The Limit of Stationary Distributions of Many-Server Queues in the Halfin–Whitt Regime

References

• [1] Adams RA (1975) Sobolev Spaces, Pure and Applied Mathematics, vol. 140 (Academic Press, New York).Google Scholar
• [2] Aghajani R, Ramanan K (2019) Ergodicity of an SPDE associated with a many-server queue. Ann. Appl. Probab. 29(2):994–1045.Google Scholar
• [3] Aghajani R, Ramanan K (2016) The limit of stationary distributions of many-server queues in the Halfin-Whitt regime, extended version. Preprint, submitted October 4, https://arxiv.org/abs/1610.01118.Google Scholar
• [4] Aghajani R, Ramanan K (2017) The hydrodynamic limit of a randomized load balancing network, extended version. Preprint, submitted October 11, https://arxiv.org/abs/1707.02005.Google Scholar
• [5] Aghajani R, Li X, Ramanan K (2017) The PDE method for the analysis of randomized load balancing networks. Proc. ACM Measurement Anal. Comput. Systems 1(2):1–28.Google Scholar
• [6] Asmussen S (2003) Applied Probability and Queues, 2nd ed. (Springer-Verlag, New York).Google Scholar
• [7] Berger D, Daily B, Dunn P, Gamarnik D, Levi R, Levine WC, Sandberg W, Schoenmeyr T (2009) A model for understanding the impacts of demand and capacity on waiting time to enter a congested recovery room. Anesthesiology 110(6):1293–1304.Crossref, Google Scholar
• [8] Billingsley P (1968) Convergence of Probability Measures (John Wiley & Sons, New York).Google Scholar
• [9] Brezis H (2011) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (Springer, New York).Crossref, Google Scholar
• [10] Brown L, Gans N, Mandelbaum A, Sakov A, Shen H, Zeltyn S, Zhao L (2005) Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc. 100(469):36–50.Crossref, Google Scholar
• [11] Cayirli T, Veral E (2003) Outpatient scheduling in healthcare: A review of literature. Production Oper. Management 12(4):519–549.Crossref, Google Scholar
• [12] Chen H, Yao D (2001) Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization, Applications of Mathematics: Stochastic Modelling and Applied Probability (Springer-Verlag, New York).Crossref, Google Scholar
• [13] Chen S, Sun Y, Kozat UC, Huang L, Sinha P, Liang G, Liu X, Shroff NB (2014) When queueing meets coding: Optimal-latency data retrieving scheme in storage clouds. Proc. IEEE INFOCOM. (IEEE, Piscataway, NJ), 1042–1050.Google Scholar
• [14] Dai JG, Harrison JM (1992) Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2(1):65–86.Google Scholar
• [15] Dai JG, Dieker AB, Gao X (2014) Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. Queueing Systems 78(1):1–29.Google Scholar
• [16] Decreusefond L, Moyal P (2008) A functional central limit theorem for the M/GI/\infty queue. Ann. Appl. Probab. 18(6):2156–2178.Google Scholar
• [17] Dibenedetto E (2002) Real Analysis, Birkhäuser Advanced Texts Basler Lehrbücher Series (Birkhauser Verlag, Boston).Crossref, Google Scholar
• [18] Evans L (1998) Partial Differential Equations, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI).Google Scholar
• [19] Gamarnik D, Goldberg D (2013) Steady-state GI/GI/n queue in the Halfin-Whitt regime. Ann. Appl. Probab. 23(6):2382–2419.Google Scholar
• [20] Gamarnik D, Momčilović P (2008) Steady-state analysis of a multiserver queue in the Halfin-Whitt regime. Adv. Appl. Probab. 40(2):548–577.Crossref, Google Scholar
• [21] Goldberg D, Li Y (2017) Heavy-tailed queues in the Halfin-Whitt regime. Preprint, submitted July 25, https://arxiv.org/abs/1707.07775.Google Scholar
