Motivated by the desire to understand the performance of service-oriented call centers, which often provide low-to-moderate quality of service, this paper investigates the efficiency-driven (ED) limiting regime for many-server queues with abandonments. The starting point is the realization that, in the presence of substantial customer abandonment, call-center service-level agreements (SLA’s) can be met in the ED regime, where the arrival rate exceeds the maximum possible service rate. Mathematically, the ED regime is defined by letting the arrival rate and the number of servers increase together so that the probability of abandonment approaches a positive limit. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ> 1 (so that the arrival rate exceeds the maximum possible service rate). Even though the probability of delay necessarily approaches 1 in the ED regime, the ED regime can be realistic because, due to the abandonments, the delays need not be excessively large. This paper establishes ED many-server heavy-traffic limits and develops associated ap-proximations for performance measures in the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces and exponential abandon times (the final +M). In the ED regime, essentially the same limiting behavior occurs when the abandonment rate α approaches 0 as when the number of servers s approaches ∞; in-deed, it suffices to assume that s/α → ∞. The ED approximations are shown to be useful by comparing them to exact numerical results for the M/M/s/r + M model obtained using an algorithm developed in Whitt (2003), which exploits numerical transform inversion.

Call centers are an increasingly important part of today’s business world, employing millions of agents across the globe and serving as a primary customer-facing channel for firms in many different industries. Call centers have been a fertile area for operations management researchers in several domains, including forecasting, capacity planning, queueing, and personnel scheduling. In addition, as telecommunications and information technology have advanced over the past several years, the operational challenges faced by call center managers have become more complicated. Issues associated with human resources management, sales, and marketing have also become increasingly relevant to call center operations and associated academic research. In this paper, we provide a survey of the recent literature on call center operations management. Along with traditional research areas, we pay special attention to new management challenges that have been caused by emerging technologies, to behavioral issues associated with both call center agents and customers, and to the interface between call center operations and sales and marketing. We identify a handful of broad themes for future investigation while also pointing out several very specific research opportunities.

An algorithm is developed to rapidly compute approximations for all the standard steady-state performance measures in the basic call-center queueing model M/GI/s/r+GI, which has a Poisson arrival process, IID service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies of call centers indicate that the service-time and abandon-time distributions often are not nearly exponential, so that it is important to go beyond the Markovian M/M/s/r + M special case, but the general service-time and abandon-time distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an ap-propriate Markovian M/M/s/r + M(n) queueing model, where M(n) denotes state-dependent abandonment rates. After making an additional approximation, steady-state waiting-time dis-tributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the aban-donment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.

We establish heavy-traffic stochastic-process limits for queue-length, waiting-time and over-flow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. In order to capture the impact of the service-time distribution beyond its mean within a Markovian framework, we consider a special class of service-time distribu-tions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These service-time distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981), Puhalskii and Reiman (2000) and Garnett, Mandelbaum and Reiman (2000), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities ρn so that √ n(1 − ρn) → β for − ∞ < β < ∞. To treat finite waiting rooms, we let mn / √ n → κ for 0 < κ ≤ ∞. With the special H ∗ 2 service-time distribution, the limit processes are one-dimensional Markov processes, behaving like diffusion processes with different drift and diffusion functions in two different re-gions, above and below zero. We also establish a limit for the G/M/n/m + M model, having exponential customer abandonments.

informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queue-length 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 steady-state distribution of that diffusion process to obtain approximations for steady-state performance measures of the queueing model, focusing especially upon the steady-state delay probability. The approximations are based on heavy-traffic 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 queue-length 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 service-time 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

Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform time-dependent staffing for many-server queues. Our aim is to achieve time-stable performance in face of general time-varying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce time-stable performance for several other operational measures. Motivated by telephone call centers, we focus on many-server models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential time-to-abandon 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 simulation-based 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 square-root-staffing rule, proposed by Jennings et al., which is based on the associated infinite-server 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 infinite-server 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

Call centers are an increasingly important part of today’s business world, employing millions of agents across the globe and serving as a primary customer-facing channel for firms in many different industries. Call centers have been a fertile area for operations management researchers in several areas, including forecasting, capacity planning, queueing, and personnel scheduling. In addition, as telecommunications and information technology have advanced over the past several years, the operational challenges faced by call center managers have become more complicated as a result. Issues associated with human resources management, sales, and marketing have also become increasingly relevant to call center operations and associated academic research. In this paper, we provide a survey of the recent literature on call center operations management. Along with traditional research areas, we pay special attention to new management challenges that have been caused by emerging technologies, to behavioral issues associated with both call center agents and customers, and to the interface between call center operations and sales and marketing. We identify a handful of broad themes for future investigation while also pointing out several very specific research opportunities.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

We establish heavy-traffic limits for “nearly deterministic” queues, such as the G/D/n many-server queue. Waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn/D/1 model, where the Gn denotes “cyclic thinning ” of order n, indicating that the original (possibly general) point process of arrivals is thinned to contain only every n th point. We thus focus on the Gn/D/1 model and the generalization to Gn/Gn/1, where “cyclic thinning ” is applied to both the arrival and service processes. As n → ∞, the Gn/Gn/1 models approach the deterministic D/D/1 model. The classical example is the Erlang En/En/1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a “base ” M/M/1 model. We establish different limits in two cases: (i) when (1−ρn) √ n → β as n → ∞ and (ii) (1 − ρn)n → β as n → ∞, where ρn is the traffic intensity in model n, and 0 < β < ∞. The nearly deterministic feature leads to interesting nonstandard scaling. We also establish revealing heavy-traffic limits for the stationary waiting times and other performance measures in the Gn/Gn/1 queues, by letting ρn ↑ 1 as n → ∞.

Martingale proofs of many-server heavy-traffic limits for Markovian queues