Open Access
December 2015 Ergodic control of multi-class $M/M/N+M$ queues in the Halfin–Whitt regime
Ari Arapostathis, Anup Biswas, Guodong Pang
Ann. Appl. Probab. 25(6): 3511-3570 (December 2015). DOI: 10.1214/14-AAP1081


We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin–Whitt (QED) regime, that is, the number of servers $n$ and the total offered load $\mathbf{r}$ scale like $n\approx\mathbf{r}+\hat{\rho}\sqrt{\mathbf{r}}$ for some constant $\hat{\rho}$. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084–1134, Section 5.2].

The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.


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Ari Arapostathis. Anup Biswas. Guodong Pang. "Ergodic control of multi-class $M/M/N+M$ queues in the Halfin–Whitt regime." Ann. Appl. Probab. 25 (6) 3511 - 3570, December 2015.


Received: 1 April 2014; Revised: 1 November 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1330.60108
MathSciNet: MR3404643
Digital Object Identifier: 10.1214/14-AAP1081

Primary: 60K25
Secondary: 68M20 , 90B22 , 90B36

Keywords: asymptotic optimality , diffusion scaling , ergodic control , Halfin–Whitt (QED) regime , long time-average control , Multi-class Markovian queues , reneging/abandonment , spatial truncation , stable Markov optimal control

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.25 • No. 6 • December 2015
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