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

References

• [1] Aghajani R, Ramanan K (2020) The limit of stationary distributions of many-server queues in the Halfin–Whitt regime. Math. Oper. Res. 45(3):1016–1055.Link, Google Scholar
• [2] Arapostathis A, Hmedi H, Pang G (2021) On uniform exponential ergodicity of Markovian multiclass many-server queues in the Halfin–Whitt regime. Math. Oper. Res. 46(2):772–796.Link, Google Scholar
• [3] Arapostathis A, Pang G, Sandrić N (2019a) Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues. Ann. Appl. Probab. 29(2):1070–1126.Crossref, Google Scholar
• [4] Arapostathis A, Hmedi H, Pang G, Sandrić N (2019b) Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues. Yin G, Zhang Q, eds. Modeling, Stochastic Control, Optimization, and Applications, The IMA Volumes in Mathematics and its Applications, vol. 164 (Springer, Cham, Switzerland), 1–20.Crossref, Google Scholar
• [5] Benaïm M, Bouguet F, Cloez B (2017) Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories. Ann. Appl. Probab. 27(5):3004–3049.Crossref, Google Scholar
• [6] Braverman A, Dai J (2017) Stein’s method for steady state diffusion approximations of M/Ph/n+M systems. Ann. Appl. Probab. 27(1):550–581.Crossref, Google Scholar
• [7] Braverman A, Dai JG, Fang X (2022) High-order steady-state diffusion approximations. Oper. Res., ePub ahead of print September 8, https://doi.org/10.1287/opre.2022.2362.Google Scholar
• [8] Braverman A, Dai J, Feng J (2017) Stein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models. Stochastic Systems 6(2):301–366.Link, Google Scholar
• [9] Brosse N, Durmus A, Moulines É, Sabanis S (2019) The tamed unadjusted Langevin algorithm. Stochastic Processes Their Appl. 129(10):3638–3663.Crossref, Google Scholar
• [10] Browne S, Whitt W (1995) Piecewise-linear diffusion processes. Dshalalow JH, ed. Advances in Queueing: Theory, Methods, and Open Problems (CRC Press, Boca Raton, FL), 463–480.Google Scholar
• [11] Budhiraja A, Lee C (2009) Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34(1):45–56.Link, Google Scholar
• [12] Budhiraja A, Chen J, Rubenthaler S (2014) A numerical scheme for invariant distributions of constrained diffusions. Math. Oper. Res. 39(2):262–289.Link, Google Scholar
• [13] Cattiaux P, Chafai D, Guillin A (2012) Central limit theorems for additive functionals of ergodic Markov diffusions processes. ALEA: Latin Amer. J. Probab. Math. Statist. 9(2):337–382.Google Scholar
• [14] Chatterjee S, Meckes E (2008) Multivariate normal approximation using exchangeable pairs. ALEA: Latin Amer. J. Probab. Math. Statist. 4:257–283.Google Scholar
• [15] Da Prato G, Zabczyk J (1996) Ergodicity for Infinite Dimensional Systems, vol. 229 (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [16] Dai J, Harrison JM (1992) Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2(1):65–86.Crossref, Google Scholar
• [17] Dai J, He S (2013) Many-server queues with customer abandonment: Numerical analysis of their diffusion model. Stochastic Systems 3(1):96–146.Link, Google Scholar
• [18] Dai J, Dieker A, Gao X (2014) Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. Queueing Systems 78(1):1–29.Crossref, Google Scholar
• [19] Dai J, He S, Tezcan T (2010) Many-server diffusion limits for G/Ph/n+GI queues. Ann. Appl. Probab. 20(5):1854–1890.Crossref, Google Scholar
• [20] Dieker AB, Gao X (2013) Positive recurrence of piecewise Ornstein–Uhlenbeck processes and common quadratic Lyapunov functions. Ann. Appl. Probab. 23(4):1291–1317.Crossref, Google Scholar
• [21] Douc R, Fort G, Moulines E, Soulier P (2004) Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14(3):1353–1377.Crossref, Google Scholar
• [22] Durmus A, Moulines E (2017) Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. Ann. Appl. Probab. 27(3):1551–1587.Crossref, Google Scholar
• [23] Durmus A, Moulines E (2019) High-dimensional Bayesian inference via the unadjusted Langevin algorithm. Bernoulli 25(4A):2854–2882.Crossref, Google Scholar
• [24] Fang W, Giles MB (2020) Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift. Ann. Appl. Probab. 30(2):526–560.Crossref, Google Scholar
• [25] Fang X, Shao QM, Xu L (2019) Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula. Probab. Theory Related Fields 174(3):945–979.Crossref, Google Scholar
• [26] Fukasawa M, Obłój J (2020) Efficient discretisation of stochastic differential equations. Stochastics 92(6):833–851.Crossref, Google Scholar
• [27] Gamarnik D, Goldberg DA (2013a) On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23(5):1879–1912.Crossref, Google Scholar
• [28] Gamarnik D, Goldberg DA (2013b) Steady-state GI/G/n queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23(6):2382–2419.Crossref, Google Scholar
• [29] Gamarnik D, Stolyar AL (2012) Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: Asymptotics of the stationary distribution. Queueing Systems 71(1):25–51.Crossref, Google Scholar
