Abstract
We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by a-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin–Whitt regime that have heavy–tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound on the rate of convergence (under no abandonment). On the other hand, when all abandonment rates are positive, we show that the SDEs are exponentially ergodic uniformly over the above-mentioned class of controls. Analogous results are obtained for Lévy–driven SDEs arising from multiclass many-server queues under asymptotically negligible service interruptions. For these equations, we show that the aforementioned ergodic properties are uniform over all stationary Markov controls. We also extend a key functional central limit theorem concerning diffusion approximations so as to make it applicable to the models studied here.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albeverio, S., Brzeźniak, Z.,Wu, J.L.: Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. J. Math. Anal. Appl. 371(1), 309–322 (2010). https://doi.org/10.1016/j.jmaa.2010.05.039
Arapostathis, A., Biswas, A., Caffarelli, L.: The Dirichlet problem for stable-like operators and related probabilistic representations. Comm. Partial Differential Equations 41(9), 1472–1511 (2016). https://doi.org/10.1080/03605302.2016.1207084
Arapostathis, A., Hmedi, H., Pang, G.: On uniform exponential ergodicity of Markovian multiclass many-server queues in the Halfin–Whitt regime. ArXiv e-prints 1812.03528 (2018)
Arapostathis, A., Pang, G., Sandrić, N.: Ergodicity of a Lévy–driven SDE arising from multiclass many-server queues. Ann. Appl. Probab. 29(2), 1070–1126 (2019). https://doi.org/10.1214/18-aap1430
Dieker, A.B., Gao, X.: Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions. Ann. Appl. Probab. 23(4), 1291–1317 (2013). https://doi.org/10.1214/12-aap870
Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl. 119(3), 897–923 (2009). https://doi.org/10.1016/j.spa.2008.03.007
Gamarnik, D., Stolyar, A.L.: Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution. Queueing Syst. 71(1-2), 25–51 (2012). https://doi.org/10.1007/s11134-012-9294-x
Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 105(2), 143–158 (1996). https://doi.org/10.1007/bf01203833
Hairer, M.: Convergence of Markov Processes. Lecture Notes, University ofWarwick (2016). Available at http://www.hairer.org/notes/Convergence.pdf
Li, C.W.: Lyapunov exponents of nonlinear stochastic differential equations with jumps. In: Stochastic inequalities and applications, Progr. Probab., vol. 56, pp. 339–351. Birkh¨auser, Basel (2003)
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25(3), 487–517 (1993). https://doi.org/10.2307/1427521
Pang, G., Whitt, W.: Heavy-traffic limits for many-server queues with service interruptions. Queueing Syst. 61(2-3), 167–202 (2009). https://doi.org/10.1007/s11134-009-9104-2
Pang, G., Whitt,W.: Continuity of a queueing integral representation in the M1 topology. Ann. Appl. Probab. 20(1), 214–237 (2010). https://doi.org/10.1214/09-aap611
Skorokhod, A.V.: Asymptotic methods in the theory of stochastic differential equations, Translations of Mathematical Monographs, vol. 78. American Mathematical Society, Providence, RI (1989). Translated from the Russian by H. H. McFaden
Whitt, W.: Stochastic-process limits. An introduction to stochastic-process limits and their application to queues. Springer Series in Operations Research. Springer-Verlag, New York (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Arapostathis, A., Hmedi, H., Pang, G., Sandrić, N. (2019). Uniform Polynomial Rates of Convergence for A Class of Lévy-Driven Controlled SDEs Arising in Multiclass Many-Server Queues. In: Yin, G., Zhang, Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its Applications, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-25498-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-25498-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25497-1
Online ISBN: 978-3-030-25498-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)