### Abstract

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 r scale like n ≈ r + ρ√r for some constant ρ. 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.

Original language | English (US) |
---|---|

Pages (from-to) | 3511-3570 |

Number of pages | 60 |

Journal | Annals of Applied Probability |

Volume | 25 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Applied Probability*,

*25*(6), 3511-3570. https://doi.org/10.1214/14-AAP1081

}

*Annals of Applied Probability*, vol. 25, no. 6, pp. 3511-3570. https://doi.org/10.1214/14-AAP1081

**Ergodic control of multi-class M/M/N + M queues in the Halfin-Whitt regime.** / Arapostathis, Ari; Biswas, Anup; Pang, Guodong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ergodic control of multi-class M/M/N + M queues in the Halfin-Whitt regime

AU - Arapostathis, Ari

AU - Biswas, Anup

AU - Pang, Guodong

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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 r scale like n ≈ r + ρ√r for some constant ρ. 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.

AB - 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 r scale like n ≈ r + ρ√r for some constant ρ. 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.

UR - http://www.scopus.com/inward/record.url?scp=84943257095&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84943257095&partnerID=8YFLogxK

U2 - 10.1214/14-AAP1081

DO - 10.1214/14-AAP1081

M3 - Article

AN - SCOPUS:84943257095

VL - 25

SP - 3511

EP - 3570

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 6

ER -