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An Approximation to the Invariant Measure of the Limiting Diffusion of G / Ph / n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

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  • Xinghu Jin

    (School of Mathematics, Hefei University of Technology, Hefei, Anhui 230601, China; and Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau 999078, China; and The University of Macau’s Zhuhai Research Institute, Zhuhai, Guangdong 519000, China)

  • Guodong Pang

    (Department of Computational Applied Mathematics and Operations Research, George R. Brown School of Engineering, Rice University, Houston, Texas 77005)

  • Lihu Xu

    (Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau 999078, China; and The University of Macau’s Zhuhai Research Institute, Zhuhai, Guangdong 519000, China)

  • Xin Xu

    (School of Mathematical Sciences, South China Normal University, Guangdong 510631, China)

Abstract

In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of G / P h / n + G I queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of G / P h / n + G I queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role.

Suggested Citation

  • Xinghu Jin & Guodong Pang & Lihu Xu & Xin Xu, 2025. "An Approximation to the Invariant Measure of the Limiting Diffusion of G / Ph / n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 783-812, May.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:2:p:783-812
    DOI: 10.1287/moor.2021.0241
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    References listed on IDEAS

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    1. Hassan Hmedi & Ari Arapostathis & Guodong Pang, 2022. "Uniform stability of some large-scale parallel server networks," Queueing Systems: Theory and Applications, Springer, vol. 102(3), pages 509-552, December.
    2. Heng-Qing Ye & David D. Yao, 2016. "Diffusion Limit of Fair Resource Control—Stationarity and Interchange of Limits," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1161-1207, November.
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    4. Ari Arapostathis & Hassan Hmedi & Guodong Pang, 2021. "On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 772-796, May.
    5. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
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    7. Amarjit Budhiraja & Jiang Chen & Sylvain Rubenthaler, 2014. "A Numerical Scheme for Invariant Distributions of Constrained Diffusions," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 262-289, May.
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