Pooling strategies for call center agent cross-training

IIE Transactions (009) 41, 546 561 Copyright C IIE ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/074081708051586 Pooling strategies for call center agent cross-training EYLEM TEKIN 1,, WALLACE J. HOPP and MARK P. VAN OYEN 3 1 Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843-3131, USA E-mail: eylem@tamu.edu Stephen M. Ross School of Business and 3 Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA E-mail: {whopp,vanoyen}@umich.edu Received April 007 and accepted July 008 Downloaded By: [University of Michigan] At: 18:55 1 June 009 The efficiency benefits achievable via cross-training in call and service center environments where agents serve distinct customer types are investigated. This is achieved by first considering specialized agents grouped into N departments according to the customer type they serve. Then, cross-training policies that pool a set of departments into a single larger department that serves all of the pooled call types according to either a first-come-first-served or non-pre-emptive priority service discipline are examined. The impact of system parameters, such as the number of servers, mean service times and service time coefficient of variation, on the decision of which departments to pool in order to minimize the expected delay in the system are characterized by comparing the proposed queueing models via standard queueing approximations and numerical analysis. The results show that if the mean service times of the departments that will be pooled are similar, pooling the departments with the highest service time coefficient of variation reduces the expected delay the most. Sufficient conditions for the mean service times to be considered similar are also provided. [Supplementary materials are available for this article. Go to the publisher s online edition of IIE Transactions for the following supplemental resource: Appendix of proofs for all results developed in the paper] Keywords: Multi-class queueing systems, pooling, cross-training, flexibility, call centers 1. Introduction The proliferation of call and service centers within the rapidly expanding service economy has elevated interest in their efficient design and management. A practice that is well suited to improving call centers is agent cross-training. Cross-training can improve customer service by offering more choices for matching agents to customer service requests and enhance efficiency by utilizing the existing workforce more effectively to handle a given call load. Effective use of cross-training requires two steps: (i) a design that determines which agents are trained to handle which customer types; and (ii) a control mechanism that dynamically assigns customers to agents. In this paper, we concentrate primarily on the design step by considering situations in which the control step is straightforward due to the structure of the system. A call center has a departmental structure if the agents can be divided into groups such that each customer type is unambiguously assigned to a single group (department) Corresponding author for service. The customer type to group assignment can be one-to-one, or if agents are cross-trained to serve more than one type, many-to-one. For example, in a multilingual call center, departments could correspond to groups of agents that handle customers from a specific set of languages (Brigandi et al., 1994). If all French-speaking agents also speak English and vice versa, then we could pool the French and English departments and route both call types to the combined department. However, if some, but not all French speakers speak English, then the system no longer has a departmental structure, since French/English agents cannot be assigned to a unique department. When the cross-training pattern follows a non-departmental structure, the control problem of assigning customers to agents become more complex (see Gans et al. (003) for a discussion). However, in many call centers, such as financial services support, where customers can be typed according to the products about which they are calling, grouping agents into merged departments is a common practice (Evenson et al., 1999). In such environments, it is important to know which departments to pool via crosstraining of agents. This paper provides insight into this problem. 0740-817X C 009 IIE

Pooling strategies in call centers 547 Downloaded By: [University of Michigan] At: 18:55 1 June 009 To understand call centers with departmental structures, we consider environments in which agents are initially grouped into departments based on the customer type(s) they are assigned. We assume that departments differ with respect to parameters such as arrival rate, mean service time, variability in service time and number of agents. Then we examine the impact of pooling, namely combining two or more departments into a larger department with the agents in the pooled department cross-trained to handle all of the call types of those departments. This involves two fundamental issues: (i) how many departments to pool; and (ii) which departments to pool. The specific focus of this paper is to determine how the various system parameters arrival rates, mean service times, variability in service times and the number of agents affect these pooling decisions. Pooling has been widely studied in the literature (for example, see Mandelbaum and Reiman (1998)), usually by comparing the two extreme scenarios of a dedicated system versus a fully pooled system. In a dedicated system, each group of servers is responsible for a specific customer type, whereas in a fully pooled system all servers are merged into a single group that serves all customer types. However, in many real-life situations, and particularly in call centers, cross-training all agents to handle all call types may not be feasible due to cost and/or quality penalties arising from cross-training, excessive agent stress and/or scarcity of agents capable of handling all of the call types. Therefore, while we consider fully pooled systems in this paper, we are particularly interested in partial pooling scenarios consisting of a mixture of pooled and dedicated departments. Specifically, we consider a call center that initially consists of N dedicated departments, and seek to pool k N departments into a larger department so as to minimize the expected waiting time of customers in the queue. However, because we recognize that using expected waiting time as the obective can lead to policies that degrade service to some customer types, we also derive sufficient conditions for pooling to reduce expected waiting of all customer types. Our results show that mean service times and the service time coefficient of variations are the maor factors that affect the decision on what departments to pool when the utilization is the same across all departments. The remainder of this paper is organized as follows: Section presents a literature review. Section 3 develops models of the pooling strategies we consider. Section 4 evaluates the relative effectiveness of these strategies and draws simple principles on the structure of the pooling decisions. In Section 5, we present a numerical analysis that addresses the impact of the simple principles derived in Section 4 for general systems. Section 6 concludes the paper. All proofs are given in the online Appendix.. Literature review The literature on call centers has grown rapidly in recent years. Gans et al. (003) and Aksin et al. (007) provide excellent surveys of this literature. Most of the analytic call center literature has been based on queueing models where the customers and agents are homogenous (see Koole and Mandelbaum (00) for a thorough review). A large body of the queueing literature on call centers has focused on dynamic routing of calls to available agents with the right skills. Perry and Nilsson (199) considered a system in which two classes of calls are served by a single pool of agents (V-design) and determined both the number of agents and the assignment policy needed to yield specified expected waiting times. Bhulai and Koole (003) also modeled a variant of this system. Stanford and Grassman (000) considered a call center with two call types and two agent groups where one group can handle both call types and the other can handle only one of them (N-design), and used fixed priority policies and matrix-geometric methods for performance and staffing analysis. Shumsky (004) considered the same problem and proposed an approximate analysis. Borst and Seri (000) proposed a dynamic routing rule that prioritizes the call type that is farthest behind schedule, and determined bounds on the number of fully cross-trained agents needed to provide a given service level. Koole and Talim (000) modeled a multi-skill call center as a network of queues and approximated each queue as an M/M/r loss system to minimize the number of unanswered calls. Gans and Zhou (003) studied a call routing problem with two call types where one call always has priority over the other and used a Markov decision process model to achieve service level constraints. Another stream of queueing-based call center research has relied on heavy traffic analysis to obtain asymptotic results. Harrison and Lopez (1999) considered the optimal dynamic assignment of n non-identical servers working in parallel to serve m types of customers with minimal waiting costs. They showed that if server skills overlap in an appropriate manner, then in the limiting Brownian control problem all servers merge into a single service pool. Bell and Williams (001) proved the asymptotic optimality of threshold controls for the N-design discussed above. In contrast, Van Mieghem (1995) proved the asymptotic optimality of a simple generalized cµ rule for a V-design with convex waiting costs. Harrison and Zeevi (005) studied the problem of staffing large call centers using stochastic fluid models. Armony and Maglaras (004a, 004b) considered multi-class, multi-server call centers with a call-back option, and proposed asymptotically optimal routing and staffing policies. Bassamboo et al. (006) investigated a similar problem with abandonments for large call centers and proposed a method for staffing and routing based on linear programming. It is important to note that our scope does not restrict attention solely to highly utilized systems and also we do not restrict attention to systems with a large number of servers. The literature most closely related to this paper has focused on choosing appropriate skill sets for servers, usually by comparing dedicated and fully pooled systems. The basic pooling models on which this work is based are described

