Annals of Operations Research

, Volume 160, Issue 1, pp 69–82 | Cite as

Second order fluid models with general boundary behaviour



A crucial property of second order fluid models is the behaviour of the fluid level at the boundaries. Two cases have been considered: the reflecting and the absorbing boundary. This paper presents an approach for the stationary analysis of second order fluid models with any combination of boundary behaviours. The proposed approach is based on the solution of a linear system whose coefficients are obtained from a matrix exponent. A practical example demonstrates the suitability of the technique in performance modeling.


Second order fluid models Matrix exponent Stationary distribution Numerical analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agapie, M., & Sohraby, K. (2001). Algorithmic solution to second order fluid flow. In Proc. of IEEE infocom, Anchorage, Alaska, Usa, Apr. 2001. Google Scholar
  2. Ahn, S., & Ramaswami, V. (2003). Fluid flow models and queues—a connection by stochastic coupling. Communications in Statistics. Stochastic Models, 19(3), 325–348. CrossRefGoogle Scholar
  3. Ahn, S., & Ramaswami, V. (2004). Transient analysis of fluid flow models via stochastic coupling. Communications in Statistics. Stochastic Models, 20(1), 71–101. CrossRefGoogle Scholar
  4. Ang, E.-J., & Barria, J. (2000). The Markov modulated regulated Brownian motion: a second-order fluid flow model of a finite buffer. Queueing Systems, 35, 263–287. CrossRefGoogle Scholar
  5. Anick, D., Mitra, D., & Sondhi, M. M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Technical Journal, 61(8), 1871–1894. Google Scholar
  6. Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stochastic Models, 11, 1–20. CrossRefGoogle Scholar
  7. Bean, N. G., O’Reilly, M. M., & Taylor, P. G. (2005a). Algorithms for the first return probabilities for stochastic fluid flows. Stochastic Models, 21(1). Google Scholar
  8. Bean, N. G., O’Reilly, M. M., & Taylor, P. G. (2005b). Hitting probabilities and hitting times for stochastic fluid flows. Stochastic Processes and their Applications, 115, 1530–1556. CrossRefGoogle Scholar
  9. Chen, D.-Y., Hong, Y., & Trivedi, K. S. (2002). Second order stochastic fluid flow models with fluid dependent flow rates. Performance Evaluation, 49(1–4), 341–358. CrossRefGoogle Scholar
  10. Cox, D. R., & Miller, H. D. (1972). The theory of stochastic processes. London: Chapman & Hall. Google Scholar
  11. da Silva Soares, A., & Latouche, G. (2002). Further results on the similarity between fluid queues and QBDs. In G. Latouche & P. Taylor (Eds.), Proc. of the 4th int. conf. on matrix-analytic methods (pp. 89–106). Adelaide, 2002. Singapore: World Scientific. Google Scholar
  12. da Silva Soares, A., & Latouche, G. (2006). Matrix-analytic methods for fluid queues with finite buffers. Performance Evaluation, 63(4), 295–314. CrossRefGoogle Scholar
  13. German, R., Gribaudo, M., Horváth, G., & Telek, M. (2003). Stationary analysis of FSPNs with mutually dependent discrete and continuous parts. In International conference on petri net performance models—PNPM 2003 (pp. 30–39). Urbana, IL, USA, Sept. 2003. New York: IEEE CS Press. CrossRefGoogle Scholar
  14. Gribaudo, M., & German, R. (2001). Numerical solution of bounded fluid models using matrix exponentiation. In Proc. 11th GI/ITG conference on measuring, modelling and evaluation of computer and communication systems (MMB). Aachen, Germany, Sep. 2001. VDE Verlag. Google Scholar
  15. Horton, G., Kulkarni, V. G., Nicol, D. M., & Trivedi, K. S. (1998). Fluid stochastic Petri nets: theory, application, and solution techniques. European Journal of Operations Research, 105(1), 184–201. CrossRefGoogle Scholar
  16. Igelnik, B., Kogan, Y., Kriman, V., & Mitra, D. (1995). A new computational approach for stochastic fluid models of multiplexers with heterogeneous sources. Queueing Systems—Theory and Applications, 20, 85–116. CrossRefGoogle Scholar
  17. Karandikar, R. L., & Kulkarni, V. G. (1995). Second-order fluid flow models: reflected Brownian motion in a random environment. Operations Research, 43, 77–88. CrossRefGoogle Scholar
  18. Kobayashi, H., & Ren, Q. (1992). A mathematical theory for transient analysis of communication networks. IEICE Transactions Communications, E75-B(12), 1266–1276. Google Scholar
  19. Kosten, L. (1984). Stochastic theory of data-handling systems with groups of multiple sources. In Proceedings of the IFIP WG 7.3/TC 6 second international symposium on the performance of computer-communication systems (pp. 321–331). Zurich, Switzerland. Google Scholar
  20. Mitra, D. (1987). Stochastic fluid models. In Proceedings of performance’87 (pp. 39–51). Brussels, Belgium. Google Scholar
  21. Mitra, D. (1988). Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Advances in Applied Probability, 20, 646–676. CrossRefGoogle Scholar
  22. Rabehasaina, L., & Sericola, B. (2003). Stability analysis of second order fluid flow models in a stationary ergodic environment. Annals of Applied Probability, 13(4). Google Scholar
  23. Ramaswami, V. (1996). Matrix analytic methods for stochastic fluid flows. In D. Smith & P. Hey (Eds.), Proc. ITC 16 (pp. 1019–1030). Edinburgh. Amsterdam: Elsevier. Google Scholar
  24. Ren, Q., & Kobayashi, H. (1995). Transient solutions for the buffer behavior in statistical multiplexing. Performance Evaluation, 23, 65–87. CrossRefGoogle Scholar
  25. Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Annals of Applied Probability, 4(2), 390–413. CrossRefGoogle Scholar
  26. Rogers, L. C. G., & Shi, Z. (1994). Computing the invariant law of a fluid model. Journal of Applied Probability, 31(4), 885–896. CrossRefGoogle Scholar
  27. Sericola, B. (1998). Transient analysis of stochastic fluid models. Performance Evaluation, 32(4). Google Scholar
  28. Stern, T. E., & Elwalid, A. I. (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Advances in Applied Probability, 23, 105–139. CrossRefGoogle Scholar
  29. Tanaka, T., Hashida, O., & Takahashi, Y. (1995). Transient analysis of fluid models for ATM statistical multiplexer. Performance Evaluation, 23, 145–162. CrossRefGoogle Scholar
  30. Wolter, K. (1997). Second order fluid stochastic petri nets: an extension of GSPNs for approximate and continuous modelling. In Proc. of world congress on system simulation (pp. 328–332). Singapore, Sep. 1997. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Dip. di InformaticaUniversità di TorinoTorinoItaly
  2. 2.INRIA–IRISARennesFrance
  3. 3.Dept. of TelecommunicationsTechnical University of BudapestBudapestHungary

Personalised recommendations