The Annals of Applied Probability

Ergodicity of an SPDE associated with a many-server queue

Reza Aghajani and Kavita Ramanan

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Abstract

We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a first-come-first-serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1-\beta N^{-1/2}+o(N^{-1/2})$ for some $\beta>0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Itô equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [Aghajani and Ramanan (2016)] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in [Oper. Res. 29 (1981) 567–588]. The methods introduced here are more generally applicable for the analysis of a broader class of networks.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 994-1045.

Dates
Received: October 2016
Revised: December 2017
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298935

Digital Object Identifier
doi:10.1214/18-AAP1419

Mathematical Reviews number (MathSciNet)
MR3910022

Zentralblatt MATH identifier
07047443

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G10: Stationary processes 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Stochastic partial differential equations Itô process stationary distribution ergodicity asymptotic coupling asymptotic equivalent coupling GI/GI/N queue Halfin–Whitt regime diffusion approximations

Citation

Aghajani, Reza; Ramanan, Kavita. Ergodicity of an SPDE associated with a many-server queue. Ann. Appl. Probab. 29 (2019), no. 2, 994--1045. doi:10.1214/18-AAP1419. https://projecteuclid.org/euclid.aoap/1548298935


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References

  • [1] Aghajani, R. and Ramanan, K. (2015). Ergodicity of an SPDE associated with a many-server queue. Extended version. Preptint. Available at arXiv:1512.02929 [math.PR].
    arXiv: 1512.02929
  • [2] Aghajani, R. and Ramanan, K. (2016). The limit of stationary distributions of many-server queues in the Halfin–Whitt regime. Preprint. Available at arXiv:1610.01118 [math.PR].
    arXiv: 1610.01118
  • [3] Asmussen, S. (2003). Applied Probability and Queues: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
    Zentralblatt MATH: 1029.60001
  • [4] Bakhtin, Y. and Mattingly, J. C. (2005). Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7 553–582.
    Mathematical Reviews (MathSciNet): MR2175090
    Zentralblatt MATH: 1098.34063
    Digital Object Identifier: doi:10.1142/S0219199705001878
  • [5] Bogachev, V. I. (2007). Measure Theory, Vol. I, II. Springer, Berlin.
    Zentralblatt MATH: 1120.28001
  • [6] Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2759829
    Zentralblatt MATH: 1220.46002
  • [7] Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S. and Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc. 100 36–50.
    Zentralblatt MATH: 1117.62303
    Digital Object Identifier: doi:10.1198/016214504000001808
  • [8] Burton, T. A. (2005). Volterra Integral and Differential Equations. Mathematics in Science and Engineering 167. Academic Press, Orlando, FL.
    Zentralblatt MATH: 1075.45001
  • [9] Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Y. G. (1982). Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 245. Springer, New York.
  • [10] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
    Zentralblatt MATH: 0849.60052
  • [11] Decreusefond, L. and Moyal, P. (2008). A functional central limit theorem for the $M/GI/\infty$ queue. Ann. Appl. Probab. 18 2156–2178.
    Mathematical Reviews (MathSciNet): MR2473653
    Zentralblatt MATH: 1154.60347
    Digital Object Identifier: doi:10.1214/08-AAP518
    Project Euclid: euclid.aoap/1227708915
  • [12] Doytchinov, B., Lehoczky, J. and Shreve, S. (2001). Real-time queues in heavy traffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. 11 332–378.
    Zentralblatt MATH: 1015.60086
    Digital Object Identifier: doi:10.1214/aoap/1015345295
    Project Euclid: euclid.aoap/1015345295
  • [13] Weinan, E., Mattingly, J. C. and Sinai, Y. (2001). Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys. 224 83–106.
    Zentralblatt MATH: 0994.60065
    Digital Object Identifier: doi:10.1007/s002201224083
  • [14] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
    Zentralblatt MATH: 0902.35002
  • [15] Gamarnik, D. and Goldberg, D. A. (2013). Steady-state $GI/GI/n$ queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23 2382–2419.
    Zentralblatt MATH: 1285.60090
    Digital Object Identifier: doi:10.1214/12-AAP905
    Project Euclid: euclid.aoap/1382447692
  • [16] Goldberg, D. A. (2013). On the steady-state probability of delay and large negative deviations for the ${GI}/{GI}/n$ queue in the Halfin–Whitt regime. Preprint. Available at arXiv:1307.0241.
    arXiv: 1307.0241
  • [17] Gamarnik, D. and Momčilović, P. (2008). Steady-state analysis of a multiserver queue in the Halfin–Whitt regime. Adv. in Appl. Probab. 40 548–577.
    Zentralblatt MATH: 1148.60070