• [30] Gamarnik D, Zeevi A (2006) Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16(1):56–90.Crossref, Google Scholar
• [31] Gloter A, Honoré I, Loukianova D (2020) Approximation of the invariant distribution for a class of ergodic jump diffusions. ESAIM Probab. Statist. 24:883–913.Crossref, Google Scholar
• [32] Gurvich I (2014a) Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. Ann. Appl. Probab. 24(6):2527–2559.Crossref, Google Scholar
• [33] Gurvich I (2014b) Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Math. Oper. Res. 39(1):121–162.Link, Google Scholar
• [34] Gurvich I (2023) Erratum: Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. Ann. Appl. Probab. 33(6B):5810–5815.Google Scholar
• [35] Hairer M (2024) An introduction to Stochastic PDEs. https://www.hairer.org/notes/SPDEs_Course.pdf.Google Scholar
• [36] Hairer M, Mattingly JC (2008) Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36(6):2050–2091.Crossref, Google Scholar
• [37] Harrison JM, Williams RJ (1987) Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15(1):115–137.Crossref, Google Scholar
• [38] Hmedi H, Arapostathis A, Pang G (2022) Uniform stability of some large-scale parallel server networks. Queueing Systems 102:509–552.Crossref, Google Scholar
• [39] Hmedi H, Arapostathis A, Pang G (2023) On system-wide safety staffing of large-scale parallel server networks. Oper. Res. 71(2):415–432.Link, Google Scholar
• [40] Jones GL (2004) On the Markov chain central limit theorem. Probab. Surveys 1:299–320.Crossref, Google Scholar
• [41] Komorowski T, Landim C, Olla S (2012) Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation, vol. 345 (Springer Science & Business Media, New York).Crossref, Google Scholar
• [42] Kusuoka S, Tudor CA (2012) Stein’s method for invariant measures of diffusions via Malliavin calculus. Stochastic Processes Their Appl. 122(4):1627–1651.Crossref, Google Scholar
• [43] Lamberton D, Pages G (2002) Recursive computation of the invariant distribution of a diffusion. Bernoulli 8(3):367–405.Google Scholar
• [44] Lu J, Tan Y, Xu L (2022) Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2):937–964.Google Scholar
• [45] Mattingly JC, Stuart AM, Higham DJ (2002) Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Processes Their Appl. 101(2):185–232.Crossref, Google Scholar
• [46] McLeish DL (1974) Dependent central limit theorems and invariance principles. Ann. Probab. 2(4):620–628.Crossref, Google Scholar
• [47] Meyn SP, Tweedie RL (1993) Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3):518–548.Crossref, Google Scholar
• [48] Panloup F (2008a) Computation of the invariant measure for a Lévy driven SDE: Rate of convergence. Stochastic Processes Their Appl. 118(8):1351–1384.Crossref, Google Scholar
• [49] Panloup F (2008b) Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process. Ann. Appl. Probab. 18(2):379–426.Crossref, Google Scholar
• [50] Partington JR (2004) Linear Operators and Linear Systems: An Analytical Approach to Control Theory (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [51] Peszat S, Zabczyk J (1995) Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1):157–172.Crossref, Google Scholar
• [52] Puhalskii AA, Reiman MI (2000) The multiclass GI/PH/N queue in the Halfin–Whitt regime. Adv. Appl. Probab. 32(2):564–595.Crossref, Google Scholar
• [53] Rebolledo R (1980) Central limit theorems for local martingales. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 51(3):269–286.Crossref, Google Scholar
• [54] Reinert G, Röllin A (2009) Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37(6):2150–2173.Crossref, Google Scholar
• [55] Sethuraman S (2002) A martingale central limit theorem. Accessed March 12, 2024, https://www.math.arizona.edu/~sethuram/notes/wi_mart1.pdf.Google Scholar
• [56] Stolyar AL (2015) Tightness of stationary distributions of a flexible-server system in the Halfin–Whitt asymptotic regime. Stochastic Systems 5(2):239–267.Link, Google Scholar
• [57] Talay D (2002) Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Processes Related Fields 8(2):163–198.Google Scholar
• [58] Tuominen P, Tweedie RL (1994) Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Probab. 26(3):775–798.Crossref, Google Scholar
• [59] Wang FY, Xiong J, Xu L (2016) Asymptotics of sample entropy production rate for stochastic differential equations. J. Statist. Phys. 163(5):1211–1234.Crossref, Google Scholar
• [60] Wu L (1995) Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23(2):20–445.Google Scholar
• [61] Wu L (2001) Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Processes Their Appl. 91(2):205–238.Crossref, Google Scholar
• [62] Ye HQ, Yao DD (2016) Diffusion limit of fair resource control-stationarity and interchange of limits. Math. Oper. Res. 41(4):1161–1207.Link, Google Scholar
• [63] Ye HQ, Yao DD (2018) Justifying diffusion approximations for multiclass queueing networks under a moment condition. Ann. Appl. Probab. 28(6):3652–3697.Crossref, Google Scholar