Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T08:23:34.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal selection of stochastic intervals under a sum constraint

Published online by Cambridge University Press:  01 July 2016

E. G. Coffman Jr. ,
L. Flatto  and
R. R. Weber
E. G. Coffman Jr.*
Affiliation:
AT &T Bell Laboratories
L. Flatto*
Affiliation:
AT & T Bell Laboratories
R. R. Weber*
Affiliation:
University of Cambridge
*
∗Postal address: AT & T Bell Laboratories Murray Hill, New Jersey 07974, USA.
∗Postal address: AT & T Bell Laboratories Murray Hill, New Jersey 07974, USA.
∗∗ Postal address: Queen&s College, Cambridge, CB3 9ET, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.

We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

Keywords

DECISION RULEON-LINE DECISIONSELECTION POLICY
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 454 - 473
Copyright
Copyright © Applied Probability Trust 1987 

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

1. Bennett, G. (1962) Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57, 33–45.Google Scholar
2. Bruno, J. L. and Downey, P. J. (1986) Probabilistic bounds for dual bin-packing. Acta Informatica. Google Scholar
3. Campbell, G. and Samuels, S. M. (1981) Choosing the best of the current crop. Adv. Appl. Prob. 15, 510–532.Google Scholar
4. Chen, R. W., Nair, V. N., Odlyzko, A. M., Shepp, L. A. and Vardi, Y. (1984) Optimal sequential selection of N random variables under a constraint. J. Appl. Prob. 21, 537–547.Google Scholar
5. Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.Google Scholar
6. Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
7. Glick, N. (1978) Breaking records and breaking boards. Amer. Math. Monthly 85, 2–26.Google Scholar
8. Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, (1963), 13–29.Google Scholar
9. Logan, B. F. and Shepp, L. A. (1977) A variational problem for random Young tableaux. Adv. Math. 26, 206–222.Google Scholar
10. Mallows, C. L., Nair, V. N., Shepp, L. A. and Vardi, Y. (1985) Optimal sequential selection of optical fibers and secretaries. Math. Operat. Res. 10, 709–715.CrossRefGoogle Scholar
11. Samuels, S. M. and Steele, J. M. (1981) Optimal sequential selection of a monotone sequence from a random sample. Ann. Prob. 9, 937–947.CrossRefGoogle Scholar