Abstract
This article reviews various models within the queueing framework which have been suggested for teletraffic data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on- and off-periods and long-range dependence in teletraffic transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) influence within teletraffic models. We demonstrate some of the above effects in an explorative data analysis of Munich Universities’ intranet data.
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References
S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).
S. Asmussen, Subexponential asymptotics for stochastic processes: Extremal behaviour, stationary distributions and first passage probabilities, Ann. Appl. Probab. 8 (1998) 354–374.
S. Asmussen, L. Fløe Henriksen and C. Kl¨ uppelberg, Large claims approximations for risk processes in a Markovian environment, Stochastic Process. Appl. 54 (1994) 29–43.
S. Asmussen and C. Kl¨ uppelberg, Stationary M=G=1 excursions in the presence of heavy tails, J. Appl. Probab. 34 (1997) 208–212.
S. Asmussen, C. Kl¨ uppelberg and K. Sigman, Sampling at subexponential times, with queueing applications, Stochastic Process. Appl. 79 (1999) 265–286.
S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps, Adv. in Appl. Probab. 31 (1999), to appear.
J. Beran, R. Sherman, M.S. Taqqu and W. Willinger, Long-range dependence in variable-bit-rate video traffic, IEEE Trans. Commun. 43 (1995) 1566–1579.
N.H. Bingham, C.M. Goldie and J. Teugels, Regular Variation(Oxford Univ. Press, 1987).
A.A. Borovov, Stochastic Processes in Queueing Theory (Springer, New York, 1976).
O.J. Boxma, Fluid queues and regular variation, Performance Evaluation 29 (1996) 699–712.
G.L. Choudhury and W. Whitt, Long-tail buffer-content distributions in broadband networks, Per-formance Evaluation 30 (1997) 177–190.
J.W. Cohen, Some results on regular variation for distributions in queueing and fluctuation theory, J. Appl. Probab. 10 (1973) 343–353.
P. Embrechts, C. Kl ¨ uppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997).
P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics 1 (1982) 55–72.
A. Feldmann and W. Whitt, Fitting mixtures of exponentials to long-tailed distributions to analyze netwok performance models, Performance Evaluation 31 (1998) 247–278.
W. Feller, An Introduction to Probability Theory and its Applications, Vol.II (Wiley, New York, 1971).
R. Garg, Characterization of video traffic, Technical Report, International Computer Science Insti-tute, Berkeley, CA (1995).
M.W. Garret and W. Willinger, Analysis, modeling and generation of self-similar VBR video traffic, in:ACM SigComm '94 (ACM, London, September 1994).
H. Gogl, Data acquisition and data analysis in ATM based networks, Technical Report, De-partment of Computer Science, Munich University of Technology, Germany (1998). http:// www.jessen.informatik.tu-muenchen.de/forschung/leistung/ATM Project/ atm index.html.
C.M. Goldie and C. Kl ¨ uppelberg, Subexponential distributions, in: A Practical Guide To Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions,eds.R. Adler, R. Feldman and M.S. Taqqu (Birkh¨ auser, Boston, 1998) pp. 435–459.
M. Greiner, M. Jobmann and L. Lipsky, The importance of power-tail distributions for modeling queueing systems, Oper. Res. 47 (1999) 313–326.
D. Heath, S. Resnick and G. Samorodnitsky, Patterns of buffer overflow in a class of queues with long memory in the input stream, Ann. Appl. Probab. 7 (1997) 1021–1057.
D. Heath, S. Resnick and G. Samorodnitsky, Heavy tails and long range dependence in on/off processes and associated fluid models, Math. Oper. Res. 23 (1998) 145–165.
P.R. Jelenkovi´ c and A.A. Lazar, A network multiplexer with multiple time scale and subexponential arrivals, in: Stochastic Networks, Stability and Rare Events, Lecture Notes in Statistics (Springer, New York, 1996) pp. 215–235.
P.R. Jelenkovi´ c and A.A. Lazar, Subexponential asymptotics of a Markov-modulated random walk with queueing applications, J. Appl. Probab. 35 (1998) 325–347.
