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Telephone call centers: Tutorial, review, and research prospects
 Mgmt
, 2003
"... Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments trad ..."
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Cited by 150 (6 self)
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Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value – and at the same time fundamentally limited – in their ability to characterize system performance. We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research. Acknowledgments The authors thank Lee Schwarz, Wallace Hopp and the editorial board of M&SOM for initiating this project, as well as the referees for their valuable comments. Thanks are also due to L. Brown, A. Sakov, H. Shen, S. Zeltyn and L. Zhao for their approval of importing pieces of [36, 112].
Server Staffing to Meet TimeVarying Demand
 Management Science
, 1996
"... We consider a multiserver service system with general nonstationary arrival and servicetime processes in which s(t), the number of servers as a function of time, needs to be selected to meet projected loads. We try to choose s(t) so that the probability of a delay (before beginning service) hits or ..."
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Cited by 51 (22 self)
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We consider a multiserver service system with general nonstationary arrival and servicetime processes in which s(t), the number of servers as a function of time, needs to be selected to meet projected loads. We try to choose s(t) so that the probability of a delay (before beginning service) hits or falls just below a target probability at all times. We develop an approximate procedure based on a timedependent normal distribution, where the mean and variance are determined by infiniteserver approximations. We demonstrate that this approximation is effective by making comparisons with the exact numerical solution of the Markovian M t / M / s t model.
A staffing algorithm for call centers with skillbased routing
 Manufacturing and Service Operations Management
, 2005
"... informs ® doi 10.1287/msom.1050.0086 © 2005 INFORMS Call centers usually handle several types of calls, butitis usually notpossible or costeffective to have every agent be able to handle every type of call. Thus, the agents tend to have different skills, in different combinations. In such an environ ..."
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Cited by 32 (7 self)
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informs ® doi 10.1287/msom.1050.0086 © 2005 INFORMS Call centers usually handle several types of calls, butitis usually notpossible or costeffective to have every agent be able to handle every type of call. Thus, the agents tend to have different skills, in different combinations. In such an environment, it is challenging to route calls effectively and determine the staff requirements. This paper addresses both of these routing and staffing problems by exploiting limited crosstraining. Consistent with the literature on flexible manufacturing, we find that minimal flexibility can provide great benefits: Simulation experiments show that when (1) the servicetime distribution does not depend on the call type or the agent and (2) each agent has only two skills, in appropriate combinations, the performance is almost as good as when each agent has all skills. We apply this flexibility property to develop an algorithm for both routing and staffing, aiming to minimize the total staff subject to perclass performance constraints. With appropriate flexibility, it suffices to use a suboptimal routing algorithm. Simulation experiments show that the overall procedure can be remarkably effective: The required staff with limited crosstraining can be nearly the same as if all agents had all skills. Hence, the overall algorithm is nearly optimal for that scenario.
Staffing of timevarying queues to achieve timestable performance
, 2005
"... Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probab ..."
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Cited by 28 (19 self)
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Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce timestable performance for several other operational measures. Motivated by telephone call centers, we focus on manyserver models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential timetoabandon distribution (the +M), an exponential servicetime distribution and a nonhomogeneous Poisson arrival process. We develop three different methods for staffing, with decreasing generality and decreasing complexity: First, we develop a simulationbased iterativestaffing algorithm (ISA), and conduct experiments showing that it is effective. The ISA is appealing because it applies to very general models and is automatically validating: we directly see how well it works. Second, we extend the squarerootstaffing rule, proposed by Jennings et al., which is based on the associated infiniteserver model. The rule dictates that the staff level at time t be st = mt + β √ mt, where mt is the offered load (mean number of busy servers in the infiniteserver model) and the constant β reflects the service grade. We show that the service grade β in the staffing formula can be represented as a function of the target delay probability α by
Piecewiselinear diffusion processes
 Advances in Queueing
, 1995
"... Diffusion processes are often regarded as among the more abstruse stochastic processes, but diffusion processes are actually relatively elementary, and thus are natural first candidates to consider in queueing applications. To help demonstrate the advantages of diffusion processes, we show that ther ..."
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Cited by 27 (8 self)
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Diffusion processes are often regarded as among the more abstruse stochastic processes, but diffusion processes are actually relatively elementary, and thus are natural first candidates to consider in queueing applications. To help demonstrate the advantages of diffusion processes, we show that there is a large class of onedimensional diffusion processes for which it is possible to give convenient explicit expressions for the steadystate distribution, without writing down any partial differential equations or performing any numerical integration. We call these tractable diffusion processes piecewise linear; the drift function is piecewise linear, while the diffusion coefficient is piecewise constant. The explicit expressions for steadystate distributions in turn yield explicit expressions for longrun average costs in optimization problems, which can be analyzed with the aid of symbolic mathematics packages. Since diffusion processes have continuous sample paths, approximation is required when they are used to model discretevalued processes. We also discuss strategies for performing this approximation, and we investigate when this approximation is good for the steadystate distribution of birthanddeath processes. We show that the diffusion approximation tends to be good when the differences between the birth and death rates are small compared to the death rates.
