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Scheduling Flexible Servers with Convex Delay Costs in Many-Server Service Systems

Columbia Business School, New York, New York 10027
IEOR Department, Columbia University, New York, New York 10027
ig2126{at}columbia.edu
ww2040{at}columbia.edu

In a recent paper we introduced the queue-and-idleness ratio (QIR) family of routing rules for many-server service systems with multiple customer classes and server pools. A newly available server serves the customer from the head of the queue of the class (from among those the server is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. Under fairly general conditions, QIR produces an important state-space collapse as the total arrival rate and the numbers of servers increase in a coordinated way. That state-space collapse was previously used to delicately balance service levels for the different customer classes. In this sequel, we show that a special version of QIR stochastically minimizes convex holding costs in a finite-horizon setting when the service rates are restricted to be pool dependent. Under additional regularity conditions, the special version of QIR reduces to a simple policy: linear costs produce a priority-type rule, in which the least-cost customers are given low priority. Strictly convex costs (plus other regularity conditions) produce a many-server analogue of the generalized-cµ (Gcµ) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.

Key Words: queues; many-server queues; heavy-traffic limits for queues; service systems; cost minimization in many-server queues; skill-based routing; generalized-cµ rule; queue-and-idleness-ratio control
History: Received: June 29, 2007; accepted: November 22, 2007.

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