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.

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.

Our note 1 is dedicated to the Palm/Erlang-A Queue. This is the simplest practiceworthy queueing model, that accounts for customers ’ impatience while waiting. The model is gaining importance in support of the staffing of call centers, which is a central step in their Service-Engineering. We discuss computations of performance measures, both theoretical and software-based (via the 4CallCenter software). Then several examples of Palm/Erlang-A applications are presented, mostly motivated by and based on real call center data. Acknowledgements. The research of both authors was supported by ISF (Israeli Science Foundation) grants 388/99, 126/02 and 1046/04, by the Niderzaksen Fund and by the Technion funds for the promotion of research and sponsored research. 1 Parts of the text are adapted from [8], [15], [17] and [22]

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI many-server queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We characterize performance in the fluid model over alternating intervals in which the system is overloaded and underloaded (including critically loaded). For each t ≥ 0 and y ≥ 0, we determine the amount of fluid that is in service (in queue) at time t and has been so for time at most y. We obtain the service content density by applying the Banach contraction fixed point theorem. We also determine the time-varying potential waiting time, i.e., the virtual waiting time of a quantum of fluid arriving at a specified time, assuming that it will not abandon. The potential waiting time is determined by an ordinary differential equation. We show that a time-varying service capacity can be chosen to stabilize delays at any fixed target. Key words: queues with time-varying arrivals; nonstationary queues; many-server queues; deterministic fluid model; fluid approximation; queues with abandonment; non-Markovian queues.

Motivated by healthcare systems and customer contact centers, we introduce and analyze a deterministic fluid model that can be used to show how queue lengths and waiting times depend on model parameters in a large-scale service system that experiences periods of overloading. The main feature of the model is time-varying arrival rate and staffing, but the model also includes the realistic feature of customer abandonment with a non-exponential patience distribution. Our key assumptionsare(i)thatthescaleislarge(therearemanyservers)and(ii) thatthesystemalternates between overloaded intervals and underloaded intervals. We develop algorithms to describe the time-dependent performance. For example, we determine, at each time, the content in queue that has beensofor at most a specifiedduration, as afunction of thetwo parameters: time andduration. Wealsodeterminethetime-varyingpotential waitingtime, i.e., thevirtualwaitingtimeofanarrival at a specified time, assuming that it will not abandon. We conduct simulations to confirm that the algorithm and the approximation are effective. Keywords: Large-scaleservicesystems; queueswithtime-varyingarrivals; nonstationaryqueues; many-server queues; deterministic fluid model; fluid approximation; queues with abandonment; non-Markovian queues. 1

In this paper, we studied multi-server queue with Poisson arrivals, general service time distribution, and deterministic reneging times. This work was motivated by the timeout mechanism used in managing application servers in transaction processing environments. In such systems, a customer who stays in the queue longer than the timeout period is lost. We proposed a scaling approach, and a fast and accurate approximation for the expected waiting time in the queue.

Abstract This paper introduces a deterministic fluid model that approximates the many-server Gt/GI/st + GI queueing model, and determines the time-dependent performance functions. The fluid model has time-varying arrival rate and service capacity, abandonment from queue, and non-exponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixed-point equation for the time-varying rate that fluid enters service and the solution of an ordinary differential equation for the time-varying head-ofline waiting time, during each overloaded interval. Simulations are conducted to confirm that the algorithm and the approximation are effective. Keywords queues with time-varying arrivals · nonstationary queues · transient behavior · many-server queues · deterministic fluid model · customer abandonment · non-Markovian queues 1

The G t /GI/s t +GI many-server fluid queue