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Queue-and-Idleness-Ratio Controls in Many-Server Service Systems

Kellogg School of Management, Northwestern University, Evanston, Illinois 60208
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
i-gurvich{at}kellogg.northwestern.edu
ww2040{at}columbia.edu

Motivated by call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called queue-and-idleness-ratio (QIR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified state-dependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified state-dependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important state-space collapse (SSC) result in the quality-and-efficiency-driven (QED) many-server heavy-traffic limiting regime. The SSC result is applied here to prove stochastic-process limits and in subsequent papers to solve important staffing and control problems for large-scale service systems.

Key Words: heavy-traffic; Halfin-Whitt regime; QED regime; state-space collapse; diffusion limits; control of queueing systems
History: Received: October 21, 2007; revision received: October 26, 2008;