Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T07:44:48.618Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of palm-stationary buffer content distributions in fluid flow queues

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

Published online by Cambridge University Press:  01 July 2016

Tomasz Rolski ,
Sabine Schlegel  and
Volker Schmidt
Tomasz Rolski*
Affiliation:
Wroclaw University
Sabine Schlegel*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Mathematical Institute, Wroclaw University, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland.
∗∗ Postal address: Institute of Stochastics, University of Ulm, Helmholtzstraße 18, D-89069 Ulm, Germany.
∗∗ Postal address: Institute of Stochastics, University of Ulm, Helmholtzstraße 18, D-89069 Ulm, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a fluid flow queueing system with m independent sources alternating between periods of silence and activity; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying. We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. when at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotic representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996; 1997) who considered the special case of regular variation.

Keywords

Regular variationsubexponential distributionaggregatingtail asymptoticsrandom walkon-off fluid flow

MSC classification

Primary: 60K25: Queueing theory 60F10: Large deviations 68M20: Performance evaluation; queueing; scheduling 90B15: Network models, stochastic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 235 - 253
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Footnotes

Supported by KBN grant 2 PO3A 046 08 (1995-97).

References

Anantharam, V. (1995). On the sojourn time of sessions at an ATM buffer with longe-range dependent input traffic. In Proc. 34th IEEE CDC, Vol 1. IEEE Control Systems Society, New York, pp. 859–864.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. CUP, Cambridge, UK.CrossRefGoogle Scholar
Boxma, O. J. (1996). Fluid queues and regular variation. Performance Evaluation 27/28, 699–712.CrossRefGoogle Scholar
Boxma, O. J. (1997). Regular variation in a multi-source fluid queue. In Teletraffic Contributions for the Information Age (Proc. ITC 15), ed. Ramaswami, V. and Wirth, P. E. Elsevier, Amsterdam, pp. 391–402 Google Scholar
Choudhury, G. L. and Whitt, W. (1997). Long-tail buffer-content distributions in broadband networks. Performance Evaluation 30, 177–190.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Rel. Fields 72, 529–557.Google Scholar
Cline, D. B. H. (1994). Intermediate regular and Π variation. Proc. London Math. Soc. 68, 594–616.Google Scholar
Cohen, J. W. (1976). On Regenerative Processes in Queueing Theory. Lecture Notes in Econ. Math. Syst. 121, Springer, Berlin.Google Scholar
Cohen, J. W. (1994). On the effective bandwidth in buffer design for the multi-server channels. Report BS-R9406. CWI Amsterdam.Google Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Proc. Appl. 13, 263–278.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1996a). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Operat. Res. 23, 145–165.Google Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1996b). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob. 4, 1021–1057.Google Scholar
Heyman, D.P. and Lakshman, T. V. (1996). Source models for VBR broadcast-video traffic. IEEE J. Sel. Areas Commun. 4, 40–48.Google Scholar
Iosifescu, M. (1980). Finite Markov Chains and Their Applications. Wiley, New York.Google Scholar
Jelenković, P. R. and Lazar, A. A. (1996). Asymptotic results for multiplexing subexponential on-off sources. Technical Report 457-96-23. Columbia University, New York.Google Scholar
Pakes, A. G. (1975). On the tails of waiting-time distribution. J. Appl. Prob. 12, 555–564.Google Scholar
Palmowski, Z. and Rolski, T. (1996). A note on martingale inequalities for fluid models. Statist. Prob. Lett. 31, 13–21.Google Scholar
Pitman, E. J. G. (1980). Subexponential distribution functions. J. Austral. Math. Soc. (A) 29, 337–347.Google Scholar
Whitt, W. (1993). Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommun. Systems 2, 71–107.Google Scholar