548 Tekin et al. Downloaded By: [University of Michigan] At: 18:55 1 June 009 in Kleinrock (1976, pp. 7 90). Smith and Whitt (1981) and Benaafar (1995) showed that pooling reduces the average delay if arrivals and service times have the same distribution, but it can increase the average delay when different classes of customers require different service times. Buzacott (1996) considered a serial system with n stages where each customer is served by a distinct server at each stage, and transformed this system into a parallel system with n servers in which each server can perform the operations required for all n stages. He showed that the pooled system where all servers can perform all tasks is superior to the unpooled alternative, and the higher the task variability the greater the advantage. Mandelbaum and Reiman (1998) considered pooling of a Jackson network into an M/PH/1 queue. They compared the pooled and unpooled systems in terms of the steady-state mean soourn times and showed that depending on the system parameters, pooling can be either good or bad. Argon and Andradóttir (006) studied the effects of pooling several adacent stations in tandem lines where servers are cross-trained and two or more servers can work on the same ob simultaneously. They showed that the benefits of pooling can be substantial and that the bottleneck station should be among the pooled stations to obtain the greatest improvement. Our research differs from this literature in that: (i) we consider parallel service environments (e.g., call centers); and (ii) by comparing the performance of many (partially) pooled systems we provide insights for managerial decision making. Finally, some researchers have proposed alternatives to pooling. For example, Sheikhzadeh et al. (1998) and Jordan et al. (004) studied chaining of servers where each customer can be routed to one of two adacent servers and each server can process customers from two adacent classes. These studies showed that chaining has the potential to achieve most of the benefits of pooling with respect to performance measures such as the expected time spent in the system and throughput. Hopp et al. (004) investigated the value of chaining in CONWIP serial production lines, and showed that the impact of forming a complete chain of skills sets can be substantial in increasing throughput. Iravani et al. (005) developed a structural flexibility index for systems with cross-trained servers that quantifies the flexibility inherent in an arbitrary system structure and also elucidated the virtues of chaining. Wallace and Whitt (005) addressed routing and staffing problems in call centers under limited cross-training of agents (e.g., chaining) and demonstrated that when each agent has only two skills, in appropriate combinations, the performance is almost as good as when each agent has all skills. 3. Pooling strategies We consider a call or service center that services N customer types. We assume that the system initially has a departmental structure, in which customers of type i are served Fig. 1. Pooling of two departments. by agents in department i with c i servers, i = 1,,...,N. We further assume that type i customers arrive according to a Poisson process with rate λ i, and require a service time drawn from an independent and identicaly distributed (i.i.d.) sequence with finite mean T i and squared coefficient of variation (SCV) νi. Hence, the initial system can be modeled as N parallel M/G/c queues. We assume that this original system is stable, which requires ρ i = λ i T i /c i < 1 for i = 1,...,N. We define a pooling strategy by a set K, which represents the set of departments where all servers are cross-trained to handle all customer types in K (see Fig. 1 for an illustration of a case where N = 3andK ={1, }). Our obective is to determine which set of departments to pool in order to achieve the largest reduction in expected long-run average customer waiting time in queue (i.e., expected delay). Because standard performance measures cannot be expressed in closed form for the M/G/c queue, a variety of approximation methods have been proposed. A number of authors (Krampe et al., 1973; Stoyan, 1976; Hokstad, 1978; Nozaki and Ross, 1978) have independently proposed the following approximation for the expected delay in queue: Consider an M/G/c queue with arrival rate λ, mean service time T, utilization ρ = λt/c and service time SCV ν. Then, the standard M/G/c approximation for expected delay, denoted by W,is W = 1 + ν λ where ρ(ρc) c c!(1 ρ) p 0(c,ρ) = 1 + ν g(c,ρ), (1) λ g(c,ρ) = ρ(ρc)c c!(1 ρ) p 0(c,ρ) [ c 1 = ρ(ρc)c (ρc) n c!(1 ρ) n! n=0 ] 1 + (ρc)c, c!(1 ρ) is the expected number of customers in queue for an M/M/c queue with utilization ρ and p 0 (c,ρ) is the probability of an empty system.

Pooling strategies in call centers 549 Downloaded By: [University of Michigan] At: 18:55 1 June 009 Equation (1) is a heavy-traffic approximation and is exact for the M/G/1andM/M/c queues. Whitt (1993) reported that it is usually an excellent approximation, especially when ρ is high. In our analyses, we will use Equation (1) to approximate expected delay. Hence, for the original system with N departments, where each customer type is served by its own department, the standard approximation of the expected delay in department i is given by W i = ((1 + νi )/λ i)g(c i,ρ i ), and the standard approximation of the expected delay in the system across all customer typesisgivenby D 0 = 1 ( N ν i + 1 ) g(c i,ρ i ), () i=1 where = N i=1 λ i. If we pool departments in the set K ={i 1, i,...,i k } (where K {1,,...,N}), the expected delay in the system is D K = 1 λ W K + λ W, (3) K / K where W K is the approximate expected delay for the pooled departments in set K. Our problem is to choose the set K to minimize D K subect to the constraint K =k. Thatis, we seek the maximum impact from pooling k departments for a given k. Approximating the expected delay in the pooled queue, W K, requires specification of the policy used to serve the k customer types. We consider two policies: First-Come- First-Served (FCFS) and Non-Pre-Emptive Priority (NPP) service. Many service systems use FCFS to achieve fairness or because they cannot distinguish customer types until service has begun. However, it is well known that FCFS can increase the expected waiting time when heterogenous service distributions are pooled. Thus, we also consider the NPP discipline to represent systems in which customer types are identified prior to service (e.g., via an interactive voice response phone menu in call centers) and this information is used to prioritize calls. In order to obtain analytical results and clear insights, in the following we will use the above queueing approximations as acceptably accurate models of system behavior. 3.1. FCFS service discipline Suppose departments in the set K ={i 1, i,...,i k } are pooled so that these customer types are served on a FCFS basis by K c servers. The SCV for the aggregate service time distribution (νk ) of the pooled department is given by ( )( νk = K λ K λ ( T ν + 1 )) ( K λ ) 1. (4) T Since arrivals of each customer type follow a Poisson process, using Equation (1), we can approximate the expected delay for customers in the pooled department by W FCFS K = ν K + 1 ( K λ ) g c, K K c ρ K c, (5) and hence, expected system delay can be approximated as DK FCFS = 1 ν K + 1 g K c, c ρ K K c + ν + 1 g(c,ρ ). (6) / K Because customer arrivals are Poisson and service times are i.i.d., all customers within the pooled department experience the same mean wait in queue under the FCFS service discipline. 3.. NPP service discipline We now consider systems in which customers in the pooled queue are served (non-pre-emptively) in priority order. Suppose departments in the set K ={i 1, i,...,i k } are pooled and the indices in set K are ordered so that customer type i is given non-pre-emptive priority in service over customer type i l if < l. There is no exact expression for the expected delay in queueing systems with multiple Poisson arrival streams, multiple servers and general service time distributions. Therefore, we use the following approximation (Buzacott and Shanthikumar 1993): W NPP K,n = WFCFS K W M/G/1:FCFS K W M/G/1:NPP K,n. (7) In Equation (7), WK,n NPP denotes the approximate delay for customer type i n (i.e., the customer with priority n) in the pooled queue that serves k customer types in the set K ={i 1, i,...,i k } based on the NPP service discipline. WK FCFS is given by Equation (5) and denotes the approximate expected delay in the pooled department when the service discipline is FCFS. WK M/G/1:FCFS is the expected delay for the pooled queue when the original K c servers are replaced by a single server that serves customer type i n with a mean service time of T in / K c, i n K and n = 1,...,k, and FCFS is used. Similarly, WK,n M/G/1:NPP is the expected delay of customers with priority n in the pooled queue when the original K c servers are replaced by a single server that serves customer type i n with a mean service time of T in / K c, i n K and n = 1,...,k, and NPP is used. Hence, the first term in Equation (7) is the ratio by which the expected delay changes when we shift from one server to K c servers under the FCFS service discipline. This approximation is based on the notion