P.R. Jelenkovi´ c and A.A. Lazar, Asymptotic results for multiplexing subexponential on-off sources, Adv. in Appl. Probab. 31 (1999) to appear.
C. Kl¨ uppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988) 132–141.
C. Kl¨ uppelberg and T. Mikosch, Delay in claim settlement and ruin probability approximations, Scand. Actuar. J. (1995) 154–168.
C. Kl¨ uppelberg and T. Mikosch, Explosive Poisson shot noise processes with applications to risk reserves, Bernoulli 1 (1995) 125–147.
C. Kl¨ uppelberg and A. Sch¨ arf, Stable limits of Poisson shot noise processes, preprint (1999). Center for Mathematical Science, M¨ unich University of Technology, Germany. http://www.ma.tum. de/stat/Papers/papers.html.
T. Kurtz, Limit theorems for workload input models, in: Stochastic Networks: Theory and Appli-cations, eds. F.P. Kelly, S. Zachary and I. Ziedins, Royal Statistical Society Lecture Note Series, Vol. 4 (1996).
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(1) (1994) 1–15.
T. Lindvall, Lectures of the Coupling Method (Wiley, New York, 1992).
L. Lipsky, Queueing Theory: A Linear Algebraic Approach (MacMillan, New York, 1992).
B.B. Mandelbrot, Self-similar error clusters in communication systems and the concept of condi-tional stationarity, IEEE Trans. Commun. Technology 13 (1965) 71–90.
M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models.An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences (Johns Hopkins Univ. Press, Baltimore, MD, 1981).
A.G. Pakes, On the tails of waiting time distributions, J. Appl. Probab. 12 (1975) 555–564.
K. Park, G. Kim and M. Crovella, On the relationship between file sizes, transport protocols, and self-similar network traffic, Technical Report BU-CS-96–016, Computer Science Department, Boston University (1996).
K. Park, G. Kim and M. Crovella, On the effect of traffic self-similarity on network performance, in: Proc.of the 1997 SPIE Internat.Conf.on Performance and Control of Network Systems (1997).
V. Paxson and S. Floyd, Wide area traffic: The failure of Poisson modeling, IEEE/ACM Trans. Networking 3(3) (1995) 226–244.
S.I. Resnick, Adventures in Stochastic Processes (Birkh¨ auser, Boston, 1992).
S.I. Resnick, Heavy tail modeling in teletraffic data, Ann. Statist. 25 (1997) 1805–1848.
S.I. Resnick and G. Samorodnitsky, Performance decay in a single server exponential queueing model with long range dependence, Oper. Res. 45 (1997) 235–243.
S.I. Resnick and G. Samorodnitsky, Fluid queues, leaky buckets, on-off processes and teletraffic modeling with highly variable and correlated inputs, to appear in: Selfsimilar Network Traffic and Performance Evaluation, eds. K. Park and W. Willinger (Wiley, New York).
G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes (Chapman and Hall, New York, 1994).
K. Sigman, Appendix: A primer on heavy-tailed distributions, Queueing Systems (this issue) 33 (1999) 261–275.
W.L. Smith, On the tails of queueing time distributions, Technical Report, Mimeo Series No. 830,Department of Statistics, University of North Carolina, Chapel Hill (1972).
M.S. Taqqu and J. Levy, Using renewal processes to generate long-range dependence and high vari-ability, in: Dependence in Probability and Statistics, eds. E. Eberlein and M.S. Taqqu (Birkh¨ auser, Boston, 1986) pp. 73–89.
M.S. Taqqu, W. Willinger and R. Sherman, Proof of a fundamental result in self-similar traffic modeling, Computer Commun. Rev. 27 (1997) 5–23.
W. Willinger, M.S. Taqqu, R. Sherman and D.V. Wilson, Self-similarity through high-variability: Statistical analysis of Ethernet LAN traffic at the source level, IEEE/ACM Trans. Networking 5(1) (1997) 71–86.
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Greiner, M., Jobmann, M. & Klüppelberg, C. Telecommunication traffic, queueing models, and subexponential distributions. Queueing Systems 33, 125–152 (1999). https://doi.org/10.1023/A:1019120011478
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DOI: https://doi.org/10.1023/A:1019120011478