority policy. Two recent paper are by Kim et al. [7] and Dudin et al. [8]. Kim et al. [7] conider a call center model with a callback option where the queue capacity for inbound i finite. A in our modeling, cutomer balking and abandonment are allowed. The author provide an efficient algorithm for calculating the tationary probabilitie of the ytem tate. Moreover, they derive the Laplace Stieltje tranform of the ojourn time ditribution of virtual cutomer. Dudin et al. [8] conider a lightly different modeling, where lot cutomer are called back. There are two agent team, one that handle in priority inbound, and another one that handle in priority outbound. They compute the tationary probabilitie, and deduce the ytem performance meaure. They alo numerically addre the taffing iue for the two team. Our approach differ from thoe in [0,6 8] ince we allow for agent reervation trategie. We alo allow to control the propoition of the callback offer, wherea in all above reference thi option i propoed to all cutomer. Other paper conidering finite amount of outbound tak are Armony and Ward [] and Gurvich et al. [9]. They tudy call center that exercie cro-elling. The cro-elling phae i initiated by the agent and can thu be conidered a a type of outbound work in finite number. However, thee are le related to our pecific context of callback. Structure of the paper. The remainder of thi paper i tructured a follow. In Section 2, we decribe the call center model with a callback option. In Section 3, we addre the optimal routing problem for outbound call. In Section 4, we evaluate the performance meaure under the optimal reervation policy. In Section 5, we ue the optimization and performance meaure reult to examine the impact of the policy parameter on performance. We then provide concluion and highlight future reearch direction. Part of the proof of the reult of the main paper are given in the Appendice and the online upplement (ee Appendix H). 2. Model decription We conider a call center modeled a a multi-erver queueing ytem with identical, parallel erver (agent). The call center handle two type of job: inbound call (type job or inbound) initiated by cutomer, and outbound call (type 2 job or outbound) initiated by agent. Each agent can handle both type of job. Type job requet for a real-time ervice, while type 2 job are cutomer with a potponed ervice. A job 2 cutomer i originally a job cutomer that ha choen to be called back. The real-time ervice i more important in the ene that the waiting time of an inbound call hould be in the order of econd or minute, wherea the potponed ervice could be delayed for everal hour. Thi i the attractive apect for uing the callback option. It allow to create a flexibility by delaying ome of the workload for future proceing, which would improve the ytem performance. The arrival proce of inbound i aumed to be a homogeneou Poion proce with rate λ. Inbound call arrive at a dedicated firt come, firt erved (FCFS) queue with infinite capacity, denoted by queue. We aume that the ervice time for inbound are i.i.d. and exponentially ditributed with rate µ. Cutomer in queue can be impatient. After entering the

4 4 B. Legro et al. / Performance Evaluation 95 (206) 40 Fig. 2. The callback option model. queue, a cutomer will wait a random length of time for ervice to begin. If ervice ha not begun by thi time, the cutomer will abandon. Time before abandonment for inbound are aumed to be i.i.d. and exponentially ditributed with rate β. Becaue of the flexibility of type 2 job, the ytem manager allocate more capacity to real-time ervice. Type job have therefore a trict non-preemptive priority over type 2 job, which mean that if an agent i buy with a job 2, the agent will finih firt thi job before turning to a newly arrived job. The non-preemption priority rule i coherent with the common call center practice, where it i not appropriate to interrupt a converation with a low priority cutomer. In addition, we allow for agent reervation policie for inbound. In other word, we allow an agent to remain idle when queue i empty and queue 2 i not. Thi may reduce the waiting time of future inbound arrival. For imilar multi-channel call center ituation, agent reervation policie have been hown to be efficient [9,2]. If a cutomer accept to be called back, he virtually join a FCFS queue, denoted by queue 2. Due to the nature of the outbound demand, we conider for thi cutomer, the three poibilitie a follow. With probability r, he ha exactly the ame need a the one he had when he firt made her call. In thi cae, the ervice time i aumed to be exponentially ditributed with rate µ (imilarly to an inbound cutomer). With probability r 2 (r + r 2 > 0), he ha already reolved her problem or a part of it. Hence, her ervice time may be horter. We aume in thi cae that the ervice time i exponentially ditributed with rate µ 2 (µ 2 µ ). Finally, with the remaining probability r r 2, the outbound cutomer i not available, and an agent will try again to call her back later on. To handle uch a ituation, we aume that the agent pend a random duration aumed to be exponentially ditributed with rate µ 3. Thi duration correpond to the required time to leave a meage to the cutomer, and to place her back in the queue at the lat poition (he will be called back when he will again reach the firt poition under the FCFS rule). Decription of the call back option. The tate of the ytem at a given time t i defined by four variable: x, y, 2, 3, where x i the number of inbound in queue or in ervice plu the number of outbound in ervice with the ame ervice time requirement a inbound (ervice rate µ ), y i the number of outbound in queue 2, 2 i the number of agent buy with outbound that require a fat ervice (ervice rate µ 2 ), and 3 i the number of agent handling non-available outbound ituation (rate µ 3 ), for x, y 0 and 0 2, 3. Conider a newly arriving inbound call. If at leat one agent i available, the cutomer immediately tart ervice. If all agent are buy and the number of waiting call in queue i trictly lower than a given threhold, denoted by k N, a delay information i announced to the cutomer. The delay information i baed on the ytem tate. We do not retrict the model to a pecific type of information: it could be the length of queue, the expected value or ome quantile of the waiting time, etc. The new inbound cutomer then react to the delay information. She either balk (immediately leave the ytem) with probability α x,2, 3, or join queue with probability α x,2, 3 where he may abandon or tart ervice after ome time duration. We aume that the probability α x,2, 3 increae in the announced delay, i.e., α x+,2, 3 α x,2, 3, for x < + k, 0 2, 3. Note that the probability α x,2, 3 could be choen contant for the cae with no delay information. If the number of waiting call in queue i higher than or equal to k, the ytem provide a delay information a well a a callback option. Exceeding the threhold k capture the fact that cutomer are likely to experience too long waiting time in cae they would requet for a real-time ervice. The delay information i ytem tate-dependent. Concretely, the new inbound cutomer have the following three poibilitie upon her arrival: he balk (immediately leave the ytem) with probability α x,2, 3, or he chooe the callback option and virtually join queue 2 with probability q x,2, 3, or he join queue with probability q x,2, 3 α x,2, 3, for x k, 0 2, 3. Again, we aume that α x+,2, 3 α x,2, 3 and q x+,2, 3 q x,2, 3 for x k and 0 2, 3. Alo, the quantitie α x,2, 3 and q x,2, 3 could be choen contant for x k, 0 2, 3. In uch a cae, we will then imply write them a α or q to implify the preentation. An illutration of the model i given in Fig. 2. Problem formulation. Let u firt define the performance meaure of interet. We denote by W, W 2 and W the random variable meauring the tationary waiting time of erved inbound in queue, the tationary waiting time of outbound in queue 2, and the unconditional tationary waiting time in the queue of an arbitrary job (inbound or outbound), repectively. We alo denote by P a the tationary proportion of inbound that leave the ytem without ervice either by abandoning queue, or by balking upon arrival. The tationary proportion of inbound that balk upon arrival i defined a P b. We finally denote by ψ the tationary probability that a new inbound call become an outbound one.

5 B. Legro et al. / Performance Evaluation 95 (206) 40 5 Table Model notation. Sytem tate decription x Number of inbound in queue or in ervice plu number of outbound (with the ame ervice requirement a inbound) in ervice y Number of outbound in queue 2 2 Number of agent handling fat-erved outbound 3 Number of agent handling non-available outbound ituation Exogenou parameter λ Arrival rate of inbound Number of agent r Probability that an outbound call ha the ame ervice requirement a an inbound one r 2 Probability that an outbound call ha a horter ervice requirement than an inbound one r r 2 Probability that an outbound call in queue 2 i not available µ Service rate of inbound, and alo a part of outbound with the ame ervice requirement µ 2 Service rate for fat-erved outbound µ 3 Service rate for handling non-available outbound β Abandonment rate for each inbound call in queue α x,2, 3 Probability that a new inbound call balk upon arrival q x,2, 3 Probability that an inbound call accept the callback offer upon arrival Control parameter k Threhold on the length of queue, at which we tart to propoe the callback offer c(x, y, 2, 3 ) Curve for the agent reervation policy Performance meaure Ψ Proportion of inbound that accept the callback offer P a Proportion of inbound that leave the ytem without ervice (after a balking or an abandonment) E(W ), E(W 2 ), Expected waiting time for erved inbound in queue, expected waiting time for outbound in queue 2, and unconditional waiting E(W) time in the queue of an arbitrary job (inbound or outbound), repectively We conider an economic framework baed on the holding cot of job and 2, and the cot of lot call (becaue of balking or abandonment). The objective of the ytem manager i to characterize the optimal routing policy which minimize the expected ytem cot, denoted by SC, and given by SC = γ E(W ) + γ 2 E(W 2 ) + γ 3 P a, where γ, γ 2 and γ 3 are the cot parameter, and where E(Z) i the expected value of a given random variable Z. We aume that γ > γ 2 to give more importance to the waiting time of inbound than that of outbound. The control parameter for the call center manager are the threhold k for queue which characterize the callback option, and the agent reervation policy for inbound. For a given tate (x, y, 2, 3 ) (0 x < and y > 0), there are two poible action: the firt one i to erve an outbound call and move to tate (x +, y, 2, 3 ) with probability r, or to tate (x, y, 2 +, 3 ) with probability r 2, or to tate (x, y, 2, 3 + ) with probability r r 2 ; the econd one i to keep the firt outbound in line in queue 2 and tay at tate (x, y, 2, 3 ). The knowledge of the optimal action at each tate define a function denoted by c(x, y, 2, 3 ). The curve of thi function eparate the tate where the optimal action i to erve an outbound call from thoe where it i optimal to keep an outbound call in queue 2. The function c(x, y, 2, 3 ) define therefore the agent reervation policy. It will be characterized in Section 3. A ummary of the model notation i given in Table. The call center model decribed above i referred to a Model G (general model). Becaue of it complexity, we define ubmodel that correpond to variou pecial cae, for which it i eaier to oberve and prove inight. We denote by Model A the ubmodel where outbound have the ame ervice rate a inbound and thee are available when they are called back (r = and r 2 = 0), by Model B a ubmodel of Model A where inbound are infinitely patient (β = 0), by Model C a particular cae of Model B where the balking and callback parameter are aumed to be contant (for example when no information i given to arriving cutomer). We alo define Model NI (non-idling model) a ubmodel of Model G where idling i not allowed (i.e., the firt outbound call in queue 2 tart ervice a oon a an agent become available and queue i empty). An illutration of the ubmodel i depicted in Fig. 3. Markov deciion proce approach. For Model G, we formulate the routing problem a a Markov deciion proce (MDP). Since we are conidering long-term average performance, it i optimal to chedule job at arrival, ervice completion or abandonment time. If it i optimal to keep a erver idle at a given time, then the action remain optimal until the next event in the ytem. Thi reult follow directly from the continuou-time Bellman equation [20, Chapter ]. Therefore, it uffice to conider the ytem only at arrival, ervice completion or abandonment time. Due to the call abandonment in queue, the total event rate i not bounded. We therefore ue the traditional approach where we aume that queue ha a limited capacity N (N 0). The parameter N i choen high enough to approximate the real ytem. The total event rate i then uniformly bounded by λ + max(µ, µ 2, µ 3 ) + Nβ, and without lo of generality, we aume that it i equal to one. We next ue the well known uniformization technique [20, Chapter 8], which allow to apply dicrete-time dynamic programming to characterize the optimal routing policy. The poible action for an agent jut after a ervice completion (and queue i empty) are either to remain idle, or to erve an outbound call if queue 2 i not empty. We chooe to formulate a 2-tep value function, in order to eparate

6 6 B. Legro et al. / Performance Evaluation 95 (206) 40 Fig. 3. The ubmodel. tranition and action and implify the involved expreion. We define the equence U n (x, y, 2, 3 ) and V n (x, y, 2, 3 ) over n tep, for n, x, y 0 and 0 2, 3. For n 0, we have U n+ (x, y, 2, 3 ) = γ (x ) + + γ 2 y + λ (0 x <)V n (x +, y, 2, 3 ) + ( x <+k) ( αx,2, 3 )V n (x +, y, 2, 3 ) + α x,2, 3 (V n (x, y, 2, 3 ) + γ 3 ) + (+k x <+N)(q x,2, 3 V n (x, y +, 2, 3 ) + α x,2, 3 (V n (x, y, 2, 3 ) + γ 3 ) + ( q x,2, 3 α x,2, 3 )V n (x +, y, 2, 3 )) + (x =+N)(q N,2, 3 V n (x, y +, 2, 3 ) + ( q N,2, 3 )(V n (x, y, 2, 3 ) + γ 3 )) + β(x ) + (V n (x, y, 2, 3 ) + γ 3 ) + min( 2 3, x)µ V n (x, y, 2, 3 ) + 2 µ 2 V n (x, y, 2, 3 ) + 3 µ 3 V n (x, y +, 2, 3 ) + λ β(x ) + min( 2 3, x)µ 2 µ 2 3 µ 3 Vn (x, y, 2, 3 ), for x, y 0, and 0 2, 3, () where (x A) i the indicator function of a ubet A, and V n+ (x, y, 2, 3 ) = min(r U n+ (x +, y, 2, 3 ) + r 2 U n+ (x, y, 2 +, 3 ) + ( r r 2 )U n+ (x, y, 2, 3 + ), U n+ (x, y, 2, 3 )), for y > 0 and 0 x < and V n+ (x, y, 2, 3 ) = U n+ (x, y, 2, 3 ) in the remaining cae. We chooe V 0 (x, y, 2, 3 ) = U 0 (x, y, 2, 3 ) = 0, for x, y 0, and The tranition at boundary tate x = N are choen uch that the monotonicity propertie of the value function are maintained. The value of thi choice i proven in the proof of Theorem in Section 3.2. Another poibility to maintain the monotonicity propertie i to ue the moothed rate truncation a propoed by Bhulai et al. [2], however, thi would imply a more complicate expreion of the value function in our etting. The long-term average optimal action can be obtained through value iteration, by recurively evaluating V n uing Eq. (), for n 0. A n tend to infinity, the minimizing action converge to the optimal one [20]. For 0 x < and y > 0, the minimizing action i choen between keeping an outbound call in queue 2 or tarting the ervice of thi call. For x , we do not conider any control action becaue of the priority for inbound (i.e., no poibility of having an idle agent while a call i waiting in queue ). 3. Optimal agent reervation policy We conider the ingle, the two-erver and the multi-erver cae. For the multi-erver cae of Model G, we firt prove a preliminary reult tating that when all agent are idling and queue 2 i not empty, then it i optimal to erve at leat the firt outbound call in line. A corollary of thi reult i that non-idling i optimal in the ingle-erver cae. In the twoerver cae, we prove in Theorem the optimal reervation policy for Model A. It i a threhold policy on the number of waiting outbound in queue 2. For the multi-erver cae of Model A and G, we conjecture that the optimal routing follow a tate-dependent threhold policy, i.e., a witching curve. For Model A, the witching curve i only baed on the number of outbound in queue 2 and the number of buy agent. In addition to that, for Model G, the optimal policy depend on the number of each job type in ervice. The reult for the multi-erver cae i intuitive and a tandard extenion, in MDP problem, of the proved reult in the ingle and two-erver cae. It i however very hard to obtain a proof becaue of the growing dimenionality of the

7 B. Legro et al. / Performance Evaluation 95 (206) 40 7 underlying tate pace and the problem et down by the departure term. Thi proof i related to a well known fundamental queueing control problem, for which no rigorou proof doe exit yet. We believe that our proof for the two-erver cae hould give ome indication that would motivate future reearch. Thi open quetion conit in howing the propagation of a monotonicity relation through the minimizing operator. In Remark inide the proof of Theorem in Appendix A, we provide the mathematical detail of what hould be proven to rigorouly obtain the multi-erver reult. It reduce to that for the well known routing problem in the heterogeneou multi-erver queue, where the objective i to find a non-preemptive routing policy that minimize the long run average time in the ytem [22 24]. For a background on thi quetion, we refer the reader to Koole [25]. 3.. Preliminary reult Propoition provide a preliminary reult for Model G. Propoition. In the multi-erver cae of Model G, if all agent are idling and queue 2 i not empty, then it i optimal to erve at leat an outbound call. Proof. For γ 2 > 0, it i clear that an outbound call in queue 2 ha to be erved at one point. Otherwie, queue 2 would contain an infinite number of outbound due to the FCFS rule. Therefore, a policy which would not erve an outbound call cannot be optimal. We next prove that the bet ituation for the ervice of an outbound call i when all agent are idling. Serving an outbound call alway improve the performance of outbound whether thi outbound call i erved when all agent are idling or in another ituation. An outbound taken in ervice would deteriorate the performance of inbound if new inbound arrive at a buy ytem while thi outbound call i till in ervice. The lowet value of the probability of uch an event i reached in the cae thi outbound call ha been taken in ervice when all agent are idling. Moreover, an outbound call ervice duration doe not depend on the ytem tate. Thu, erving an outbound call when all agent are idling improve the performance of outbound and ha the mallet probability to deteriorate the performance meaure of inbound. Since all outbound ha to be erved at one point, an optimal tate-dependent policy force the ervice of outbound, if any, when all agent are idle. We next deduce the optimal agent reervation policy for the ingle-erver cae of Model G. Corollary. In the ingle-erver cae of Model G, the optimal agent reervation policy i the non-idling policy. The proof of Corollary directly follow from Propoition. In Section of the online upplement (ee Appendix H), we propoe another proof of thi corollary for Model A uing an MDP approach Two-erver reult for Model A In the two-erver cae, uing Propoition, we never encounter ituation for the optimal policy where the two agent are idling and at leat one outbound call i in queue 2. When one erver i buy, we prove in Theorem that the optimal policy in Model A i of threhold type for the reervation of the other erver. Theorem. In the two-erver cae for Model A, when one agent i buy, there exit a threhold on the number of outbound in queue 2, at and beyond which it i optimal to erve the firt waiting outbound in line, and it i optimal to not erve outbound in the remaining cae. The proof i given in Appendix A. It i baed on the propagation of monotonicity reult of the value function a defined in Section 2. Thi type of proof i tandard in MDP problem [25]. Yet, our reult cannot directly follow from [25] for the following reaon. The exiting reult concern motly the ingle-erver-one-dimenional cae. Le i doable in the multidimenional cae for the propagation of the reult through the minimizing operator. Moreover, abandonment from queue i allowed here, a feature that often break the monotonicity propertie when pace truncation i required. We how in our proof that the monotonicity propertie are maintained. Finally, the complexity of the proof come from the arrival term, which i pecific in our model and require a pecial conideration, becaue the two queue are involved and the cutomer reaction i tate-dependent Multi-erver conjecture Let u now comeback to the multi-erver cae. Uing the value function defined in Section 2, we conjecture that the optimal policy i of witch type. For both Model A and G, we conduct a numerical tudy from which we deduce the witching curve which eparate tate where it i optimal to erve an outbound call from thoe where it i not. We alo examine the impact of the ytem parameter on the reervation policy.

8 8 B. Legro et al. / Performance Evaluation 95 (206) Switching curve for Model A For Model A, we do not need to ditinguih between inbound and outbound in ervice. Let u rewrite the value function for Model A (µ = µ 2 = µ, r = ). We have for n 0, with U n+ (x, y) = γ (x ) + + γ 2 y + λ (0 x<) V n (x +, y) + ( x<+k) (( α x )V n (x +, y) + α x (V n (x, y) + γ 3 )) + (+k x<+n) (q x V n (x, y + ) + α x (V n (x, y) + γ 3 ) + ( q x α x )V n (x +, y)) + (x=+n) (q N V n (x, y + ) + ( q N )(V n (x, y) + γ 3 )) + β(x ) + (V n (x, y) + γ 3 ) + min(, x)µv n (x, y) + λ β(x ) + min(, x)µ Vn (x, y), for x, y 0, V n+ (x, y) = min(u n+ (x +, y ), U n+ (x, y)), for y > 0 and 0 x < and V n+ (x, y) = U n+ (x, y) in the remaining cae. We chooe V 0 (x, y) = U 0 (x, y) = 0, for x, y 0. We conjecture that the optimal policy i a function of x (number of call in ervice plu number of inbound in queue ) and y (number of outbound in queue 2). Fig. 4 give variou optimal witching curve to illutrate the impact of the ytem parameter on the optimal policy. The abcia axi in each figure repreent the overall number of job in the ytem (number of outbound in queue 2 plu number of call in ervice) and the ordinate axi repreent the number of call in ervice. We only conider tate where 0 x <. For the remaining tate, the only poible action i to keep outbound in the queue. The optimal action can be read from the figure. Conider a given point (x + y, x) (0 x < and y > 0). If thi point i trictly under the curve, then it i optimal to erve an outbound call and therefore move from (x + y, x) to (x + + y, x + ) = (x + y, x + ). If thi new point i trictly under the curve then the optimal action i to erve another outbound call. We continue to take the deciion to erve by moving on a vertical line until we reach the curve. On the witching curve or above, the optimal action i to keep outbound in the queue. The value to chooe x + y in abcia intead of y i to oberve the evolution from a non-optimal point to the optimal one on a vertical line intead of a diagonal one. The curve in dahed line repreent the non-idling policy. We oberve that when x = 0 and y > 0, the optimal action i alway to erve an outbound call (thi hold from Propoition ). Given that the witching curve i increaing in x + y, it i an increaing tep function. It i given by c(x + y) = min(y 0, x + y) + (x+y y ) + (x+y y2 ) + + (x+y y y0 ), (2) where y 0 < y < y 2 < < y y0. The parameter y 0,..., y y0 are the level that repreent the changing point of the witching curve. Uing Propoition, we have y 0 0. Eq. (2) can be interpreted a follow. Aume we have x + y job in the ytem (x buy agent and y outbound in queue 2). If x + y < y, then it i optimal to have at mot y 0 tak in ervice, i.e., if x < y 0 we move from tate (x, y) to tate (min(y 0, x + y), y (min(y 0, x + y) x)), and if x y 0 we tay in tate (x, y). If y x + y < y 2, then at mot y 0 + job hould be in ervice, i.e., if x < y 0 + we move from tate (x, y) to tate (min(y 0 +, x+y), y (min(y 0 +, x+y) x)), and if x y 0 + we tay in tate (x, y), and o on. Finally, if y y y0, then at mot y 0 + y 0 = job hould be in ervice. In other word, when x + y y y0, no agent are reerved for inbound and it i optimal to move from tate (x, y) to tate (min(, x + y), y (min(, x + y) x)). A qualitative interpretation of Eq. (2) i that the more numerou queued outbound and the le buy are the agent, the more likely the optimal deciion would be to erve an outbound call. Thi witch type policy in the multi-erver cae i a tandard extenion of the threhold policy in the two-erver cae. The new element in the multi-erver cae i that the deciion to erve an outbound call hould no longer only depend on the length of queue 2, ince more than one agent might be involved. For a given ituation with x buy agent and x idle agent, the optimal policy i a threhold policy on the length of queue 2. Thi lead, a a conequence, to a witch type policy. We next examine the impact of the parameter on the reervation policy. In Propoition 2, we prove that the more importance i given to inbound and the le cutomer are likely to accept the callback offer, the higher hould be the reervation for inbound. Propoition 2. Conider two ituation with identical arrival and departure parameter (λ, α x for x, β, and µ). The firt ituation ha the cot parameter γ, γ 2 and γ 3 and the econd one ha γ, γ 2 and γ 3. The callback parameter are contant for both ituation. They are q and q + q for the firt and econd ituation, repectively. If γ γ, γ 2 γ 2, γ 3 γ 3, q 0, then the firt ituation require more reervation than the econd one. In other word, the witching curve i lower for the firt ituation. The proof of thi propoition i given in Appendix B. The impact of the cot parameter γ 2 i illutrated in Fig. 4(a), i.e., the witching curve increae (the reervation decreae) in γ 2. The oppoite i true when γ or γ 3 increae. Fig. 4(b) illutrate the impact of a contant callback parameter (q x = q for x + k). It how that the more cutomer are likely to accept the callback option, the higher i the witching curve (le reervation for inbound). The ame obervation hold when q x i not contant (Fig. 4(c)). The key factor, whether the callback parameter i contant or not, i the proportion of outbound.

9 B. Legro et al. / Performance Evaluation 95 (206) 40 9 (a) Impact of γ 2 (λ = 4, q = 40%, α = β = 0). (b) Impact of q (γ 2 = 0.05, λ = 4, α = β = 0). (c) Impact of q +k+x (λ = 4, γ 2 = 0.05, α = β = 0, x 0). (d) Impact of λ (γ 2 = 0.05, q = 40%, α = β = 0). (e) Impact of α (λ = 4, γ 2 = 0.05, q = 40%, β = 0.). (f) Impact of α +x (λ = 4, γ 2 = 0.05, q = 40%, β = 0, x 0). Fig. 4. Optimal witching curve (µ = 0.2, r =, = 28, γ =, k = 5, γ 3 = 0.5). A le intuitive obervation i that the witching curve i not monotone in the workload, defined a λ/µ (Fig. 4(d)). We oberve that reervation doe not happen in the extreme ituation of light or heavy workload. For light workload ituation, the ytem capacity i high enough, uch that both call type experience mall waiting time. Then, the reervation for inbound call doe not need to be ubtantial. For high workload ituation, queue i often long. Thu, a high proportion of cutomer would chooe the callback option and join queue 2. Given that queue 2 i alo long, the ytem hould not further deteriorate the waiting of outbound by reerving agent for job. However, for an intermediate ituation, with a moderate workload, job 2 are le numerou, and do not therefore need to have acce to all agent. The ytem may then conider agent reervation for job. Fig. 4(e) reveal that the impact of the balking parameter α x and the abandonment parameter β are not imilar to that of the workload. For high value of α x or β, the ytem capacity i high enough to achieve mall waiting time. However, the proportion of abandonment i high, o, the reervation for inbound need to be important to avoid too much abandonment. For low value of α x or β, the reervation policy mainly depend on the workload λ/µ (ee Fig. 4(e) and (f)) Switching curve for Model G We now conider Model G. Fig. 5 and 6 illutrate the witching curve for the optimal policy in Model G. Again, the curve in dahed line repreent the non-idling policy.

10 0 B. Legro et al. / Performance Evaluation 95 (206) 40 Fig. 5. Optimal witching curve (λ = 3.8, q = 40%, α = β = 0, γ =, γ 2 = 0.05, k = 5, µ = 0.2, µ 2 =, µ 3 = 0, r = r 2 = /3, = 28). A expected, we oberve that the optimal deciion are not only baed on the number of outbound in queue 2 and the number of buy agent a for Model A, but alo the identity of the job in ervice. We ditinguih three different zone delimited by two witching curve. A firt witching curve i defined for the cae where all buy agent are buy with rate µ ( 2 = 3 = 0). Thi ituation i the wort for the occupancy of the agent, becaue µ 3 µ 2 µ. Thu, under thi firt witching curve, for any tate with le buy agent or more outbound in queue 2, the optimal deciion i to erve an outbound call (if any), i.e., we move from tate (x+y , x ) to tate (x++y , x ) = (x + y , x ). A econd witching curve i defined for the cae were all buy agent are buy with rate µ 3 (x = 2 = 0). Thi ituation i the bet for the occupancy of the agent. On and above thi econd witching curve, for any tate with more buy agent and le outbound in queue 2, the optimal deciion i to keep all outbound in queue 2. The ordering µ 3 µ 2 µ jutifie that the firt witching curve i below the econd one. Even in the cae µ 3 = µ 2, the econd witching curve (x = 2 = 0) i till higher than a witching curve where all buy agent are buy with rate µ 2 (x = 3 = 0). The reaon i the high need of erving outbound when all agent are buy with rate µ 3. If the agent are all handling a non-available outbound ituation, they would not reduce the number of outbound in the ytem, o, the need for erving outbound doe not reduce. Yet, for ituation with mall number of cutomer in the ytem or high number of cutomer in queue 2, the two extreme witching curve (correponding to 2 = 3 = 0 and x = 2 = 0) coincide. Therefore, there only exit a finite number of tate where the optimal deciion depend on the identity of the job in ervice. Fig. 6(a) reveal that the two extreme witching curve get cloer to one another a r, µ 2, or µ 3 increae. The reaon i the imilarity between the ervice requirement of inbound and outbound. Fig. 6(b) reveal that a r + r 2 decreae, the two extreme witching curve get higher, i.e., le agent reervation. The reaon i related to the difficulty of erving an outbound call. When agent are often handling non-available outbound ituation, it i difficult to reduce the length of queue 2, therefore, outbound hould benefit from more availability of the agent. Similarly to Model A, ince the witching curve i increaing in x + y , it i an increaing tep function. Given that agent handle 3 different type of job, we define the 3 variable increaing function b(x, 2, 3 ) which give the buyne of the agent team. Becaue the number of agent i finite, we aume without lo of generality that 0 b(x, 2, 3 ). Thi buyne function correct the witching curve, defined for Model A, into c(x + y ) = min(y 0, x + y ) (b(x,2, 3 ) b 0 ) + (x+y y ) (b(x,2, 3 ) b ) + (x+y y 2 ) (b(x,2, 3 ) b 2 ) + + (x+y y y0 ) (b(x,2, 3 ) b y0 ) + (x+y y y0 ), where y 0 < y < y 2 < < y y0 and 0 < b 0 b b y0. The parameter y i, 0 i y 0, have the ame ignification a thoe for Model A. The parameter b i, 0 i y 0, are the level of change of the buyne of the agent team. The value of the b i can be determined uing value iteration. From the numerical experiment, we oberve that the value of the b i are different than one only for mall value of i. Thi implie that the buyne of the agent team affect the optimal deciion only when the number of buy agent i low. The reaon i related to the blocking rik for an inbound call. When mot of the agent are idling, the deciion to erve an outbound call would mot likely not block the agent team. In uch a ituation, what affect the deciion i then the identity of job in ervice. In the oppoite cae, when mot of the agent are buy, the ervice of an outbound call could eaily lead to a blocking ituation (waiting time for inbound call). In uch a ituation, what affect the deciion i then the total number of buy agent (x ) and the length of queue 2 (y), more than the identity of the job in ervice. 4. Performance analyi We compute the tationary performance meaure. In Section 4., we profit from the contant tranition rate and propoe an exact algorithm for Model C. In Section 4.2, we provide a controlled approximation baed on value iteration

11 B. Legro et al. / Performance Evaluation 95 (206) 40 (a) Example with r = 80%, r 2 = 5%, µ = 0.2, µ 2 = 0.5 and µ 3 = 0. (b) Example with r = 0%, r 2 = 0%, µ = 0.2, µ 2 = and µ 3 = 0. Fig. 6. Optimal witching curve (λ = 3.8, q = 40%, α = β = 0, γ =, γ 2 = 0.05, k = 5, = 28). Fig. 7. Markov chain for Model C (q = q + α). for Model A and G. In Section 4.3, we conider pecial cae of agent reervation for Model C (Section 4.3.) and the nonidling cae for Model A (Section 4.3.2). Thi allow to obtain cloed-form expreion for the bound of the performance meaure of Model A and C. 4.. Model C We compute here E(W ), E(W 2 ), P b and Ψ. Our approach i baed on the analyi of the underlying Markov chain. We compute the tationary probabilitie of the ytem tate by olving a ytem of linear difference equation. We do o by olving the involved homogeneou equation defined on the et of complex number. Although ome quantitie contain infinite ummation, we provide a method that allow to do the exact computation within a finite number of calculation. Conider the tochatic proce {(x(t), y(t)), t 0}, where x(t) denote the number of call in queue (job ) or in ervice (job or 2); and y(t) denote that in queue 2 (job 2) at a given time t 0. We have x(t), y(t) {0,, 2,...}, for t 0. A inter-arrival and ervice time are exponentially ditributed, {(x(t), y(t)), t 0} i a Markov chain. An illutration of thi Markov chain in given in Fig. 7. We denote by p x,y the tationary probability to be in tate (x, y), for x, y N. In what follow, we compute the tationary probabilitie, from which we thereafter deduce the ytem performance meaure of interet. To implify the preentation of the analyi, we divide it into the following 7 tep:

12 2 B. Legro et al. / Performance Evaluation 95 (206) 40 Step. We provide the et of equilibrium equation relating the tationary probabilitie. Step 2. We implify the expreion of p x,y, for x + k and y 0, by expreing them a a function of only two tate probabilitie from the row y in the Markov chain. Step 3. We how how p x,y, for x + k and y 0, can be computed a a function of p +k,0, p +k,,..., p +k,y. Step 4. We evaluate all tationary probabilitie for x 0 and y = 0 a a function of p 0,0. Step 5. For y 0, we develop a recurrence method to compute all tationary probabilitie of row y + in the Markov chain a a function of the previou row. Thu all tationary probabilitie can be derived a a function of p 0,0. Step 6. Although p 0,0 involve an infinite ummation, we provide a method to compute it within a finite number of calculation. Step 7. We finally derive the ytem performance meaure a a function of the tationary probabilitie. The detail for each tep are given in Appendix C Model A and G We compute here E(W ), E(W 2 ), P a, and Ψ. We propoe a numerical method baed on the iterative computation of the dynamic programming operator. For Model A, auming the witch policy a defined in Section 3.3., the value function can be rewritten, for n 0, a V n+ (x, y) = γ (x ) + + γ 2 y + λ (0 x<) V n (x +, y) + ( x<+k) (( α x )V n (x +, y) + α x (V n (x, y) + γ 3 )) + (+k x<+n) (q x (V n (x, y + ) + γ 4 ) + α x (V n (x, y) + γ 3 ) + ( q x α x )V n (x +, y)) + (x=+n) (q N (V n (x, y + ) + γ 4 ) + ( q N )(V n (x, y) + γ 3 )) + β(x ) + (V n (x, y) + γ 3 ) + min(, x)µ (y>0) ( (x+y y,x y 0 ) + (y <x+y y 2,x y 0 +) + (y2 <x+y y 3,x y 0 +2) + + (y y0 <x+y,x ))V n (x, y ) + (y>0) ( (x+y y,x y 0 ) + (y <x+y y 2,x y 0 +) + (y2 <x+y y 3,x y 0 +2) + + (y y0 <x+y,x )) V n (x, y) + λ β(x ) + min(, x)µ Vn (x, y), for x, y 0, with V 0 (x, y) = 0, for x, y 0. For Model G, auming the witch policy a defined in Section 3.3.2, the value function can be rewritten, for n 0, a U n+ (x, y, 2, 3 ) = γ (x ) + + γ 2 y + λ (0 x <)V n (x +, y, 2, 3 ) + ( x <+k) ( αx,2, 3 )V n (x +, y, 2, 3 ) + α x,2, 3 (V n (x, y, 2, 3 ) + γ 3 ) + (+k x <+N)(q x,2, 3 (V n (x, y +, 2, 3 ) + γ 4 ) + α x,2, 3 (V n (x, y, 2, 3 ) + γ 3 ) + ( q x,2, 3 α x,2, 3 )V n (x +, y, 2, 3 )) + (x =+N)(q N,2, 3 (V n (x, y +, 2, 3 ) + γ 4 ) + ( q N,2, 3 )(V n (x, y, 2, 3 ) + γ 3 )) + β(x ) + (V n (x, y, 2, 3 ) + γ 3 ) + min( 2 3, x)µ (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x ))(r V n (x, y, 2, 3 ) + r 2 V n (x, y, 2 +, 3 ) + ( r r 2 )V n (x, y, 2, 3 + )) + (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x )) V n (x, y, 2, 3 ) + 2 µ 2 (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x ))(r V n (x +, y, 2, 3 ) + r 2 V n (x, y, 2, 3 ) + ( r r 2 )V n (x, y, 2, 3 + )) + (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x )) V n (x, y, 2, 3 ) + 3 µ 3 (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x ))(r V n (x +, y, 2, 3 ) + r 2 V n (x, y, 2 +, 3 ) + ( r r 2 )V n (x, y, 2, 3 ))

13 B. Legro et al. / Performance Evaluation 95 (206) (y>0) ( (x+y y,x y 0,b(x, 2, 3 ) b 0 ) + (y <x+y y 2,x y 0 +,b(x, 2, 3 ) b ) + + (y y0 <x+y ,x )) V n (x, y, 2, 3 ) + λ β(x ) + min( 2 3, x)µ 2 µ 2 3 µ 3 Vn (x, y, 2, 3 ), for x, y 0, and 0 2, 3, with V 0 (x, y, 2, 3 ) = 0, for x, y 0 and 0 2, 3. In both cae (Model A and G), the tandard way of obtaining the long-term performance meaure i through value iteration, by recurively evaluating V n, for n 0. A n tend to infinity, the difference V n+ (x, y, 2, 3 ) V n (x, y, 2, 3 ) converge to the deired metric. Thu, we top the iteration until the following criterion i met max {V n+ (x, y, 2, 3 ) V n (x, y, 2, 3 )} min {V n+ (x, y, 2, 3 ) V n (x, y, 2, 3 )} < ϵ, x,y, 2, 3 x,y, 2, 3 for ome given mall ϵ. In what follow we precie the parameter in the value function which allow to compute the deired performance meaure. One can calculate the expected number of cutomer in queue, ay E(N ), by letting γ =, γ 2 = 0, γ 3 = 0, γ 4 = 0 in the value function; the expected number of cutomer in queue 2, ay E(N 2 ), by letting γ = 0, γ 2 =, γ 3 = 0, γ 4 = 0; the proportion of cutomer who abandon the ytem, P a, by letting γ = 0, γ 2 = 0, γ 3 = /λ, γ 4 = 0; the proportion of cutomer who chooe the callback offer, Ψ, by letting γ = 0, γ 2 = 0, γ 3 = 0, γ 4 = /λ. Uing next the Little law, we obtain the expected waiting time for erved cutomer in queue, E(W ) = E(N ) ; and the expected waiting λ( P a Ψ ) time in queue 2, E(W 2 ) = E(N 2) λψ Special cae We conider here ome pecial cae of agent reervation for Model C and the non-idling cae for Model A Special reervation cae for Model C We define for Model C the threhold y 0 on the number of buy agent. If the number of buy agent i lower than or equal to y 0 ( y 0 ) and at leat one outbound call i in queue 2, then we erve thi outbound call. In the remaining cae, we do not erve outbound. Therefore, the witching curve of thi policy i c(x + y) = min(x + y, y 0 ). Since the optimal action i to erve an outbound call when all agent are idling (Propoition ), the wort policy for outbound (the bet cae for inbound) conit of erving an outbound call only when all agent are idling. We refer to the latter a the highet reervation policy. It correpond to the cae y 0 =. A for the non-idling policy, it correpond to the cae y 0 =. The analyi of thi policy i a deduced from that of Section 4.. In Corollary 2, we give cloed-form expreion for E(W ), P b and Ψ a a function of y 0. The proof i given in Section 2 of the online upplement (ee Appendix H). Corollary 2. For y 0, we have k q a! Ψ = a( q α) with P b = E(W ) = α a! p 0,0 q a y 0 a y 0 y 0!! y 0 p 0,0 = q a y 0 a y 0 y 0!! k a! λ( Ψ P b ) a x x! + + p 0,0 k a( q α) k a( q α) y 0 k p 0,0 x, k, a( q α) x + q a a y 0 y 0! y 0! k k a( q α) + a( q α) a( q α) 2 k a( q α) a x+y 0 (y 0 +x)! + k a () x + a () k! x! q a a y 0 y 0! y 0! () k k a( q α) k a( q α).,

14 4 B. Legro et al. / Performance Evaluation 95 (206) 40 In Appendix D, further implification of the above expreion are given for the multi-erver pecial cae: y 0 = (highet reervation) and y 0 = (non-idling) Non-idling cae for Model A We provide in Propoition 3 cloed-form expreion for E(W ), P b, P a and Ψ. The proof i given in Section 3 of the online upplement (ee Appendix H). Propoition 3. For the non-idling cae, we have a k x! λ x αi + µ+iβ a p 0,0 = x i= x= x! + k a ( α i ) and Ψ = P b = P a = i= q k+x P +k+x = a (P p,0 ) = α x P +x = a! E(W ) = k a! α x + x β λ k x α x λ x P a P b β( Ψ P a ). λ x x i= i= a! a αi µ+iβ a αi µ+iβ a k λ x+k i= ( α i ) x+k ( α i q i ) i=k+ x+k (µ+iβ) i= q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= k ( α i ) i= k ( α i ) i= + α k+x x= i= k ( α i ) i= x= q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= k ( α i ) q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= + α x+k + (x + k) β k ( α λ i ) k ( α i ) i= i= q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i=, p 0,0, q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= p 0,0, q k+x λ x+k x+k ( α i q i ) i=k+ x+k (µ+iβ) i= p 0,0, 5. Numerical experiment We invetigate the benefit of the callback offer and the impact of the policy parameter on the ytem performance. The policy parameter are the tate-dependent number of agent reerved for inbound, and the threhold k for the callback propoition. Becaue of the analyi complexity, the concluion we derive are mainly baed on numerical obervation. For ome particular cae, we develop analytical reult that provide better undertanding and upport the concluion. 5.. Benefit of the callback offer We evaluate the benefit of the callback offer on the performance meaure, in relation with the ytem workload. We conider the ingle-erver non-idling cae (optimal policy, Corollary ) of Model C with k = 0 (optimal k, Propoition 5). The objective i to provide a imple cloed-form expreion of the difference between the ytem cot in two ituation; with and without the callback offer. Uing Corollary 2, for y 0 = = and k = 0 (ituation with the callback option), we obtain

15 B. Legro et al. / Performance Evaluation 95 (206) 40 5 (a) α = 0% and q = 40%. (b) α = 60% and q = 0%. Fig. 8. Sytem cot with and without the callback offer (non-idling cae of Model C, =, k = 0, SC = E(W ) + 0.E(W 2 )). +a(α/q)( α)( ( q α)a) P b = αa, Ψ = qa, E(W +αa +αa ) = ( q α)a µ ( ( q α)a)( qa), and E(W 2) =. For y µ ( )( ( q α)a) 0 = = and k = (ituation without the callback option), we obtain after implification P b = αa, Ψ = 0 and E(W +αa ) = ( α)a. The lat µ ( ( α)a) ituation reduce to an M/M/ queue with balking. We do not provide for it the expreion of E(W 2 ) becaue outbound do not exit when the callback option i not offered (it i imply conidered a zero). The difference in ytem cot between the two ituation (without the offer minu with the offer), denoted by (a), i then (a) = γ ( α)a µ ( ( α)a) ( q α)a γ 2 + a(α/q)( α)( ( q α)a). ( ( q α)a)( qa) µ ( a( α))( ( q α)a) Thi difference can be either poitive or negative. In Fig. 8, we illutrate the impact of the arrival rate on the ytem cot in the two ituation (with and without the callback offer). Fig. 8(a) reveal that the callback option can improve the ytem cot when the arrival rate i high. Since we have (0) = γ 2 µ < 0, under light workload, the callback option hould not be provided. Roughly peaking, if a call goe to queue 2, then thi call would looe priority. Thi i not ueful becaue call in queue have anyway hort waiting time. Under a heavy workload ituation, the preference i not alway for uing the callback option. A a tend to α, (a) become equivalent to γ +αa γ 2 qa. Thi expreion can either be poitive or negative depending on the ign of it numerator. µ( ( α)a) The higher γ or q are in comparion with γ 2 and α, the more likely thi expreion would be poitive. More qualitatively, thi implie that the callback offer ha a poitive effect only if a high importance i given to inbound and if cutomer would eaily accept the callback offer (Fig. 8(a)). In the cae where cutomer are more likely to balk than to accept the callback offer (Fig. 8(b)), providing a callback offer would deteriorate the ytem cot. Finally, we numerically oberve that the function (a) i increaing in a. Thi induce that the benefit or the lo due to the ue of the callback option would be more apparent under a heavy workload ituation. Thi i preciely the value of the callback offer, that could better manage congeted ituation Impact of agent reervation We examine here the impact of the reervation on the ytem performance. We firt conider the two-erver cae in Section 5.2., and econd the multi-erver cae in Section Two-erver cae The reaon for conidering the two-erver cae i to allow the reervation policy to be only dependent on one parameter; the threhold y 0 that define the limit on the length of queue 2 at and above which no agent hould be reerved for inbound. An illutration of the effect of y 0 on the performance meaure i given for Model C in Fig. 9. The reervation for inbound increae in y 0. Therefore, E(W ) and P a decreae in y 0 (Fig. 9(a) and (c)), and E(W 2 ) increae in y 0 (Fig. 9(b)). We oberve from Fig. 9(b) that reervation deteriorate the overall expected waiting time, E(W). Thi i related to two reaon. The firt one i that agent reervation create unproductive idling ituation with one agent idle while queue 2 i not empty. The latter deteriorate the overall performance of the ytem. The econd reaon i related to the reduction of balking and abandonment of inbound. Since reervation induce more availability of agent for inbound, it reduce the proportion of lot inbound (P a ). Agent have then to treat more tak (recall that outbound do not abandon) a hown in Fig. 9(d). Thi deteriorate the overall expected waiting time. It i the negative effect for reducing the proportion of abandonment.

16 6 B. Legro et al. / Performance Evaluation 95 (206) 40 (a) E(W ). (b) E(W 2 ) and E(W). (c) P a. (d) Agent utilization. Fig. 9. Effect of y 0 ( = 2, λ =.9, µ = µ 2 =, α = q = 30%, r = 00%, k = 0, β = 0) Impact of the call center ize We invetigate the impact of the call center ize on the performance meaure and it relation with the reervation policy. Conider Model C with k = 0 (optimal k, Propoition 5) and let u define ρ a ρ = λ. Propoition 4 provide µ convexity reult jutifying that reervation policie bring higher improvement in large call center than in mall one (recall that non-idling i optimal in the ingle-erver cae). For large call center, thee reult upport therefore the well known notion that only limited erver pooling/flexibility/availability i needed [26,27]. Propoition 4. Conider the non-idling cae of Model C. For the optimal threhold on queue (k = 0), P b, Ψ, E(W ) and E(W) are decreaing and convex in, when ρ = a/ and λ are held contant. The proof of the propoition i given in Appendix E. Table 2 illutrate for Model C the behavior of the performance meaure a a function of, when ρ i held contant and equal to In the econd and third column, we give the upper and lower bound of E(W ) uing the reult of Section 4.3., repectively. The upper bound i obtained in the non-idling cae and the lower bound i obtained in the highet reervation cae. In the fourth column, we compute the relative difference between the upper and lower bound of E(W ) o a to ae the poibilitie of performance improvement for inbound. We oberve that the higher i, the more it i poible to improve E(W ). In the fifth column, we give the lower bound of E(W 2 ) obtained in the non-idling cae. We do not give upper bound of E(W 2 ) from the extreme reervation cae, ince thee are too high and do not provide intereting ituation for the optimization problem. In the ixth column, we give the total expected ytem cot in the non-idling cae. We oberve that the expected total cot decreae in. In the eventh, eight and ninth column, we give the optimal performance meaure obtained via the algorithm propoed in Section 4., under the optimal reervation policy. In the lat column, we compute the relative difference between the optimal ytem cot and that obtained in the non-idling cae. We oberve that the larger i the call center, the higher i the agent reervation for inbound. For mall call center (for 5 in Table 2), non-idling i optimal. A increae, we oberve for the optimal reervation policy that E(W ) move from the neighborhood of it upper bound to that of it lower bound, and that E(W 2 ) remain relatively cloe to it lower bound. Thi implie that the relative difference between the ytem cot in the optimal cae and that in the non-idling cae increae in the call center ize. In ummary, the main concluion of thi ection i that reervation ha more potential of improvement in large call center, ince large call center allow for le flexibility than mall one.