550 Tekin et al. Downloaded By: [University of Michigan] At: 18:55 1 June 009 that this ratio is relatively insensitive to the service discipline (see, e.g., Buzacott and Shanthikumar (1993, p. 88) for ustification and further details.) To complete our description of the approximation in Equation (7), we describe the computations of WK M/G/1:FCFS and WK,n M/G/1:NPP. 3..1. Computation of WK M/G/1:FCFS Expected delay for an M/G/1 queue is well known to be λe(s ) (1 ρ), (8) where λ is the arrival rate, E(S ) is the second moment of the service time and ρ is the utilization (see, e.g., Kulkarni (1995, p. 379)). To compute WK M/G/1:FCFS, first note that when departments in the set K ={i 1, i,...,i k } are pooled, the pooled system has an arrival rate of K λ and K c servers. To model this system as an M/G/1 queue, for = T in / K c and E((S i n ) ) = E(Si n )/( K c ) as the new mean service time and the corresponding second moment, respectively. Then the second moment of the service time in the pooled queue is each i n K, n = 1,...,k, we define T i n E ( SK ) 1 = = K λ λ E ( S ) = K K λ T (ν + 1) ( K λ )( K c ). K λ E ( ) S )( K c ( K λ Hence, from Equation (8) the single server pooled system has expected delay: WK M/G/1:FCFS K = λ ( T ν + 1 ) (1 ρ ) ( K c ), (9) where ρ K = λ T K c. (10) 3... Computation of WK,n M/G/1:NPP For a M/G/1 queue with multiple customer types and NPP service discipline, using T i n = T in / K c and E((S i n ) ) = E(Si n )/( K c ), i n K and n = 1,...,k, as the new mean service time and the corresponding second moment, respectively, the expected delay for a customer with priority n (i.e., customer type i n ) is given by (see, e.g., Gelenbe and Mitrani (1980, pp. 35 40)): W M/G/1:NPP K,n = ) W 0 (1 ρ Hn )(1 ρ Hn+1 ), (11) where ρ H1 = 0, ρ Hn+1 = ρ i 1 + ρ i +...+ ρ i n 1 + ρ i n, ρ i n = λ i n T in K K c and W 0 = λ ( T ν + 1 ) ( K c ). Note that both WK M/G/1:FCFS and WK,n M/G/1:NPP expressions are exact. Using Equations (5), (9) and (11) and substituting into Equation (7), we obtain the approximate expected delay of a customer with priority n as WK,n NPP = 1 ρ (1 ρ Hn )(1 ρ Hn+1 ) K λ ( T ν + 1 ) ( ( K λ ) g c,ρ ). (1) T K Hence, the approximate expected system delay under the NPP service policy is [ k ( v + 1 ) D NPP K λ in (1 ρ ) K λ T n=1 ( K λ ) (1 T ρhn )(1 ρ Hn+1 ) g ( ) c,ρ + ν + 1 g(c,ρ ), (13) K / K = 1 where ρ is given by Equation (10). 4. Analysis We now use the above models to examine the impact of the following system parameters on the choice of which departments to pool: (i) mean service times (T i ); (ii) number of servers (c i ) for each department; and (iii) the SCV of the service times (νi ). The system utilization has a first-order effect on performance, so we must carefully control for this effect. To eliminate utilization as a factor, we assume that all N departments are staffed so that their utilizations are equal to ρ (i.e., ρ i = ρ for i = 1,...,N). Koole and Mandelbaum (00) reported that most call centers operate at a utilization level of approximately 90 95%. As a result, the utilization does not vary significantly across departments. Furthermore, having uniform utilization is fair to agents from different departments. Thus, we make the assumption of uniform utilization across departments, and we investigate the impact of T i, c i and ν i, i = 1,...,N on the pooling decision. We perform all of our analyses under both FCFS and NPP service disciplines. In the following, we use the approximations to expected delays that are presented in the previous section. Hence, whenever we say expected delay, it is shorthand for approximate expected delay. 4.1. Effect of mean service time To investigate the role of service time while isolating utilization, we consider systems where ρ i = ρ, c i = c, ν i = ν, cρ = λ i T i for i = 1,,...,N. Note that this scales arrival rates so that utilization is held constant as service times vary between departments. As a consequence, the departments with long mean service times have low demand arrival rates.

Pooling strategies in call centers 551 Downloaded By: [University of Michigan] At: 18:55 1 June 009 From Equation (6), the expected system delay under FCFS is given by D FCFS K = 1 ν [ ( + 1 1 T k i /T )g(kc,ρ) K ] + (N k)g(c,ρ). (14) Smith and Whitt (1981) showed that when service rates are different, pooling can be counterproductive under the FCFS service discipline. By considering an example where two M/M/1 queues are pooled, they demonstrated that the expected waiting time can be arbitrarily large if the two service rates differ greatly from each other. In Proposition 1, we provide a sufficient condition for pooling to be advantageous in systems with uniform utilization and service time variability under the FCFS service discipline. Proposition 1. When ρ i = ρ, c i = c, ν i = ν, cρ = λ i T i for i = 1,,...,N, if there exists a set K {1,,...,N} such that K =k and k 3 T i /T, (15) K then DK FCFS D 0 (i.e., the system with k pooled departments has an approximately smaller expected delay than the original system). As an example, consider k = and assume that we pool departments i and. Condition (15) reduces to (Ti 6T i T + T )/T i T 0 which implies that Ti 6T i T + T 0. Without loss of generality, we can assume T i T and define T i = at where a R and a 1. Then, the last inequality can be written as a 6a + 1 0. The root that satisfies the constraint a 1 for this quadratic function is a = 3 +, and the inequality is satisfied when a 3 +. Thus, if there exist two departments for which the expected service time of one does not exceed approximately six times that of the other, pooling is always advantageous under the FCFS service discipline. Assuming that there is at least one k-tuple that satisfies the condition in Proposition 1, the k departments to pool to minimize the expected system delay given by Equation (14) are the ones that minimize K T i/t.thatis, there exist N choose k k-tuples for N departments. Let k denote the set of all k-tuples. Then, the set K k such that K = arg min{k k : K T i/t } minimizes the expected system delay. Suppose that, without loss of generality, departments are numbered such that T 1 T... T N, i.e., the mean service time of department 1 is the smallest and the mean service time of department N is the largest. The following proposition shows that the k departments to pool to minimize the expected system delay should be consecutive based on this ordering. Proposition. Let T ={T 1, T,...,T N } where T i R + and finite, N 3 and the elements of T are ordered such that T i T if i <. Denote by k the collection of all subsets of T with k elements. If the set K k is defined as K = arg min{k k : K T i/t }, then K is a set of k consecutive elements of T. This proposition implies that instead of computing K T i/t for each K k which results in N choose k possible combinations, it is sufficient to compute this function for only sets of size k where the elements are ordered in the sense described above. This results in N + 1 k computations. In cases where more than one set of k departments achieve the minimum, ties can be broken arbitrarily. Note that K T i/t is minimal when all service times are equal, and it implies that the k departments to pool should be the ones with mean service times closest to each other. For instance, for k =, the best departments to pool are the pair that satisfy (i, ) = arg min i {T /T i : T T i, i, = 1,,...,N, i}. The intuition behind this result is the following; if two departments, one with very short and the other with very long service times, are pooled, then under FCFS the customers with short processing times will wait longer than they did in the original system because of the customers with long service times who arrive before them and tie up the server. To examine the equity (fairness) issues involved in pooling under FCFS, we can compare the difference in expected delay experienced by customer type n K without and with pooling; given by [ ( ) ] W n WK FCFS = ν + 1 T n g c,ρ) 1 cρ k ( T i g(kc,ρ) [ ( )] ν + 1 cρ g(kc,ρ) T n 1 T k i. Hence, if k ( T i)/t n, the expected delay for type n customers declines; otherwise, pooling may increase the expected delay of these customers. To see the impact of pooling under the NPP service discipline when ρ i = ρ, c i = c, ν i = ν, cρ = λ i T i for i = 1,,...,N, note that Equation (13) reduces to [( DK NPP = 1 ν + 1 k 1 ρ T ) T K in (k (n 1)ρ)(k nρ) n=1 ] g(kc,ρ) + (N k)g(c,ρ). (16) From Equation (16), the expected delay in the system is minimized when priority is given to the customer type with smaller mean service time. This can be shown by a simple interchange argument once the customers that are served by the pooled department are sequenced in priority order so that T i1... T ik. Pooling k departments and serving the customers in the pooled queue with a NPP service discipline may also result in a longer expected system delay.

55 Tekin et al. Downloaded By: [University of Michigan] At: 18:55 1 June 009 The following proposition presents a sufficient condition for DK NPP D 0. Proposition 3. When ρ i = ρ, c i = c, ν i = ν, cρ = λ i T i for i = 1,,...,N, if there exists a set K {1,,...,N} such that K =k and ( k k ) 1 ρ T )(, (17) T K in (k (n 1)ρ)(k nρ) n=1 then D NPP D 0 (i.e., the system with k pooled departments K has an approximately smaller expected delay than the original system). Let us define B(n) = (k (n 1)ρ)(k nρ) for n = 1,,...,k. If there is at least one k-tuple that satisfies the condition in Proposition 3, the best k departments to pool in order to minimize expected system delay (given by Equation (16)) are the ones that minimize: ( k ) 1 T )(. (18) B(n)T K in n=1 Hence, under NPP, the decision of which k departments to pool depends on the mean service times as well as the utilization,which impacts B(n). Equation (18) also implies that the mean service times in the pooled system should be similar. In general, it is possible for pooling to increase expected delay for some customer types. However, since for i n K: W in WK,n NPP ν + 1 cρ g(kc,ρ) 1 ρ T in T, (19) (k (n 1)ρ)(k nρ) K it follows that if 1 1 ρ (k (n 1)ρ)(k nρ) K T T in, (0) then W in WK,n NPP. Hence, if the condition given in Equation (0) holds for all n, then the sufficiency condition in Equation (17) is satisfied, and pooling improves service for all customer types. We would like to note that the sufficiency condition provided for NPP in Proposition 3 is weaker than the condition provided for FCFS in Proposition 1. We can easily show this by comparing the expressions on the Right-Hand Side (RHS) of inequalities (15) and (17). That is, RHS of Equation (15) k RHS of Equation (17) = ( T i k ) k T i)( (1 ρ) T K T in B(n) n=1 ( ) = T i F(ρ) 0, where F(ρ) := k n=1 1 T in Note that F(0) = 0and F (ρ) = k 1 n=1 [ ] k(k (n 1)) k(k n) 1 +. k (n 1)ρ k nρ [ k(k n)n 1 1 ] 0, (k nρ) T in T in+1 since T in T in+1, n = 1,...,k 1. This implies that F(ρ) 0forall0 ρ<1, and hence the RHS of Equation (15) is larger than k times the RHS of Equation (17). We use the results of this section to provide managerial insights on: (i) how many departments to pool (i.e., how to choose k); and (ii) which departments to pool, via two examples. First, we consider a four-department system (N = 4) in which two customer types have short mean service times, while the others require much longer service times. For example, type 1 calls may be arriving from remote repair technicians at independent shops who are requesting simple firmware information, type calls are for product registration and types 3 and 4 are complex technical support and problem solving tasks. Figure presents the minimal expected system delay as we successively pool departments when T 1 = 1, T = 1., T 3 = 50, T 4 = 55, ρ = 0.8, c = andν = 1. For each k, we find the optimal set of departments to pool, so that expected system delay is minimized. The case for k = 1 corresponds to the specialist system. Observe that for k = pooling decreases expected system delay under both FCFS and NPP. However, when we further increase the number of pooled departments to k = 3, performance becomes worse so that the resulting expected delay exceeds that of the specialist system. We note that for k = 3, the sufficiency conditions of Propositions 1 and 3 are not satisfied. Finally, when all departments are pooled (k = 4), the expected delay is slightly less than in the specialist system, but only under NPP. Under FCFS, Fig.. Expected delay in queue for ρ = λ i T i /c for i = 1,, 3, 4.

Pooling strategies in call centers 553 Downloaded By: [University of Michigan] At: 18:55 1 June 009 the delay from pooling all departments is better than pooling three departments, but is still worse than that of the specialist system. We can explain these observations by investigating the expressions for D 0, DK FCFS and DK NPP in this section. From Equations (), (14) and (16), we can write the difference in the expected system delay between the specialist and the pooled systems as follows: D 0 DK i = 1 ν + 1 [kg(c,ρ) VF i g(kc,ρ)], (1) for i = FCFS, NPP, where ( ) VF FCFS = 1 T k i /T and VF NPP K ( k 1 ρ = T i) T K in (k (n 1)ρ)(k nρ). n=1 We refer to VF as the variability factor since this term corresponds to the SCV of the aggregate service time distribution. As VF increases, the benefit of pooling in reducing expected system delay decreases. In the above example, when two departments are pooled, VF FCFS = 1and VF NPP = 0.95. As a result, pooling reduces the expected system delay. However, when three departments are pooled, VF FCFS = 10.8 and VF NPP = 3.8. As a result of this steep increase in VF values, pooling three departments performs worse than the specialist system, even though the customers are served by a larger pool of servers. The reason for the increase in VF values is that the mean service time of type customers is much less than that of type 3 and type 4 customers. When all departments are pooled, VF FCFS = 1.54 and VF NPP = 4.4. In this case, VF values also increase; however, the amount of increase in this step is less than the one in the previous step. The expected system delay for the pooled system decreases because of the increase in the number of servers. Note that the variability factor under NPP is always less than the VF under FCFS because customers with shorter mean service times are given priority under NPP, and hence, they do not have to wait for customers with long mean service times. To summarize, the pooling decision is affected by the trade-off between two factors. As more departments are pooled, the variability factor increases. On the other hand, since pooling increases the number of servers, as more departments are pooled, the value of g(kc,ρ) decreases. Hence, the benefit of pooling depends on whether the increase in the variability factor can be compensated by pooling service capacities. When mean service times of the customers in the pooled system differ greatly (i.e., the sufficiency conditions of Propositions 1 and 3 are not satisfied), the variability factor can be too high and take away the benefit of pooling servers. Second, we consider a system with heterogenous mean service times (T 1 = 1, T = 3, T 3 = 4.5, T 4 = 10, T 5 = 1 Fig. 3. Expected delay in queue for ρ = λ i T i /c for i = 1,,...,6. and T 6 = 4 with c = andν = 1) where the sufficiency conditions are always satisfied as we successively pool departments i = 1,,...,6. Figure 3 shows that this results in a steady decrease in expected system delay as k increases under both FCFS and NPP. For each value of k in the figure, the optimal set of departments to pool is also indicated. Note that, under FCFS, the optimal set of departments to pool does not change with utilization. However, under NPP, the optimal set does depend on utilization, as we expect from Equation (16). We observe that as k increases, the optimal set of departments to pool does not change in a greedy way. For instance, under FCFS with ρ = 0.95, it is optimal to pool departments 4, 5 and 6 when k = 3, but it is optimal to pool departments, 3, 4 and 5 when k = 4, and department 6 is no longer in the optimal set. Since all departments have the same number of servers in this example, the departments to include in the pooled system as k increases depend on the value of the variability factor induced by pooling these departments, which in turn depend on the values of mean service times. Moreover, Fig. 3 indicates that as utilization increases, the incremental benefit of pooling increases more rapidly with k. 4.. Effect of department size To consider the impact of department size, we set ρ i = ρ, T i = T and ν i = ν, andρ/t = λ i /c i, i = 1,...,N. That is, departments see different arrival rates but are initially staffed such that their utilization is the same. Expected customer delay in the system under FCFS is given by D K = DK FCFS = DK NPP [ ( ) = 1 ν + 1 g c i,ρ + ] g(c i,ρ). () i / K

