Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-12T19:31:28.607Z Has data issue: false hasContentIssue false

On an elementary characterization of the increasing convex ordering, by an application

Published online by Cambridge University Press:  14 July 2016

Armand M. Makowski*
Affiliation:
University of Maryland, College Park
*
Postal address: Electrical Engineering Department and Institute for Systems Research, University of Maryland, College Park, MD 20742.

Abstract

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

MSC classification

Type
Short Communications
Copyright
Copyright © Applied Probability Trust 1994 

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

The work of this author was supported through NSF Grants NSFD CDR-88-03012 and ASC-88-02764.

References

[1] Baccelli, F. and Makowski, A. M. (1989) Queueing models for systems with synchronization constraints. Proc. IEEE 77, Special Issue on Dynamics of Discrete Event Systems , 138161.Google Scholar
[2] Chang, C.-S. (1994) Smoothing point processes as a means to increase throughput. Operat. Res. Google Scholar
[3] Daley, D. J. and Rolski, T. (1984) Some comparison results for waiting times in single- and many-server queues. J. Appl. Prob. 21, 887900.CrossRefGoogle Scholar
[4] Karamata, J. (1932) Sur une inégalité relative aux fonctions convexes. Publ. Math. Univ. Belgrade 1, 145148.Google Scholar
[6] Kiefer, J. and Wolfowitz, A. (1956) On the characteristics of the general queueing process, with applications to random walks. Ann. Math. Statist. 27, 147161.CrossRefGoogle Scholar
[5] Kleinrock, L. (1976) Queueing Systems I: Theory. Wiley, New York.Google Scholar
[7] Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
[8] Rolski, T. and Stoyan, D. (1976) On the comparison of waiting times in GI/G/1 queues. Operat. Res. 24, 197200.CrossRefGoogle Scholar
[9] Ross, S. (1984) Stochastic Processes. Wiley, New York.Google Scholar
[10] Rüschendorf, L. (1981) Ordering of distributions and rearrangements of functions. Ann. Prob. 9, 276283.Google Scholar
[11] Stoyan, D. (1984) Comparison Methods for Queues and Other Stochastic Models, English translation ed. Daley, D. J., Wiley, New York.Google Scholar
[12] Strassen, V. (1965) The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423439.Google Scholar