17 B. Legro et al. / Performance Evaluation 95 (206) 40 7 Table 2 Effect of (ρ = 0.99, µ = µ 2 =, r =, α = β = 0, q = 30%, k = 0, SC = E(W ) + 0.0E(W 2 )). E(W ) max E(W ) min rd E(W 2 ) min SC NI E(W ) op E(W 2 ) opt SC opt rd % % % % % % % % % % % % % % % % 5.3. Impact of the threhold k We examine the optimization of the threhold k. We alo invetigate the relation between reervation and k. Finally, the policy of a fixed threhold k i evaluated in comparion with a tate-dependent k Exogenou parameter and threhold k Propoition 5 give, for the non-idling cae for Model C, firt order monotonicity reult in k. Propoition 5. In the non-idling cae for Model C, P b i inenitive to k, Ψ i decreaing in k, E(W ) and E(W 2 ) are increaing in k, for k 0. The proof of the propoition i given in Appendix F. A conequence of the monotonicity reult of E(W ) and E(W 2 ) i that k = 0 i optimal for non-idling Model C. Yet, k = 0 i not the optimal value for Model A, B and G becaue of inbound balking, abandonment and/or the poible non-availability of a called back cutomer. Balking and call acceptance parameter for Model B. We conider the impact of α x and q x (x ) on the monotonicity of the performance meaure in k for Model B. In Fig. 0, we conider three numerical cae: Cae : q x = 0.4 and α x = min(0.5, 0.05x), Cae 2: q x = min(0.4, 0.05x) and α x = 0.5, Cae 3: q x = 0. and α x = min(0.5, 0.05x 2 ), for x. Cae and 3 illutrate ituation with non contant balking parameter and Cae 2 illutrate a ituation with non contant callback acceptance parameter. The monotonicity reult in k in Cae and 2 are identical to thoe derived for the non-idling cae of Model C. However, in Cae 3, E(W ) i non-increaing in k. When α x i trongly increaing in x (Cae 3), the inbound expected waiting time can be non-increaing in k (Fig. 0(a)). The proportion of inbound increae in k. Therefore, inbound arrive more often at a long queue (large value of x), and the balking would then be more important (large value of α x ). Although increaing k ha the negative effect of increaing balking, it alo ha the poitive effect of reducing the ytem workload by reducing arrival that enter the ytem. Thi can improve the expected waiting time of inbound. From the numerical experiment, we however oberve that q x do not impact the firt order monotonicity reult in k. Thi i related to the fact that q x ha no effect on the ytem workload, and that the callback offer i only propoed for x k. Abandonment for Model A. Fig. and 2 illutrate the impact of k on the performance meaure, for different value of the abandonment rate β. We oberve that the abandonment in queue only affect the monotonicity propertie of E(W ) and E(W 2 ). Thi explain why k = 0 i no longer necearily optimal. Two phenomenon are in competition when β > 0. From the one hand, increaing k reduce the number of callback and increae thu the proportion of inbound, which would in turn increae E(W ) and E(W 2 ). From the other hand, the increaing of the number of cutomer in queue increae alo the departure rate (after abandonment or ervice) of inbound from the ytem, which make the ytem more efficient and may decreae E(W ) and E(W 2 ). The firt (econd) phenomenon i predominant for mall (large) value of β. We oberve that the non-increaing of E(W 2 ) require higher arrival or abandonment rate than the non-increaing of E(W ) (Fig. 2). The behavior of the other performance meaure i more intuitive; the proportion of abandonment increae in k, and the proportion of callback decreae in k. Outbound ervice proce for Model G. Let u define T, a random variable, repreenting the total time pent by the ytem capacity to erve an outbound call. For a given outbound call, thi correpond to the ummation of the duration pent by agent to handle eventually it non-availability ituation plu it ervice duration. The cae k = 0 i alo not necearily optimal when the overall expected time pent to erve an outbound call i larger than the expected time to erve an inbound one. In Propoition 6, we give the expected value and the tandard deviation of the time pent by the ytem capacity to erve an outbound call.

18 8 B. Legro et al. / Performance Evaluation 95 (206) 40 (a) E(W ). (b) E(W 2 ). (c) P b. (d) Ψ. Fig. 0. Impact of balking and call acceptance parameter ( =, λ = 0.5, µ =, r = 00%, non-idling cae). (a) E(W ). (b) E(W 2 ). (c) P a. (d) Ψ. Fig.. Impact of abandonment ( =, λ =.2, µ =, α = q = 30%, r = 00%).

19 B. Legro et al. / Performance Evaluation 95 (206) 40 9 (a) E(W ). (b) E(W 2 ). Fig. 2. Impact of abandonment ( = 0, λ = 2, µ =, β = 3, α = 0%, q = 30%, r = 00%). (a) E(W ). (b) E(W 2 ). Fig. 3. Impact of the ervice proce ( =, λ = 0.75, µ =, µ 2 =.5, µ 3 = 0, α = β = 0, q = 30%, r = 0%, r 2 = 7%). Propoition 6. The random time T ha a phae type ditribution with expected value E(T) = r r + r 2 µ + r 2 r + r 2 µ 2 + r r 2 r + r 2 µ 3, and tandard deviation r (4 3r 2r 2 ) σ (T) = µ 2 (r + r 2 )(2 r r 2 ) + r 2 (4 3r 2 2r ) µ 2 2 (r + r 2 )(2 r r 2 ) + ( r r 2 )(4 r r 2 ) µ 2 3 (r + r 2 )(2 r r 2 ). The proof of thi propoition i given in Appendix G. From Propoition 6, we deduce that outbound require a larger expected time of treatment than that of inbound if and only if r r 2 > r 2. (3) µ 3 µ µ 2 Inequality (3) imply tate that if the time lot in handling a non-available ituation i larger than the time aved due to fat outbound (thoe who have already reolved a part of their problem), then outbound require a larger expected time of treatment. Fig. 3 illutrate a ituation where the overall expected time of an outbound treatment, E(T), i larger than that of the ervice time of an inbound, /µ. We oberve that the monotonicity propertie in k of the performance meaure E(W ), P a and Ψ are not affected by the parameter of ervice of outbound, becaue of the higher priority given to inbound. The reaon i that, during their ojourn in the queue, the latter will only ait at ervice duration that are exponentially ditributed with rate µ i (i =, 2, 3). We oberve that E(W 2 ) i either trictly increaing in k or decreaing then increaing. The econd ituation occur when outbound are treated within a much larger time than that of inbound. Two phenomenon are in competition; the firt one already mentioned earlier i that increaing k reduce the number of outbound which would uffer from the high proportion of prioritized inbound. The econd one i that if k i too mall, the proportion of outbound can be too important for the ytem capacity. It might then take too long time to erve them.

20 20 B. Legro et al. / Performance Evaluation 95 (206) 40 Table 3 Impact of k (λ = 49.5, = 50, µ =, µ 2 =.5, µ 3 = 0, r = 50%, r 2 = 30%, α = 0%, q = 30%, β = 0.5, SC = E(W ) + 0.0E(W 2 ) + P a ). k E(W ) max E(W ) min rd P amax P amin rd E(W 2 ) SC NI % 4.85% 4.04% 20.3% % 5.02% 4.20% 9.62% % 5.2% 4.44% 7.50% % 5.32% 4.56% 6.87% % 5.43% 4.68% 6.7% % 5.52% 4.77% 5.80% % 5.56% 4.82% 5.22% % 5.69% 4.96% 4.78% % 5.74% 5.08% 3.06% % 5.82% 5.28% 0.23% k E(W ) opt E(W 2 ) opt P aopt SC opt rd % % % % % % % % % % % % % % % % % % % % Reervation and threhold k We invetigate here the relation between the agent reervation policy and the choice for the threhold k. We proved in Propoition 2 that for Model A, the higher i q, the le agent hould be reerved for inbound. The reaon i the low proportion of inbound. The impact of k i imilar to that of q. Increaing k i equivalent to decreaing q, therefore the higher k i the more agent hould be reerved for inbound. Thi obervation agree with the claical idea in control problem tating that the longet queue hould be preferred: through the choice of the reervation level in our model. However, Table 3 reveal that thi obervation i no longer true when Model G i conidered. In thi table, we provide the performance meaure for different value of k. Similarly to Table 2, we provide the upper and lower bound for E(W ) and P a to examine the poibilitie of improvement. We alo compute the lower bound for E(W 2 ). In the preented numerical illutration, the two extreme ituation are again the non-idling cae and the extreme reervation cae. In the lat five column, we give the optimal value of the performance meaure. We alo compute the relative difference found in the comparion between the non-idling cae and the optimal cae. On the contrary to what one would expect, we oberve here that agent reervation decreae in k. For example in Table 3, when k 6, non-idling i optimal. The reaon i related to two phenomenon. The firt one i the poible nonavailability of outbound (20% are not available). The econd one i the maller impact of outbound in ervice on inbound performance when r < than when r = (50% of outbound occupy agent a horter time than inbound). The low priority of outbound together with their non-full availability make queue 2 difficult to reduce, epecially when inbound are numerou in the ytem (i.e., when k i high). Therefore, the increaing of E(W 2 ) in k i trong (ee column 8) and reervation for inbound hould not be provided when k i high. Becaue outbound occupy agent a horter time than inbound when r <, outbound have le impact on E(W ) in Model G than in Model A. Thu, the effect of k and the agent reervation on E(W ) i weaker for Model G than for Model A (ee column 2 and 3). Increaing reervation when k i high ha a trong impact on E(W 2 ) but a mall one on E(W ), which advocate for a non-idling policy. The deterioration of E(W 2 ) with reervation i weaker when k i mall, o, reervation hould be provided in thi cae to reduce E(W ) Value of a fixed threhold k We have defined the threhold parameter k on the number of call in queue to control the deciion of propoing or not the callback offer. We have hown that k = 0 i optimal for the non-idling cae of Model C. In other word, the callback offer hould be propoed to all delayed cutomer. It i alo the cae for Model A, B and G in mot cae. Yet, with ignificant abandonment or large treatment time for outbound, k = 0 may not be any longer optimal. In the modeling, the value of a fixed threhold k come from it implicity and from the analyi tractability for the performance evaluation. However, a fixed threhold k may not be optimal. It i then alo intereting to evaluate the performance of our fixed-k policy in comparion with a tate-dependent-k policy for the propoition of the callback offer upon arrival. For Model A with contant balking and callback acceptance parameter, the value function defined in Section 2 can be rewritten, for n 0, including the deciion to propoe or not the callback offer through the operator W n, a U n+ (x, y) = γ (x ) + + γ 2 y + λw n (x, y) + β(x ) + (V n (x, y) + γ 3 ) + min(, x)µv n (x, y) + λ β(x ) + min(, x)µ Vn (x, y), for x, y 0,

21 B. Legro et al. / Performance Evaluation 95 (206) 40 2 Fig. 4. Optimal witching curve for the callback offer (λ =.2, µ =, q = α = 30%, β =, SC = E(W ) E(W 2 ) + 0.P a, r =, = ). Table 4 Comparion between the two threhold modeling (µ =, r = ). Cae Parameter λ β α q SC.2 30% 30% E(W ) E(W 2 ) + 0.P a % 80% E(W ) E(W 2 ) + 0.0P a % 40% E(W ) E(W 2 ) + 0.0P a % 60% E(W ) E(W 2 ) + 0.2P a % 60% E(W ) E(W 2 ) Cae Optimal fixed-k policy Optimal tate-dependent-k policy k E(W ) E(W 2 ) P a SC E(W ) E(W 2 ) P a SC % % % % % % % % % % 0.04 with V n+ (x, y) = min(u n+ (x +, y ), U n+ (x, y)), for y > 0 and 0 x < and V n+ (x, y) = U n+ (x, y) in the remaining cae, and W n+ (x, y) = min(α(u n+ (x, y) + γ 3 ) + ( α)u n+ (x +, y), α(u n+ (x, y) + γ 3 ) + qu n+ (x, y + ) + ( q α)u n+ (x +, y)), for x and W n+ (x, y) = U n+ (x, y) in the remaining cae. We chooe W 0 (x, y) = V 0 (x, y) = U 0 (x, y) = 0, for x, y 0. In Fig. 4, we preent the optimal deciion found through value iteration. We only preent the tate where an action on the callback offer ha to be taken (x ). We oberve that the optimal deciion for the callback offer if of witch type. The optimal deciion for the point on the curve i not to propoe the callback offer. To the contrary to the reervation policy found in Section 3, the witching curve i not monotonou in x or in y. We oberve that if the optimal deciion in a given tate (x, y) i not to propoe the callback offer, then the ame deciion hould be taken in tate (x, y + ). The reaon i related to the congetion of queue 2. The deciion not to propoe the offer i taken in order to ue call abandonment in queue which decreae the ytem workload. Therefore, if the ytem i too congeted with y outbound in queue 2, it would alo be with y + outbound in queue 2. The deciion a a function of x i more complex. For mall value of x, the deciion i more likely to give the offer o a to reduce the number of cutomer in queue. Thi deciion can be taken becaue the ytem i not congeted. For higher value of x, the offer can be interrupted to reduce the workload in the ytem by letting cutomer abandon from queue. For even higher value of x, the proportion of abandonment and the waiting time in queue can be o ignificant that the deciion i again to propoe the callback offer even if it would increae the ytem workload. One can compare between the two modeling, with a fixed or a tate-dependent k uing imulation. The optimal fixed-k i aumed to be a real number in order to achieve a lower ytem cot than if k would be an integer (in practice, thi mean that randomization between two adjacent threhold i allowed). For variou etting, Table 4 reveal that the difference between the optimal ytem cot and the cot found with a fixed-k i not important. However, it i notable that the optimal tate-dependent policy improve E(W 2 ) and almot do not affect the other metric. For Model G, the optimal deciion for the callback offer can be obtained uing the ame approach. However, further aumption hould be made on the balking parameter when the callback option would be propoed or not. For intance, it

22 22 B. Legro et al. / Performance Evaluation 95 (206) 40 Table 5 Impact of the parameter. E(W ) E(W 2 ) P a Increaing Increaing the agent reervation + + Increaing k Increaing k with a high balking/abandonment parameter + +, Increaing k with a high difficulty to erve outbound +, eem appropriate to aume that the callback offer would reduce the balking behavior. In thi cae, the concluion derived above are till valid. The callback offer reduce then at the ame time balking, abandonment and the waiting time in queue, but it increae the ytem workload. For Model G, either the treatment time of outbound i horter than that of inbound and k = 0 i thu optimal, or it i not and the concluion derived above are alo till valid. To conclude Section 5.3, k = 0 i optimal when the balking parameter are contant (α x = α, x ), no abandonment i conidered (β = 0), or the treatment time of an outbound call i lower or equal than that of an inbound one (E(T) /µ ). Increaing k increae the ize of queue. When α x i trongly increaing in x, thi alo increae the balking proportion which reduce the effective arrival rate. When β > 0, call abandonment help to reduce the length of queue. If much more importance i given to the waiting time in queue than that to abandonment (γ γ 3 ), then k > 0 i ueful to dicharge the ytem. If the treatment time of an outbound call i large, it i alo ueful to have k > 0 in order to avoid too high proportion of outbound. The relation between the optimal reervation policy and the optimal k depend on the ervice proce of outbound. If thi one i identical to that of inbound (Model A), then more agent hould be reerved for inbound a k increae. Summary of Section 5 reult. Table 5 ummarie the impact of the parameter on the objective function component. We ue the ign + for a poitive effect and the ign for a negative one. In mot oberved ituation, k = 0 i optimal and the reervation policy can be obtained via the MDP approach from Section 3. In the remaining cae, a finite number of tep hould be done to find the optimal value of k with it correponding reervation policy (by tarting from the cae k = 0 and by incrementing k by one at each tep). The number of tet i finite becaue the deterioration of E(W 2 ) in k after a given value of k i much fater than the eventual improvement of E(W ) in k. Beyond thi value of k, any reervation policy would anyway further deteriorate E(W 2 ). Moreover, P a deteriorate with k. Hence, after a given value of k, the total expected ytem cot only increae in k and the earch for the optimal value of k hould be topped at that point. 6. Concluion and future reearch We conidered a call center that offer two channel: real-time telephone ervice and potponed (callback) ervice. Cutomer chooe which channel to ue baed on a probabilitic choice model. We demontrated the operational advantage of agent reervation in thi context. The key operational finding of thi paper are that () the value of the callback option i more ignificant under heavily loaded ituation, (2) the benefit of agent reervation are more apparent in large call center than in mall one, (3) reervation increae the agent utilization due to the abandonment reduction, (4) reervation i not likely to be ued under light or heavily loaded ituation, (5) the callback offer hould be propoed to all delayed cutomer except when the abandonment i ignificant or when the overall treatment time of an outbound i much larger than that of an inbound. Thee operational finding came together with theoretical contribution. The major one are () the proof that non-idling i optimal in the ingle-erver cae, (2) the proof of the optimality of a threhold policy in the two-erver cae, (3) the algorithm propoed for the performance evaluation when tranition rate are aumed to be contant. Several intereting area of future reearch arie. It would be ueful to empirically validate the cutomer reaction model to the callback offer through real data analyi. It i alo intereting to extend the proof of the optimal policy for the two-erver cae to that for the multi-erver cae. Another reearch avenue i to conider other optimization problem formulation, for example in term of quantile on the waiting time ditribution of inbound and outbound call. Finally, it would be intereting to conider non-tationary arrival parameter and invetigate it impact on job cheduling. Acknowledgment Thi work wa upported by Agence Nationale de la Recherche under the project ANR-JCJC-SIMI3-202-OPERA. We alo want to expre our gratitude to two anonymou reviewer and the aociate editor for their ueful comment, that ignificantly improved thi paper. Appendix A. Proof of Theorem We firt rewrite the value function in the two-erver cae for Model A (µ = µ 2 = µ, r = ). So a to implify the preentation of the proof, we redefine the tate a follow. The parameter z denote the tate of the agent team (z = 0

