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

Abstract

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-c\mu (Gc\mu ) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.queues, many-server queues, heavy-traffic limits for queues, service systems, cost minimization in many-server queues, skill-based routing, generalized-c\mu rule, queue-and-idleness-ratio control

Centralized vs. Decentralized Ambulance Diversion: A Network Perspective

“…Some relevant examples in the context of parallel server networks with multiple servers per server group (as the one we study here) Atar [2005], Gurvich and Whitt [2009], Tezcan [2008], Stolyar [2005], Adan et al…”

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“…More recently [20] proposed the Fixed Queue Ratio (FQR) routing rule together with corresponding staffing rule for general parallel server systems. Under certain additional conditions, the FQR rule provides an asymptotically optimal solution; in [19] the authors showed that it is asymptotically optimal when the holding costs are convex. The main difference of our approach from this body of literature is that all of the papers above assume that basic and non-basic activities are known a priori.…”

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“…The system of equations in (16) represents the balance equations of the underlying DTMC, keeping in mind that the action choice only affects the chain transitions immediately following a phase 1 service completion. In particular, for any fixed (i, j) the right hand side in (16) lists the possible transitions from other states into (i, j, 0) and (i, j, 1), with the corresponding probabilities.…”