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Asymptotic behavior of a multiplexer fed by a long-range dependent process

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Zhen Liu ,
Philippe Nain ,
Don Towsley  and
Zhi-Li Zhang
Zhen Liu*
Affiliation:
INRIA
Philippe Nain*
Affiliation:
INRIA
Don Towsley*
Affiliation:
University of Massachusetts
Zhi-Li Zhang*
Affiliation:
University of Minnesota
*
∗Postal address: INRIA, 2004, route de Lucioles, B.P. 93, 06902 Sophia Antipolis, France.
∗Postal address: INRIA, 2004, route de Lucioles, B.P. 93, 06902 Sophia Antipolis, France.
∗∗∗Postal address: Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA.
∗∗∗Postal address: Department of Computer Science and Engineering, University of Minnesota, 200 Union St. S.E., Minneapolis, MN 55455, USA.
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Abstract

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ < 1, where ρ is the arrival rate at the buffer.

Keywords

Asymptotic self-similar processlong-range dependencesubexponential distributionsPareto distributionlarge deviationsqueues

MSC classification

Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 105 - 118
Copyright
Copyright © Applied Probability Trust 1999 

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References

Anantharam, V. (1995). On the sojourn time of sessions at an ATM buffer with long-range dependent input traffic. In Proc. 34th IEEE Conf. Decision & Control, Vol. 1. IEEE Control Systems Society, New York, pp. 859–864.Google Scholar
Beran, J. (1994). Statistics for Long-memory Processes. Chapman and Hall, New York.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. CUP, Cambridge, UK.Google Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.Google Scholar
Boxma, O. J. (1996). Fluid queues and regular variation. Performance Evaluation 27/ 28, 699–712.Google Scholar
Boxma, O. J., and Dumas, V. (1998). Fluid queues with long-tailed activity period distributions. Computer Communications 21, 1509–1529.Google Scholar
Brichet, F., Roberts, J. W., Simonian, A., and Veitch, D. (1996). Heavy traffic analysis of a storage model with long range dependent on/off sources. QUESTA 23, 197–216.Google Scholar
Chistakov, V. P. (1964). A theorem on sums of independent positive random variables and its application to branching random processes. Theory Prob. Appl. 9, 640–648.Google Scholar
Choudhury, G. L., and Whitt, W. (1997). Long-tail buffer-content distributions in broadband networks. Performance Evaluation 30, 177–190.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Rel. Fields, 72, 529–557.CrossRefGoogle Scholar
Cox, D. R., and Isham, V. (1980). Point Processes. Chapman and Hall, New York.Google Scholar
Duffield, N. G. (1996). On the relevance of long-tailed durations for the statistical multiplexing of large aggregations. In Proc. 34th Annual Allerton Conf. Communication, Control and Computing.Google Scholar
Duffield, N. G., and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue with applications. Math. Proc. Cam. Phil. Soc. 118, 363–374.Google Scholar
Embrechts, P., and Omey, E. (1984). A property of longtailed distributions. J. Appl. Prob. 21, 80–87.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Garrett, M., and Willinger, W. (1994). Analysis, modeling and generation of self-similar VBR video traffic. In Proc. SIGCOMM '94. Communications, architechtures, protocols and applications (Computer Communication Review 24). ACM Press, New York, pp. 269–280.Google Scholar
Goldie, C. M. (1978). Subexponential distributions and dominated-variation tails. J. Appl. Prob. 15, 440–442.Google Scholar
Heath, D., Resnick, S., and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob, 7, 1021–1057.Google Scholar
Jelenkovic, P. R., and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on-off sources. To appear in Adv. Appl. Prob. 31.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Leland, W., Taqqu, M., Willinger, W., and Wilson, D. (1994). On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15.CrossRefGoogle Scholar
Likhanov, N., Tsybakov, B., and Georganas, N. D. (1995). Analysis of an ATM buffer with self-similar (fractal) input traffic. In Proc. INFOCOM'95. IEEE Computer Society, New York, pp. 985–992.Google Scholar
Liu, Z., Nain, P., Towsley, D., and Zhang, Z.-L. (1997). Asymptotic behavior of a multiplexer fed by a long-range dependent process. INRIA report No. 3230.Google Scholar
Neveu, J. (1979). Bases Mathématiques du Calcul des Probabilités. Masson & Cie, Paris.Google Scholar
Norros, I. (1994). A storage model with self-similar input. QUESTA 16, 387–396.Google Scholar
Pakes, A. G. (1975). On the tails of waiting time distributions. J. Appl. Prob. 12, 555–564.Google Scholar
Parulekar, M., and Makowski, A. M. (1996). Tail probabilities for a multiplexer with self-similar traffic. In Proc. INFOCOM'96. IEEE Computer Society, New York, pp. 1452–1459.Google Scholar
Parulekar, M., and Makowski, A. M. (1997). ail probabilities for M/G/∞ input processes (I): Preliminary asymptotics. QUESTA, 27, 271–296.Google Scholar
Parulekar, M., and Makowski, A. M. (1997). M/G/∞ input processes: A versatile class of models for network traffic. In Proc. INFOCOM'97 IEEE Computer Society, New York.Google Scholar
Paxson, V., and Floyd, S. (1993). Wide area traffic: The failure of Poisson modeling. IEEE/ACM Trans. on Networking 3, 226–244.Google Scholar
Rolski, T., Schelgel, S., and Schmidt, V. (1996). Asymptotics of Palm-stationary buffer content distributions in fluid flow queues. Adv. Appl. Prob. 31, 235–253.Google Scholar
Shiryayev, A. N. (1984). Probability. Springer, New York.Google Scholar
Takács, L. (1962). Theory of queues. OUP, New York.Google Scholar