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Convex majorization with an application to the length of critical paths

Published online by Cambridge University Press:  14 July 2016

Isaac Meilijson  and
Arthur Nádas
Isaac Meilijson*
Affiliation:
Tel-Aviv University
Arthur Nádas*
Affiliation:
IBM System Products Division
*
∗ Postal address: Department of Statistics, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel.
∗∗ Postal address: IBM System Products Division, Hopewell Junction, NY 12575, U.S.A.
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Abstract

  1. 1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .

  2. 2. Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.

  3. 3. Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.

Keywords

CONVEX AND STOCHASTIC INEQUALITYHARDY–LITTLEWOOD MAXIMAL FUNCTIONPERT NETWORK
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 671 - 677
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research carried out while this author was Visiting Scientist at the IBM Thomas J. Watson Research Center, Yorktown Heights.

References

[1] Dubins, L. E. and Meilijson, I. (1974) On stability for optimization problems. Ann. Prob. 2, 243–255.Google Scholar
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[3] Hardy, G. H. and Littlewood, J. E. (1930) A maximal theorem with function–theoretic applications. Acta Math. 54, 81–116.Google Scholar
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[6] Robillard, P. and Trahan, M. (1977) The completion time of PERT networks. Opns Res. 25, 15–29.Google Scholar