Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T18:18:49.883Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic convexity and its applications

Published online by Cambridge University Press:  01 July 2016

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked*
Affiliation:
University of Arizona
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Mathematics, Building 89, The University of Arizona, Tucson, AZ 85721, USA.
∗∗ Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady-state waiting time in a GI/D/c queue, and (as a function of the arrival or service rates) in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case (as a function of the arrival rate) is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff in the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In non-parametric statistics, the stochastic concavity (or convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

Keywords

STOCHASTIC MONOTONICITYCLASSES OF DISTRIBUTIONS WITH THE SEMIGROUP PROPERTYPOISSON PROCESSQUEUESPROPORTIONAL HAZARDIMPERFECT REPAIRBRANCHING PROCESSESEMPIRICAL DISTRIBUTIONSCONVEX PARAMETRIZATIONMAJORIZATION AND SCHUR-CONVEXITYRELIABILITY THEORYCUMULATIVE DAMAGE SHOCK MODELS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 427 - 446
Copyright
Copyright © Applied Probability Trust 1988 

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 the Air Force Office of Scientific Research, USAF, under Grant AFOSR-84–0205. Reproduction in whole or in part is permitted for any purpose of the United States Government.

References

Berg, M. and Cleroux, R. (1982) A marginal cost analysis for an age replacement policy with minimal repair. Infor 20, 258–263.Google Scholar
Brown, M. and Proschan, F. (1983) Imperfect repair. J. Appl. Prob. 20, 851–859.Google Scholar
Cambanis, S. and Simons, G. (1982) Probability and expectation inequalities. Z. Wahrscheinlichkeitsth. 59, 1–25.Google Scholar
Cleroux, R., Dubuc, S. and Tilquin, C. (1979) The age replacement problem with minimal repair and random repair costs. Operat. Res. 27, 1158–1167.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Grassmann, W. (1983) The convexity of the mean queue size of the M/M/c queue with respect to the traffic intensity. J. Appl. Prob. 20, 916–919.Google Scholar
Harel, A. and Zipkin, P. (1987) Strong convexity results for queueing systems. Operat. Res. 35, 405–418.Google Scholar
Jagers, A. A. and Van Doorn, E. A. (1986) On the continued Erlang loss functions. Operat. Res. Lett. 5, 43–46.Google Scholar
Keilson, J. (1979) Markov Chain Models-Rarity and Exponentiality. Springer-Verlag, New York.CrossRefGoogle Scholar
Lee, H. L. and Cohen, M. A. (1983) A note on the convexity of performance measures of M/M/c queueing systems. J. Appl. Prob. 20, 920–923.Google Scholar
Lindley, D. V. (1952) The theory of queues with single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Rolfe, A. J. (1971) A note on marginal allocation in multiple-server service systems. Management Sci. 17, 656–658.Google Scholar
Schweder, T. (1982) On the dispersion of mixtures. Scand. J. Statist. 9, 165–169.Google Scholar
Shaked, M. (1980) On mixtures from exponential families. J. R. Statist. Soc. B 42, 192–198.Google Scholar
Shanthikumar, J. G. (1987) Stochastic majorization of random variables with proportional equilibrium rates. Adv. Appl. Prob. 19, 854–872.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
Tu, H. Y. and Kumin, H. (1983) A convexity result for a class of GI/G/1 queueing systems. Operat. Res. 31, 948–950.CrossRefGoogle Scholar
Weber, R. R. (1980) On the marginal benefit of adding servers to G/GI/m queues. Management Sci. 26, 946–951.Google Scholar
Weber, R. R. (1983) A note on waiting times in single server queues. Operat. Res. 31, 950–951.Google Scholar