On a Queueing Model with Group Services

  • Alexander Zeifman
  • Anna Korotysheva
  • Yakov Satin
  • Galina Shilova
  • Tatyana Panfilova
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 356)


An analogue of M t /M t /S/S Erlang loss system for a queue with group services is introduced and considered. Weak ergodicity of the model is studied. We obtain the bounds on the rate of convergence to the limiting characteristics and consider two concrete queueing models with finding of their main limiting characteristics.


Markovian queueing models group service weak ergodicity bounds 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Van Doorn, E.A., Zeifman, A.I.: On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst. 63, 241–252 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Erlang, A.K.: Løsning af nogle Problemer fra Sandsynlighedsregningen af Betydning for de automatiske Telefoncentraler. Elektroteknikeren 13, 5–13 (1917)Google Scholar
  3. 3.
    Fricker, C., Robert, P., Tibi, D.: On the rate of convergence of Erlang’s model. J. Appl. Probab. 36, 1167–1184 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Gnedenko, B.V., Makarov, I.P.: Properties of a problem with losses in the case of periodic intensities. Diff. Equations 7, 1696–1698 (1971) (in Russian)Google Scholar
  5. 5.
    Granovsky, B., Zeifman, A.: Nonstationary queues: estimation of the rate of convergence. Queueing Syst. 46, 363–388 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kijima, M.: On the largest negative eigenvalue of the infinitesimal generator associated with M/M/n/n queues. Oper. Res. Let. 9, 59–64 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Massey, W.A., Whitt, W.: On analysis of the modified offered-load approximation for the nonstationary Erlang loss model. Ann. Appl. Probab. 4, 1145–1160 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Satin, Y.A., Zeifman, A.I., Korotysheva, A.V., Shorgin, S.Y.: On a class of Markovian queues. Informatics and Its Applications 5(4), 6–12 (2011) (in Russian)Google Scholar
  9. 9.
    Voit, M.: A note of the rate of convergence to equilibrium for Erlang’s model in the subcritical case. J. Appl. Probab. 37, 918–923 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Zeifman, A.I.: Stability for contionuous-time nonhomogeneous Markov chains. Lect. Notes Math. 1155, 401–414 (1985)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zeifman, A.I.: Properties of a System with Losses in the Case of Variable Rates. Autom. Remote Control (1), 82–87 (1989)Google Scholar
  12. 12.
    Zeifman, A.I., Isaacson, D.: On strong ergodicity for nonhomogeneous continuous-time Markov chains. Stoch. Proc. Appl. 50, 263–273 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Zeifman, A.I.: Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Proc. Appl. 59, 157–173 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zeifman, A., Leorato, S., Orsingher, E., Satin, Y., Shilova, G.: Some universal limits for nonhomogeneous birth and death processes. Queueing Systems 52, 139–151 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Zeifman, A. I., Bening, V. E., Sokolov, I. A.: Markov Chains and Models in Continuous Time. Elex-KM, Moscow (2008) (in Russian)Google Scholar
  16. 16.
    Zeifman, A.I.: On the nonstationary Erlang loss model. Autom. Rem. Contr. 70, 2003–2012 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Zeifman
    • 1
  • Anna Korotysheva
    • 2
  • Yakov Satin
    • 2
  • Galina Shilova
    • 2
  • Tatyana Panfilova
    • 2
  1. 1.Institute of Informatics Problems RASVologda State Pedagogical UniversityRussia
  2. 2.ISEDT RASVologda State Pedagogical UniversityRussia

Personalised recommendations