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Matrix-analytic solution of infinite, finite and level-dependent second-order fluid models

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Abstract

This paper presents a matrix-analytic solution for second-order Markov fluid models (also known as Markov-modulated Brownian motion) with level-dependent behavior. A set of thresholds is given that divide the fluid buffer into homogeneous regimes. The generator matrix of the background Markov chain, the fluid rates (drifts) and the variances can be regime dependent. The model allows the mixing of second-order states (with positive variance) and first-order states (with zero variance) and states with zero drift. The behavior at the upper and lower boundary can be reflecting, absorbing, or a combination of them. In every regime, the solution is expressed as a matrix-exponential combination, whose matrix parameters are given by the minimal nonnegative solution of matrix quadratic equations that can be obtained by any of the well-known solution methods available for quasi birth death processes. The probability masses and the initial vectors of the matrix-exponential terms are the solutions of a set of linear equations. However, to have the necessary number of equations, new relations are required for the level boundary behavior, relations that were not needed in first-order level dependent and in homogeneous (non-level-dependent) second-order fluid models. The method presented can solve systems with hundreds of states and hundreds of thresholds without numerical issues.

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Notes

  1. The implementation can be downloaded from http://www.hit.bme.hu/~ghorvath/software.

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Acknowledgements

The authors thank the exceptional efforts of the anonymous reviewers, their comments helped to improve the manuscript significantly. The authors thank the help of Balázs Ráth as well, for the useful advises in revising Sect. 4. This research is partially supported by the OTKA-123914 project.

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Authors and Affiliations

  1. Department of Networked Systems and Services, Budapest University of Technology and Economics, Magyar tudósok krt 2., Budapest, 1117, Hungary

    Gábor Horváth

  2. MTA-BME Information Systems Research Group, Magyar tudósok krt 2., Budapest, 1117, Hungary

    Miklós Telek

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  1. Gábor Horváth
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  2. Miklós Telek
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Correspondence to Gábor Horváth.

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Horváth, G., Telek, M. Matrix-analytic solution of infinite, finite and level-dependent second-order fluid models. Queueing Syst 87, 325–343 (2017). https://doi.org/10.1007/s11134-017-9544-z

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