Downloaded By: [University of Michigan] At: 18:55 1 June 009 554 Tekin et al. The expected delay in the system is the same under FCFS and NPP because mean service times are equal for all departments as shown in the following proposition. Proposition 4. When ρ = λ i T/c i and ν i = ν for i = 1,,...,N, DK NPP = DK FCFS. Pooling decreases the expected system delay for both FCFS and NPP since D 0 D K = ((ν + 1)/ )[ g(c i,ρ) g( c i,ρ)] > 0, and g(c,ρ) is a non-increasing function of c when ρ is fixed. It can also be shown that g(c,ρ)isconvexinc when ρ is fixed. This enables us to prove the following theorem. Theorem 1. When ρ i = ρ, T i = T,ν i = ν and ρ/t = λ i /c i, i = 1,,...,N, pooling the k smallest departments minimizes the expected delay in the system under both FCFS and NPP. Under NPP, the expected system delay is not affected by the priority order of each customer type served. On the other hand, the priority order affects the individual expected delay of customers. To see what type of priority policy should be used, for K ={i 1,...,i k } we define C(n) = n =1 c i, n = 1,...,k, andnotethatexpecteddelayofa customer with priority n under NPP is W NPP K,n = (ν + 1)T ρc(k) 1 ρ (1 ρc(n 1)/C(k))(1 ρc(n)/c(k)) g ( ) c i,ρ. (3) By a simple interchange argument in Equation (3), we see that priority should be given to the customer with the smallest arrival rate (equivalently, the smallest number of servers) in order to minimize expected delay in the system. Under both FCFS and NPP, the expected delay for each customer type in the pooled queue is less than or equal to the delay in the specialist system since for n K: W n W FCFS K = ν + 1 [ T g(c n,ρ) g ρ c n ( c i,ρ ) ] c > 0, i and [ W in WK,n NPP = ν + 1 T g(cin,ρ) ρ c in (1 ρ)/c(k) [1 ρc(n 1)/C(k)][1 ρc(n)/c(k)] ( )] g c i,ρ > ν + 1 [ T g(cin,ρ) ρ c in ( ) 1 ] [1 C(n 1)/C(k)]C(k) g c i,ρ [ = ν + 1 T g(cin,ρ) g( c ] i,ρ) ρ c in C(k) C(n 1) [ ( > ν + 1 T )] g(c in,ρ) g c i,ρ > 0 ρc in for i n K. To summarize, when ρ = λ i T/c i and ν i = ν for i = 1,,...,N, pooling is always beneficial. It is optimal to pool the k smallest departments. The expected system delay and the expected delay for each customer type are decreasing in the number of pooled departments (k). Hence, complete pooling is the best. 4.3. Effect of service time coefficient of variation Next, we consider the case where departments differ only in terms of their service time SCV values (νi ). Since λ i = λ, T i = T, c i = c and ρ i = ρ = λt/c for i = 1,,...,N, it follows that: [ D K = DK FCFS = DK NPP = 1 ( ν i + 1 ) g(kc,ρ) k + ( ν i + 1 ) ] g(c,ρ). (4) i / K As in the previous section, DK FCFS following proposition. = D NPP K as shown in the Proposition 5. When ρ i = ρ = λt/cfori = 1,,...,N and departments may have different service time SCV values, = D FCFS D NPP K K. Furthermore, since D 0 D K = ( (ν i + 1)/ ) [g(c,ρ) g(kc,ρ)/k] > 0, pooling is always advantageous under both FCFS and NPP when departments differ only in service time SCV. The following result shows how to pool for maximum effect. Theorem. When λ i = λ, T i = T, c i = c and ρ i = ρ = λt/c and ν i is the service time SCV for department i, i = 1,,...,N, pooling the k departments with the highest service time SCV values minimizes the expected system delay under both FCFS and NPP. Under the selection rule of Theorem, the expected delay in the system decreases as the number of pooled departments increases. To observe this, relabel the departments so that ν i ν whenever i <, and denote K ={1,...,k} and K ={1,...,k + 1}. Then, using Equation (4), we can show that D K D K by the following proposition. Proposition 6. When ρ = λt/c for all departments, and departments may have different service time SCV values the

Pooling strategies in call centers 555 Downloaded By: [University of Michigan] At: 18:55 1 June 009 expected system delay (D K ) decreases as the number of pooled departments (k) increases. However, for n K: W n WK FCFS = ν n + 1 λ g(c,ρ) (ν i + 1) g(kc,ρ) [ λk g(kc,ρ) ( (ν n λ + 1) ν i + 1 ) ], k so that the expected waiting times of priority n customers in the pooled departments will decrease under FCFS if k [ (ν i + 1)]/(νn + 1). When ν i k 1fori K, this condition is satisfied for all customer types served by the pooled department. For example, if service times are exponentially distributed, then νi = 1fori = 1,...,N, and the condition is satisfied for all customer types. On the other hand, under NPP: W in W NPP K,n [ g(kc,ρ) (ν i λ n + 1 ) 1 ρ ( ν i + 1 )]. (5) (k (n 1)ρ)(k nρ) Hence, pooling does not increase the expected delay for customers with priority n if the term in the brackets in Equation (5) is non-negative. 5. Numerical analysis In the previous sections, we studied the effects of mean service time, number of servers and the service time SCV by varying each at a time, and determined the optimal pooling principles such as pooling the departments with similar processing times, pooling the smallest departments and pooling the departments with highest service time SCV values. In this section, we aim to address the following three questions via a numerical analysis. 1. How much do these simple principles carry over if we consider a general system with various different values for the parameters?. Can we approximately order these principles according to their impact in reducing the expected customer delay? 3. How do errors in the approximations that we used for computing the expected delay affect the pooling decisions? In our numerical study, we considered three- and fourdepartment systems and used the approximations developed in Section 3 for FCFS and NPP service disciplines to determine the best departments to pool in order to minimize the expected customer delay in the system. For each experiment, we computed the expected delay in the system for all possible combinations of departments that can be pooled. We performed this computational analysis for various different systems. Below, we present a representative subset from this analysis. For the three-department system, we considered ρ = 0.95 for all departments. We set c 1 = 5, T 1 = 3andν1 = 0. for department 1, and c = 5, T = 5andν = 0.5fordepartment. We varied c 3 = 10, 15, 0, T 3 = 10, 15, 40 and ν3 = 0.8, 1, resulting in 7 experiments in total as shown in the first four columns of Table 1. We chose these parameter values so that in all experiments if one uses the principle of pooling the two departments with similar mean service times, then pooling departments 1 and is optimal. If one uses the principle of pooling the two departments with the least number of servers, then pooling departments 1 and 3 is optimal. Finally, if one uses the principle of pooling the two departments with the highest service time SCV values, then pooling departments and 3 is optimal. Column 5 (i.e., D 0 ) of Table 1 gives the expected delay in the system where there is no pooling. Columns 6 and 7 (columns 9 and 10) in Table 1 give the minimum expected delay in the system when two departments are pooled (i.e., K =), and the departments to pool for the minimum expected delay under FCFS (NPP), respectively. Column 8 (column 11) in Table 1 gives the expected delay in the system if all three departments are pooled (i.e., K =3) under FCFS (NPP) service discipline. As can be seen from column 7 in Table 1, the best strategy is either to pool the two departments with similar mean service times or to pool the two departments with the highest service time SCV values. When T 3 = 40, pooling department 3 with any of the other two departments violates the sufficiency condition given in Proposition 1. In other words, D 0 > D{1,3} FCFS and D 0 > D{,3} FCFS, because the mean service time of department 3 is too long compared to the others. Hence, this department is not pooled with any of the other two departments even if it has the highest service time SCV. As a result, note that when T 3 = 40, increasing the number of pooled departments from k = tok = 3 increases the expected delay. Otherwise, pooling more departments results in a smaller expected delay. In all other cases (i.e., when T 3 40), except for experiments 13, and 3, pooling the two departments with the highest service time SCV values is the best. In experiments 13, and 3, the service time SCV values for all three departments can be considered as, at most, moderate variability. However, when T 3 = 15, this mean service time is significantly larger than the mean service times of the other two departments. Hence, pooling the two departments with similar mean service times (i.e., departments 1 and ) is better. It is also important to mention that if departments and 3 were pooled,the expected delayswouldbe 3.30, 3.15 and 3.30 for experiments 13, and 3, respectively. These delays are close to the ones obtained by pooling departments 1 and. That is, the percentage deviations of expected delay are 1.5, 3.6and.% when departments and 3 are pooled rather than pooling departments 1 and for experiments 13, and 3, respectively. As a result, the principle of pooling the two departments with the highest service time SCV values is a good general rule when the sufficiency condition

