Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T04:23:02.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Bivariate characterization of some stochastic order relations

Published online by Cambridge University Press:  01 July 2016

J. George Shanthikumar  and
David D. Yao
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
David D. Yao*
Affiliation:
Columbia University
*
∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Bivariate (and multivariate) functional representations for the following stochastic order relations are established: the likelihood ratio ordering, hazard rate ordering, and the usual stochastic ordering. The motivation of the study is (i) to provide a general approach that supports the ‘pairwise interchange' arguments widely used in various settings, and (ii) to develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared. Applications are illustrated through problems in stochastic scheduling, closed queueing network and reliability.

Keywords

FUNCTIONAL REPRESENTATIONLIKELIHOOD RATIO ORDERINGHAZARD RATE ORDERINGSTOCHASTIC ORDERING
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 642 - 659
Copyright
Copyright © Applied Probability Trust 1991 

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 in part by NSF under grant ECS-8811234.

The major part of this research was undertaken while David D. Yao was affiliated with the division of Applied Sciences, Harvard University. He has also been supported in part by NSF under grant ECS-88013183, and by ONR under contract N00014-84-K-0465.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Reinhart and Winston, New York.Google Scholar
Brown, M. and Solomon, H. (1973) Optimal issuing policies under stochastic field lives. J. Appl. Prob. 10, 761–768.CrossRefGoogle Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing networks with exponential servers. Operat. Res. 15, 252–267.Google Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.Google Scholar
Karlin, S. (1957) Pólya-type distributions, II. Ann. Math. Statist. 28, 281–308.Google Scholar
Karlin, S. and Proschan, F. (1960) Pólya-type distributions of convolutions. Ann. Math. Statist. 31, 721–736.Google Scholar
Keilson, J. and Sumita, U. (1982) Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–198.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications . Academic Press, New York.Google Scholar
Pinedo, M. L. and Weber, R. R. (1984) Inequalities and bounds for stochastic shops. SIAM J. Appl. Math. 44, 869–879.Google Scholar
Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Shanthikumar, J. G., Yamazaki, G. and Sakasegawa, H. (1991) Characterization of optimal order of servers in a tandem queue with blocking. Operat. Res. Lett. 10, 17–22.Google Scholar
Shanthikumar, J. G. and Yao, D. D. (1986a) The preservation of likelihood ratio ordering under convolution. Stoch. Proc. Appl. 23, 259–267.CrossRefGoogle Scholar
Shanthikumar, J. G. and Yao, D. D. (1986b) The effect of increasing service rates in a closed queueing network. J. Appl. Prob. 23, 474–483.Google Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar