Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-06T04:25:33.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic bounds for the fluid queue fed by sub-exponential On/Off sources

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

V. Dumas  and
A. Simonian
V. Dumas*
Affiliation:
MAB, Université Bordeaux I
A. Simonian*
Affiliation:
France Télécom CNET
*
∗ Postal address: MAB, Université Bordeaux I, 351, cours de la Libération, 33405 Talence Cedex, France.
∗∗ Postal address: France Télécom/CNET, 38/40 rue du Général Leclerc, 92794 Issy-les-Moulineaux Cedex 9, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a fluid queue fed by a superposition of a finite number of On/Off sources, the distribution of the On period being subexponential for some of them and exponential for the others. We provide general lower and upper bounds for the tail of the stationary buffer content distribution in terms of the so-called minimal subsets of sources. We then show that this tail decays at exponential or subexponential speed according as a certain parameter is smaller or larger than the ouput rate. If we replace the subexponential tails by regularly varying tails, the upper bound and the lower bound are sharp in that they differ only by a multiplicative factor.

Keywords

Fluid queueOn/Off sourcesbuffer contentstationaritytail-behavioursubexponential distributionsregular variationlong-range dependence

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 244 - 255
Copyright
Copyright © Applied Probability Trust 2000 

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] Agrawal, R., Makowski, A. M. and Nain, Ph. (1999). On a reduced load equivalence for fluid queues under subexponentiality. INRIA Rept No. 3466. QUESTA (Special Volume on Queues with Heavy-Tailed Distributions) 33, 5–41.Google Scholar
[2] Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
[3] Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at subexponential times with queueing applications. Stoch. Proc. Appl. 79, 265–286. Tail behaviour of M/G/1 queue via independent sampling. To appear in Stoch. Proc. Appl.Google Scholar
[4] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
[5] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory (Palm-Martingale Calculus and Stochastic Recurrences). Applications of Mathematics, No. 26, Springer, New York.Google Scholar
[6] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[7] Boxma, O. (1997). Regular variation in a muti-source fluid queue. In Teletraffic Contributions for the Information Age, eds Ramaswami, V. and Wirth, P. E.. ITC-15, North Holland, Amsterdam, pp. 391–402.Google Scholar
[8] Boxma, O. and Dumas, V. (1998). Fluid queues with long-tailed activity period distributions. CWI Rept PNA-R9705. To appear in Computer Commun. 21, 1509–1529.Google Scholar
[9] Brichet, F., Roberts, J., Simonian, A. and Vei-tch, D. (1996). Heavy traffic analysis of a storage model with long-range dependent on/off sources. Queueing System 23, 197–215.Google Scholar
[10] Chistyakov, V. (1964). A theorem on sums of independent, positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.CrossRefGoogle Scholar
[11] Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2nd edn, volume II. John Wiley, New York.Google Scholar
[12] Jelenković, P. and Lazar, A. (1999). Aymptotic results for multiplexing subexponential on/off sources. Adv. App. Prob. 31, 394–421.Google Scholar
[13] Kella, O. and Whitt, W. (1992). A storage model with a two-state random environment. Operat. Res. 40, S257–S262.CrossRefGoogle Scholar
[14] Kingman, J. (1970). Inequalities in the theory of queues. J. R. Statist. Soc., B 32, 102–110.CrossRefGoogle Scholar
[15] Klüppelberg, C., (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Rel. Fields 82, 259–269.Google Scholar
[16] Pakes, A. G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.CrossRefGoogle Scholar
[17] Roberts, J., Mocci, U. and Virtamo, J. (1996). Broadband Network Teletraffic: Performance Evaluation and Design of Broadband Multiservice Networks. Final rept Action COST 242. Springer, New York.Google Scholar
[18] Rolski, T,. Schlegel, S. and Schmidt, V. (1996). Asymptotics of palm-stationary buffer content distributions in fluid flow queues. Adv. Appl. Prob. 31, 235–253.Google Scholar