Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T17:47:04.024Z Has data issue: false hasContentIssue false
  • English
  • Français

On Maxima and Minima of Partial Sums of Strongly Interchangeable Random Variables

Published online by Cambridge University Press:  27 July 2009

Teunis J. Ott  and
J. George Shanthikumar
Teunis J. Ott
Affiliation:
Belicore Morristown, New Jersey 07960
J. George Shanthikumar
Affiliation:
University of California at Berkeley, Berkeley, California 94720
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the concept of “strong interchangeability” of random vectors. Strongly interchangeable random vectors arise naturally in packetized voice channels, M/G/1 queues, symmetric queueing networks, and other standard symmetric distributions. We study some properties of strongly interchangeable random vectors. We show that if (X1, …, XN) is a strongly interchangeable random vector, then even though there is no Markov property, taboo probabilities can be used to compute the joint distribution of ŽN = min1≤n≤N σnk=IXk and ZN = max1≤n≤N σnk=1Xk. For a special instance of this problem that arises in packetized voice communication, it is shown that the resulting algorithm essentially has a complexity of order N4. When ( σnk=1Xk, n = 1,… N) is an associated random vector bound for the joint distribution of ŽN and ZN are obtained and applied to the packetized voice communication problem.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 4 , Issue 3 , July 1990 , pp. 319 - 332
Copyright
Copyright © Cambridge University Press 1990

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

Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing–Probability models. New York: Holt, Rinehart and Winston.Google Scholar
Block, H., Savits, T. & Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10: 765–772.CrossRefGoogle Scholar
Block, H., Savits, T. & Shaked, M. (1985). A concept of negative dependence using stochastic ordering. Statist. Probab. Lett. 3: 81–86.CrossRefGoogle Scholar
Efron, B. (1965). Increasing properties of Polya frequency functions. Ann. Math. Stat. 36: 272–279.CrossRefGoogle Scholar
Kamae, T., Krengel, U. & O'Brien, G. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5: 899–912.CrossRefGoogle Scholar
Latouche, G. (1989). A study of deterministic cycles in packet queues subject to periodic traffic. Technical report, Université Libre de Bruxelles, Bruxelles, Belgium.Google Scholar
Ott, T.J. & Shanthikumar, J.G. (1989). On a buffer problem for packetized voice with an N-periodic interchangeable input process. Technical report, Bellcore, Morristown, N.J.Google Scholar
Ott, T.J. & Shanthikumar, J.G. (1989). Structural properties and stochastic bounds for a buffer problem in packetized voice transmission. Technical report, Bellcore, Morristown, N.J.Google Scholar
Ramamurthy, G. & Sengupta, B. (1989). Delay analysis of a packet voice multiplexer by the D1/D/l queue. Technical report, AT&T Bell Laboratories.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1988). Parametric stochastic convexity and concavity of stochastic processes. Ann. Inst. Stat. Math. (to appear).Google Scholar
Sriram, K. & Whitt, W. (1986). Characterizing superposition arrival processes in packet multiplexers for voice and data. IEEE J. SAC. 4: 833–846.Google Scholar
Tchen, A.H. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8: 814–827.CrossRefGoogle Scholar