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Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment

Author

Listed:
  • Rajeeva L. Karandikar

    (Indian Statistical Institute, New Delhi, India)

  • Vidyadhar G. Kulkarni

    (University of North Carolina, Chapel Hill, North Carolina)

Abstract

This paper considers a stochastic fluid model of a buffer content process { X ( t ), t ≥ 0} that depends on a finite-state, continuous-time Markov process { Z ( t ), t ≥ 0} as follows: During the time-intervals when Z ( t ) is in state i , X ( t ) is a Brownian motion with drift μ i , variance parameter σ i 2 and a reflecting boundary at zero. This paper studies the steady-state analysis of the bivariate process {( X ( t ), Z ( t )), t ≥ 0} in terms of the eigenvalues and eigenvectors of a nonlinear matrix system. Algorithms are developed to compute the steady-state distributions as well as moments. Numerical work is reported to show that the variance parameter has a dramatic effect on the buffer content process.

Suggested Citation

  • Rajeeva L. Karandikar & Vidyadhar G. Kulkarni, 1995. "Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment," Operations Research, INFORMS, vol. 43(1), pages 77-88, February.
    • Handle: RePEc:inm:oropre:v:43:y:1995:i:1:p:77-88
    DOI: 10.1287/opre.43.1.77
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    Cited by:

    1. Berkelmans, Wouter & Cichocka, Agata & Mandjes, Michel, 2020. "The correlation function of a queue with Lévy and Markov additive input," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1713-1734.
    2. M. Gribaudo & D. Manini & B. Sericola & M. Telek, 2008. "Second order fluid models with general boundary behaviour," Annals of Operations Research, Springer, vol. 160(1), pages 69-82, April.
    3. Gábor Horváth & Miklós Telek, 2017. "Matrix-analytic solution of infinite, finite and level-dependent second-order fluid models," Queueing Systems: Theory and Applications, Springer, vol. 87(3), pages 325-343, December.
    4. Nail Akar & Omer Gursoy & Gabor Horvath & Miklos Telek, 2021. "Transient and First Passage Time Distributions of First- and Second-order Multi-regime Markov Fluid Queues via ME-fication," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1257-1283, December.
    5. Horton, Graham & Kulkarni, Vidyadhar G. & Nicol, David M. & Trivedi, Kishor S., 1998. "Fluid stochastic Petri nets: Theory, applications, and solution techniques," European Journal of Operational Research, Elsevier, vol. 105(1), pages 184-201, February.
    6. Bean, Nigel G. & Nguyen, Giang T. & Nielsen, Bo F. & Peralta, Oscar, 2022. "RAP-modulated fluid processes: First passages and the stationary distribution," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 308-340.
    7. Marco Gribaudo & Illés Horváth & Daniele Manini & Miklós Telek, 2020. "Modelling large timescale and small timescale service variability," Annals of Operations Research, Springer, vol. 293(1), pages 123-140, October.
    8. Guy Latouche & Matthieu Simon, 2018. "Markov-Modulated Brownian Motion with Temporary Change of Regime at Level Zero," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1199-1222, December.
    9. Ivanovs, Jevgenijs & Boxma, Onno & Mandjes, Michel, 2010. "Singularities of the matrix exponent of a Markov additive process with one-sided jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1776-1794, August.
    10. Nigel Bean & Angus Lewis & Giang T. Nguyen & Małgorzata M. O’Reilly & Vikram Sunkara, 2022. "A Discontinuous Galerkin Method for Approximating the Stationary Distribution of Stochastic Fluid-Fluid Processes," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2823-2864, December.
    11. Nicole Bäuerle, 1998. "The advantage of small machines in a stochastic fluid production process," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(1), pages 83-97, February.

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