Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T11:59:48.322Z Has data issue: false hasContentIssue false
  • English
  • Français

WHEN ARE ON–OFF SOURCES SIS?: CONDITIONS AND APPLICATIONS

Published online by Cambridge University Press:  01 October 2004

Sarut Vanichpun  and
Armand M. Makowski
Sarut Vanichpun
Affiliation:
Department of Electrical and Computer Engineering, Institute for Systems Research, University of Maryland, College Park, MD 20742, E-mail: [email protected]
Armand M. Makowski
Affiliation:
Department of Electrical and Computer Engineering, Institute for Systems Research, University of Maryland, College Park, MD 20742, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent advances from the theory of multivariate stochastic orderings can be used to formalize the “folk theorem” that positive correlations lead to larger buffer levels at a discrete-time infinite capacity multiplexer queue. In particular, it is known that if the input traffic is larger than its independent version in the supermodular (sm) ordering, then their corresponding buffer contents are similarly ordered in the increasing convex (icx) ordering.

A sufficient condition for the aforementioned sm comparison is the stochastic increasingness in sequence (SIS) property of the input traffic. In this article, we provide conditions for the stationary on–off source to be SIS. We then use this result to find conditions under which the superposition of independent on–off sources and the M|G|∞ input model are each sm greater than their respective independent version. Similar but weaker SIS conditions are also obtained for renewal on–off processes.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 4 , October 2004 , pp. 423 - 443
Copyright
© 2004 Cambridge University Press

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

REFERENCES

Agrawal, R., Makowski, A.M., & Nain, Ph. (1999). On a reduced load equivalence for fluid queues under subexponentiality. Queueing Systems: Theory & Applications 33: 5–41.Google Scholar
Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. International Series in Decision Processes. New York: Holt, Rinehart and Winston.
Bäuerle, N. (1997). Monotonicity results for MR|GI|1 queues. Journal of Applied Probability 34: 514–524.Google Scholar
Bäuerle, N. & Rolski, T. (1998). A monotonicity result for the workload in Markov-modulated queues. Journal of Applied Probability 35: 741–747.Google Scholar
Duffield, N.G. & O'Connell, N. (1995). Large deviations and overflow probabilities for the general single server queue, with applications. Proceedings of the Cambridge Philosophical Society 118: 363–374.Google Scholar
Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics 38: 1466–1474.Google Scholar
Glynn, P.W. & Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. Journal of Applied Probability 31: 131–159.Google Scholar
Likhanov, N., Tsybakov, B., & Georganas, N.D. (1995). Analysis of an ATM buffer with self-similar (“fractal”) input traffic. In Proceedings of IEEE INFOCOM 1995, pp. 985–992.
Livny, M., Melamed, B., & Tsiolis, A.K. (1993). The impact of autocorrelation on queuing systems. Management Science 39: 322–339.Google Scholar
Makowski, A.M. & Parulekar, M. (2000). Buffer asymptotics for M|G|∞ input processes. In K. Park & W. Willinger (eds.), Self-similar network traffic and performance evaluation. New York: Wiley, pp. 215–248.
Meester, L.E. & Shanthikumar, J.G. (1993). Regularity of stochastic processes: A theory of directional convexity. Probability in the Engineering and Informational Sciences 7: 343–360.Google Scholar
Müller, A. & Scarsini, M. (2000). Some remarks on the supermodular order. Journal of Multivariate Analysis 73: 107–119.Google Scholar
Parulekar, M. & Makowski, A.M. (1997). Tail probabilities for a multiplexer driven by M|G|∞ input processes (I): Preliminary asymptotics. Queueing Systems: Theory & Applications 27: 271–296.Google Scholar
Resnick, S. & Samorodnitsky, G. (2000). Fluid queues, on/off processes, and teletraffic modeling with highly variable and correlated inputs. In K. Park & W. Willinger (eds.), Self-similar network traffic and performance evaluation. New York: Wiley, pp. 171–192.
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. New York: Academic Press.
Shaked, M. & Shanthikumar, J.G. (1997). Supermodular stochastic orders and positive dependence of random vectors. Journal of Multivarate Analysis 61: 86–101.Google Scholar
Stoyan, D. (1984). Comparison methods for queues and other stochastic models. English Translation: D.J. Daley, ed. New York: Wiley.
Szekli, R. (1995). Stochastic ordering and dependence in applied probability. Lecture Notes in Statistics Vol. 97. New York: Springer-Verlag.
Toyoizumi, H., Shanthikumar, J.G., & Wolff, R.W. (1997). Two extremal autocorrelated arrival processes. Probability in the Engineering and Informational Sciences 11: 441–450.Google Scholar
Vanichpun, S. (2001). The effects of positive correlations on buffer occupancy: Comparisons and lower bounds via supermodular ordering. M.S. thesis, Department of Electrical and Computer Engineering, University of Maryland, College Park, MD. Also available as ISR Technical Report No. 2001-4, Institute for Systems Research, University of Maryland, College Park, MD.
Vanichpun, S. & Makowski, A.M. (2002). The effects of positive correlations on buffer occupancy: Lower bounds via supermodular ordering. In Proceedings of IEEE INFOCOM 2002, pp. 1298–1306.