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

Xinghu Jin (), Guodong Pang (), Lihu Xu () and Xin Xu ()
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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

Mathematics of Operations Research, 2025, vol. 50, issue 2, 783-812

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.

Keywords: Primary: 60H15; secondary: 60G51; 60G52; G / Ph / n + GI queues; Halfin–Whitt regime; multidimensional diffusion with piecewise-linear drift; Euler–Maruyama scheme; central limit theorem; moderate deviation principle; Stein’s equation; Malliavin calculus; weighted occupation time (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:2:p:783-812

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