Queueing Systems

, Volume 82, Issue 1–2, pp 43–73 | Cite as

The analysis of cyclic stochastic fluid flows with time-varying transition rates

  • Barbara MargoliusEmail author
  • Małgorzata M. O’Reilly


We consider a stochastic fluid model (SFM) \(\{(\widehat{X}(t),J(t)),t\ge 0\}\) driven by a continuous-time Markov chain \(\{J(t),t\ge 0 \}\) with a time-varying generator \(T(t)\) and cycle of length 1 such that \(T(t)=T(t+1)\) for all \(t\ge 0\). We derive theoretical expressions for the key periodic measures for the analysis of the model, and develop efficient methods for their numerical computation. We illustrate the theory with numerical examples. This work is an extension of the results in Bean et al.  (Stoch. Models 21(1):149–184, 2005) for a standard SFM with time-homogeneous generator, and suggests a possible alternative approach to that developed by Yunan and Whitt (Queueing Syst. 71(4):405–444, 2012).


Nonstationary queues Queues with time-varying arrivals Stochastic fluid model Cyclic stochastic fluid model 

Mathematics Subject Classification

76M35 60H99 60J99 



The authors would like to thank the anonymous referees for their helpful suggestions. Also, Małgorzata O’Reilly would like to thank the Australian Research Council for funding this research through Discovery Project DP110101663.


  1. 1.
    Ahn, S., Ramaswami, V.: Fluid flow models and queues—a connection by stochastic coupling. Stoch. Models 19(3), 325–348 (2003)CrossRefGoogle Scholar
  2. 2.
    Ahn, S., Ramaswami, V.: Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch. Models 20(1), 71–101 (2004)CrossRefGoogle Scholar
  3. 3.
    Ahn, S., Ramaswami, V.: Efficient algorithms for transient analysis of stochastic fluid flow models. J. Appl. Probab. 42(2), 531–549 (2005)CrossRefGoogle Scholar
  4. 4.
    Anick, D., Mitra, D., Sondhi, M.M.: Stochastic theory of data handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894 (1982)CrossRefGoogle Scholar
  5. 5.
    Asmussen, S.: Stationary distributions for fluid flow models with or without Brownian noise. Commun. Stat. Stoch. Models 11(1), 21–49 (1995)CrossRefGoogle Scholar
  6. 6.
    Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Hitting probabilities and hitting times for stochastic fluid flows. Stoch. Process. Appl. 115, 1530–1556 (2005)CrossRefGoogle Scholar
  7. 7.
    Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Algorithms for the first return probabilities for stochastic fluid flows. Stoch. Models 21(1), 149–184 (2005)CrossRefGoogle Scholar
  8. 8.
    Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Algorithms for the Laplace-Stieltjes transforms of the first return probabilities for stochastic fluid flows. Methodol. Comput. Appl. Probab. 10(3), 381–408 (2008)CrossRefGoogle Scholar
  9. 9.
    Bhatia, R., Rosenthal, P.: How and why to solve the operator equation \(AX-XB=Y\). Bull. Lond. Math. Soc. 29, 1–21 (1997)CrossRefGoogle Scholar
  10. 10.
    Liu, Y., Whitt, W.: A network of time-varying many-server fluid queues with customer abandonment. Oper. Res. 59(4), 835–846 (2011)CrossRefGoogle Scholar
  11. 11.
    Liu, Y., Whitt, W.: Large-time asymptotics for the Gt/Mt/st+GIt many-server fluid queue with abandonment. Queueing Syst. 67(2), 145–182 (2011)CrossRefGoogle Scholar
  12. 12.
    Liu, Y., Whitt, W.: The Gt/GI/st+GI many-server fluid queue. Queueing Syst. 71(4), 405–444 (2012)CrossRefGoogle Scholar
  13. 13.
    Liu, Y., Whitt, W.: Algorithms for time-varying networks of many-server fluid queues. INFORMS J. Comput. Artic. Adv. 1–15 (2014)Google Scholar
  14. 14.
    da Silva Soares, A., Latouche, G.: Further results on the similarity between fluid queues and QBDs. In: Latouche, G., Taylor, P. (eds.) Matrix-Analytic Methods Theory and Applications, pp. 89–106. World Scientific Press, River Edge (2002)CrossRefGoogle Scholar
  15. 15.
    Margolius, B.H.: The matrices \({ R}\) and \({ G}\) of matrix analytic methods and the time-inhomogeneous periodic Quasi-Birth-and-Death process. Queueing Syst. 60, 131–151 (2008)CrossRefGoogle Scholar
  16. 16.
    Ramaswami, V.: Matrix analytic methods: a tutorial overview with some extensions and new results. In: Matrix-analytic Methods in Stochastic Models (Flint, MI). Lecture Notes in Pure and Appl. Math., vol. 183, pp. 261–296. Dekker, New York (1997)Google Scholar
  17. 17.
    Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Proceedings of the 16th International Teletraffic Congress, Edinburgh, pp. 1019–1030 (7–11 June 1999)Google Scholar
  18. 18.
    Rogers, L.C.: Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Probab. 4(2), 390–413 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Cleveland State UniversityClevelandUSA
  2. 2.University of TasmaniaHobartAustralia

Personalised recommendations