23 B. Legro et al. / Performance Evaluation 95 (206) when both agent are idle; z = when only one agent i buy with an inbound or an outbound call; and z = 2 when both agent are buy), x i redefined here a the number of inbound in queue and y i the number of outbound in queue 2. We have for n 0, U n+ (0, 0, y) = γ 2 y + λv n (, 0, y) + ( λ)v n (0, 0, y), for y 0, U n+ (, 0, y) = γ 2 y + λv n (2, 0, y) + µv n (0, 0, y) + ( λ µ)v n (, 0, y), for y 0, U n+ (2, x, y) = γ x + γ 2 y + λ (0 x<k) (( α x )V n (2, x +, y) + α x (V n (2, x, y) + γ 3 )) + (k x<n) (q x V n (2, x, y + ) + α x (V n (2, x, y) + γ 3 ) + ( q x α x )V n (2, x +, y)) + (x=n) (q N V n (2, x, y + ) + ( q N )(V n (2, x, y) + γ 3 )) + βx(v n (2, x, y) + γ 3 ) + 2µ () V n (, 0, y) + (x>0) V n (2, x, y) + ( λ βx 2µ) V n (2, x, y), for x, y 0, with V n+ (0, 0, y) = U n+ (, 0, y ) for y > 0 (recall that we aume that it i optimal to erve an outbound call when all agent are idling); V n+ (, 0, y) = min(u n+ (2, 0, y ), U n+ (, 0, y)) for y > 0; and V n+ (z, x, y) = U n+ (z, x, y) in the remaining cae. We chooe V 0 (z, x, y) = U 0 (z, x, y) = 0, for z = 0,, 2 and x, y 0. We define a cla of function F from {0,, 2} N 2 to R a follow: f F if f (2, x +, y) f (2, x, y), f (, 0, y) f (0, 0, y), f (2, 0, y) f (, 0, y), f (2, x, y + ) f (2, x, y), f (0, 0, y + ) f (0, 0, y), f (, 0, y + ) f (, 0, y), f (2, x, y) + f (2, x +, y + ) f (2, x +, y) + f (2, x, y + ), (0) f (0, 0, y) + f (, 0, y + ) f (, 0, y) + f (0, 0, y + ), () f (, 0, y) + f (2, 0, y + ) f (2, 0, y) + f (, 0, y + ), (2) f (2, x, y + 2) + f (2, x +, y) f (2, x, y + ) + f (2, x +, y + ), (3) f (0, 0, y + 2) + f (, 0, y) f (0, 0, y + ) + f (, 0, y + ), (4) f (, 0, y + 2) + f (2, 0, y) f (, 0, y + ) + f (2, 0, y + ), (5) for x, y 0. Relation (4) and (7) define a cla of increaing function in x and in y. Relation (0) define upermodularity for z = 2. By umming up Relation (0) and (3) we obtain f (2, x, y) + f (2, x, y + 2) 2f (2, x, y + ), by umming up Relation () and (4) we obtain f (0, 0, y) + f (0, 0, y + 2) 2f (0, 0, y + ), and by umming up Relation (2) and (5) we obtain f (, 0, y) + f (, 0, y + 2) 2f (, 0, y + ). Thu if f F, then f i convex in y. Relation (3) mean that the function f (2, x, y + ) f (2, x +, y) i increaing in y. Remark. For the multi-erver cae of Model G, we need to add another relation to the cla of function defined below. The additional relation i f (x + 2, y, 2, 3 ) + f (x, y +, 2, 3 ) f (x +, y, 2, 3 ) + f (x +, y +, 2, 3 ). It i required to prove that the relation f (x, y + 2, 2, 3 ) + f (x +, y, 2, 3 ) f (x, y +, 2, 3 ) + f (x +, y +, 2, 3 ) propagate through the minimizing operator. The proof through value iteration i hard to do for the arrival term if x = + k 2, and for the ervice term if 0 x It i however doable for the remaining cae. To implify the preentation, we denote by erve the deciion action to erve an outbound call, and by keep the deciion action to keep an outbound call in queue 2. The proof of the optimality of the threhold policy reduce to how that Relation (3) i true for U n, n 0. We next prove by induction on n that both V n and U n are in F. We divide the proof into the following 5 tep: Step. We prove that V 0, U 0 F. Step 2. We prove that if U n F, then V n F, for n 0. Step 3. We prove that the cot term G(z, x, y) = γ x + γ 2 y i in F. Step 4. We prove for a given n 0 that if V n F, then the following arrival term i alo in F : A n (2, x, y) = (0 x<k) (( α x )V n (2, x +, y) + α x (V n (2, x, y) + γ 3 )) + (k x<n) (q x V n (2, x, y + ) + α x (V n (2, x, y) + γ 3 ) + ( q x α x )V n (2, x +, y)) + (x=n) (q N V n (2, x, y + ) + ( q N )(V n (2, x, y) + γ 3 )), for x, y 0, A n (, 0, y) = V n (2, 0, y) and A n (0, 0, y) = V n (, 0, y) for y 0. (4) (5) (6) (7) (8) (9)

24 24 B. Legro et al. / Performance Evaluation 95 (206) 40 Step 5. We prove for a given n 0 that if V n F, then the following departure term i alo in F : D n (2, x, y) = βx(v n (2, x, y) + γ 3 ) + 2µ () V n (, 0, y) + (x>0) V n (2, x, y) + ( λ βx 2µ) V n (2, x, y), for x, y 0, D n (, 0, y) = µv n (0, 0, y) + ( λ µ)v n (, 0, y) and D n (0, 0, y) = ( λ)v n (0, 0, y) for y 0. The proof for the previou five tep are given below. Step. For x, y 0 and z = 0,, 2, V 0 (z, x, y) = U 0 (z, x, y) = 0. Then V 0, U 0 F. Step 2. Aume that for a given n 0, U n F. We only conider the non-trivial cae where z = 0 or z = and y > 0. In the other cae U n = V n. Therefore we only need to how Relation (5), (6), (8), (9), (), (2), (4) and (5). For Relation (5) and (8), we have V n (0, 0, y) = U n (, 0, y ), for y > 0. (6) If keep i optimal in (, 0, y), then V n (, 0, y) = U n (, 0, y). Combining Eq. (6) with Relation (9) for U n lead to V n (0, 0, y) V(, 0, y) and prove Relation (5) for V n. If erve i optimal in (, 0, y), then V n (, 0, y) = U n (2, 0, y ). Combining Eq. (6) with Relation (6) for U n lead to V n (0, 0, y) V(, 0, y) and prove Relation (5) for V n. We have V n (0, 0, y+) = U n (, 0, y). Combining Inequality (6) with Relation (9) for U n lead to V n (0, 0, y) V(0, 0, y+). Therefore in all cae, Relation (5) and (8) hold for V n. For Relation (6) and (9), we have V n (, 0, y) U n (2, 0, y ), for y > 0, (7) V n (, 0, y) U n (, 0, y), for y 0. (8) Oberve that V n (2, 0, y) = U n (2, 0, y). Combining Inequality (7) with Relation (7) for U n prove Relation (6). If keep i optimal in (, 0, y + ), then V n (, 0, y + ) = U n (, 0, y + ). Combining equality (8) with Relation (9) for U n prove Relation (9) for V n. If erve i optimal in (, 0, y + ), then V n (, 0, y + ) = U n (2, 0, y). Combining equality (7) with Relation (7) for U n prove (9) for V n. Therefore in all cae, Relation (6) and (9) hold for V n. For Relation (), we have V n (, 0, y) + V n (0, 0, y + ) 2U n (, 0, y) for y 0, (9) V n (, 0, y) + V n (0, 0, y + ) U n (2, 0, y ) + U n (, 0, y) for y 0. (20) If keep i the optimal action in tate (, 0, y + ), for y > 0, then V n (0, 0, y) + V n (, 0, y + ) = U n (, 0, y ) + U n (, 0, y + ). Thu, combining the convexity in y of U n and Inequality (9) prove Relation () for V n, for y 0. If erve i the optimal action in tate (, 0, y + ), for y > 0, then V n (0, 0, y) + V n (, 0, y + ) = U n (, 0, y ) + U n (2, 0, y). Combining Relation (2) for U n and Inequality (20) prove Relation () for V n, for y > 0. In all cae, Relation () then hold for V n. For Relation (2), we have V n (2, 0, y) + V n (, 0, y + ) U n (2, 0, y) + U n (, 0, y + ) for y 0, (2) V n (2, 0, y) + V n (, 0, y + ) 2U n (2, 0, y) for y 0. (22) If keep i the optimal action in tate (, 0, y), for y > 0, then V n (, 0, y) + V n (2, 0, y + ) = U n (, 0, y) + U n (2, 0, y + ). Thu, Relation (2) for U n and Inequality (2) prove Relation (2) for V n, for y 0. If erve i the optimal action in tate (, 0, y), for y > 0, then V n (, 0, y) + V n (2, 0, y + ) = U n (2, 0, y ) + U n (2, 0, y + ). Combining the convexity in y of U n and Inequality (22) prove Relation (2) for V n, for y > 0. In all cae, Relation (2) then hold for V n. For Relation (4), we have V n (0, 0, y + ) + V n (, 0, y + ) U n (, 0, y) + U n (, 0, y + ) for y 0, (23) V n (0, 0, y + ) + V n (, 0, y + ) U n (, 0, y) + U n (2, 0, y) for y 0. (24) If keep i the optimal action in tate (, 0, y), for y > 0, then V n (0, 0, y + 2) + V n (, 0, y) = U n (, 0, y + ) + U n (, 0, y). Inequality (23) prove Relation (4) for V n, for y 0. If erve i the optimal action in tate (, 0, y), for y 0, then V n (0, 0, y + 2) + V n (, 0, y) = U n (, 0, y + ) + U n (2, 0, y ). Combining next Relation (5) for U n and Inequality (24) prove Relation (4) for V n, for y 0. Finally in all cae, Relation (4) i true for V n. For Relation (5), we have V n (, 0, y + ) + V n (2, 0, y + ) U n (, 0, y + ) + U n (2, 0, y + ) for y 0, (25) V n (, 0, y + ) + V n (2, 0, y + ) U n (2, 0, y) + U n (2, 0, y + ) for y 0. (26) If keep i the optimal action in tate (, 0, y+2), for y 0, then V n (, 0, y+2)+v n (2, 0, y) = U n (, 0, y+2)+u n (2, 0, y). Combining next Relation (5) for U n and Inequality (25) prove Relation (5) for V n, for y 0. If erve i the optimal action in tate (, 0, y + 2), for y 0, then V n (, 0, y + 2) + V n (2, 0, y) = U n (2, 0, y + ) + U n (2, 0, y). Inequality (26) prove Relation (5) for V n, for y 0. Finally in all cae, Relation (5) i true for V n.

25 B. Legro et al. / Performance Evaluation 95 (206) Step 3. The tep i eay to prove and directly follow from [25, page 33]. Step 4. Aume that V n F, for a given n 0. We now prove that A n F. In Relation (5), (7) (9), () and (4), x i contant and the arrival of a new call ha the ame effect on each term of the relation (either increaing the number of cutomer in queue by one, or changing z into z + ). Moreover, ince the tranition rate are contant, the induction from V n to A n i traightforward (ee [25, page 35]). Next, the other relation have to be hown to prove the induction from V n to A n. For Relation (4), (7), (0) and (3), the cae x < k i a implification of the cae k x < N becaue the poibility of going to queue 2 i not conidered. We therefore only how the cae k x < N. For Relation (4), if x = k, then A n (2, x +, y) A n (2, x, y) = q k V n (2, x +, y + ) + α k (V n (2, x +, y) + γ 3 ) + ( α k q k )V n (2, x + 2, y) ( α k )V n (2, x +, y) α k (V n (2, x, y) + γ 3 ) = q k (V n (2, x +, y + ) V n (2, x +, y)) + α k (V n (2, x +, y) V n (2, x, y)) + ( α k q k )(V n (2, x + 2, y) V n (2, x +, y)) + γ 3 (α k α k ) 0, ince V n i increaing in x and in y. If k x < N, then A n (2, x +, y) A n (2, x, y) = q x+ V n (2, x +, y + ) + α x+ (V n (2, x +, y) + γ 3 ) + ( α x+ q x+ )V n (2, x + 2, y) q x V n (2, x, y + ) α x (V n (2, x, y) + γ 3 ) ( α x q x )V n (2, x +, y) = q x (V n (2, x +, y + ) V n (2, x +, y)) + (q x+ q x )V n (2, x +, y + ) + α x (V n (2, x +, y) V n (2, x, y)) + (α x+ α x )V n (2, x +, y) + γ 3 (α x+ α x ) + ( α x+ q x+ )(V n (2, x + 2, y) V n (2, x +, y)) + (α x + q x α x+ q x+ )V n (2, x +, y) (q x+ q x )(V n (2, x +, y + ) V n (2, x +, y)) 0, ince V n i increaing in y and q x i increaing in x. If x = N, then A n (2, x +, y) A n (2, x, y) = q x V n (2, x +, y + ) + ( q x )(V n (2, x +, y) + γ 3 ) q x V n (2, x, y + ) α x (V n (2, x, y) + γ 3 ) ( α x q x )V n (2, x +, y) = α x (V n (2, x +, y + ) V n (2, x, y)) + q x (V n (2, x +, y + ) V n (2, x, y + )) + γ 3 ( q x α x ) 0, ince V n i increaing in x and in y. Finally in all cae, Relation (4) i true for A n. For Relation (6), we may write A n (2, 0, y) A n (, 0, y) = ( α 0 )V n (2,, y) + α 0 (V n (2, 0, y) + γ 3 ) V n (2, 0, y) = ( α 0 )(V n (2,, y) V n (2, 0, y)) + α 0 γ 3 0, ince Relation (4) i true for V n. Hence, Relation (6) i true for A n. For the following relation, we do not write the term in γ 3 ince the do diappear in the conidered difference. For Relation (0), if x = k, then A n (2, x, y) + A n (2, x +, y + ) A n (2, x, y + ) A n (2, x +, y) = α k V n (2, x, y) + ( α k )V n (2, x +, y) + q k V n (2, x +, y + 2) + α k V n (2, x +, y + ) + ( α k q k )V n (2, x + 2, y + ) α k V n (2, x, y + ) ( α k )V n (2, x +, y + ) q k V n (2, x +, y + ) α k V n (2, x +, y) ( α k q k )V n (2, x + 2, y) = α k (V n (2, x +, y + ) + V n (2, x, y) V n (2, x +, y) V n (2, x, y + )) + q k (V n (2, x +, y + 2) + V n (2, x + 2, y) V n (2, x + 2, y + ) V n (2, x +, y + )) + ( α k )(V n (2, x + 2, y + ) + V n (2, x +, y) V n (2, x +, y + ) V n (2, x + 2, y)). The term proportional to α k i poitive ince Relation (0) hold for V n, the term proportional to q k i poitive ince Relation (3) hold for V n, the term proportional to α k i poitive ince Relation (0) hold for V n. Hence, Relation (0) i true for A n, for x = k. If k x < N, then A n (2, x, y) + A n (2, x +, y + ) A n (2, x, y + ) A n (2, x +, y) = q x V n (2, x, y + ) + α x V n (2, x, y) + ( q x α x )V n (2, x +, y) + q x+ V n (2, x +, y + 2) + α x+ V n (2, x +, y + ) + ( q x+ α x+ )V n (2, x + 2, y + )

