Advances in Applied Probability

Admission control for multidimensional workload input with heavy tails and fractional Ornstein-Uhlenbeck process

Amarjit Budhiraja, Vladas Pipiras, and Xiaoming Song

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Abstract

The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 2 (2015), 476-505.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236984

Digital Object Identifier
doi:10.1239/aap/1435236984

Mathematical Reviews number (MathSciNet)
MR3360386

Zentralblatt MATH identifier
1321.60196

Subjects
Primary: 60G18: Self-similar processes 60G57: Random measures
Secondary: 60G15: Gaussian processes 90B15: Network models, stochastic

Keywords
Poisson random measure Gaussian random measure self-similarity heavy-tailed distribution fractional Brownian motion fractional Ornstein-Uhlenbeck process admission control

Citation

Budhiraja, Amarjit; Pipiras, Vladas; Song, Xiaoming. Admission control for multidimensional workload input with heavy tails and fractional Ornstein-Uhlenbeck process. Adv. in Appl. Probab. 47 (2015), no. 2, 476--505. doi:10.1239/aap/1435236984. https://projecteuclid.org/euclid.aap/1435236984


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References

  • Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Prob. 29, 766–801.
    Mathematical Reviews (MathSciNet): MR1849177
    Zentralblatt MATH: 1015.60047
    Digital Object Identifier: doi:10.1214/aop/1008956692
    Project Euclid: euclid.aop/1008956692
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
    Mathematical Reviews (MathSciNet): MR838085
  • Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Operat. Res. 23, 145–165.
    Mathematical Reviews (MathSciNet): MR1606462
    Zentralblatt MATH: 0981.60092
    Digital Object Identifier: doi:10.1287/moor.23.1.145
  • Kaj, I. and Taqqu, M. S. (2008). Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In In and out of Equilibrium, 2 (Progress. Prob. 60), Birkhäuser, Basel, pp. 383–427.
    Mathematical Reviews (MathSciNet): MR2477392
    Zentralblatt MATH: 1154.60020
    Digital Object Identifier: doi:10.1007/978-3-7643-8786-0_19
  • Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28, 215–243.
    Mathematical Reviews (MathSciNet): MR1628422
    Digital Object Identifier: doi:10.1023/A:1019190821105
  • Kurtz, T. G., Kelly, F. P., Zachary, S. and Ziedins, I. (1996). Limit theorems for workload input models. In Stochastic Networks, Clarendon Press, Oxford, pp. 119–140.
    Zentralblatt MATH: 06157196
  • Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15.
  • Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion. Ann. Appl. Prob. 12, 23–68.
    Mathematical Reviews (MathSciNet): MR1890056
    Digital Object Identifier: doi:10.1214/aoap/1015961155
    Project Euclid: euclid.aoap/1015961155
  • Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2200233
  • Pipiras, V., Taqqu, M. S. and Levy, J. B. (2004). Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10, 121–163.
    Mathematical Reviews (MathSciNet): MR2044596
    Digital Object Identifier: doi:10.3150/bj/1077544606
    Project Euclid: euclid.bj/1077544606
  • Protter, P. E. (2004). Stochastic Integration and Differential Equations (Appl. Math. (New York) 21). Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2020294
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
    Mathematical Reviews (MathSciNet): MR1280932
    Zentralblatt MATH: 0925.60027
  • Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.
    Mathematical Reviews (MathSciNet): MR1555421
    Digital Object Identifier: doi:10.1007/BF02401743
  • Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333–374.
    Mathematical Reviews (MathSciNet): MR1640795
    Zentralblatt MATH: 0918.60037
    Digital Object Identifier: doi:10.1007/s004400050171

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