Stochastic Fluid Model with Jumps: The Bounded Model

Abstract

As one of the important hybrid models, stochastic fluid models (SFMs, in short) have many applications. In this work, we discuss a class of SFMs modulated by Markovian chain with lower and upper jumps, where the fluid level has a positive upper bound and a lower bound zero. The joint limiting distribution of the buffer level and the environment state is expressed by a series of differential equations. An example is give to illustrate the theory obtained.

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References

  1. 1.

    Ahn, S.: A transient analysis of Markov fluid models with jumps. J. Korean Stat. Soc. 38, 351–366 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ahn, S., Ramaswami, V.: Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch. Models 20, 71–104 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ahn, S., Jeon, J., Ramaswami, V.: Steady state analysis of finite fluid flow models using finite QBDs. Queueing Syst. 49, 223–259 (2005)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ahn, S., Badescu, A.L., Ramaswami, V.: Time dependent analysis of finite buffer fluid flows and risk models with a divident barrier. Queueing Syst. 55, 207–222 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Anick, D., Mitra, D.D., Sondhi, M.M.: Stochastic theory of data handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894 (1982)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bean, N.G., O’Reilly, M.M.: Performance measures of a multi-layer Markovian fluid model. Ann. Op. Res. 160, 99–120 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

    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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Algorithms for return probabilities for stochastic fluid flows. Stoch. Models 21, 149–184 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    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. Prob. 10, 381–408 (2008)

    Article  Google Scholar 

  10. 10.

    Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Hitting probabilities and hitting times for stochastic fluid flows: the bounded model. Prob. Eng. Inf. Sci. 23, 121–47 (2009)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bean, N.G., O’Reilly, M.M., Ren, Y.: Second-order Markov reward models driven by QBD processes. Perform. Eval. 69, 440–455 (2012)

    Article  Google Scholar 

  12. 12.

    da Silva Soares, A., Latouche, G.: Further results on the similarity between fluid queues and QBDs. In G. Latouche, P. Taylor (eds.) Matrix-analytic methods: theory and applications. Singapore:World Scientific, 89–106 (2002)

  13. 13.

    Elwalid, A.I., Mitra, D.: Analysis and design of rate-based congestion control of high-speed networks, I: stochastic fluid models, access regulation. Queueing Syst. Theory Appl. 9, 19–64 (1991)

    Article  Google Scholar 

  14. 14.

    Kulkarni, V.: Fluid models for single buffer systems. In: Dshalalow, J.H. (ed.) Frontiers in queueing, pp. 321–338. CRC Press, Boca Raton, Models and Applications in Science and Engineering (1997)

  15. 15.

    Kulkarni, V., Yan, K.: A fluid model with upward jumps at the boundary. Queueing Syst. 56, 103–117 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Prob. 20(3), 646–676 (1988)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Miyazawa, M., Tanada, H.: A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps. J. Appl. Prob. 39, 604–618 (2002)

    MathSciNet  Article  Google Scholar 

  18. 18.

    O’Reilly, M.M., Scheinhardt, W.: Stationary distributions for a class of Markov-modulated tandem fluid queues. Stoch. Models 33, 524–550 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In Proceedings of the 16th international teletraffic congress, 1019–1030 (1999)

  20. 20.

    Simonian, A., Virtamo, J.: Transient and stationary distributions for fluid queues and input processes with a density. SIAM J. Appl. Math. 51, 1732–1739 (1991)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Soares, A., Latouche, G.: Matrix-analytic methods for fluid queues with finite buffers. Perform. Eval. 63, 295–314 (2006)

    Article  Google Scholar 

  22. 22.

    Stern, T.E., Elwalid, A.I.: Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23, 105–139 (1991)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Tzenova, E., Adan, I., Kulkarni, V.: Fluid models with jumps. Stoch. Models 21, 37–55 (2005)

    MathSciNet  Article  Google Scholar 

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Correspondence to Yong Ren.

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This work is supported by the National Natural Science Foundation of China (11871076).

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Hu, L., Ren, Y. Stochastic Fluid Model with Jumps: The Bounded Model. Int. J. Appl. Comput. Math 7, 1 (2021). https://doi.org/10.1007/s40819-020-00933-z

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Keywords

  • Stochastic fluid flow model
  • Limiting distribution
  • Jump