Scheduling flexible servers with convex delay costs in many-server service systems

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, QIRproduces 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 QIRstochastically 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 QIRreduces 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.