On Gaussian Approximation of Multi-Channel Networks with Input Flows of General Structure

  • E. O. Lebedev
  • H. V. LivinskaEmail author
  • J. Sztrik

In this paper, a multi-channel queueing network with input flow of a general structure is considered. The multi-dimensional service process is introduced as the number of customers at network nodes. In the heavy-traffic regime, a functional limit theorem of diffusion approximation type is proved under the condition that the input flows converge to their limits in the uniform topology. A limit Gaussian process is constructed and its correlation characteristics are represented explicitly via the network parameters. A network with nonhomogeneous Poisson input flow is studied as a particular case of the general model, and a correspondent Gaussian limit process is built.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. V. Anisimov and E.A. Lebedev, Stochastic Queueing Networks. Markov Models, Lybid, Kyiv (1992).Google Scholar
  2. 2.
    V. Anisimov and J. Sztrik, “Asymptotic analysis of some controlled finite-source queueing systems,” Act. Cyber., 9, 27–39 (1989).MathSciNetzbMATHGoogle Scholar
  3. 3.
    V. V. Anisimov, Switching Processes in Queueing Models, ISTE Ltd (2008).Google Scholar
  4. 4.
    V. V. Anisimov, “Switching queueing networks,” in: Lecture Notes in Computer Sciences, Vol. 5233, Springer-Verlag (2011), pp. 258–283.Google Scholar
  5. 5.
    G. P. Basharin, P. P. Bocharov, and Y.A. Kogan, Analysis of Queues in Calculating Networks, Nauka, Moscow (1989).Google Scholar
  6. 6.
    A. Dvurechenskij, L.A. Kyljukina, and G.A. Ososkov, “On a problem of the busy-period determination in queues with infinitely many servers,” J. Appl. Probab., No. 21, 201–206 (1984).MathSciNetCrossRefGoogle Scholar
  7. 7.
    I. I. Gikhman and A. V. Skorohod, Theory of Stochastic Processes, Vol. 1, Nauka, Moscow (1971).Google Scholar
  8. 8.
    B. I. Grigelionis, “On convergence of sums of stepwise processes to Poisson one,” Theor. Probab. Appl., 8, No. 2, 189–194 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. A. Horn and Ch.R. Johnson, Matrix Analysis, Cambridge University Press (1985).Google Scholar
  10. 10.
    V. S. Korolyuk and V.V. Korolyuk, Stochastic Models of Systems, Kluwer Acad. Press, Dordrecht (1999).CrossRefzbMATHGoogle Scholar
  11. 11.
    E.O. Lebedev, “A limit theorem for stochastic networks and its applications,” Theor. Probab. Math. Stat., No. 68, 81–92 (2003).MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. A. Lebedev and I. A. Makushenko, Risk Optimisation for Multi-channel Stochastic Network, National Library of Ukraine, Kyiv (2007).Google Scholar
  13. 13.
    E. Lebedev, A. Chechelnitsky, and A. Livinska, “Multi-channel network with interdependent input flows in heavy traffic,” Theor. Probab. Math. Stat., No. 97, 109–119 (2017).Google Scholar
  14. 14.
    H. V. Livinska and E.A. Lebedev, “Conditions of Gaussian non-Markov approximation for multichannel networks,” in: Proceedings of the ECMS-2015, Albena (Varna), Bulgaria (2015), pp. 642–649.Google Scholar
  15. 15.
    A. V. Livinska and E.A. Lebedev, “Gaussian and diffusion limits for multi-channel stochastic networks,” J. Math. Sci., 218, No. 3, 106–113 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Livinska and E. Lebedev, “On transient and stationary regimes for multi-channel networks with periodic inputs,” Appl. Stat. Comput., No. 319, 13–23 (2018).MathSciNetGoogle Scholar
  17. 17.
    H.V. Livinska, “A limit theorem for non-Markovian multi-channel networks under heavy traffic conditions,” Theor. Probab. Math. Stat., No. 93, 113–122 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    W.A. Massey and W. Whitt, “Networks of infinite-server queues with nonstationary Poisson input,” Queueing Syst., No. 13, 183–250 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    W. A. Massey and W. Whitt, “A stochastic model to capture and time dynamics in wireless communication systems,” Probab. Eng. Inform. Sci., No. 8, 541–569 (1994).CrossRefGoogle Scholar
  20. 20.
    J. Matis and T. E. Wehrly, “Generalized stochastic compartmental models with Erlang transit times,” J. Pharm. Biopharm., No. 18, 589–607 (1990).CrossRefGoogle Scholar
  21. 21.
    A. A. Nazarov and S.P. Moiseeva, Method of Asymptotic Analysis in Queueing Theory, Sc.-Techn. Litr. Publ., Tomsk (2006).Google Scholar
  22. 22.
    A. A. Nazarov and I. L. Lapatin, “Asymptotically Poisson MAP-flows,” Bull. Tomsk State Univ. Ser. Control Comput. Tech. Inform., 4, No. 13, 72–78 (2010).Google Scholar
  23. 23.
    A. Nazarov, J. Sztrik, and A. Kvach, “Some features of a finite-source M/GI/1 retrial queuing system with collisions of customers,” in: Distributed Computer and Communication Networks DCCN 2017, Communications in Computer and Information Science,Vol. 700, Vishnevskiy, K. Samouylov, and D. Kozyrev (eds.), Springer Verlag, 186–200 (2017).Google Scholar
  24. 24.
    J. Sztrik, “Finite-source queueing systems and their applications,” in: Formal Methods in Computing Chap. 7, M. Ferenczi, A. Pataricza, and L. Ronyai (eds.), Akademia Kiado, Budapest, Hungary (2005).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Taras Shevchenko University of KyivKyivUkraine
  2. 2.University of DebrecenDebrecenHungary

Personalised recommendations