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Tandem fluid queue with long-range dependent inputs: sticky behaviour and heavy traffic approximation

Abstract

Empirical studies have shown that traffic in communication networks exhibits long-range dependence. Cumulative network traffic is often thought to be well modelled by a fractional Brownian motion (fBm). FBm approximations of queueing systems have also been well justified by theoretical results, and most of this achievements are based on one Hurst parameter. However, just as pointed out by Konstantopoulos and Lin (in: Glasserman, Sigman, and Yao (eds) Stochastic Networks: Stability and Rare Events. Lecture Notes in Statistics, Springer-Verlag, New York, 1996), various Hurst parameters may be more appropriate. At the same time, sticky Brownian motions on the half-line have many applications in queueing theory, and could be obtained as heavy traffic limits of queueing systems with exceptional service (arrive) mechanisms. In this paper, reflected operator fractional Brownian motion, sticky operator fractional Brownian motion, and a d-node tandem fluid queue with long-range dependent inputs and sticky boundaries are constructed. Moreover, in heavy traffic environment, it is shown that the scaled buffer-content process of the d-node tandem queue converges weakly to a sticky operator Brownian motion.

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Acknowledgements

We thank the two anonymous reviewers for providing constructive comments/suggestions, which have significantly improved the quality of this paper. We also would like to thank Prof. Yiqiang, Q. Zhao, Carleton University, for stimulating discussions. This work was supported by Shandong Provincial Natural Science Foundation (Nos. ZR2019MA035 and ZR2020MA036).

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Correspondence to Hongshuai Dai.

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Dai, H. Tandem fluid queue with long-range dependent inputs: sticky behaviour and heavy traffic approximation. Queueing Syst 101, 165–196 (2022). https://doi.org/10.1007/s11134-022-09798-z

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  • DOI: https://doi.org/10.1007/s11134-022-09798-z

Keywords

  • Tandem fluid queue
  • Long-range dependence
  • Sticky Brownian motion
  • Heavy traffic limits
  • Infinite source Poisson model

Mathematics Subject Classification

  • 60K25
  • 60G15