    Digital Object Identifier: doi:10.1239/aap/1214950216
    Project Euclid: euclid.aap/1214950216
  • [18] Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345–380.
    Zentralblatt MATH: 1032.60056
    Digital Object Identifier: doi:10.1007/s004400200216
  • [19] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 223–259.
    Zentralblatt MATH: 1238.60082
    Digital Object Identifier: doi:10.1007/s00440-009-0250-6
  • [20] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
    Zentralblatt MATH: 0455.60079
    Digital Object Identifier: doi:10.1287/opre.29.3.567
  • [21] Harrison, J. M. (2013). Brownian Models of Performance and Control. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR3157450
    Zentralblatt MATH: 1308.60003
  • [22] Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53–69.
    Mathematical Reviews (MathSciNet): MR2074672
    Digital Object Identifier: doi:10.1023/B:QUES.0000032800.52494.51
  • [23] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
    Zentralblatt MATH: 0734.60060
  • [24] Kaspi, H. and Ramanan, K. (2011). Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 33–114.
    Zentralblatt MATH: 1208.60095
    Digital Object Identifier: doi:10.1214/09-AAP662
    Project Euclid: euclid.aoap/1292598028
  • [25] Kaspi, H. and Ramanan, K. (2013). SPDE limits of many-server queues. Ann. Appl. Probab. 23 145–229.
    Zentralblatt MATH: 1271.60098
    Digital Object Identifier: doi:10.1214/11-AAP821
    Project Euclid: euclid.aoap/1359124384
  • [26] Kazamaki, N. and Sekiguchi, T. (1983). Uniform integrability of continuous exponential martingales. Tohoku Math. J. (2) 35 289–301.
    Zentralblatt MATH: 0519.60043
    Digital Object Identifier: doi:10.2748/tmj/1178229055
    Project Euclid: euclid.tmj/1178229055
  • [27] Kruk, Ł., Lehoczky, J., Ramanan, K. and Shreve, S. (2008). Double Skorokhod map and reneging real-time queues. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz. Inst. Math. Stat. (IMS) Collect. 4 169–193. IMS, Beachwood, OH.
    Zentralblatt MATH: 1170.60314
  • [28] Kruk, Ł., Lehoczky, J., Ramanan, K. and Shreve, S. (2011). Heavy traffic analysis for EDF queues with reneging. Ann. Appl. Probab. 21 484–545.
    Zentralblatt MATH: 1220.60053
    Digital Object Identifier: doi:10.1214/10-AAP681
    Project Euclid: euclid.aoap/1300800980
  • [29] Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
    Zentralblatt MATH: 0850.60019
  • [30] Mandelbaum, A. and Momčilović, P. (2008). Queues with many servers: The virtual waiting-time process in the QED regime. Math. Oper. Res. 33 561–586.
    Zentralblatt MATH: 1213.60150
    Digital Object Identifier: doi:10.1287/moor.1070.0310
  • [31] Maslowski, B. and Seidler, J. (1998). Invariant measures for nonlinear SPDE’s: Uniqueness and stability. Arch. Math. (Brno) 34 153–172.
    Zentralblatt MATH: 0914.60028
  • [32] Mattingly, J. C. (2003). On recent progress for the stochastic Navier Stokes equations. In Journées “Équations aux Dérivées Partielles” Exp. No. XI, 52. Univ. Nantes, Nantes.
  • [33] Puhalskii, A. A. and Reed, J. E. (2010). On many-server queues in heavy traffic. Ann. Appl. Probab. 20 129–195.
    Zentralblatt MATH: 1201.60088
    Digital Object Identifier: doi:10.1214/09-AAP604
    Project Euclid: euclid.aoap/1262962320
  • [34] Puhalskii, A. A. and Reiman, M. I. (2004). Correction: “The multiclass $GI/PH/N$ queue in the Halfin–Whitt regime” [Adv. in Appl. Probab. 32 (2000), no. 2, 564–595; MR1778580]. Adv. in Appl. Probab. 36 971.
    Zentralblatt MATH: 0962.60089
    Digital Object Identifier: doi:10.1239/aap/1013540179
    Project Euclid: euclid.aap/1013540179
  • [35] Reed, J. (2009). The $G/GI/N$ queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 2211–2269.
    Zentralblatt MATH: 1181.60137
    Digital Object Identifier: doi:10.1214/09-AAP609
    Project Euclid: euclid.aoap/1259158771
  • [36] Reed, J. and Talreja, R. (2015). Distribution-valued heavy-traffic limits for the $G/GI/\infty$ queue. Ann. Appl. Probab. 25 1420–1474.
    Mathematical Reviews (MathSciNet): MR3325278
    Zentralblatt MATH: 1315.60102
    Digital Object Identifier: doi:10.1214/14-AAP1027
    Project Euclid: euclid.aoap/1427124133
  • [37] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1: Foundations. Cambridge Univ. Press, Cambridge.
    Zentralblatt MATH: 0949.60003
  • [38] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2: Itô Calculus. Cambridge Univ. Press, Cambridge.
  • [39] Scheutzow, M. (2009). Chaining techniques and their application to stochastic flows. In Trends in Stochastic Analysis. London Mathematical Society Lecture Note Series 353 35–63. Cambridge Univ. Press, Cambridge.
    Zentralblatt MATH: 1180.37077
  • [40] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • [41] Whitt, W. (2005). Heavy-traffic limits for the $G/H_{2}^{*}/n/m$ queue. Math. Oper. Res. 30 1–27.

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