556 Tekin et al. Table 1. Best pooling strategies for a three-department system FCFS ( K =) NPP ( K =) FCFS ( K =3) NPP ( K =3) Experiment c 3 T 3 ν3 D 0 D K K D K D K K D K Downloaded By: [University of Michigan] At: 18:55 1 June 009 1 10 10 0.8 4.74.98, 3 1.60.41, 3 0.95 10 10 1 4.95 3.07, 3 1.68.47, 3 1.00 3 10 10 6.03 3.50, 3.10.75, 3 1.5 4 10 15 0.8 4.95 3.41, 3 1.97.43, 3 0.91 5 10 15 1 5.18 3.54, 3.10.49, 3 0.97 6 10 15 6.30 4.18, 3.7.8, 3 1.6 7 10 40 0.8 5.5 3.70 1, 3.84.66, 3 1.08 8 10 40 1 5.49 3.94 1, 4.17.76, 3 1.18 9 10 40 6.68 5.13 1, 5.83 3.6, 3 1.64 10 15 10 0.8 4.36.80, 3 1.5.9, 3 0.9 11 15 10 1 4.55.90, 3 1.61.36, 3 0.98 1 15 10 5.5 3.39, 3.09.68, 3 1.7 13 15 15 0.8 4.65 3.5 1, 1.95.36, 3 0.91 14 15 15 1 4.85 3.44, 3.09.44, 3 0.98 15 15 15 5.88 4.17, 3.81.81, 3 1.31 16 15 40 0.8 5.06 3.53 1, 4.10.64, 3 1.09 17 15 40 1 5.8 3.76 1, 4.48.75, 3 1.0 18 15 40 6.40 4.88 1, 6.39 3.30, 3 1.71 19 0 10 0.8 4.04.63, 3 1.4.18, 3 0.88 0 0 10 1 4..73, 3 1.5.5, 3 0.95 1 0 10 5.09 3.4, 3.03.60, 3 1.6 0 15 0.8 4.38 3.04 1, 1.88.9, 3 0.90 3 0 15 1 4.57 3.3 1,.03.37, 3 0.97 4 0 15 5.5 4.06, 3.78.77, 3 1.33 5 0 40 0.8 4.89 3.39 1, 4.14.6, 3 1.10 6 0 40 1 5.10 3.60 1, 4.54.73, 3 1.0 7 0 40 6.16 4.66 1, 6.56 3.30, 3 1.74 in Proposition 1 is satisfied. To support this further, note that if departments 1 and 3 (i.e., two departments with the least number of servers) are pooled, the expected delay would be 4.0, 3.80 and 4.06 with percentage deviations from the best (i.e., pooling departments 1 and ) as 9., 5 and 5.7% for experiments 13, and 3, respectively, which is clearly worse than pooling departments 1 and, or and 3. Under NPP, pooling the two departments with the highest service time SCV values is the best in all cases. By prioritizing the customers with respect to their mean service times so that higher priority is given to customers with shorter mean service times, the NPP service discipline already reduces the variability caused by different mean service times in the system, and by pooling the two departments with the highest service time SCV values it can further reduce the variability in the system, and hence the expected delay is reduced. Expected delay also reduces as the number of pooled departments increase as a result of prioritization. Table presents the best pooling strategies for a fourdepartment system under FCFS for ρ = 0.9. In our experimental design, we fixed the number of servers, mean service times and service time SCV values for departments 1, and 3asc 1 = c 3 = 4, c = 10, T 1 = T 3 = 5, T = 10, ν 1 = 0.5, ν = 1andν 3 = 1.5. We varied c 4 =, 8, 3, T 4 = 3, 15, 30 and ν4 = 0., 1, 3 for department 4 resulting in 7 cases. Our observations are similar to what we have observed for the three-department systems. When two out of four departments are pooled, pooling the two with the highest SCV values is the best strategy for all experiments except for experiments 6, 15, 18, 4 and 7 in Table. When three out of four departments are pooled, pooling the three departments with the highest SCV values is the best strategy for all experiments except for experiments 11, 17,, 6 and 7 in Table. In all of the cases where pooling based on the highest service time SCV values is not optimal, the departments with the highest service time SCV values have either very short or very long mean service times with respect to each other. Therefore, the optimal strategy is to pool the two departments with similar mean service times, but still with high service time SCV values. Table 3 presents the best pooling strategies for a fourdepartment system under NPP for the same experiments that we have used for FCFS. In Table 3, the departments that should be pooled to minimize the expected delay are ordered based on the optimal priority of service to their corresponding customer types. For example, in experiment 1 in Table 3, when K =, it is optimal to pool the

Pooling strategies in call centers 557 Table. Best pooling strategies for a four-department system under FCFS K = K =3 K =4 Experiment c 4 T 4 ν D 0 D K K D K K D K Downloaded By: [University of Michigan] At: 18:55 1 June 009 1 3 0. 8.44 5.53, 3 3.53 1,, 3 1.98 3 1 9.48 6.58, 3 4.09, 3, 4.01 3 3 3 1.09 7.7 3, 4 4.0, 3, 4.09 4 15 0. 10.08 6.61, 3 4. 1,, 3.18 5 15 1 11.33 7.86, 3 4.69, 3, 4.34 6 15 3 14.45 8.71, 4 5.18, 3, 4.75 7 30 0. 10.33 6.78, 3 4.3 1,, 3.4 8 30 1 11.61 8.06, 3 5.1, 3, 4.75 9 30 3 14.81 9.8, 4 6.18, 3, 4 3.58 10 8 3 0. 5.06 3.6, 3.0 1,, 3 1.14 11 8 3 1 5.59 3.79, 3.47 1, 3, 4 1.1 1 8 3 3 6.93 4.53 3, 4.74, 3, 4 1.39 13 8 15 0. 8.51 5.48, 3 3.39 1,, 3 1.57 14 8 15 1 9.40 6.38, 3 3.89, 3, 4 1.9 15 8 15 3 11.64 7.96, 4 4.88, 3, 4.81 16 8 30 0. 9.30 5.99, 3 3.70 1,, 3.10 17 8 30 1 10.8 6.97, 3 4.68 1,, 3.81 18 8 30 3 1.7 9.4, 3 6.86, 3, 4 4.59 19 3 3 0. 1.90 1.18, 3 0.69 1,, 3 0.9 0 3 3 1.04 1.30 3, 4 0.8, 3, 4 0.34 1 3 3 3.38 1.58 3, 4 0.98, 3, 4 0.48 3 15 0. 5.3 3.31, 3 1.87, 3, 4 0.6 3 3 15 1 5.71 3.71, 3.16, 3, 4 0.90 4 3 15 3 6.68 4.67, 3.87, 3, 4 1.60 5 3 30 0. 6.87 4.8, 3.50 1,, 3 1.04 6 3 30 1 7.37 4.78, 3 3.00 1,, 3 1.60 7 3 30 3 8.6 6.03, 3 4.5 1,, 3 3.00 departments that serve customer types 3 and, and customer type 3 has NPP over customer type. We also make similar observations for NPP. When two departments out of four departments are pooled, pooling the two departments with the highest SCV values is the best strategy for all experiments except for experiments 3, 9, 18 and 7 in Table 3. When three out of four departments are pooled, pooling the three departments with the highest SCV values is the best strategy for all experiments except for experiments 7, 16 and 3 in Table 3. For all cases in Tables and 3, the expected delay decreases as the number of pooled departments increase. In the systems that we consider, there are two types of variability. The first type is the variability caused by the differences in mean service times. The second type is the service time variability (i.e., random fluctuations in service durations). As the first type of variability increases, the advantages of pooling decreases as shown in Section 4.1. Mandelbaum and Reiman (1998) made the same conclusion for different systems under heavy traffic. On the other hand, Section 4.3 shows that as the second type of variability increases, pooling becomes more advantageous. This result was also reported in Buzacott (1996) for a different setting. Our numerical analysis provides guidelines in managing this trade-off between the two types of variability. In general, pooling departments with the highest service time SCV values reduces the expected delay in the system the most given that the sufficiency conditions in Propositions 1 and 3 are satisfied for FCFS and NPP, respectively. In other words, making the pooling decision based on the second type of variability improves the system performance the most if the variability caused by the differences in mean service times is kept under a threshold. However, it is possible that the departments with the highest service time SCV values are the ones with very different mean service times, and hence, the sufficiency conditions are not satisfied. In such cases, it is better to pool the departments with similar mean service times (so that the sufficiency conditions are satisfied) rather than pooling the departments with the highest service time SCV values. All of our observations summarized above are based on the approximations for the expected delay developed for pooled systems in Section 3. In order to assess the performance of these approximations, we simulated the systems tabulated in Tables 1, and 3. The simulations were performed using a computer code written in C. Each

