## The Annals of Applied Probability

### Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic

Rami Atar

#### Abstract

A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system’s state. We examine two versions of the problem: “nonpreemptive,” where service is uninterruptible, and “preemptive,” where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V. The construction of these policies is based on a Hamilton–Jacobi–Bellman equation associated with V.

#### Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2606-2650.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1133965774

Digital Object Identifier
doi:10.1214/105051605000000601

Mathematical Reviews number (MathSciNet)
MR2187306

Zentralblatt MATH identifier
1098.60083

#### Citation

Atar, Rami. Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15 (2005), no. 4, 2606--2650. doi:10.1214/105051605000000601. https://projecteuclid.org/euclid.aoap/1133965774

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