Queueing Systems

, Volume 80, Issue 1–2, pp 105–125 | Cite as

Fluid approach to two-sided reflected Markov-modulated Brownian motion



We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has been shown recently that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at \(b >0\), and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. In so doing, we obtain a new representation for the stationary distribution. It is factorised into a vector determined by the phase behaviour when the fluid is either at the level 0 or the level \(b\), and a matrix expression characteristic of the process when the fluid is in the open interval \((0,b)\).


Markov-modulated linear fluid models Reflected two-sided Markov-modulated Brownian motion Weak convergence  Stationary distribution 

Mathematics Subject Classification

60J25 60J65 60B10 



The authors thank the anonymous referees for their constructive criticism of an earlier version of the paper. They acknowledge the financial support of the Ministère de la Communauté française de Belgique through the ARC grant AUWB-08/13–ULB 5, and of the Australian Research Council through the Discovery Grant DP110101663


  1. 1.
    Asmussen, S.: Stationary distributions for fluid flow models with or without Brownian noise. Commun. Stat. 11(1), 21–49 (1995)Google Scholar
  2. 2.
    Asmussen, S., Kella, O.: A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab. 32, 376–393 (2000)CrossRefGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1999)CrossRefGoogle Scholar
  4. 4.
    Breuer, L.: First passage times for Markov additive processes with positive jumps of phase-type. J. Appl. Probab. 45, 778–799 (2008)CrossRefGoogle Scholar
  5. 5.
    da Silva Soares, A., Latouche, G.: Matrix-analytic methods for fluid queues with finite buffers. Perfor. Eval. 63, 295–314 (2005)CrossRefGoogle Scholar
  6. 6.
    da Silva Soares, A., Latouche, G.: Fluid queues with level dependent evolution. Eur. J. Oper. Res. 196, 1041–1048 (2009)CrossRefGoogle Scholar
  7. 7.
    D’Auria, B., Ivanovs, J., Kella, O., Mandjes, M.: Two-sided reflection of Markov-modulated Brownian motion. Stoch. Models 28(2), 316–332 (2012)CrossRefGoogle Scholar
  8. 8.
    Ivanovs, J.: Markov-modulated Brownian motion with two reflecting barriers. J. Appl. Probab. 47(4), 1034–1047 (2010)CrossRefGoogle Scholar
  9. 9.
    Karandikar, R.L., Kulkarni, V.: Second-order fluid flow models: reflected Brownian motion in a random environment. Oper. Res. 43, 77–88 (1995)CrossRefGoogle Scholar
  10. 10.
    Kruk, L., Lehoczky, J., Ramanan, K., Shreve, S.: An explicit formula for the Skorokhod map on \([0, a]\). Ann. Probab. 35, 1740–1768 (2007)CrossRefGoogle Scholar
  11. 11.
    Latouche, G., Nguyen, G.T.: The morphing of fluid queues into Markov-modulated Brownian motion. Submitted (2013)Google Scholar
  12. 12.
    Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Smith, D., Hey, P. (eds.) Teletraffic Engineering in a Competitive World (Proceedings of the 16th International Teletraffic Congress), pp. 1019–1030. Elsevier Science B.V., Edinburgh UK (1999)Google Scholar
  13. 13.
    Ramaswami, V.: A fluid introduction to Brownian motion and stochastic integration. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M., Yao, D. (eds.) Matrix-Analytic Methods in Stochastic Models, volume 27 of Springer Proceedings in Mathematics & Statistics, Chapter 10, pp. 209–225. Springer Science, New York NY, (2013)Google Scholar
  14. 14.
    Rogers, L.C.G.: Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Probab. 4, 390–413 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Département d’InformatiqueUniversité libre de BruxellesBrussels Belgium
  2. 2.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia

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