Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating socio-technical 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].

We study an iterative cutting-plane algorithm on an integer program, for minimizing the staffing costs of a multiskill call center subject to service-level requirements which are estimated by simulation. We solve a sample average version of the problem, where the service-levels are expressed as functions of the staffing for a fixed sequence of random numbers driving the simulation. An optimal solution of this sample problem is also an optimal solution to the original problem when the sample size is large enough. Several difficulties are encountered when solving the sample problem, especially for large problem instances, and we propose practical heuristics to deal with these difficulties. We report numerical experiments with examples of different sizes. The largest example corresponds to a real-life call center with 65 types of calls and 89 types of agents (skill groups). 1

Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating socio-technical 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.

A (customer) contact center is a collection of resources providing an interface between a service provider and its customers. The classical contact center is a call center, containing a collection of service representatives (reps) who talk to customers over the telephone. In a call center, the service reps are supported by quite elaborate information-and-communication-technology (ICT) equipment, such as a private branch exchange (PBX), an interactive voice response (IVR) unit, an automatic call distributor (ACD), a personal computer (PC) and assorted databases. With the rapid growth of e-commerce, contact between the service provider and its customers if often made via e-mail or the Internet instead of by telephone. Thus the general interface between a service provider and its customers is now often called a contact center. The design and management of contact centers is important, and worthy of research, because contact centers comprise a large, growing part of the economy and because they are quite complicated. Classic call centers are complicated because

The performance analysis of multiserver systems is notoriously hard, especially when the system involves resource sharing or prioritization. We provide two new analytical tools for the performance analysis of multiserver systems: moment matching algorithms and dimensionality reduction of Markov chains (DR). Moment matching algorithms allow us to approximate a general distribution with a phase type (PH) distribution. Our moment matching algorithms improve upon existing ones with respect to the computational efficiency (we provide closed form solutions) as well as the quality and generality of the solution (the first three moments of almost any nonnegative distribution are matched). Approximating job size and interarrival time distributions by PH distributions enables modeling a multiserver system by a Markov chain, so that the performance of the system is given by analyzing the Markov chain. However, when the multiserver system involves resource sharing or prioritization, the Markov chain often has a multidimensionally infinite state space, which makes the analysis computationally hard. DR allows us to closely approximate a multidimensionally infinite Markov chain with a Markov

We examine and compare simulation-based algorithms for solving the agent scheduling problem in a multiskill call center. This problem consists in minimizing the total costs of agents under constraints on the expected service level per call type, per period, and aggregated. We propose a solution approach that combines simulation with integer or linear programming, with cut generation. In our numerical experiments with realistic problem instances, this approach performs better than all other methods proposed previously for this problem. We also show that the two-step approach, which is the standard method for solving this problem, sometimes yield solutions that are highly suboptimal and inferior to those obtained by our proposed method.

Abstract. The paper studies a multiserver retrial queueing system with m servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter µ1. A time between retrials is exponentially distributed with parameter µ2 for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as µ2 increases to infinity. As µ2 → ∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated ‘usual’ multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided. Keywords: Multiserver retrial queues, Queue-length distribution, Stochastic calculus, Martingales and semimartingales

We develop stochastic models of time-dependent arrivals, with focus on the application to call centers. Our models reproduce essential features of call center arrivals observed in recent empirical studies, namely, a time-varying arrival intensity over the course of a day, and nonzero correlation between the arrival counts in different time periods within the same day. For each of the new models, we characterize the joint distribution of the vector of arrival counts with particular focus on characterizing how the new models are more flexible than standard or previously proposed models. We report empirical results from a study on arrival data from a real-life call center, including the essential features of the arrival process, the goodness-of-fit of the estimated models, and the sensitivity of various simulated performance measures of the call center to the choice of arrival process model.

Abstract — We consider a finite buffer queueing model with several key features of call centers, such as retrials, feedbacks, and impatience. In addition, because we do not completely understand the customer impatience behavior, we use a general distribution for the maximum waiting time before abandoning the call. We develop a QBD process with infinite state space for the queue in a call center situation. To solve for the stationary performance measures, we introduce an effective approximation method, and numerical examples have been presented to show the effectiveness of our method. Index Terms — Multi-server queues, call centers, feedbacks, QBD process. retrials, I.