Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T17:30:02.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Economies of scale in queues with sources having power-law large deviation scalings

Part of: Stochastic processes Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

N. G. Duffield
N. G. Duffield*
Affiliation:
AT&T Laboratories
*
∗Postal address: AT&T Laboratories, Room 2D-113, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[Wt/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions limL→xL–1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function limb→xb–u/a(I(b) – δbv/a) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

Keywords

LARGE DEVIATIONSSCALING LIMITSATM MULTIPLEXERSFRACTIONAL BROWNIAN MOTIONEFFECTIVE BANDWIDTH APPROXIMATION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F10: Large deviations 60G18: Self-similar processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 840 - 857
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Borell, C. (1975) The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 205–216.CrossRefGoogle Scholar
[2] Borovkov, A. A. (1984) Asymptotic Methods in Queueing Theory. Wiley, Chichester.Google Scholar
[3] Botvich, D. D. and Duffield, N. G. (1995) Large deviations, the shape of the loss curve, and economies of scale in large multiplexers. Queueing Systems. 20, 293–320.Google Scholar
[4] Buffet, E. and Duffield, N. G. (1994) Exponential upper bounds via martingales for multiplexers with Markovian arrivals. J. Appl. Prob. 31, 1049–1061.CrossRefGoogle Scholar
[5] Chang, C. S. (1994) Stability, queue length and delay of deterministic and stochastic queueing networks. IEEE Trans. Automatic Control. 39, 913–931.CrossRefGoogle Scholar
[6] Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1996) Squeezing the most out of ATM. IEEE Trans. Commun. 44, 203–217.Google Scholar
[7] Courcoubetis, C. and Weber, R. (1996) Buffer overflow asymptotics for a buffer handling many traffic sources. J. Appl. Prob. 33, 886–903.CrossRefGoogle Scholar
[8] Dembo, A. and Zeitouni, O. (1993) Large Deviation Techniques and Applications. Jones and Bartlett, Boston.Google Scholar
[9] 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
[10] Ellis, R. S. (1985) Entropy, Large Deviations, and Statistical Mechanics , Springer. New York.CrossRefGoogle Scholar
[11] Gibbens, R. J. and Hunt, P. J. (1991) Effective bandwidths for the multi-type UAS channel. Queueing Systems 9, 17–28.Google Scholar
[12] Glynn, P. W. and Whitt, W. (1993) Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Prob. 31A, 131–159.Google Scholar
[13] Hui, J. Y. (1988) Resource allocation for broadband networks. IEEE J. Selected Areas in Commun. 6, 1598–1608.CrossRefGoogle Scholar
[14] Kelly, F. P. (1991) Effective bandwidths at multi-type queues. Queueing Systems 9, 5–16.CrossRefGoogle Scholar
[15] Kesidis, G., Walrand, J. and Chang, C. S. (1993) Effective bandwidths for multiclass Markov fluids and other ATM sources. IEEE/ACM Trans. Networking 1, 424–428.CrossRefGoogle Scholar
[16] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993) On the self-similar nature of Ethernet traffic. ACM SIGCOMM Comput. Commun. Rev. 23, 183–193.CrossRefGoogle Scholar
[17] Lewis, J. T. and Pfister, C.-E. (1995) Thermodynamic probability theory: some aspects of large deviations. Russian Math. Surveys 50, 279–317.CrossRefGoogle Scholar
[18] Mandelbrot, B. and Van Ness, J. W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.CrossRefGoogle Scholar
[19] Norros, I. (1994) A storage model with self-similar input. Queueing Systems 16, 387–396.CrossRefGoogle Scholar
[20] Rockafellar, R. T. (1970) Convex Analysis. Princeton University Press, Princeton.Google Scholar
[21] Samorodnitsky, G. and Taqqu, M. S. (1993) Linear models with long-range dependence and with finite or infinite variance. In New Directions in Time-Series Analysis, part II. (IMA Vol. Math. Appl. 46, 325–340.) Springer, New York.Google Scholar
[22] Simonian, A. and Guibert, J. (1994) Large deviations approximation for fluid queues fed by a large number of on-off sources. Proc. ITC 14, Antibes, 1994. pp. 1013–1022.CrossRefGoogle Scholar
[23] Weiss, A. (1986) A new technique for analysing large traffic systems. J. Appl. Prob. 18, 506–532.Google Scholar
[24] Whitt, W. (1993) Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommun. Syst. 2, 71–107.Google Scholar