Application of \(\varphi\)-Sub-Gaussian Random Processes in Queueing Theory

Chapter

Abstract

The chapter is devoted to investigation of the class \(V (\varphi,\psi )\) of \(\varphi\)-sub-Gaussian random processes with application to queueing theory. This class of stochastic processes is more general than the Gaussian one; therefore, all results obtained in general case are valid for Gaussian processes by selection of certain Orlicz N-functions \(\varphi\) and ψ. We consider different queues filled by an aggregate of such independent sources and obtain estimates for the tail distribution of some extremal functionals of incoming random processes and their increments which describe behavior of the queue. We obtain the upper bound for the buffer overflow probability for the corresponding storage process and apply obtained result to the aggregate of sub-Gaussian generalized fractional Brownian motion processes.

References

  1. 1.
    Boulongne, P., Pierre-Loti-Viaud, D., Piterbarg, V.: On average losses in the ruin problem with fractional Brownian motion as input. Extremes 12, 77–91 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Buldygin, V.V., Kozachenko, Yu.V.: Metric Characterization of Random Variables and Random Processes. AMS, Providence, RI (2000)MATHGoogle Scholar
  3. 3.
    Giuliano-Antonini, R., Kozachenko, Yu., Nikitina, T.: Spaces of ϕ-subgaussian random variables. Rendiconti, Academia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni, 121o, XXVII, fasc.1, 95–124 (2003)Google Scholar
  4. 4.
    Kozachenko, Yu.V., Kovalchuk, Yu.A.: Boundary value problems with random initial conditions, and functional series from \(\text{Sub}_{\varphi }(\varOmega )\). I. Ukr. Math. J. 50(4), 504–515 (1998)MATHMathSciNetGoogle Scholar
  5. 5.
    Kozachenko, Yu.V., Ostrovskij, E.I.: Banach spaces of random variables of sub-Gaussian type. Theory Probab. Math. Stat. 32, 45–56 (1986)MATHGoogle Scholar
  6. 6.
    Kozachenko, Yu., Vasylyk, O.: Random processes from classes \(V (\varphi,\psi )\). Theory Probab. Math. Stat. 63, 100–111 (2000)MATHMathSciNetGoogle Scholar
  7. 7.
    Kozachenko, Yu., Sottinen, T., Vasilik, O.: Weakly self-similar stationary increment processes from the space \(SSub_{\varphi }(\varOmega )\). Theory Probab. Math. Stat. 65, 77–88 (2002)MathSciNetGoogle Scholar
  8. 8.
    Kozachenko, Yu., Vasylyk, O., Yamnenko, R.: Upper estimate of overrunning by \(\text{Sub}_{\varphi }(\varOmega )\) random process the level specified by continuous function. Random Oper. Stoch. Equ. 13(2), 111–128 (2005)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Krasnoselskii, M.A., Rutitskii, Ya.B.: Convex Functions in the Orlicz spaces. Noordhoff, Gr\(\ddot{o}\) ningen (1961)Google Scholar
  10. 10.
    Massoulie, L., Simonian, A.: Large buffer asymptotics for the queue with fractional Brownian input. J. Appl. Prob. 36, 894–906 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Norros, I.: A storage model with self-similar input. Queueing Syst. 16(3–4), 387–396 (1994)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Norros, I.: On the use of fractional Brownian motions in the theory of connectionless networks. IEEE J. Sel. Areas Commun. 13(6), 953–962 (1995)CrossRefGoogle Scholar
  13. 13.
    Vasylyk, O.I., Kozachenko, Yu.V., Yamnenko R.E.: \(\varphi\)-Subgaussovi vypadkovi protsesy: monographia (in Ukrainian). VPC “Kyiv University”, Kyiv (2008)Google Scholar
  14. 14.
    Yakovenko, T., Yamnenko, R.: Generalized fractional Brownian motion in Orlicz spaces. Theory Stoch. Process. 14(3–4), 174–188 (2008)MATHMathSciNetGoogle Scholar
  15. 15.
    Yamnenko, R.: Ruin probability for generalized \(\varphi\)-sub-Gaussian fractional Brownian motion. Theory Stoch. Process. 12(28), part no. 3–4, 261–275 (2006)Google Scholar
  16. 16.
    Yamnenko, R.E.: Bounds for the distribution of some functionals of processes with \(\varphi\)-sub-Gaussian increments. Theory Probab. Math. Stat. 85, 181–197 (2012)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Yamnenko, R.E., Shramko, O.S.: On the distribution of storage processes from the class \(V (\varphi,\psi )\). Theory Probab. Math. Stat. 83, 191–206 (2011)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Yamnenko, R., Vasylyk, O.: Random process from the class \(V (\varphi,\psi )\): exceeding a curve. Theory Stoch. Process. 13(29), part no. 4, 219–232 (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Taras Shevchenko National University of KyivKyivUkraine

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