• [22] Goldberg D, Li Y (2017) Simple and explicit bounds for multi-server queues with universal 1 / (1 − rho) scaling. Preprint, submitted June 14, https://arxiv.org/abs/1706.04628.Google Scholar
• [23] Hairer M, Mattingly JC, Scheutzow M (2011) Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149(1–2):223–259.Google Scholar
• [24] Halfin S, Whitt W (1981) Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3):567–588.Google Scholar
• [25] Harrison JM, Williams RJ (1987) Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15(1):115–137.Google Scholar
• [26] Jelenković P, Mandelbaum A, Momčilović P (2004) Heavy traffic limits for queues with many deterministic servers. Queueing Systems 47(1–2):53–69.Google Scholar
• [27] Kallenberg O (1983) Random Measures, 3rd ed. (Akademie-Verlag, Berlin).Google Scholar
• [28] Kang W, Ramanan K (2010) Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20(6):2204–2260.Google Scholar
• [29] Kang W, Ramanan K (2012) Asymptotic approximations for stationary distributions of many-server queues with abandonment. Ann. Appl. Probab. 22(2):477–521.Crossref, Google Scholar
• [30] Kang W, Ramanan K (2014) Characterization of stationary distributions of reflected diffusions. Ann. Appl. Probab. 24(4):1329–1374.Google Scholar
• [31] Karatzas I, Shreve S (1991) Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics Series (Springer-Verlag, New York).Google Scholar
• [32] Kaspi H, Ramanan K (2011) Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21(1):33–114.Google Scholar
• [33] Kaspi H, Ramanan K (2013) SPDE limits of many-server queues. Ann. Appl. Probab. 23(1):145–229.Google Scholar
• [34] Kolesar P (1984) Stalking the endangered CAT: A queueing analysis of congestion at automatic teller machines. Interfaces 14(6):16–26.Link, Google Scholar
• [35] Krichagina EV, Puhalskii AA (1997) A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Systems Theory Appl. 25(1–4):235–280.Google Scholar
• [36] Liang G, Kozat UC (2014) TOFEC: Achieving optimal throughput-delay trade-off of cloud storage using erasure codes. Proc. IEEE INFOCOM. 2014 (IEEE, Piscataway, NJ), 826–834.Google Scholar
• [37] Mandelbaum A, Momčilović P (2008) Queues with many servers: The virtual waiting-time process in the QED regime. Math. Oper. Res. 33(3):561–586.Link, Google Scholar
• [38] Puhalskii AA, Reed J (2010) On many-server queues in heavy traffic. Ann. Appl. Probab. 20(1):129–195.Google Scholar
• [39] Puhalskii AA, Reiman MI (2004) The multiclass GI/PH/N queue in the Halfin-Whitt regime. Adv. Appl. Probab. 32(2):564–595.Google Scholar
• [40] Reed J (2009) The G/GI/N queue in the Halfin-Whitt regime. Ann. Appl. Probab. 19(6):2211–2269.Google Scholar
• [41] Reed J, Talreja R (2015) Distribution-valued heavy-traffic limits for the G/GI/∞ queue. Ann. Appl. Probab. 25(3):1420–1474.Google Scholar
• [42] Skorokhod AV (1974) Integration in Hilbert space, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, New York).Crossref, Google Scholar
• [43] Walnut DF (2013) An Introduction to Wavelet Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser, Boston).Google Scholar
• [44] Walsh JB (1986) An introduction to stochastic partial differential equations. Hennequin P, ed. École d’été de Probabilités de Saint Flour XIV – 1984, Lecture Notes in Mathematics, vol. 1180 (Springer, Berlin, Heidelberg), 265–439.Crossref, Google Scholar
• [45] Whitt W (2002) Stochastic-Process Limits, Springer Series in Operations Research (Springer-Verlag, New York).Crossref, Google Scholar
• [46] Whitt W (2005) Heavy-traffic limits for the G/H_{2}^*/n/m queue. Math. Oper. Res. 30(1):1–27.Google Scholar