26 26 B. Legro et al. / Performance Evaluation 95 (206) 40 q x V n (2, x, y + 2) α x V n (2, x, y + ) ( q x α x )V n (2, x +, y + ) q x+ V n (2, x +, y + ) α x+ V n (2, x +, y) ( q x+ α x+ )V n (2, x + 2, y) = q x (V n (2, x, y + ) + V n (2, x +, y + 2) V n (2, x, y + 2) V n (2, x +, y + )) + α x (V n (2, x, y) + V n (2, x +, y + ) V n (2, x, y + ) V n (2, x +, y)) + ( α x+ q x+ )(V n (2, x +, y) + V n (2, x + 2, y + ) V n (2, x +, y + ) V n (2, x + 2, y)) + (q x+ q x )(V n (2, x +, y + 2) + V n (2, x +, y) 2V n (2, x +, y + )). The term proportional to q x, α x and q x+ α x+ are poitive ince Relation (0) i true for V n, the term proportional to q x+ q x i alo poitive ince V n i convex in y. Hence Relation (0) i true for A n, for k x < N. If x = N, then A n (2, x, y) + A n (2, x +, y + ) A n (2, x, y + ) A n (2, x +, y) = q x V n (2, x, y + ) + α x V n (2, x, y) + ( q x α x )V n (2, x +, y) + q x V n (2, x +, y + 2) + ( q x )V n (2, x +, y + ) q x V n (2, x, y + 2) α x V n (2, x, y + ) ( q x α x )V n (2, x +, y + ) q x V n (2, x +, y + ) ( q x )V n (2, x +, y) = q x (V n (2, x, y + ) + V n (2, x +, y + 2) V n (2, x, y + 2) V n (2, x +, y + )) + α x (V n (2, x, y) + V n (2, x +, y + ) V n (2, x, y + ) V n (2, x +, y)). The term proportional to q x and α x are poitive ince Relation (0) i true for V n. Hence Relation (0) i true for A n, for x = N. For Relation (2), we have for y 0, A n (, 0, y) + A n (2, 0, y + ) A n (2, 0, y) A n (, 0, y + ) = V n (2, 0, y) + α 0 V n (2, 0, y + ) + ( α 0 )V n (2,, y + ) α 0 V n (2, 0, y) ( α 0 )V n (2,, y) V n (2, 0, y + ) = ( α 0 )(V n (2, 0, y) + V n (2,, y + ) V n (2,, y) V n (2, 0, y + )) 0, ince Relation (0) hold for V n. Hence Relation (2) i true for A n. For Relation (3), if x < k the tranition rate are contant and the induction from V n to A n follow. If x = k, then A n (2, x, y + 2) + A n (2, x +, y) A n (2, x, y + ) A n (2, x +, y + ) = α k V n (2, x, y + 2) + ( α k )V n (2, x +, y + 2) + q k V n (2, x +, y + ) + α k V n (2, x +, y) + ( α k q k )V n (2, x + 2, y) α k V n (2, x, y + ) ( α k )V n (2, x +, y + ) q k V n (2, x +, y + 2) α k V n (2, x +, y + ) ( α k q k )V n (2, x + 2, y + ) = α k (V n (2, x, y + 2) + V n (2, x +, y + ) V n (2, x +, y + 2) V n (2, x, y + )) + α k (V n (2, x +, y) + V n (2, x + 2, y + ) V n (2, x + 2, y) V n (2, x +, y + )) + q k (V n (2, x +, y + ) + V n (2, x + 2, y + ) V n (2, x + 2, y) V n (2, x +, y + 2)) + V n (2, x + 2, y) + V n (2, x +, y + 2) V n (2, x +, y + ) V n (2, x + 2, y + ) = (α k α k )(V n (2, x +, y) + V n (2, x +, y + 2) 2V n (2, x +, y + )) + α k (V n (2, x, y + 2) + V n (2, x +, y) V n (2, x, y + ) V n (2, x +, y + )) ( q k α k )(V n (2, x + 2, y) + V n (2, x +, y + 2) +V n (2, x +, y + ) V n (2, x + 2, y + )). The term proportional to α k α k i poitive ince V n i convex in y, the term proportional to α k i poitive ince Relation (3) i true for V n, the term proportional to q k α k i poitive ince Relation (3) i true for V n. Hence Relation (3) i true for A n, for x = k. If k x < N, then A n (2, x, y + 2) + A n (2, x +, y) A n (2, x, y + ) A n (2, x +, y + ) = q x V n (2, x, y + 3) + α x V n (2, x, y + 2) + ( q x α x )V n (2, x +, y + 2) + q x+ V n (2, x +, y + ) + α x+ V n (2, x +, y) + ( q x+ α x+ )V n (2, x + 2, y) q x V n (2, x, y + 2) α x V n (2, x, y + ) ( q x α x )V n (2, x +, y + ) q x+ V n (2, x +, y + 2) α x+ V n (2, x +, y + ) ( q x+ α x+ )V n (2, x + 2, y + ) = q x (V n (2, x, y + 3) + V n (2, x +, y + ) V n (2, x +, y + 2) V n (2, x, y + 2)) + α x (V n (2, x, y + 2) + V n (2, x +, y) V n (2, x +, y + ) V n (2, x, y + )) + ( α x+ q x+ )(V n (2, x +, y + 2) + V n (2, x + 2, y) V n (2, x +, y + ) V n (2, x + 2, y + )) + (α x+ α x )(V n (2, x +, y + 2) + V n (2, x +, y) 2V n (2, x +, y + )).

27 B. Legro et al. / Performance Evaluation 95 (206) The term proportional to q x, α x and q x+ α x+ are poitive ince Relation (3) i true for V n, the term proportional to α x+ α x i alo poitive ince V n i convex in y. Hence Relation (3) i true for A n, for k x < N. If x = N, then A n (2, x, y + 2) + A n (2, x +, y) A n (2, x, y + ) A n (2, x +, y + ) = q x V n (2, x, y + 3) + α x V n (2, x, y + 2) + ( q x α x )V n (2, x +, y + 2) + q x V n (2, x +, y + ) + ( q x )V n (2, x +, y) q x V n (2, x, y + 2) α x V n (2, x, y + ) ( q x α x )V n (2, x +, y + ) q x V n (2, x +, y + 2) ( q x )V n (2, x +, y + ) = q x (V n (2, x, y + 3) + V n (2, x +, y + ) V n (2, x +, y + 2) V n (2, x, y + 2)) + α x (V n (2, x, y + 2) + V n (2, x +, y) V n (2, x, y + ) V n (2, x +, y + )) + ( q x α x )(V n (2, x +, y + 2) + V n (2, x +, y) 2V n (2, x +, y + )). The term proportional to q x and α x are poitive ince Relation (3) i true for V n, the term proportional to q x α x i alo poitive ince V n i convex in y. Hence Relation (3) i true for A n, for x = N. For Relation (5), we have A n (, 0, y + 2) + A n (2, 0, y) A n (, 0, y + ) A n (2, 0, y + ) = V n (2, 0, y + 2) + α 0 V n (2, 0, y) + ( α 0 )V n (2,, y) V n (2, 0, y + ) α 0 V n (2, 0, y + ) ( α 0 )V n (2,, y + ) = V n (2, 0, y + 2) + V n (2,, y) V n (2, 0, y + ) V n (2,, y + ) + α 0 (V n (2, 0, y) + V n (2,, y + ) V n (2, 0, y + ) V n (2,, y)). The term proportional to i poitive ince Relation (3) i true for V n, the term proportional to α 0 i alo poitive ince Relation (0) i true for V n. Hence Relation (5) i true for A n. Step 5. Aume that V n F, for a given n 0. We now how that D n F. For Relation (4), if x = 0, then D n (2,, y) D n (2, 0, y) = βv n (2, 0, y) + βγ 3 + 2µ(V n (2, 0, y) V n (, 0, y)) + ( λ β 2µ)(V n (2,, y) V n (2, 0, y)) βv n (2, 0, y) 0, ince V n i increaing in x and Relation (6) i true for V n. If x > 0, then D n (2, x +, y) D n (2, x, y) = βx(v n (2, x, y) V n (2, x, y)) + βγ 3 + βv n (2, x, y) + 2µ(V n (2, x, y) V n (2, x, y)) + ( λ β(x + ) 2µ)(V n (2, x +, y) V n (2, x, y)) βv n (2, x, y) 0, ince V n i increaing in x. Hence Relation (4) i true for D n. For Relation (5), we have D n (, 0, y) D n (0, 0, y) = µv n (0, 0, y) + ( λ µ)(v n (, 0, y) V n (0, 0, y)) µv n (0, 0, y) 0. Hence Relation (5) i true for D n. For Relation (6), we have D n (2, 0, y) D n (, 0, y) = µ(v n (, 0, y) V n (0, 0, y)) + µv n (, 0, y) + ( λ 2µ)(V n (2, 0, y) V n (, 0, y)) µv n (, 0, y) 0. Hence Relation (6) i true for D n. For Relation (7), if x 0, then D n (2, x, y + ) D n (2, x, y) = βx(v n (2, x, y + ) V n (2, x, y)) + 2µ () (V n (0, 0, y + ) V n (0, 0, y)) + ( λ βx 2µ)(V n (2, x, y + ) V n (2, x, y)) 0, ince V n i increaing in y. Hence Relation (7) hold for D n. Relation (8) and (9) are obviouly alo true for D n. For Relation (0), if x, y 0, then D n (2, x, y) + D n (2, x +, y + ) D n (2, x +, y) D n (2, x, y + ) = βx(v n (2, x, y) + V n (2, x, y + ) V n (2, x, y) V n (2, x, y + )) + β(v n (2, x, y + ) V n (2, x, y)) + 2µ () (V n (, 0, y) + V n (2, 0, y + ) V n (2, 0, y) V n (, 0, y + )) + 2µ (x>0) (V n (2, x, y) + V n (2, x, y + ) V n (2, x, y) V n (2, x, y + )) + ( λ β(x + ) 2µ)(V n (2, x, y) + V n (2, x +, y + ) V n (2, x +, y) V n (2, x, y + )) + β(v n (2, x, y) V n (2, x, y + )) 0, ince Relation (0) and (2) are true for V n.

28 28 B. Legro et al. / Performance Evaluation 95 (206) 40 For Relation (), we have for y 0, D n (0, 0, y) + D n (, 0, y + ) D n (, 0, y) D n (0, 0, y + ) = µ(v n (0, 0, y + ) V n (0, 0, y)) + ( λ µ)(v n (0, 0, y) + V n (, 0, y + ) V n (, 0, y) V n (0, 0, y + )) + µ(v n (0, 0, y) V n (0, 0, y + )) 0, ince Relation () i true for V n. For Relation (2), we have for y 0, D n (, 0, y) + D n (2, 0, y + ) D n (2, 0, y) D n (, 0, y + ) = µ(v n (0, 0, y) V n (0, 0, y + )) + 2µ(V n (, 0, y + ) V n (, 0, y)) + ( λ 2µ)(V n (, 0, y) + V n (2, 0, y + ) V n (2, 0, y) V n (, 0, y + )) + µ(v n (, 0, y) V n (, 0, y + )) µ(v n (0, 0, y) + V n (, 0, y + ) V n (, 0, y) V n (0, 0, y + )). The term proportional to µ i poitive ince Relation () i true for V n. Therefore Relation (2) i true for D n. For Relation (3), if x, y 0, then D n (2, x, y + 2) + D n (2, x +, y) D n (2, x, y + ) D n (2, x +, y + ) = βx(v n (2, x, y + 2) + V n (2, x, y) V n (2, x, y + ) V n (2, x, y + )) + β(v n (2, x, y) V n (2, x, y + )) + 2µ () (V n (, 0, y + 2) + V n (2, 0, y) V n (, 0, y + ) V n (2, 0, y + )) + 2µ (x>0) (V n (2, x, y + 2) + V n (2, x, y) V n (2, x, y + ) V n (2, x, y + )) + ( λ β(x + ) 2µ)(V n (2, x, y + 2) + V n (2, x +, y) V n (2, x, y + ) V n (2, x +, y + )) + β(v n (2, x, y + 2) V n (2, x, y + )) β(v n (2, x, y + 2) + V n (2, x, y) 2V n (2, x, y + )) 0, ince Relation (3) and (5) are true for V n and V n i convex in y. Therefore, Relation (3) i true for D n. For Relation (4), we have for y 0, D n (0, 0, y + 2) + D n (, 0, y) D n (0, 0, y + ) D n (, 0, y + ) = µ(v n (0, 0, y) V n (0, 0, y + )) + ( λ µ)(v n (0, 0, y + 2) + V n (, 0, y) V n (0, 0, y + ) V n (, 0, y + )) + µ(v n (0, 0, y + 2) V n (0, 0, y + )) µ(v n (0, 0, y + 2) + V n (0, 0, y) 2V n (0, 0, y + )) 0, ince Relation (4) i true for V n and ince V n i convex in y. Hence, Relation (4) i true for D n. For Relation (5), we have for y 0, D n (, 0, y + 2) + D n (2, 0, y) D n (, 0, y + ) D n (2, 0, y + ) = µ(v n (0, 0, y + 2) V n (0, 0, y + )) + 2µ(V n (, 0, y) V n (, 0, y + )) + ( λ 2µ)(V n (, 0, y + 2) + V n (2, 0, y) V n (, 0, y + ) V n (2, 0, y + )) + µ(v n (, 0, y + 2) V n (, 0, y + )) µ(v n (, 0, y + 2) + V n (, 0, y) 2V n (, 0, y + )) + µ(v n (, 0, y) + V n (0, 0, y + 2) V n (0, 0, y + ) V n (, 0, y + )). The two term proportional to µ are poitive, the firt one becaue V n i convex in y and the econd one becaue Relation (4) hold for V n. Therefore Relation (5) i true for D n. The proof i completed. Appendix B. Proof of Propoition 2 To prove Propoition 2, we need to prove by induction on n (n 0) that, for x, y 0, V (x, y) + n V n(x +, y) V n (x, y) + V n (x +, y), (27) U (x, y) + n U n(x +, y) U n (x, y) + U n (x +, y), (28) V n (x, y + ) + V n(x, y) V n (x, y + ) + V n(x, y), (29) U n (x, y + ) + U n(x, y) U n (x, y + ) + U n(x, y), (30) where V n (x, y), U n (x, y) and V (x, n y), U (x, y) n are the value function aociated with the parameter γ, γ 2, γ 3 and q for x + k, and the parameter γ, γ 2, γ 3 and q + q for x + k, repectively. Summing up Relation (27) and (29) prove that V (x, n y + ) + V n(x +, y) V n (x, y + ) + V n (x +, y). Thi implie that ituation require more reervation than ituation 2. We have U 0 = V 0 = U = 0 V 0 = 0. Thu, Relation (27) (30) hold for n = 0.

