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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn, S.: A transient analysis of Markov fluid models with jumps. J. Korean Stat. Soc. 38, 351–366 (2009)
Ahn, S., Ramaswami, V.: Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch. Models 20, 71–104 (2004)
Ahn, S., Jeon, J., Ramaswami, V.: Steady state analysis of finite fluid flow models using finite QBDs. Queueing Syst. 49, 223–259 (2005)
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)
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)
Bean, N.G., O’Reilly, M.M.: Performance measures of a multi-layer Markovian fluid model. Ann. Op. Res. 160, 99–120 (2008)
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)
Bean, N.G., O’Reilly, M.M., Taylor, P.G.: Algorithms for return probabilities for stochastic fluid flows. Stoch. Models 21, 149–184 (2005)
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)
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)
Bean, N.G., O’Reilly, M.M., Ren, Y.: Second-order Markov reward models driven by QBD processes. Perform. Eval. 69, 440–455 (2012)
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)
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)
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)
Kulkarni, V., Yan, K.: A fluid model with upward jumps at the boundary. Queueing Syst. 56, 103–117 (2007)
Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Prob. 20(3), 646–676 (1988)
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)
O’Reilly, M.M., Scheinhardt, W.: Stationary distributions for a class of Markov-modulated tandem fluid queues. Stoch. Models 33, 524–550 (2017)
Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In Proceedings of the 16th international teletraffic congress, 1019–1030 (1999)
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)
Soares, A., Latouche, G.: Matrix-analytic methods for fluid queues with finite buffers. Perform. Eval. 63, 295–314 (2006)
Stern, T.E., Elwalid, A.I.: Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23, 105–139 (1991)
Tzenova, E., Adan, I., Kulkarni, V.: Fluid models with jumps. Stoch. Models 21, 37–55 (2005)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the National Natural Science Foundation of China (11871076).
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-020-00933-z