Abstract
Stochastic fluid models have been widely used to model the level of a resource that changes over time, where the rate of variation depends on the state of some continuous-time Markov process. Latouche and Taylor (Queueing Syst 63:109–129, 2009) introduced an approach, using matrix analytic methods and the reduced load approximation for loss networks, to analyse networks of fluid models all driven by the same modulating process where the buffers are infinite. We extend the method to networks involving finite buffer models and illustrate the approach by deriving performance measures for a simple network as characteristics such as buffer size are varied. Our results provide insight into the situations where the infinite buffer model is a reasonable approximation to the finite buffer model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn, S., Badescu, A., Ramaswami, V.: Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Syst. 55, 207–222 (2007)
Ahn, S., Ramaswami, V.: Fluid flow models and queues: a connection by stochastic coupling. Stoch. Models 19(3), 325–348 (2003)
Asmussen, S.: Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11(1), 21–49 (1995)
Balázs, M., Horváth, G., Kolumbán, S., Kovács, P., Telek, M.: Fluid level dependent Markov fluid models with continuous zero transition. Perform. Eval. 68(11), 1149–1161 (2011)
Bean, N., O’Reilly, M.: A stochastic two-dimensional fluid model. Stoch. Models 29(1), 31–63 (2013)
Bean, N., O’Reilly, M., Taylor, P.G.: Hitting probabilities and hitting times for stochastic fluid flows. Stoch. Proc. Appl. 115, 1530–1556 (2005)
Bean, N., O’Reilly, M., Taylor, P.G.: Hitting probabilities and hitting times for stochastic fluid flows: the bounded model. Probab. Eng. Inf. Sci. 23(1), 121–147 (2009)
Bean, N., O’Reilly, M.: The stochastic fluid-fluid model: a stochastic fluid model driven by an uncountable-state process, which is a stochastic fluid model itself. Stoch. Process. Appl. 124(5), 1741–1772 (2014)
Çinlar, E.: Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ (1975)
da Silva Soares, A.: Fluid queues: building upon the analogy with QBD processes. Ph.D. thesis, Université Libre de Bruxelles (2005)
Govorun, M., Latouche, G., Remiche, M.: Stability for fluid queues: characteristic inequalities. Stoch. Models 29(1), 64–88 (2013)
Kelly, F.: Blocking probabilities in large circuit-switched networks. Adv. Appl. Probab. 18(2), 473–505 (1986)
Kelly, F.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991)
Latouche, G., Nguyen, G.: Fluid approach to two-sided Markov-modulated Brownian motion. Queueing Syst. 80(1), 105–125 (2015)
Latouche, G., Nguyen, G.: Slowing time: Markov-modulated Brownian motions with a sticky boundary. Stoch. Models 33(2), 297–321 (2017)
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia (1999)
Latouche, G., Taylor, P.G.: A stochastic fluid model for an ad hoc mobile network. Queueing Syst. 63, 109–129 (2009)
Margolius, B., O’Reilly, M.: The analysis of cyclic stochastic fluid flows with time-varying transition rates. Queueing Syst. 82(1–2), 43–73 (2016)
Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Proceedings of the 16th International Teletraffic congress, Edinburgh, pp. 1019–1030 (1999)
Rogers, L.: Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Probab. 4, 390–413 (1994)
Samuelson, A., Haigh, A., O’Reilly, M., Bean, N.: Stochastic model for maintenance in continuously deteriorating systems. Eur. J. Oper. Res. 259(3), 1169–1179 (2017)
Sericola, B.: Transient analysis of stochastic fluid models. Perform. Eval. 32(4), 245–263 (1998)
Sonenberg, N.: Networks of interacting stochastic fluid models. Ph.D. thesis, The University of Melbourne, http://minerva-access.unimelb.edu.au/handle/11343/208775 (2017)
Telek, M., Vécsei, M.: Finite queues at the limit of saturation. In: 9th International Conference on Quantitative Evaluation of SysTems (QEST), pp. 33–42. Conference Publishing Services (CPS), London (2012)
The On-Line Encyclopedia of Integer Sequences: https://oeis.org/
Van Lierde, S., da Silva Soares, A., Latouche, G.: Invariant measures for fluid queues. Stoch. Models 24(1), 133–151 (2008)
Acknowledgements
The authors would like to acknowledge the support of the Australian Research Council (ARC) through Laureate Fellowship FL130100039 and the ARC Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sonenberg, N., Taylor, P.G. Networks of interacting stochastic fluid models with infinite and finite buffers. Queueing Syst 92, 293–322 (2019). https://doi.org/10.1007/s11134-019-09619-w
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-019-09619-w