Dynamic scheduling of a multiclass queue in the HalfinWhitt heavy traffic regime
, 2003
"... We consider a Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes. Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates and aba ..."
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Cited by 27 (4 self)
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We consider a Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes. Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates and abandonment penalties are generally different for the different classes. The problem studied is that of dynamically scheduling the various classes. We consider the HalfinWhitt heavy traffic regime, where the total arrival rate and the number of servers both become large in such a way that the system’s traffic intensity parameter approaches one. An approximating diffusion control problem is described and justified as a purely formal (i.e., non rigorous) heavy traffic limit. The HamiltonJacobiBellman equation associated with the limiting diffusion control problem is shown to have a smooth (classical) solution, and optimal controls are shown to have an extremal or “bangbang ” character. Several useful qualitative insights are derived from the mathematical analysis, including a “square root rule ” for sizing large systems and a sharp contrast between system behavior in the HalfinWhitt regime versus that observed in the “conventional ” heavy traffic regime. The latter phenomenon is illustrated by means of a numerical example having two customer classes.
A diffusion approximation for the G/GI/n/m queue
 Operations Research
"... informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra ..."
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Cited by 26 (7 self)
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informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queuelength stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra waiting spaces). We use the steadystate distribution of that diffusion process to obtain approximations for steadystate performance measures of the queueing model, focusing especially upon the steadystate delay probability. The approximations are based on heavytraffic limits in which n tends to infinity as the traffic intensity increases. Thus, the approximations are intended for large n. For the GI/M/n/ � special case, Halfin and Whitt (1981) showed that scaled versions of the queuelength process converge to a diffusion process when the traffic intensity �n approaches 1 with �1 − �n � √ n → � for 0 <�<�. A companion paper, Whitt (2005), extends that limit to a special class of G/GI/n/mn models in which the number of waiting places depends on n and the servicetime distribution is a mixture of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. Finite waiting rooms are treated by incorporating the additional limit mn / √ n → � for 0 <� � �. The approximation for the more general G/GI/n/m model developed here is consistent
Pricing and design of differentiated services: Approximate analysis and structural insights
 Operations Research
, 2005
"... We consider a Markovian service system that offers two grades of service to a market of heterogenous users: a “guaranteed ” (G) service rate to high priority users, and “besteffort” (BE) type service, in which residual capacity not allocated to Gusers is shared by the low priority users. Users, in ..."
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Cited by 14 (4 self)
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We consider a Markovian service system that offers two grades of service to a market of heterogenous users: a “guaranteed ” (G) service rate to high priority users, and “besteffort” (BE) type service, in which residual capacity not allocated to Gusers is shared by the low priority users. Users, in turn, are sensitive to both price and congestionrelated effects. The service provider’s objective is to optimally design the system so as to extract maximum revenues. This design problem consists of optimally pricing the two service classes, and determining the mechanism by which users are informed of the state of congestion in the system. Since these objectives are difficult to address using exact analysis, we pursue approximations that are tractable and lead to structural insights. Specifically, we first solve a deterministic problem to obtain a “fluidoptimal ” solution which is subsequently evaluated and refined to account for stochastic fluctuations. Using diffusion limits, we derive large capacity approximations that yield the following structural results: (i) pricing rules derived from the deterministic analysis are “almost” optimal; (ii) the optimal operational regime for the system is close to heavytraffic, and; (iii) realtime congestion notification results in increased revenues. Numerical results illustrate the accuracy of the proposed approximations and validate the aforementioned structural insights.
Telephone call centers: A tutorial and literature review
 Computer Access and Internet Use, (Working Paper at http:// www2000.ogsm.vanderbilt.edu/papers/race/science.html). Bridging the Racial Divide on the Internet, Science
, 2003
"... Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments trad ..."
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Cited by 9 (3 self)
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Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value – and at the same time fundamentally limited – in their ability to characterize system performance. We characterize the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research.
Partitioning customers into service groups
 Management Science
, 1999
"... We explore the issues of when and how to partition arriving customers into service groups that will be served separately, in a firstcome firstserved manner, by multiserver service systems having a provision for waiting, and how to assign an appropriate number of servers to each group. We assume th ..."
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Cited by 9 (1 self)
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We explore the issues of when and how to partition arriving customers into service groups that will be served separately, in a firstcome firstserved manner, by multiserver service systems having a provision for waiting, and how to assign an appropriate number of servers to each group. We assume that customers can be classified upon arrival, so that different service groups can have different servicetime distributions. We provide methodology for quantifying the tradeoff between economies of scale associated with larger systems and the benefit of having customers with shorter service times separated from other customers with longer service times, as is done in service systems with express lines. To properly quantify this tradeoff, it is important to characterize servicetime distributions beyond their means. In particular, it is important to also determine the variance of the servicetime distribution of each service group. Assuming Poisson arrival processes, we then can model the congestion experienced by each server group as an M/G/s queue with unlimited waiting room. We use previously developed approximations for M/G/s performance measures to quickly evaluate alternative partitions.