Consider a sequence of stationary GI/D/N queues indexed by N with servers' utilization 1 #/ # N , # > 0. For such queues we show that the scaled waiting times NWN converge to the (finite) supremum of a Gaussian random walk with drift -#.

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]

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

A stochastic processing network is a system that takes materials of various kinds as inputs, and uses processing resources to produce other materials as outputs. Such a network provides a powerful abstraction of a wide range of real world, complex systems, including semiconductor wafer fabrication facilities, networks of data switches, and large-scale call centers. Key performance measures of a stochastic processing network include throughput, cycle time, and holding cost. The network performance can dramatically be affected by the choice of operational policies. We propose a family of operational policies called maximum pressure policies. The maximum pressure policies are attractive in that their implementation uses minimal state information of the network. The deployment of a resource (server) is decided based on the queue lengths in its serviceable buffers and the queue lengths in their immediate downstream buffers. In particular, the decision does not use arrival rate information that is often difficult or impossible to estimate reliably. We prove that a maximum pressure policy can maximize throughput for a general class of stochastic processing networks. We also establish an asymptotic optimality of maximum pressure policies for stochastic processing networks with a unique bottleneck. The optimality is in terms of minimizing workload process. A key step in the proof of the asymptotic optimality is to show that the network processes under maximum pressure policies exhibit a state space collapse.