558 Tekin et al. Table 3. Best pooling strategies for a four-department system under NPP K = K =3 K =4 Experiment c 4 T 4 ν D 0 D K K D K K D K Downloaded By: [University of Michigan] At: 18:55 1 June 009 1 3 0. 8.44 5.1 3, 3.0 1, 3, 1.34 3 1 9.48 6.16 3, 3.40 4, 3, 1.36 3 3 3 1.09 6.74 4, 3.48 4, 3, 1.4 4 15 0. 10.08 6.1 3, 3.61 1, 3, 1.41 5 15 1 11.33 6.81 3, 4 3.91 3,, 4 1.5 6 15 3 14.45 8.0 3, 4 4.5 3,, 4 1.79 7 30 0. 10.33 6.7 3, 3.64 3,, 4 1.9 8 30 1 11.61 7.13 3, 4 3.86 3,, 4 1.47 9 30 3 14.81 8.35, 4 4.39 3,, 4 1.91 10 8 3 0. 5.06 3.00 3, 1.70 1, 3, 0.67 11 8 3 1 5.59 3.49 4, 3 1.93 4, 3, 0.7 1 8 3 3 6.93 4.03 4, 3.06 4, 3, 0.8 13 8 15 0. 8.51 5.05 3,.86 1, 3, 1.00 14 8 15 1 9.40 5.8 3, 4 3.8 3,, 4 1.3 15 8 15 3 11.64 7.31 3, 4 3.97 3,, 4 1.80 16 8 30 0. 9.30 5.5 3, 3.09 3,, 4 0.95 17 8 30 1 10.8 6.43 3, 4 3.46 3,, 4 1.7 18 8 30 3 1.7 8.11, 4 4.38 3,, 4.07 19 3 3 0. 1.90 1.08 3, 0.57 1, 3, 0.16 0 3 3 1.04 1. 3, 0.65 4, 3, 0.19 1 3 3 3.38 1.46 4, 3 0.75 4, 3, 0.7 3 15 0. 5.3 3.03 3, 1.59 1, 3, 0.46 3 3 15 1 5.71 3.4 3, 1.97 1, 3, 0.67 4 3 15 3 6.68 4.37 3, 4.56 3,, 4 1.19 5 3 30 0. 6.87 3.91 3,.05 1, 3, 0.55 6 3 30 1 7.37 4.41 3,.5 3,, 4 0.85 7 3 30 3 8.6 5.66 3, 3.33 3,, 4 1.58 simulation run began with an empty system, and was simulated for 000 000 service completions, including a warmup period of 00 000 service completions. Each run was replicated 0 times. The standard errors were computed at a confidence level of 99%. In the simulations, we generated service times from a Gamma distribution with shape parameter α and scale parameter β so that the mean and the variance of the service time are αβ and αβ, respectively. Hence, for a department with mean service time of T and service time SCV of ν,wesettheshapeparameter to α = 1/ν and the scale parameter to β = Tν.For each experiment, we simulated all possible combinations of pooling scenarios, and chose the set of departments that resulted in the minimum mean delay as the optimal set of departments to be pooled. We compared these sets and their corresponding mean delays with the ones we obtained from approximations. For three-department systems, the optimal sets of departments obtained from simulation exactly match the ones obtained from the approximations for all 108 experiments. The expected delays obtained by the approximations and the simulations are presented in Table 4 for four-department systems when K =and K =3 under both FCFS and NPP. In Table 4, each row corresponds to the experiment given in Tables and 3. D K and K represent the minimum expected delay and the corresponding set of pooled departments obtained by the approximations, respectively. K values are listed in Tables and 3. On the other hand, K denotes the optimal set of departments to pool that are determined by simulation, and D K denotes the corresponding mean delay. The sets K and K are the same for all cases except for the seven instances typed in bold in Table 4. For example, under FCFS when K =, for experiment number 18, K ={, 3} from Table. However, the optimal set we obtained by simulation for the same experiment is K ={, 4}. When we compute the expected delays for these two sets by using the approximation and simulation, respectively, we obtain D{,3} FCFS = 9.4, D{,4} FCFS = 10.01, D {,3} FCFS = 9.89 ± 0.07, D {,4} FCFS = 9.56 ± 0.09. These values are very close to each other, and the same observation is valid for the other six cases in bold in Table 4. In addition, when we compare the approximate expected delays with the mean delays obtained by simulation for all cases listed in Table 4, we see that the performances of both approximations (i.e., FCFS and NPP) are good. For FCFS, the average and maximum percentage errors are 1.39% and 11.9%, respectively, out of 54 cases. In 48 of these 54 cases, the percentage error is less

