T elephone 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.

A call center is a service network in which agents provide telephone-based services. Customers who seek these services are delayed in tele-queues. This article summarizes an analysis of a unique record of call center operations. The data comprise a complete operational history of a small banking call center, call by call, over a full year. Taking the perspective of queueing theory, we decompose the service process into three fundamental components: arrivals, customer patience, and service durations. Each component involves different basic mathematical structures and requires a different style of statistical analysis. Some of the key empirical results are sketched, along with descriptions of the varied techniques required. Several statistical techniques are developed for analysis of the basic components. One of these techniques is a test that a point process is a Poisson process. Another involves estimation of the mean function in a nonparametric regression with lognormal errors. A new graphical technique is introduced for nonparametric hazard rate estimation with censored data. Models are developed and implemented for forecasting of Poisson arrival rates. Finally, the article surveys how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations. We then survey how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations.

T he most common model to support workforce management of telephone call centers is the M/M/N/B model, in particular its special cases M/M/N (Erlang C, which models out busy signals) and M/M/N/N (Erlang B, disallowing waiting). All of these models lack a central prevalent feature, namely, that impatient customers might decide to leave (abandon) before their service begins.In this paper, we analyze the simplest abandonment model, in which customers' patience is exponentially distributed and the system's waiting capacity is unlimited (M/M/N ϩ M). Such a model is both rich and analyzable enough to provide information that is practically important for call-center managers. We first outline a method for exact analysis of the M/M/N ϩ M model, that while numerically tractable is not very insightful. We then proceed with an asymptotic analysis of the M/M/N ϩ M model, in a regime that is appropriate for large call centers (many agents, high efficiency, high service level). Guided by the asymptotic behavior, we derive approximations for performance measures and propose ''rules of thumb'' for the design of large call centers. We thus add support to the growing acknowledgment that insights from diffusion approximations are directly applicable to management practice.

We develop a framework for asymptotic optimization of a queueing system. The motivation is the sta ng problem of call centers with 100's of agents or more. Such a call center is modeled as an M M N queue, where the number of agents N is large. Within our framework, we determine the asymptotically optimal sta ng level N that trades o agents' costs with service quality: the higher the latter, the more expensive is the former. As an alternative t o this optimization, we also develop a constraint satisfaction approach where one chooses the least N that adheres to a given constraint o n w aiting cost. Either way, the analysis gives rise to three regimes of operation: quality-driven, where the focus is on service quality; e ciencydriven, which emphasizes agents' costs; and a rationalized regime that balances, and in fact uni es, the other two. Numerical experiments reveal remarkable accuracy of our asymptotic approximations: over a wide range of parameters, from the very small to the extremely large, N is exactly optimal, or it is accurate to within a single agent. We demonstrate the utility o f our approach b y revisiting the square-root safety sta ng principle, which is a long-existing ruleof-thumb for sta ng the M M N queue. In its simplest form, our rule is as follows: if c is the hourly cost of an agent, and a is the hourly cost of customers' delay, then N = R + y a c p R, where R is the o ered load, and y is a function that is easily computable.2000 Mathematics Subject Classi cation: 60K25 primary, 90B22 secondary.

We consider a queueing system with multitype customers and flexible (multiskilled) servers that work in parallel. If Q i is the queue length of type i customers, this queue incurs cost at the rate of C i Q i , where C i · is increasing and convex. We analyze the system in heavy traffic (Harrison and Lopez 1999) and show that a very simple generalized c -rule (Van Mieghem 1995) minimizes both instantaneous and cumulative queueing costs, asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive. This rule aims at myopically maximizing the rate of decrease of the instantaneous cost at all times, which translates into the following: when becoming free, server j chooses for service a type i customer such that i ∈ arg max i C i Q i ij where ij is the average service rate of type i customers by server j.An analogous version of the generalized c -rule asymptotically minimizes delay costs. To this end, let the cost incurred by a type i customer be an increasing convex function C i D of its sojourn time D. Then, server j always chooses for service a customer for which the value of C i D ij is maximal, where D and i are the customer's sojourn time and type, respectively.

This paper develops methods to determine appropriate staffing levels in call centers and other many-server queueing systems with time-varying arrival rates. The goal is to achieve targeted time-stable performance, even in the presence of significant time variation in the arrival rates. The main contribution is a flexible simulation-based iterative-staffing algorithm (ISA) for the M t/G/s t + G model--with nonhomogeneous Poisson arrival process (the M t) and customer abandonment (the + G). For Markovian M t/M/s t + M special cases, the ISA is shown to converge. For that M t/M/s t + M model, simulation experiments show that the ISA yields time-stable delay probabilities across a wide range of target delay probabilities. With ISA, other performance measures--such as agent utilizations, abandonment probabilities, and average waiting times--are stable as well. The ISA staffing and performance agree closely with the modified-offered-load approximation, which was previously shown to be an effective staffing algorithm without customer abandonment. Although the ISA algorithm so far has only been extensively tested for M t/M/s t + M models, it can be applied much more generally--to M t/G/s t + G models and beyond.contact centers, call centers, staffing, nonstationary queues, queues with time-dependent arrival rates, many-server queues, capacity planning, queues with abandonment, time-varying Erlang models

We consider a multiserver service system with general nonstationary arrival and service-time 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 time-dependent normal distribution, where the mean and variance are determined by infinite-server 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.operator staffing, queues, nonstationary queues, queues with time-dependent arrival rates, multiserver queues, infinite-server queues, capacity planning

The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate µ, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents).In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and µ held fixed:Quality-Driven (QD): n ≈ (λ/µ) · (1 + γ ), γ > 0, and Quality and Efficiency Driven (QED):In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/ √ n). Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems.Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.