29 B. Legro et al. / Performance Evaluation 95 (206) We firt prove that Relation (28) implie Relation (27). Aume now that Relation (28) hold for a given n 0. Therefore, U n (x, y) + U n(x +, y) U n (x +, y) + U n(x, y). We only conider the non-trivial cae where 0 x < and y > 0. We have V n (x +, y) + V n(x, y) U n (x +, y) + U n(x, y) for 0 x, y > 0, (3) V n (x +, y) + V n(x, y) U n (x +, y) + U n(x +, y ) for 0 x, y > 0, (32) V n (x +, y) + V n(x, y) U n (x + 2, y ) + U n(x +, y ) for 0 x 2, y > 0. (33) If keep i the optimal action in tate (x, y) and (x+, y) for ituation 2 and, repectively, then V n (x, y)+v n(x+, y) = U n (x, y) + U n(x +, y). Combining Eq. (3) with Relation (28) for U n prove Relation (27) for V n. If erve i the optimal action in tate (x, y) and (x+, y) for ituation 2 and, repectively, then V n (x, y)+v n(x+, y) = U n (x +, y ) + U n(x + 2, y ). Combining Eq. (33) with Relation (28) for U n prove Relation (27) for V n. If erve i the optimal action in tate (x, y) and keep i the optimal action in tate (x +, y) for ituation 2 and, repectively, then V n (x, y) + V n(x +, y) = U n (x +, y ) + U n(x +, y). Inequality (32) prove Relation (27) for V n. The cae where keep would be the optimal action in tate (x, y) and erve would be the optimal action in tate (x +, y) for ituation 2 and, repectively, i not conidered becaue it i in contradiction with Relation (28) for U n. We econd prove that Relation (30) implie Relation (29). Aume now that Relation (30) hold for a given n 0. Again, we only conider the non-trivial cae where 0 x < and y > 0. We have V n (x, y) + V n(x, y + ) U n (x, y) + U n(x, y + ) for 0 x, y > 0, (34) V n (x, y) + V n(x, y + ) U n (x, y) + U n(x +, y) for 0 x, y > 0, (35) V n (x, y) + V n(x, y + ) U n (x +, y ) + U n(x +, y) for 0 x, y > 0. (36) If keep i the optimal action in tate (x, y) and (x, y+) for ituation and 2, repectively, then V n (x, y)+v n (x, y+) = U n (x, y) + U (x, n y + ). Combining Eq. (34) with Relation (30) for U n prove Relation (29) for V n. If erve i the optimal action in tate (x, y) and (x, y+) for ituation and 2, repectively, then V n (x, y)+v n (x, y+) = U n (x +, y ) + U n (x +, y). Combining Eq. (36) with Relation (30) for U n prove Relation (29) for V n. If keep i the optimal action in tate (x, y) and erve i the optimal action in tate (x, y + ) for ituation and 2, repectively, then V n (x, y) + V (x, n y + ) = U n(x, y) + U n (x +, y). Inequality (35) prove Relation (29) for V n. The cae where erve would be the optimal action in tate (x, y) and keep would be the optimal action in tate (x, y + ) for ituation and 2, repectively, i not conidered becaue it i in contradiction with Relation (30) for U n. We now prove that Relation (27) and (29) for V n imply Relation (28) and (30) for U n+. The proof of Relation (27) for the departure term can be eaily done ince the term are identical in ituation and 2 except for the cot parameter related to the abandonment. Thi implie a poitive difference β(γ 3 γ 3 )((x+ )+ (x ) + ) 0 ince γ 3 γ 3. We therefore only focu on the cot and arrival term. We denote by G(x, y) and G (x, y) the cot term in ituation and 2, repectively and A(x, y) and A (x, y) the arrival term in ituation and 2, repectively. We have G (x, y) + G(x +, y) G(x, y) G (x +, y) = (γ γ )((x + )+ (x ) + ) 0, ince γ γ and G(x, y) + G (x, y + ) G (x, y) G(x, y + ) = γ 2 γ 2 0, ince γ 2 γ 2. For the arrival term we may write for x + k (the term where x < + k are implification of thi cae and are therefore omitted) A (x, y) + n A n(x +, y) A n (x, y) A n (x +, y) = α x (V (x, y) + n V n(x +, y) V n (x, y) V n (x +, y)) + q(v (x, n y + ) + V n(x +, y + ) V n (x, y + ) V n (x +, y + )) + ( α x+ q)(v n (x +, y) + V n(x + 2, y) V n (x +, y) V n (x + 2, y)) + q (V n (x + 2, y) V n (x +, y + ) + V (x, n y + ) V n (x +, y)) + (γ 3 γ )(α 3 x+ α x ). The term proportional to α x, q and α x+ q are poitive ince Relation (27) i true for V n. The term proportional to q i poitive ince thi relation define that the optimal policy in ituation 2 i of witch type. The lat term i alo poitive ince γ 3 γ 3. Therefore, Relation (28) i true for U n+. For the arrival term we alo may write for x + k (the term where x < + k are implification of thi cae and are therefore omitted) A (x, n y + ) + A n(x, y) A n (x, y + ) A n (x, y) = α x (V (x, n y + ) + V n(x, y) V n (x, y + ) V (x, y)) + n q(v (x, n y + 2) + V n(x, y + ) V n (x, y + 2) V (x, n y + )) + ( α x q)(v n (x +, y + ) + V n(x +, y) V n (x +, y + ) V n (x +, y)) + q (V (x, n y + 2) V n (x +, y + ) + V n (x +, y) V n (x, y + )).

30 30 B. Legro et al. / Performance Evaluation 95 (206) 40 The term proportional to α x, q and α x q are poitive ince Relation (29) i true for V n. The term proportional to q i poitive ince thi relation define that the optimal policy in ituation 2 i of witch type. Therefore, Relation (30) i true for U n+. Thi finihe the proof of the propoition. Appendix C. Performance analyi for Model C We provide here the detail for the tep of the performance evaluation method for Model C. Step. The tationary probabilitie are determined by the following et of equilibrium equation. For y = 0, we may write λp x,0 = (x + )µp x+,0, for 0 x < y 0, (37) λp y0,0 = (y 0 + )µp y0 +,0 + y 0 µp y0,, for x = y 0, (38) (λ + xµ)p x,0 = (x + )µp x+,0 + λp x,0, for y 0 < x <, (39) (( α)λ + µ)p,0 = µp +,0 + λp,0, for x =, (40) (λ( α) + µ)p x,0 = µp x+,0 + λ( α)p x,0, for < x + k, (4) (λ( α) + µ)p x,0 = µp x+,0 + ( q α)λp x,0, for x > + k. (42) For y = y i and i y 0, we have (λ + (y 0 + i )µ)p y0 +i,y i = (y 0 + i)µp y0 +i,y i, for x = y 0 + i, (43) (λ + (y 0 + i)µ)p y0 +i,y i = λp y0 +i,y i + (y 0 + i)µp y0 +i,y i + min(y 0 + i +, )µp y0 +i+,y i, for x = y 0 + i. (44) For 0 < y < y, y i y < y i+ and i < y 0, we get (λ + (y 0 + i)µ)p y0 +i,y = (y 0 + i)µp y0 +i,y+ + (y 0 + i + )µp y0 +i+,y, for x = y 0 + i. (45) For 0 < y y and i = 0 or y i y y i+ and i < y 0, we have (λ + xµ)p x,y = (x + )µp x+,y + λp x,y, for y 0 + i < x <, (46) (λ( α) + µ)p,y = µp +,y + λp,y, for x =, (47) (λ( α) + µ)p x,y = µp x+,y + λ( α)p x,y, for < x < + k, (48) (λ( α) + µ)p +k,y = µp +k+,y + λ( α)p +k,y + qλp +k,y, for x = + k, (λ( α) + µ)p x,y = µp x+,y + ( q α)λp x,y + qλp x,y, for x > + k. (49) Finally, for y y y0, we may write (λ( α) + µ)p,y = µp,y+ + µp +,y, for x =, (50) (λ( α) + µ)p x,y = µp x+,y + λ( α)p x,y, for < x < + k, (λ( α) + µ)p +k,y = µp +k+,y + λ( α)p +k,y + qλp +k,y, for x = + k, (λ( α) + µ)p x,y = µp x+,y + ( q α)λp x,y + qλp x,y, for x > + k. (5) Step 2. We denote by a the offered load, a = λ µ. Lemma implifie the expreion of p x,y, for x + k and y 0, by writing them a a function of only two tate probabilitie from the row y in the Markov chain a given in Fig. 7. Lemma. The following hold.. If y = y i for i y 0, then a p y0 +i,y i = y 0 + i + y 0 + i p y0 +i,y y 0 + i i. 2. For i < y 0, y i y < y i+ and 2 x y 0 i or i = 0, 0 y < y and 2 x y 0, we have x x p y0 +i+x,y = (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j. (y 0 + i + x)! 3. For 0 y < y y0, we have a( α) p +,y = + p,y a p,y. 4. For y 0 and 0 x k, we have a( α) a( α) x a( α) p +x,y = p +,y p,y j= a( α) x.

31 B. Legro et al. / Performance Evaluation 95 (206) 40 3 Proof. The proof of the firt tatement i traightforward. If y = y i for i y 0, then Eq. (43) lead to p y0 +i,y i = ( a y 0 + y 0+i )p +i y 0 +i y0 +i,y i. We now prove the econd tatement by induction on x. For i < y 0, y i y < y i+ and 2 x y 0 i or i = 0, 0 y < y and 2 x y 0, let u define the property P(x) by x x P(x) : p y0 +i+x,y = (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j, (y 0 + i + x)! for 0 i < y 0 and 2 x < y 0 i. Combining x = y 0 + i + and Eq. (46), and combining x = y 0 + and Eq. (39) prove that P(2) i true. Aume that P(x) and P(x+) are true, and let u prove that P(x+2) i alo true, for 0 i < y 0 and 2 x < y 0 i. We may write x x p y0 +i+x,y = (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j, (y 0 + i + x)! j= j= and p y0 +i+x+,y = (y 0 + i + )p y0 +i+,y (y 0 + i + x + )! x x (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j. j= Eq. (46) for i < y 0 and 2 x < y 0 i or Eq. (39) for i = 0 and 2 x < y 0 are equivalent to p y0 +i+x+2,y = a + y 0 + i + x + y 0 + i + x + 2 p a y 0 +i+x+,y y 0 + i + x + 2 p y 0 +i+x,y. We thu obtain, for 0 i < y 0 and 2 x < y 0 i, p y0 +i+x+2,y = a + y 0 + i + x + x x (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j y 0 + i + x + 2 (y 0 + i + x + )! j= a x x (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j ap y0 +i,y (y 0 + i + j)!a x j y 0 + i + x + 2 (y 0 + i + x)! j= = (y 0 + i + )p y0 x x +i+,y (a + y 0 + i + x + ) (y 0 + i + j)!a x j a(y 0 + i + x + ) (y 0 + i + j)!a x j (y 0 + i + x + 2)! ap y0 x x +i,y (a + y 0 + i + x + ) (y 0 + i + j)!a x j a(y 0 + i + x + ) (y 0 + i + j)!a x j (y 0 + i + x + 2)! j= j= x+ x+ = (y 0 + i + )p y0 +i+,y (y 0 + i + j)!a x j+ ap y0 +i,y (y 0 + i + j)!a x j+. (y 0 + i + x + 2)! We next deduce that P(x + 2) i alo true for 0 i < y 0 and 2 x < y 0 i. So, the property P(x) i true, which finihe the proof of the econd tatement. The third tatement immediately follow from Eq. (40) and (47). Let u now prove the fourth tatement. The correponding homogeneou equation to Eq. (4) and (48) i µz 2 (λ( α) + µ)z + λ( α) = 0, with z a a variable, for z C. It ha two olution, z = and z =. Thu for y 0 and 0 x k, x p +x,y = α + β with p,y = α + β and p +,y = α + β. Finally, for y 0 and 0 x k, we obtain a( α) a( α) x a( α) p +x,y = p +,y p,y Thi finihe the proof of the fourth tatement, and that of the lemma. j= a( α) x. Step 3. We how in Lemma 2 how p x,y, for x + k and y 0, can be computed a a function of p +k,0, p +k,,..., p +k,y.

32 32 B. Legro et al. / Performance Evaluation 95 (206) 40 Lemma 2. The olution of Eq. (42), (49) and (5) i given by y p x++k,y = a y,j x j z x, (52) for x 0, where z = 2 + a( α) a( α) 2 4( q α)a +, and the contant a y,j for y 0 and 0 j y are given by a y,0 = p +k,y, (53) for y 0, a y,y = a 0,0 y! qλz µz 2 + ( q α)λ y, for y > 0, and a y,j+ = ( µz 2 + ( q α)λ)(j + ) y i µz a y,i j 2 + ( ) i+j ( q α)λ + qλza y,j, (54) for 0 j < y and y >. i=j+2 Proof. Conider the ytem of equation given by Eq. (42), (49) and (5). Thi ytem can be olved analytically uing tandard reult from the theory of linear difference equation. Conider the correponding homogeneou equation to Eq. (42), (49) and (5). We have µz 2 (λ( α) + µ)z + ( q α)λ = 0, (55) with z a a variable, for z C. It ha two olution denoted by z and z and are given by z = a( α) a( α) 2 4( q α)a + +, 2 and z = 2 + a( α) a( α) 2 4( q α)a + +. We next provide the interval where z and z are ranging. We have 0 z < and z. Let u firt prove that z. Since z increae in q, we have z a( α) a( α) 2 4a( α) = a( α) a( α) =. 2 2 In what follow, we prove that 0 z <. Since z decreae in q, z a( α) a( α) 2 4a( α) = a( α) a( α) 2 a( + α) = <. 2 From Eq. (55), we may write µzz = ( q)λ. Since λ 0, 0 q and z > > 0, we obtain z 0. Becaue of the lat term in the right hand ide of Eq. (49) and (5), the tationary probabilitie p x++k,y, for x 0 and y 0, can be written a a um of two polynomial multiplied by z x and z x, repectively. Since z >, the convergence of the

33 B. Legro et al. / Performance Evaluation 95 (206) tationary probabilitie force the polynomial that i multiplied by z x to be equal to zero. We therefore obtain Eq. (52), for x 0 and y 0, that i, y p x++k,y = a y,j x j z x, with a y,j R for y 0 and 0 j y. In what follow, we compute the parameter a y,j, for y 0 and 0 j y, a a function of p +k,y, for y 0. It i traightforward to obtain Eq. (53). Uing Eq. (49), (5) and (52), we have y y (λ( α) + µ) a y,j x j z x = µ a y,j (x + ) j z x+ y + ( q α)λ a y,j (x ) j z x + qλ y a y,j x j z x, (56) for x, y > 0. Since y a y,j (x + ) j = y y i a y,i j x j, i=j and y a y,j (x ) j = y y i ( ) i a y,i j ( ) j x j. i=j Eq. (56) lead to (λ( α) + µ)a y,y z = µz 2 (a y,y + a y,y y) + ( q α)λ(a y,y a y,y y) + qλza y,y, (57) for y > 0. Since z i a root of Eq. (55), Eq. (57) can be rewritten a 0 = µz 2 a y,y y ( q α)λa y,y y + qλza y,y, for y > 0. Thi implie qλz a y,y = a y,y ( µz 2 + ( q α)λ)y, for y > 0. It thu follow that a y,y = a y 0,0 qλz, y! µz 2 + ( q α)λ for y > 0. For 0 j < y and y >, Eq. (56) alo lead to y i (λ( α) + µ)a y,j z = µz 2 a y,i j y i + ( q α)λ ( ) i a y,i j ( ) j + qλza y,j. (58) i=j Since z i a root of Eq. (55), Eq. (58) can be rewritten a y i µz 0 = a y,j+ (j + )(µz 2 ( q α)λ) + a y,i j 2 + ( ) i+j ( q α)λ + qλza y,j, i=j+2 i=j for 0 j < y and y >. Finally, thi lead to Eq. (54). We then compute a y,j, for y 0 and 0 j y, a a function of p +k,y, for y 0. Thi finihe the proof of the lemma. Step 4. Here, we evaluate all tationary probabilitie for x 0 and y = 0 a a function of p 0,0. Uing Eq. (37), we have p x,0 = ax x! p 0,0, for 0 x y 0. Uing the econd tatement of Lemma, we obtain x x p y0 +x,0 = (y 0 + )p y0 +,0 (y 0 + j)!a x j ap y0,0 (y 0 + j)!a x j, (60) (y 0 + x)! j= (59)

34 34 B. Legro et al. / Performance Evaluation 95 (206) 40 for 2 x y 0. From the econd and third tatement of Lemma, we may write a( α) a( α) x+ a a( α) x p +x,0 = p,0 p,0, (6) for 0 x k. Uing now Eq. (60) and (6), we obtain p +k,0 = (y 0 + )p y0 +,0 a( α) k a y 0 2 a( α) k+ (y 0 + j)!a y0 j + ( )!! ap y0,0 a( α) k a y 0 2 a( α) k+ (y 0 + j)!a y0 j + ( )!.! j= (62) Next, combining Eq. (38) and y 0 µp y0, = qλ p +k+x,0 provide a relation between p y0,0 and p y0 +,0. From Eq. (38) (42), we have λp y0,0 (y 0 + )µp y0 +,0 = qλ p +k,0 z. Combining the previou equation and Eq. (59) and (62) implie k qa y p y0 +,0 = a ( /)( z)! a (y 0 + j)!a y 0 j + ( )! j= y 0 + k y a (y 0 + j)!a y 0 j + ( )! qa ( /)( z)! Uing Eq. (38) and (63), we alo obtain p y0, = p 0,0 z + qa ( /)( z)! k a q a a y 0! y 0 2 k (y 0 + j)!a y 0 j + ( )! k+ ay0 k+ y 0! p 0,0.. k+ Uing Lemma and 2 together with Eq. (63) and (59), we thu have cloed-form expreion for the tationary probabilitie p x,0 for x 0 and p y0, a function of p 0,0. Step 5. We propoe in thi tep a method to compute the tationary probabilitie of a given row a a function of the tationary probabilitie in the previou row of the Markov chain. Conider y 0, and uppoe that the tationary probabilitie of row 0,,..., y are known in the Markov chain a a function of p 0,0. If for a given i (i {,..., y 0 }) we have y i y + < y i+ or 0 < y + < y, then (y 0 + i)µp y0 +i,y+ = qλ p +k+x,y, and if y + y y0 then µp,y+ = qλ p +k+x,y. Conequently, the firt tationary probability of row y + i alo known a a function of p 0,0. Oberve that uing Eq. (52) for y 0, we have y y p +k+x,y = a y,j x j z x = a y,j x j z x. For 0 j y, and x, y 0, we define the function f j in the variable t by f j (t) = xj t x with t [0, ). The function f j (t) i given by the recurive relation f n+ (t) = t (f n (t)) for n 0 and f 0 (t) = with t [0, ) [28]. Thu we can derive the t infinite um p +k+x,y, for 0 j y, and x, y 0, through a finite number of calculation. We next ditinguih three cae. Cae : If for a given i (i {,..., y 0 }) y + = y i, then uing the firt tatement of Lemma, the econd tationary probability of row y + (p y0 +i,y i ) i alo known a a function of p 0,0. Uing Lemma we evaluate p x,yi for y 0 + i x + k a a function of p 0,0 and p y0 +i+,y i. Uing Lemma 2 we evaluate p +k+x,yi for x 0 a a function of p +k,0, p +k,,..., p +k,yi. Since the tationary probabilitie of row 0,,..., y i 2 are known a a function of p 0,0 then we evaluate p +k+x,yi for x 0 a a function of p 0,0 and p y0 +i+,y i. Uing Eq. (44), we obtain (y 0 + i)µp y0 +i,y i = (λ( y0 +i α) + (y 0 + i)µ)p y0 +i,y i λp y0 +i,y i min(y 0 + i +, )µp y0 +i+,y i. Moreover, we have (y 0 + i)µp y0 +i,y i = qλ p +k+x,y i. Thu the equation (λ( y0 +i α) + (y 0 + i)µ)p y0 +i,y i λp y0 +i,y i min(y 0 + i +, )µp y0 +i+,y i = qλ p +k+x,y i provide a relation between p 0,0 and p y0 +i+,y i. A a conequence all probabilitie of row y + can be derived a a function of p 0,0. (63)

