Skip to main content
SpringerLink
Log in

Asymptotic Analysis of the Queueing Network \(SM-(GI/\infty )^K\)

  • Conference paper
  • First Online:
Information Technologies and Mathematical Modelling - Queueing Theory and Applications (ITMM 2015)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 564))

Abstract

We consider the infinite-server queueing network with semi-Markov arrivals. The system of differential equations for characteristic function of customers number at the network nodes is derived. The system is solved under asymptotic condition of high-rate arrivals. It is shown that probability distribution of customers at the network nodes can be approximated by multi-dimensional Gaussian distribution which parameters are obtained in the paper. Presented results of numerical experiments allow to determine the approximation applicability.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Walrand, J.: An Introduction to Queueing Networks. Prentice-Hall, Upper Saddle River (1988)

    MATH  Google Scholar 

  2. Heyman, D.P., Lucantoni, D.: Modelling multiple IP traffic streams with rate limits. IEEE/ACM Trans. Netw. 11, 948–958 (2003)

    Article  Google Scholar 

  3. Chakravarthy, S.R.: Markovian arrival processes. Wiley Encyclopedia of Operations Research and Management Science (2010)

    Google Scholar 

  4. Moiseev, A., Nazarov, A.: Asymptotic analysis of the infinite-server queueing system with high-rate semi-markov arrivals. In: IEEE International Congress on Ultra Modern Telecommunications and Control Systems (ICUMT 2014), pp. 507–513. IEEE Press, New York (2014)

    Google Scholar 

  5. Neuts, M.F., Chen, S.Z.: The infinite server queue with semi-Markovian arrivals and negative exponential services. J. Appl. Prob. 9, 178–184 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  6. Smith, W.: The infinitely-many-server queue with semi-Markovian arrivals and customer-dependent exponential service times. Oper. Res. 22, 907–912 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  7. Takács, L.: A storage process with semi-Markov input. Adv. Appl. Prob. 7, 830–844 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  8. Çinlar, E.: Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  9. Massey, W.A., Whitt, W.: Networks of infinite-server queues with nonstationary Poisson input. Queueing Sys. 13, 183–250 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nazarov, A., Moiseev, A.: Analysis of an open non-Markovian \(GI-(GI|\infty )^K\) queueing network with high-rate renewal arrival process. Prob. Inf. Transm. 49(2), 167–178 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kolmogorov, A.: Sulla determinazione empirica di una legge di distribuzione. Giornale dell’ Intituto Italiano degli Attuari 4, 83–91 (1933)

    MATH  Google Scholar 

Download references

Acknowledgments

The work is performed under the state order of the Ministry of Education and Science of the Russian Federation (No. 1.511.2014/K).

Author information

Authors and Affiliations

  1. Tomsk State University, Tomsk, Russia

    Alexander Moiseev

Authors
  1. Alexander Moiseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Moiseev .

Editor information

Editors and Affiliations

  1. Applied Mathematics and Computer Science, Belarusian State University, Minsk, Russia

    Alexander Dudin

  2. Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Kemerovo State University, Anzhero-Sudzhensk, Russia

    Rafael Yakupov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Moiseev, A. (2015). Asymptotic Analysis of the Queueing Network \(SM-(GI/\infty )^K\) . In: Dudin, A., Nazarov, A., Yakupov, R. (eds) Information Technologies and Mathematical Modelling - Queueing Theory and Applications. ITMM 2015. Communications in Computer and Information Science, vol 564. Springer, Cham. https://doi.org/10.1007/978-3-319-25861-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25861-4_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25860-7

  • Online ISBN: 978-3-319-25861-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation