Deterministic fluid models are developed to provide simple first-order performance descrip-tions for multi-server queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multi-server queues with a large number of servers and non-exponential service-time and time-to-abandon distributions. The first fluid model serves as an approxi-mation for the G/GI/s + GI queueing model, which has a general stationary arrival process with arrival rate λ, independent and identically distributed (IID) service times with a general distribution, s servers and IID abandon times with a general distribution. The fluid model is useful in the overloaded regime, where λ> s, which is often realistic because only a small amount of abandonment can keep the system stable. Numerical experiments, using simulation for M/GI/s + GI models and exact numerical algorithms for M/M/s + M models, show that the fluid model provides useful approximations for steady-state performance measures when the system is heavily loaded. The fluid model accurately shows that steady-state performance depends strongly upon the time-to-abandon distribution beyond its mean, but not upon the service-time distribution beyond its mean. The second fluid model is a discrete-time fluid model, which serves as an approximation for the Gt(n)/GI/s + GI queueing model, having a state-dependent and time-dependent arrival process. The discrete-time framework is exploited to prove that properly scaled queueing processes in the queueing model converge to fluid functions as s → ∞. The discrete-time framework is also convenient for calculating the time-dependent fluid performance descriptions. Subject classifications: Queues, approximations: multi-server queues with abandonment. Queues, multichannel: approximation of non-Markovian multichannel queues with customer abandon-ment.

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.

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

This paper proposes simple methods for staffing a single-class call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multi-server queue with customer abandonment, allowing non-exponential service-time and time-to-abandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customer-abandonment and customer-waiting costs, but attention is also given to the standard deviation of the return. The approach is to approximate the performance and the net return, conditional on the random model-parameter vector, and then uncondition to get the desired results. Two recently-developed approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birth-and-death model, having state-dependent death rates. Key words: model-parameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models

In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.

This is a supplement to the main paper, having the same title. In this work we develop methods to study the impact upon steady-state performance of delay announcements made to arriving customers in a many-server queue with customer abandonment. We assume that the queue is not visible to waiting customers, as in most customer contact centers, when contact is made by telephone, email or instant messaging. We propose simple robust announcement schemes: (i) the delay of the last served (DLS) customer and (ii) a fixed delay announcement (FDA) based on an appropriate long-run average delay. For any single-number delay announcement made immediately upon arrival, customers may balk or have new abandonment behavior as a function of the announced delay. To perform a rough-cut performance analysis, prior to detailed simulation, we introduce a fluid model, which provides an approximate and highly simplified description for large systems in an overloaded regime. In the fluid model, all customers are faced with the same delay and consequently can be given the same delay announcement. That property motivates considering a second approximation scheme: an equilibrium fixed delay announcement (FDA) in the stochastic model. We show that these two approximate descriptions of aggregate performance are effective by comparing to simulations of systems with state-dependent DLS announcements. Here we present additional material, supplementing the main paper.

Insight is provided into a previously developed M/M/s/r + M(n) approximation for the M/GI/s/r + GI queueing model by establishing fluid and diffusion limits for the approxi-mating model. Fluid approximations for the two models are compared in the many-server efficiency-driven (overloaded) regime. The two fluid approximations do not coincide, but they are close.

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Telephone call centers and their generalizations- customer contact centers- usually handle sev-eral types of customer service requests (calls). Since customer service representatives (agents) have different call-handling abilities and are typically cross-trained in multiple skills, contact centers exploit skill-based routing (SBR) to assign calls to appropriate agents, aiming to re-spond properly as well as promptly. Established agent-staffing and SBR algorithms ensure that agents have the required call-handling skills and that call routing is performed so that constraints are met for standard congestion measures, such as the percentage of calls of each type that abandon before starting service and the percentage of answered calls of each type that are delayed more than a specified number of seconds. We propose going beyond tra-ditional congestion measures to focus on the expected value derived from having particular agents handle various calls. Expected value might represent expected revenue or the likelihood of first-call resolution. Value might also reflect agent call-handling preferences. We show how value-based routing (VBR) and preference-based routing (PBR) can be introduced in the con-text of an existing SBR framework, based on static-priority routing using a highly-structured

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.