35 B. Legro et al. / Performance Evaluation 95 (206) Cae 2: If for a given i (i {,..., y 0 }) we have y i y+ < y i+ or 0 < y+ < y, then uing Lemma we evaluate p x,y+ for y 0 + i x + k a a function of p 0,0 and p y0 +i+,y+. Uing Lemma 2, we evaluate p +k+x,y+ for x 0 a a function of p +k,0, p +k,,..., p +k,y+. Since the tationary probabilitie of row 0,,..., y are known a a function of p 0,0 then we evaluate p +k+x,y+ for x 0 a a function of p 0,0 and p y0 +i+,y+. Uing Eq. (45) we obtain (y 0 + i)µp y0 +i,y+2 = (λ( y0 +i α) + (y 0 + i)µ)p y0 +i,y+ (y 0 + i + )µp y0 +i+,y+. Moreover we have µp,y+2 = qλ p +k+x,y+. Thu the equation (λ( y0 +i α) + (y 0 + i)µ)p y0 +i,y+ (y 0 + i + )µp y0 +i+,y+ = qλ p +k+x,y+ provide a relation between p 0,0 and p y0 +i+,y+. A a conequence all probabilitie of row y + can be derived a a function of p 0,0. Cae 3: If y + y y0, then uing Lemma we evaluate p x,y+ for x + k a a function of p 0,0 and p +,y+. In all cae uing Lemma 2 we evaluate p +k+x,y+ for x 0 a a function of p +k,0, p +k,,..., p +k,y+. Since the tationary probabilitie of row 0,,..., y are known a a function of p 0,0, then we evaluate p +k+x,y+ for x 0 a a function of p 0,0 and p +,y+. Uing Eq. (50), we obtain µp,y+2 = (λ( α) + µ)p,y+ µp +,y+. Moreover, we have (y 0 +i)µp y0 +i,y i = qλ p +k+x,y i. Thu, the equation (λ( α)+µ)p,y+ µp +,y+ = qλ p +k+x,y i provide a relation between p 0,0 and p +,y+. A a conequence all probabilitie of row y + can be derived a a function of p 0,0. Step 6. We now evaluate p 0,0. In what follow we prove that the overall um of the probabilitie can be evaluated after a finite number of calculation. We define the quantity P x a P x = y=0 p x,y for x. For x < + k we have x λ( α)p x = µp x+, then P +x = P, for 0 x k. For x + k we have ( q α)λp x = µp x+, k x then P +k+x = ( q α)a P, for x 0. Uing now y 0 µ y y= p y 0,y + (y 0 + )µ y 2 y=y p y0 +,y + + ( )µ y y 0 y=y p y0,y + µ P y y 0 y=0 p,y = k λq P +k+x, and P +k+x = P, we therefore obtain P = q a a( α) k y y0 p,y y y 0 y=0 y= a( q α) p y0,y y 0 + y 2 p y0 +,y y=y y y0 y=y y0 a( q α) Thu the quantity P can be computed after a finite number of calculation a a function of p 0,0. Since x= P x = x k P + () k P k a( q α), the overall um of the probabilitie can alo be evaluated after a finite number of calculation. Uing the fact that all probabilitie um up to one, we obtain p 0,0. Thi finihe the characterization of all tationary probabilitie. Step 7. We now ue the tationary probabilitie to derive the ytem performance meaure. The proportion of cutomer who ak for a callback, ψ, i given by k q ψ = q P x = P x=+k a( q α). The proportion of cutomer who balk the ytem, P b, i given by k a( α) x (a( α))k P b = α P x = αp + k x= a( q α). Applying the Little law lead to λ( ψ P b )E(W ) = xp +x. Therefore, k P a( α) k + (k ) E(W ) = λ( ψ P b ) 2 a( α) + k ( q α)a + k ( q α)a ( q α)a 2. Again, applying the Little law implie E(W 2 ) = y y 2 y y0 yp y0,y + yp y0 +,y + + yp,y + λψ y= y= y= y= x= k yp x,y. p,y.

36 36 B. Legro et al. / Performance Evaluation 95 (206) 40 Uing now the following relation ψe(w 2 ) + ( ψ P b )E(W ) = E(W), we obtain E(W). Thi finihe the characterization of the performance meaure in the general cae. Appendix D. Highet reervation and non-idling cae We implify here the expreion given in Corollary 2. We focu on the multi-erver etting for the pecial cae y 0 = (highet reervation) and y 0 = (non-idling). Thee reult are for example ueful for the numerical computation in Table 2. We firt conider the highet reervation policy with y 0 =. We have with with q Ψ = P b = E(W ) = k a! a( q α) α a! p 0,0 q a! p 0,0 = + q a! k a! λ( Ψ P b ) 2 + p 0,0 k a( q α) k a( q α) k a( q α) k p 0,0 x,, x + q a! a x+ (x+)! + k a () x + a () k! x! q a () k! k a( q α) k k k a( q α) k a( q α) We now conider the non-idling policy with y 0 =. We have k q a! p 0,0 Ψ = a( q α) k, q a P b = α a! p 0,0 () E(W ) = λ p 0,0 = a! p 0,0, a x a x! +! () k +(k ) k k 2 + q a k a( q α). a( q α) + a( q α) a( q α) 2 k ( Ψ P b ). ( q α)a +k ( q α)a ( q α)a 2 Uing the fact that the overall ytem i equivalent to an M/M/ queue with balking, we can alo compute E(W 2 ) in the non-idling cae a follow. E(W) = Ψ E(W 2 ) + ( Ψ P b )E(W ) = p 0,0 a! λ 2.,,

37 B. Legro et al. / Performance Evaluation 95 (206) Appendix E. Proof of Propoition 4 Since E(W ) and E(W 2 ) are both increaing in k, we chooe the optimal value of k which i k = 0. We rewrite the performance meaure a function of the Erlang Delay Lo Formulae, C and the parameter ρ = λ. Under the tability contraint λ( α) < µ, we have and C = E(W) = +! (ρ) x x! ( ρ( α)) Ψ ρ( α) λq ( ρ( α))., Ψ = qc, P b = αc, E(W ) = λq Ψ µ ( q α)ρ ( q α)ρ, Ψ P b Harel [29] how that the Erlang lo formulae i trictly decreaing in. In particular, he how that the function ϕ() =! (ρ) x i trictly increaing in. Thi expreion i in the denominator of C x! and i multiplied by the poitive coefficient ρ( α). We therefore deduce that C i decreaing in a well a Ψ, P b and E(W), becaue they are all proportional to C. Since Ψ and P b are decreaing in, P b Ψ i increaing in and E(W ) i decreaing in. Harel [29] alo how that +ϕ() i trictly convex in (convexity of the Erlang lo formula) and that (ρ) x = +ϕ()( ρ) i trictly convex in (convexity of the Erlang delay formula). Since C = +ϕ()( ρ( α)) +! x! ( ρ), one may write 2 C 2 = ( ρ( α)) 2( ρ( α))(ϕ ()) 2 ϕ ()( + ϕ()) ( + ϕ()( ρ( α))) 3. From [29], we have 2( ρ( α))(ϕ ()) 2 ϕ ()( + ϕ()) > 0 for α = 0 and α =. Since ϕ() doe not depend on α and ρ( α) i trictly increaing in α, we obtain 2 C > 0. Therefore, C 2 i trictly convex in. We next deduce that Ψ, P b and E(W) are convex in. One may ee that E(W ) i proportional to D, with D = We have D +2 + D 2D + = C +2 + C 2C + + (α + q) (C C + + C + C +2 2C C +2 ) ( (α + q)c )( (α + q)c + )( (α + q)c +2 ) = (C +2 + C 2C + )( (α + q)c + ) + 2(α + q)(c C + )(C + C +2 ). ( (α + q)c )( (α + q)c + )( (α + q)c +2 ) C (α+q)c. Since C i trictly convex, C +2 + C 2C + > 0. Since C i trictly decreaing, (C C + )(C + C +2 ) > 0. Thu, D i trictly convex in and E(W ) i alo trictly convex in. Thi finihe the proof of the propoition. Appendix F. Proof of Propoition 5 Recall that in the non-idling cae, we have k q a! p 0,0 Ψ = a( q α) k. q a Since for tability reaon Ψ k = q a! p 0,0 0, we obtain k ln a( q α) q a a( q α) Thu, Ψ i decreaing in k. We now rewrite E(W ) a ( q α)a E(W ) = f (k)g(k), λq k 2 0.

38 38 B. Legro et al. / Performance Evaluation 95 (206) 40 where f (k) = Ψ k, ( Ψ Pb ) and a( α) g(k) = k k + (k ) k a( α) 2 + k ( q α)a + k ( q α)a ( q α)a 2. Firt we how that f (k) i increaing in k. We have a( α) k Ψ P f k b Ψ ln ( Ψ P b ) (k) = k 2 0, ( Ψ Pb ) becaue Ψ i decreaing in k and g(k) = ( q α)a 2 qa < for tability reaon. We rewrite g(k) a k k 2 ( q α)a Only the numerator, ay n(k), of thi expreion depend on k. We have n (k) = qa a( α) ln a( α) k a( α) (k + ) ( q α)a a( α) ( q α)a + 0, ( q α)a ( q α)a + 2. ince <. We finally deduce that E(W ) i increaing in k, which complete the proof of the propoition. Appendix G. Proof of Propoition 6 When an idle agent conider the ervice of the firt outbound call in line, There are two poibilitie. The firt poibility (with probability r + r 2 > 0) i that the cutomer i available and will be erved within an exponential duration with parameter µ or µ 2 with probability r or r 2, repectively. The econd poibility (with probability r r 2 ) i that the cutomer i not available and the agent will be occupied a random duration exponentially ditributed with parameter µ 3. Thi cutomer will be then called back again latter according to the ame proce and independently of the fact that he ha been already called back. Let u denote by U i, a Bernoulli random variable, which take the value with probability r + r 2 and 0 otherwie for i ; by V i, a Bernoulli random variable, which take the value with probability r and 0 otherwie r +r 2 for i ; and by T i,j an exponential random variable with parameter µ j, for i and j =, 2, 3. The time duration, denoted by the random variable T, which i pent by the ytem capacity to erve an outbound call, can be written a follow. T = U (V T, + ( V )T,2 ) + ( U )(T,3 + U 2 (V 2 T 2, + ( V 2 )T 2,2 ) + ( U 2 )(T 2,3 + i i i = ( U k )U i V i T i, + ( U k )U i ( V i )T i,2 + ( U k )T i,3. i= k= i= k= i= k= We next derive the expected value of T. Since all the conidered random variable are independent, we have i i E(T) = E( U k )E(U i )E(V i )E(T i, ) + E( U k )E(U i )E( V i )E(T i,2 ) + i= k= i= k= = r r + r 2 µ + r 2 r + r 2 µ 2 + r r 2 r + r 2 µ 3. i= k= i E( U k )E(T i,3 ) We now derive the variance of T, denoted by Var(T). Again, from the independence of the random variable, we obtain i i i Var(T) = Var ( U k )U i V i T i, + Var ( U k )U i ( V i )T i,2 + Var ( U k )T i,3. i= k= i= k= i= k=

39 B. Legro et al. / Performance Evaluation 95 (206) Let u define the equence S n by S n = n Var k= ( U k), for n 0, with S 0 = 0. We have n S n = S n (Var( U n ) + E 2 ( U n )) + Var( U n )E 2 ( U k ), n for n. Since Var( U n ) = (r +r 2 )( r r 2 ), E 2 ( U n ) = ( r r 2 ) 2 and E 2 k= ( U k) = ( r r 2 ) 2n 2, we obtain k= S n = ( r r 2 )S n + (r + r 2 )( r r 2 ) 2n, (64) for n. Uing Eq. (64), it i eay to prove by induction that S n = ( r r 2 ) n ( ( r r 2 ) n ), for n 0. We next compute Var(U n V n T n, ), for n. We may write Var(U n V n T n, ) = Var(U n V n )Var(T n, ) + Var(U n V n )E 2 (T n, ) + E 2 (U n V n )Var(T n, ) = 2Var(Un V µ 2 n ) + E 2 (U n V n ) Hence Var i= = = i = 2Var(Un )Var(V µ 2 n ) + 2E 2 (U n )Var(V n ) + 2Var(U n )E 2 (V n ) + E 2 (U n )E 2 (V n ) = r 2( r r 2 )r 2 + 2r µ ( r r 2 )r + r r + r 2 r + r 2 = r (2 r ). µ 2 ( U k )U i V i T i, k= i S i (Var(U i V i T i, ) + E 2 (U i V i T i, )) + E 2 ( U k ) Var(U i V i T i, ) i= k= ( r r 2 ) i ( ( r r 2 ) i r (2 r ) ) i= = 2r µ 2 = i= µ 2 ( r r 2 ) i r 2 ( r µ 2 r 2 ) 2(i ) r (4 3r 2r 2 ) µ 2 (r + r 2 )(2 r r 2 ). i= + r 2 + ( r µ 2 r 2 ) 2i 2 r (2 r ) µ 2 Uing the ame approach, we alo obtain i= Var( i k= ( U k)u i ( V i )T i,2 ) = r 2(4 3r 2 2r ) µ 2 2 (r +r 2 )(2 r r 2 ) and i= Var( i k= ( U k )T i,3 ) = ( r r 2 )(4 r r 2 ). Thi finihe the proof of the propoition. µ 2 3 (r +r 2 )(2 r r 2 ) Appendix H. Supplementary data Supplementary material related to thi article can be found online at Reference [] ICMI. Extreme engagement in the multichannel contact center: Leveraging the emerging channel reearch Report and bet practice guide. ICMI Reearch Report, 203. [2] N. Gan, Y.-P. Zhou, A call-routing problem with ervice-level contraint, Oper. Re. 5 (2003) [3] O. Akşin, M. Armony, V. Mehrotra, The modern call-center: A multi-diciplinary perpective on operation management reearch, Prod. Oper. Manage. 6 (2007) [4] H. Bernett, M. Ficher, D. Mai, Blended call center performance analyi, IT Prof. 4 (2) (2002) [5] J. Pichitlamken, A. Delaurier, P. L Ecuyer, A. Avramidi, Modeling and imulation of a telephone call center, in: Proceeding of the 37th Conference on Winter Simulation, New Orlean, LA, 2003, pp [6] A. Delaurier, P. L Ecuyer, J. Pichitlamken, A. Ingolfon, A. Avramidi, Markov chain model of a telephone call center with call blending, Comput. Oper. Re. 34 (2007) [7] G. Pang, O. Perry, A logarithmic afety taffing rule for contact center with call blending, Manage. Sci. 6 () (204) [8] N. Gan, G. Koole, A. Mandelbaum, Telephone call center: Tutorial, review, and reearch propect, Manuf. Serv. Oper. Manage. 5 (2003) 73 4.

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