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Effective bandwidths for Markov regenerative sources

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Abstract

In this paper we consider the multiplexing of independent stochastic fluid sources onto a single buffer. The rate at which a source generates fluid is assumed to be modulated by a Markov regenerative process. We develop the exponential decay rates for the tails of the steady-state distribution of the buffer content. We also develop expressions for the effective bandwidths for such sources. All the results are in terms of the Perron-Frobenius eigenvalue of a matrix defined for the Markov regenerative source. As a special case we derive similar results for regenerative sources. We apply the results to video sources.

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References

  1. J. Abate, G.I. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structures Markov queueing models,Commun. Statist. Stoch. Models 10 (1994) 99–143.

    Google Scholar 

  2. J. Abate, G.I. Choudhury and W. Whitt, Exponential approximations for tail probabilities in queues I: waiting times,Oper. Res., to appear.

  3. J. Abate, G.I. Choudhury and W. Whitt, Exponential approximations for tail probabilities in queues II: sojourn time and workload,Oper. Res., to appear.

  4. D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data handling system with multiple sources,Bell Syst. Tech. J. 61 (1982) 1871–1894.

    Google Scholar 

  5. S. Asmussen, Stationary distributions for fluid models with or without Brownian noise,Commun. Statist. Stoch. Models 11 (1995) 21–49.

    Google Scholar 

  6. C.S. Chang, Stability, queue length and delay, Part II: stochastic queueing networks, in:Proc. IEEE CDC, Tucson, AZ (1992) pp. 1005–1010.

  7. C.S. Chang and J.A. Thomas, Effective bandwidth in high-speed networks,IEEE JSAC 13 (1995) 1091–1100.

    Google Scholar 

  8. E. Cinlar,Introduction to Stochastic Processes (Prentice-Hall, Engelwood Cliffs, NJ, 1975).

    Google Scholar 

  9. N.G. Duffield and N. O'Connell, Large deviations and overflow probabilities for the general single-server queue, with applications, Preprint (1993).

  10. H. Dupuis and B. Hajek, A simple formula for mean multiplexing delay for independent regenerative sources,Queueing Systems 16 (1994) 195–239.

    Google Scholar 

  11. A.I. Elwalid and D. Mitra, Statistical multiplexing of Markov modulated sources: theory and computational algorithms,Teletraffic and Datatrafflc in a Period of Change, ITC-13 (1991) pp. 495–500.

    Google Scholar 

  12. A.I. Elwalid and D. Mitra, Effective bandwidth of general Markovian traffic sources and admission control of high speed networks,IEEE/ACM Trans. Networking 1 (1993) 329–343.

    Google Scholar 

  13. W. Feller,An Introduction to Probability Theory and Its Applications, Vol 2, 2nd ed. (Wiley, New York, 1971).

    Google Scholar 

  14. R.J. Gibbens and P.J. Hunt, Effective bandwidths for multitype UAS channels,Queueing Systems 9 (1991) 17–28.

    Google Scholar 

  15. P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue, in:Studies in Applied Probability, eds. J. Galambos and J. Gani,J. Appl. Prob. 31A (special vol.) (1994) 131–156.

  16. G.H. Golub and C.F. van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, MD, 1983).

    Google Scholar 

  17. D.P. Heyman and M.J. Sobel,Stochastic Models in Operations Research, Vol. 1 (McGraw Hill, NY, 1982).

    Google Scholar 

  18. D.P. Heyman and T.V. Laxman, Statistical analysis and simulation study of video teleconference traffic in ATM networks,IEEE Trans. Circuits and Syst. Video Tech. 2 (1992) 49–59.

    Google Scholar 

  19. F.P. Kelly, Effective bandwidths for multiclass queues,Queueing Systems 9 (1991) 5–15.

    Google Scholar 

  20. G. Kesidis, J. Walrand and C.S. Chang, Effective bandwidths for multiclass Markov fluids and other ATM sources,IEEE/ACM Trans. Networking 1 (1993) 424–428.

