Abstract
We consider the long-range dependent cumulative traffic generated by the superposition of constant rate fluid sources having exponentially distributed inter start times and Pareto distributed durations with finite mean and infinite variance. We prove a sample path large deviation principle when the session start time intensity is increased and the processes are centered and scaled appropriately. Properties of the rate function are investigated. We derive a sample path large deviation principle for a related family of stationary queue length processes. The large deviation approximation of the steady-state queue length distribution is compared with the corresponding empirical distribution obtained by a computer simulation.
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References
R. Addie, P. Mannersalo and I. Norros, Most probable paths and performance formulae for buffers with Gaussian input traffic, {European Transactions on Telecommunications} 13(3) (2002) 183–196.
H. Bauer, Maß - und Integrationstheorie. (de Gruyter, Berlin, 1990).
J. Beran, Statistics for Long-Memory Processes (Chapman and Hall, New York, 1994).
C.-S. Chang, D.D. Yao, and T. Zajic, Large deviations, moderate deviations and queues with long-range dependent input, {Advances in Applied Probability} 31 (1999) 254–278.
C.-S. Chang, D.D. Yao, and T. Zajic, Large deviations, long-range dependence, and queues, in: Stochastic Modeling and Optimization, eds. D.D. Yao, H. Zhang and X.Y. Zhou (Springer-Verlag, New York, 2003) pp. 245–277.
D.R. Cox, Long-range dependence: A review, in: Statistics: An Appraisal; Proceedings 50th Anniversary Conference Iowa State Statistical Laboratory, eds. H.A. David and H.T. David (The Iowa State University Press, 1984) pp. 55–74.
A. de Acosta, Large deviations for vector valued L{é}vy processes, {Stochastic Processes and their Applications} 51 (1994) 75–115.
A. Dembo, S. Karlin and O. Zeitouni, Large exceedances for multidimensional Lévy processes, {The Annals of Applied Probability} 4(2) (1994) 432–447.
A. Dembo and T. Zajic, Large deviations: From empirical mean and measure to partial sums process, {Stochastic Processes and their Applications} 57 (1995) 191–224.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Jones and Bartlett, London, 1993).
J.-D. Deuschel and D.W. Stroock, Large Deviations (Academic Press, Inc., Boston, 1989).
N.G. Duffield, Queueing at large resources driven by long-tailed {M/G/∞}-modulated processes, {Queueing Systems} 28 (1998) 245–266.
A.J. Ganesh and N. O'Connell, A large deviation principle with queueing applications. {Stochastics and Stochastics Reports} 73(1–2) (2002) 25–35.
R.V. Hogg and S.A. Klugman, Loss Distributions (John Wiley & Sons, New York, 1984).
I. Kaj and M.S. Taqqu, Convergence to fractional Brownian motion and to the Telecom process: The integral representation approach. ‘Technical Report No. 2002:16, Uppsala University, Department of Mathematics’. July 2004.
J.F.C. Kingman, Poisson Processes. Number 3 in Oxford Studies in Probability (Clarendon Press, Oxford, 1993).
Yu. Kozachenko, O. Vasylyk and T. Sottinen, Path space large deviations of a large buffer with Gaussian input traffic, {Queueing Systems} 42 (2002) 113–129.
W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, On the self-similar nature of Ethernet traffic (extended version), {{IEEE/ACM} Transactions on Networking} 2(1) (1994) 1–15.
Z. Liu, P. Nain, D. Towsley and Z.-L. Zhang, Asymptotic behavior of a multiplexer fed by a long-range dependent process, {Journal of Applied Probability} 36 (1999) 105–118.
R.M. Loynes, The stability of a queue with non-independent inter-arrivals and service times, {Math. Proc. Cambridge Philos. Soc.} 58 (1962) 497–520.
K. Majewski, Large deviations for multi-dimensional reflected fractional Brownian motion, {Stochastics and Stochastics Reports} 75(4) (2003) 233–257. Corrigendum 76(5):479.
K. Majewski, Sample path moderate deviations for a family of long-range dependent traffic and associated queue length processes, {Stochastics} 77(1) (2005) 81–107.
K. Majewski, Minimizing large deviation paths for a family of long-range dependent processes and their fractional Brownian approximations. Submitted, January, 2005.
P. Mannersalo and I. Norros, A most probable path approach to queueing systems with general Gaussian input, {Computer Networks} 40(3) (2002) 399–411.
T. Mikosch, S. Resnick, H. Rootz{é}n and A. Stegeman, Is network traffic approximated by stable L{é}vy motion or fractional Brownian motion? {The Annals of Applied Probability} 12(1) (2002) 23–68.
A.A. Mogulskii, Large deviations for trajectories of multi-dimensional random walks, {Theory of Probability and its Applications} 21(2) (1976) 300–315.
I. Norros, Busy periods of fractional Brownian storage: A large deviations approach, {Advances in Performance Analysis} 2(1)(1999) 1–19.
E. Reich, On the integrodifferential equation of Tak{á}cs. I, {Annals of Mathematical Statistics} 29 (1958) 563–570.
S. Resnick and G. Samorodnitsky. Steady state distribution of the buffer content for {M/G/∞} input fluid queues, {Bernoulli} 7 (1999) 191–210.
M. Schilder, Some asymptotic formulas for Wiener integrals, {Transactions of the American Mathematical Society} 125 (1966) 63–85.
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MSC 2000 Classifications: Primary 60F10; Secondary 60K25, 68M20, 90B22
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Majewski, K. Sample path large deviations for a family of long-range dependent traffic and associated queue length processes. Queueing Syst 52, 105–118 (2006). https://doi.org/10.1007/s11134-006-4262-y
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DOI: https://doi.org/10.1007/s11134-006-4262-y