Pooling strategies in call centers 559 Table 4. Comparison of approximated expected delays of pooled systems with simulation results under FCFS and NPP for fourdepartment systems FCFS K = FCFS K =3 NPP K = NPP K =3 Experiment D K D K D K D K D K D K D K D K Downloaded By: [University of Michigan] At: 18:55 1 June 009 1 5.53 5.58 ± 0.04 3.53 3.50 ± 0.03 5.1 5.16 ± 0.03 3.0 3.07 ± 0.03 6.58 6.6 ± 0.04 4.09 4.10 ± 0.04 6.16 6.0 ± 0.03 3.40 3.47 ± 0.0 3 7.7 7.41 ± 0.07 4.0 4.0 ± 0.03 6.74 6.87 ± 0.06 3.48 3.53 ± 0.0 4 6.61 6.67 ± 0.05 4. 4.19 ± 0.03 6.1 6.16 ± 0.04 3.61 3.66 ± 0.03 5 7.86 7.9 ± 0.05 4.69 4.7 ± 0.04 6.81 6.94 ± 0.03 3.91 4.01 ± 0.0 6 8.71 8.80 ± 0.06 5.18 5.08 ± 0.05 8.0 8.14 ± 0.05 4.5 4.34 ± 0.03 7 6.78 6.84 ± 0.05 4.3 4.9 ± 0.03 6.7 6.3 ± 0.04 3.64 3.76 ± 0.03 8 8.06 8.11 ± 0.05 5.1 5.11 ± 0.06 7.13 7.33 ± 0.05 3.86 4.06 ± 0.03 9 9.8 9.59 ± 0.08 6.18 5.75 ± 0.06 8.35 8.55 ± 0.06 4.39 4.4 ± 0.03 10 3.6 3.30 ± 0.0.0.01 ± 0.0 3.00 3.04 ± 0.0 1.70 1.74 ± 0.0 11 3.79 3.8 ± 0.03.47.47 ± 0.0 3.49 3.55 ± 0.0 1.93 1.99 ± 0.01 1 4.53 4.6 ± 0.05.74.73 ± 0.03 4.03 4.11 ± 0.04.06.10 ± 0.01 13 5.48 5.55 ± 0.04 3.39 3.38 ± 0.03 5.05 5.11 ± 0.03.86.93 ± 0.03 14 6.38 6.4 ± 0.04 3.89 3.9 ± 0.04 5.8 5.88 ± 0.04 3.8 3.36 ± 0.03 15 7.96 7.96 ± 0.05 4.88 4.67 ± 0.06 7.31 7.38 ± 0.08 3.97 3.99 ± 0.05 16 5.99 6.07 ± 0.04 3.70 3.70 ± 0.03 5.5 5.58 ± 0.03 3.09 3.0 ± 0.03 17 6.97 7.0 ± 0.05 4.68 4.65 ± 0.03 6.43 6.5 ± 0.04 3.46 3.60 ± 0.03 18 9.4 9.56 ± 0.09 6.86 6.17 ± 0.08 8.11 8.14 ± 0.06 4.38 4.6 ± 0.04 19 1.18 1.0 ± 0.01 0.69 0.69 ± 0.01 1.08 1.10 ± 0.01 0.57 0.59 ± 0.01 0 1.30 1.31 ± 0.01 0.8 0.80 ± 0.01 1. 1.3 ± 0.01 0.65 0.69 ± 0.00 1 1.58 1.60 ± 0.0 0.98 0.98 ± 0.01 1.46 1.48 ± 0.01 0.75 0.77 ± 0.01 3.31 3.37 ± 0.03 1.87 1.91 ± 0.0 3.03 3.08 ± 0.0 1.59 1.64 ± 0.0 3 3.71 3.73 ± 0.03.16.18 ± 0.0 3.4 3.44 ± 0.0 1.97.00 ± 0.0 4 4.67 4.70 ± 0.074.87.75 ± 0.04 4.37 4.41 ± 0.05.56.53 ± 0.03 5 4.8 4.36 ± 0.03.50.51 ± 0.0 3.91 3.98 ± 0.03.05.1 ± 0.0 6 4.78 4.8 ± 0.03 3.00.97 ± 0.0 4.41 4.44 ± 0.03.5.58 ± 0.0 7 6.03 6.11 ± 0.09 4.5 4.1 ± 0.08 5.66 5.68 ± 0.08 3.33 3.3 ± 0.04 than %. For NPP, the average and maximum percentage errors are 1.85% and 5.40%, respectively, out of 54 cases. In 35 of these 54 cases, the percentage error is less than %. 6. Conclusions In this paper, we examined pooling strategies for call centers consisting of multiple departments. For different scenarios of system parameters, we compared expected system delay for specialist and pooled systems, and investigated the impact of different system parameters, including mean service times, service time variability and department size in deciding which departments to pool. In addition to analytical sufficient conditions derived from well-accepted queueing approximations, we are in some cases able to distill the mathematics into straightforward managerial principles. For example, when mean service times are similar, pooling the departments with the highest service time SCV values reduces the expected delay the most. When mean service times differ greatly, pooling may actually result in worse performance than the specialist system, even if with very high utilization. The pooling structure treated here is a common and relatively tractable type of cross-training structure. However, in many practical settings each server is given a set of skills, and hence a department structure may not exist. Previous studies suggest that partial cross-training can be nearly as effective as full pooling (see e.g., Hopp et al. (004) and Jordan et al. (004)). Thus, an important future research goal is to see whether analogies to the simple pooling principles identified here can be derived for systems using more elaborate patterns of cross-training. Acknowledgements The work of Mark Van Oyen was partially supported by NSF grant DMI-054063. References Aksin, Z., Armony, M. and Mehrotra, V. (007) The modern call center: a multi-disciplinary perspective on operations management research. Production and Operations Management, 16, 665 688. Argon, N.T. and Andradóttir, S. (006) Partial pooling in tandem lines with cooperation and blocking. Queueing Systems, 5(1), 5 30.

560 Tekin et al. Downloaded By: [University of Michigan] At: 18:55 1 June 009 Armony, M. and Maglaras, C. (004a) On customer contact centers with call back option: customer decisions, routing rules, and system design. Operations Research, 5(), 71 9. Armony, M. and Maglaras, C. (004b) Contact centers with a call back option and real-time delay information. Operations Research, 5(4), 57 545. Bassamboo, A., Harrison, J.M. and Zeevi, A. (006) Design and control of a large call center: asymptotic analysis of an LP-based method. Operations Research, 54(3), 419 435. Bell, S.L. and Williams, R.J. (001) Dynamic scheduling of a system with two parallel servers in heavy traffic with complete resource pooling: asymptotic optimality of a continuous review threshold policy. Annals of Applied Probability, 11, 608 649. Benaafar, S. (1995) Performance bounds for the effectiveness of pooling in multi-processing systems. European Journal of Operational Research, 87, 375 388. Bhulai, S. and Koole, G. 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(eds), AMS, pp. 31 48. Stoyan, D. (1976) Approximations for M/G/s queues. Mathematische Operationsforschung und Statistik, 7, 587 594. Van Mieghem, J.A. (1995) Dynamic scheduling with convex delay costs: the generalized cµ rule. Annals of Applied Probability, 5, 149 167. Wallace, R.B. and Whitt, W. (005) A staffing algorithm for call centers with skill based routing. Manufacturing and Service Operations Management, 7(4), 76 94. Whitt, W. (1993) Approximations for the GI/G/m queue. Production and Operations Management,, 114 161. Biographies Eylem Tekin is an Assistant Professor in the Department of Industrial and Systems Engineering at Texas A&M University. Before oining Texas A&M in 005, she held a faculty position in the Department of Statistics and Operations Research at the University of North Carolina-Chapel Hill. She received her B.S. and M.S. degrees in Industrial Engineering from Bilkent University, Turkey, and her Ph,D. degree from Northwestern University in 003. Her research interests include optimal design and control of queueing and inventory systems with a particular emphasis on modeling and analysis of operational flexibility. Her publications have appeared in IIE Transactions, European Journal of Operational Research, and Management Science. She is a member of INFORMS and IIE. Wallace J. Hopp s research focuses on the design, control and management of operations systems, with emphasis on innovation processes, manufacturing and supply chain systems, and healthcare systems. He has won a number of awards, including teaching awards at both the engineering and business schools at Northwestern, as well as the 1985 Nicholson Prize (for best student paper in Operations Research), the 1990 Scaife Award (with Mark Spearman, for the paper with the greatest potential for assisting an advance of manufacturing practice ), the 1998 IIE Joint Publishers Book-of-the-Year Award (for the book Factory Physics: Foundations of Manufacturing Management), the 001 Sargent Americanism Award from SME, the 005 IIE Technical Innovation Award, and the 006 SME Education Award. He is a Fellow of IIE, INFORMS, MSOM and POMS and is Editor-in-Chief of the ournal Management Science and a Senior Editor of POMS. He is an active industry consultant, whose clients have included Abbott Laboratories, Anixter, Bell & Howell, Black & Decker, Case, Dell, Ford, Eli Lilly, Emerson Electric, General Motors, John Deere, IBM, Intel, Motorola, Owens Corning, S&C Electric, SPX, Texas Instruments, Whirlpool, Zenith, and others.

Downloaded By: [University of Michigan] At: 18:55 1 June 009 Pooling strategies in call centers 561 Mark P. Van Oyen is presently an Associate Professor of Industrial and Operations Engineering at the University of Michigan, where he also serves as the Director of the Engineering Global Leadership Honors Program for the College of Engineering. His research focuses on the analysis, design, control and management of operations systems, with emphasis on healthcare and supply chains. He has won a number of awards, including ALCOA Manufacturing Systems Faculty Fellow, a best paper award from IIE Transactions, and Researcher of the Year from Loyola U. Chicago s School of Business. He has served as Associate Editor for Operations Research, Naval Research Logistics, andiie Transactions and Senior Editor for Flexible Services & Manufacturing. He has served as a faculty member in Northwestern University s School of Engineering (1993 005) and Loyola University of Chicago s School of Business Administration (1999 005). In industry, he worked on the research staff of GE Corporate R&D as well as in analysis and simulation for Lear Siegler s Instrument & Avionic Systems Division.