Skip to main content
SpringerLink
Log in

Two-Sided Truncations for a Class of Continuous-Time Markov Chains

  • Conference paper
  • First Online:
Information Technologies and Mathematical Modelling. Queueing Theory and Applications (ITMM 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 800))

Abstract

We consider nonstationary Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process.

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. Daleckij, J., Krein, M.G.: Stability of solutions of differential equations in Banach space. Am. Math. Soc. Transl. 43 (1974)

    Google Scholar 

  2. Granovsky, B.L., Zeifman, A.I.: Nonstationary queues: estimation of the rate of convergence. Queueing Syst. 46, 363–388 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Masuyama, H.: Continuous-time block-monotone Markov chains and their block-augmented truncations. Linear Algebr. Appl. 514, 105–150 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 28, 518–548 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Satin, Y.A., Zeifman, A.I., Korotysheva, A.V., Shorgin, S.Y.: On a class of Markovian queues. Inform. Appl. 5(4), 6–12 (2011). (in Russian)

    Google Scholar 

  6. Satin, Y.A., Zeifman, A.I., Korotysheva, A.V.: On the rate of convergence and truncations for a class of Markovian queueing systems. Theor. Probab. Appl. 57(3), 529–539 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Satin, Y., Korotysheva, A., Kiseleva, K., Shilova, G., Fokicheva, E., Zeifman, A., Korolev, V.: Two-sided truncations of inhomogeneous birth-death processes. In: ECMS 2016 Proceedings (2016). doi:10.7148/2016-0663

  8. Satin, Y., Korotysheva, A., Shilova, G., Sipin, A., Fokicheva, E., Kiseleva, K., Zeifman, A., Korolev, V., Shorgin, S.: Two-sided truncations for the \(M_t|M_t|S\) queueing model. In: ECMS 2017 Proceedings (2017)

    Google Scholar 

  9. Seneta, E.: Non-negative Matrices and Markov Chains. Springer Science & Business Media, Heidelberg (2006)

    MATH  Google Scholar 

  10. Stepanov, S.N.: Markov models with retrials: the calculation of stationary performance measures based on the concept of truncation. Math. Comput. Model. 30(3–4), 207–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tweedie, R.L.: Truncation approximations of invariant measures for Markov chains. J. Appl. Probab. 35, 517–536 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zeifman, A.I.: Truncation error in a birth and death system. USSR Comput. Math. Math. Phys. 28(6), 210–211 (1988)

    Article  Google Scholar 

  13. Zeifman, A.I.: Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Process. Appl. 59, 157–173 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zeifman, A., Leorato, S., Orsingher, E., Satin, Y., Shilova, G.: Some universal limits for nonhomogeneous birth and death processes. Queueing Syst. 52, 139–151 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zeifman, A., Korolev, V., Satin, Y., Korotysheva, A., Bening, V.: Perturbation bounds and truncations for a class of Markovian queues. Queueing Syst. 76, 205–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zeifman, A., Satin, Y., Korolev, V., Shorgin, S.: On truncations for weakly ergodic inhomogeneous birth and death processes. Int. J. Appl. Math. Comput. Sci. 24, 503–518 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zeifman, A.I., Korolev, V.Y.: Two-sided bounds on the rate of convergence for continuous-time finite inhomogeneous Markov chains. Stat. Probab. Lett. 103, 30–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zeifman, A.I., Korotysheva, A.V., Korolev, V., Satin, Y.A.: Truncation bounds for approximations of inhomogeneous continuous-time Markov chains. Theor. Probab. Appl. 61, 563–569 (2016)

    MATH  Google Scholar 

Download references

Acknowledgments

The work was supported by the Ministry of Education of the Russian Federation (the Agreement number 02.a03.21.0008 of 24 June 2016), by the Russian Foundation for Basic Research, project no. 15-01-01698.

Author information

Authors and Affiliations

  1. Vologda State University, Vologda, Russia

    Yacov Satin, Alexander Zeifman, Anna Korotysheva & Ksenia Kiseleva

  2. Institute of Informatics Problems FRC CSC RAS, Moscow, Russia

    Alexander Zeifman

  3. ISEDT RAS, Vologda, Russia

    Alexander Zeifman

  4. RUDN University, Moscow, Russia

    Ksenia Kiseleva

Authors
  1. Yacov Satin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Zeifman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna Korotysheva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ksenia Kiseleva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Zeifman .

Editor information

Editors and Affiliations

  1. Belarusian State University, Minsk, Belarus

    Alexander Dudin

  2. Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Kazan National Research Technological University, Kazan, Russia

    Alexander Kirpichnikov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Satin, Y., Zeifman, A., Korotysheva, A., Kiseleva, K. (2017). Two-Sided Truncations for a Class of Continuous-Time Markov Chains. In: Dudin, A., Nazarov, A., Kirpichnikov, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2017. Communications in Computer and Information Science, vol 800. Springer, Cham. https://doi.org/10.1007/978-3-319-68069-9_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68069-9_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68068-2

  • Online ISBN: 978-3-319-68069-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation