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Loss asymptotics in large buffers fed by heterogeneous long-tailed sources

Part of: Special processes Operations research and management science Computer system organization

Published online by Cambridge University Press:  01 July 2016

Nikolay Likhanov  and
Ravi R. Mazumdar
Nikolay Likhanov*
Affiliation:
Institute for Problems of Information Transmission, Moscow
Ravi R. Mazumdar*
Affiliation:
Purdue University
*
∗ Postal address: Institute for Problems of Information Transmission, 19 Bolshoi Karetnyi, GSP-4, Moscow 101447, Russia.
∗∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA. Email address: [email protected]
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Abstract

This paper presents the large-buffer asymptotics for a multiplexer which serves N types of heterogeneous sessions which have long-tailed session lengths. Specifically, the model considered is that sessions of type i ∈ {1,…,N} arrive as a Poisson process with rate λi. Each type of session (independently) remains active for a random duration, say τi, where P(τi > x) ~ αix-(1 + βi) for positive numbers αi and βi. While active, a session transmits at a rate ri. Under the assumption that the average load ρ = ∑Ni=1riλiE[τi] < C, where C denotes the server capacity, we show that both the tail distribution of the stationary buffer content and the loss asymptotics in finite buffers of size z behave approximately as z-κJ0, where κJ0 depends not only on the βi but also on the transmission rates ri; it is the ratio of βi to ri which determines κJ0. When specialized to the homogeneous case, i.e., when ri=r and βi = β for all i, the result coincides with results reported in the literature which have been shown under more restrictive hypotheses. Finally, it is a simple observation that light-tailed sessions only have the effect of reducing the available capacity for long-tailed sessions, but do not contribute otherwise to the definition of κJ0.

Keywords

Fluid queueslong-tailed inputsstationaryasymptotics

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1168 - 1189
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
[2] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory (Applications of Math. 26). Springer, New York.Google Scholar
[3] Botvich, A. and Duffield, N. G. (1995). Large deviations, economies of scale and the shape of the loss curve in large multiplexers. Queueing Systems 20, 293–320.Google Scholar
[4] Boxma, O. J. and Dumas, V. (1998). Fluid queues with long-tailed activity period distributions. Comput. Commun. 21, 1509–1529.Google Scholar
[5] Choe, J. and Shroff, N. B. (1999). On the supremum distribution of integrated stationary Gaussian processes with negative linear drift. Adv. Appl. Prob. 31, 135–157.Google Scholar
[6] Courcoubetis, C. and Weber, R. R. (1996). Buffer overflow asymptotics for a switch handling many traffic sources. J. Appl. Prob. 33, 886–903.CrossRefGoogle Scholar
[7] Duffield, N. G. (1996). Economies of scale in queues with sources having power-law large deviation scalings. J. Appl. Prob. 33, 840–857.Google Scholar
[8] Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue with applications. Math. Proc. Camb. Phil. Soc. 118, 363–374.Google Scholar
[9] Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York.Google Scholar
[10] Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long-range dependence in on/off processes and associated fluid models. Math. Operat. Res. 23, 145–165.Google Scholar
[11] Jelenkovic, P. R. and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on–off sources. Adv. Appl. Prob. 31, 394–421.CrossRefGoogle Scholar
[12] Kelly, F. P. (1996). Notes on effective bandwidths. In Stochastic Networks, eds Kelly, F. P., Zachary, S. and Ziedins, I.. Oxford University Press, pp. 141–168.Google Scholar
[13] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15.Google Scholar
[14] Likhanov, N. (2000). Bounds on buffer occupancy probability with self-similar input traffic. In Self-Similar Network Traffic and Performance Evaluation, eds Park, K. and Willinger, W.. John Wiley, New York, pp. 193–213.Google Scholar
[15] Likhanov, N. and Mazumdar, R. R. (1999). Cell loss asymptotics in buffers fed with a large number of independent stationary sources. J. Appl. Prob. 36, 86–96.Google Scholar
[16] Liu, Z., Nain, P., Towsley, D. and Zhang, Z-L. (1999). Asymptotic behavior of a multiplexer fed by long-range dependent process. J. Appl. Prob. 36, 105–118.Google Scholar
[17] Norros, I. (1994). A storage model with self-similar input. Queueing Systems 16, 387–396.Google Scholar
[18] Parulekar, M. and Makowski, A. M. (1997). Tail probabilities for M/G/∞ input processes. Queueing Systems 27, 271–296.CrossRefGoogle Scholar
[19] Paxon, V. and Floyd, S. (1993). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Networking 3, 226–244.Google Scholar
[20] Rainer, C. and Mazumdar, R. R. (1998). A note on the conservation law for continuous reflected processes and its application to queues with fluid inputs. Queueing Systems 28, 283–291.Google Scholar
[21] Resnick, R. and Samorodnitsky, G. (1999). Activity periods of an infinite server queue and performance of certain heavytailed fluid queues. Queueing Systems 33, 43–71.CrossRefGoogle Scholar
[22] Simonian, A. and Guibert, J. (1995). Large deviations approximation for fluid queues fed by a large number of on/off sources. IEEE J. Sel. Areas Commun. 13, 1017–1027.CrossRefGoogle Scholar