    Google Scholar 

  21. J.F.C. Kingman, A convexity property of positive matrices,Quart. J. Math., Ser. 2, 12 (1961) 283–284.

    Google Scholar 

  22. J.F.C. Kingman, Inequalities in the theory of queues,J. Roy. Statist. Soc. B32 (1970) 102–110.

    Google Scholar 

  23. V.G. Kulkarni and T. Rolski, Fluid model driven by an Ornstein Uhlenbeck process,Prob. Eng. Inform. Sci. 8 (1995) 403–417.

    Google Scholar 

  24. V.G. Kulkarni, Fluid models for single buffer systems, in:Frontiers in Queueing: Models and Applications in Science and Engineering, ed. J.H. Dshalalow (CRC Press, Boca Raton, Florida, to appear).

  25. V.G. Kulkarni,Modeling and Analysis of Stochastic Systems (Chapman-Hall, NY, 1995).

    Google Scholar 

  26. V.G. Kulkarni and R.L. Karandikar, Second order fluid flow models: reflected Brownian motion in a random environment,Oper. Res. 43 (1995) 77–88.

    Google Scholar 

  27. W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, On the self similar nature of Ethernet traffic (extended version),IEEE/ACM Trans. Networking 2 (1994) 1–15.

    Google Scholar 

  28. B. Melamed and B. Sengupta, TES Modeling of video traffic,IEICE Trans. Commun. E75-B(5) (1992) 1292–1300.

    Google Scholar 

  29. D. Mitra, Stochastic theory of a fluid model of producers and consumers coupled by a buffer,Adv. Appl. Prob. (1988) 646–676.

  30. A. Narayanan and V.G. Kulkarni, First passage times in fluid models with an application to two priority fluid systems, to appear inProc. IPDS 96.

  31. M.F. Neuts, The caudal characteristic curve of queues,Adv. Appl. Prob. 18 (1986) 221–254.

    Google Scholar 

  32. P. Ney and E. Nummelin, Markov additive processes I: eigenvalue properties and limit theorems,Ann. Prob. 15 (1987) 561–592.

    Google Scholar 

  33. P. Ney and E. Nummelin, Markov additive processes II: Large deviations,Ann. Prob. 15 (1987) 593–609.

    Google Scholar 

  34. N.U. Prabhu and A. Pacheco, A storage model for data communication systems,Queueing Systems 19 (1995) 1–40.

    Google Scholar 

  35. S.M. Ross, Bounds on the delay distribution in GI/G/1 queue,J. Appl. Prob. 11 (1974) 417–421.

    Google Scholar 

  36. E. Seneta,Non-negative Matrices and Markov Chains, 2nd ed. (Springer, NY, 1973).

    Google Scholar 

  37. A. Simonian, Stationary analysis of a fluid queue with input rate varying as an Ornstein-Uhlenbeck process,SIAMJ. Appl. Math. 51 (1991) 823–842.

    Google Scholar 

  38. A. Simonian, and J. Virtamo, Transient and stationary distributions for fluid queues and input processes with a density,SIAM J. Appl. Math. 51 (1991) 1731–1739.

    Google Scholar 

  39. K. Sohraby, On the theory of general on-off sources with applications in high speed networks,Proc. INFOCOM'93, San Fransisco (1993) pp. 401–410.

  40. T.E. Stern and A.I. Elwalid, Analysis of separable Markov modulated rate models for information handling systems,Adv. Appl. Prob. 23 (1991) 105–139.

    Google Scholar 

  41. H.C. Tijms,Stochastic Modeling and Analysis: A Computational Approach (Wiley, New York, 1986).

    Google Scholar 

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Authors and Affiliations

  1. Department of Operations Research, University of North Carolina, 27599-3180, Chapel Hill, NC, USA

    V. G. Kulkarni

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  1. V. G. Kulkarni
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This research was partially supported by NSF Grant No. NCR-9406823.

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Kulkarni, V.G. Effective bandwidths for Markov regenerative sources. Queueing Syst 24, 137–153 (1996). https://doi.org/10.1007/BF01149083

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