Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T10:57:35.894Z Has data issue: false hasContentIssue false
  • English
  • Français

From matchbox to bottle: a storage problem

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ludwig Baringhaus  and
Rudolf Grübel
Ludwig Baringhaus*
Affiliation:
Universität Hannover
Rudolf Grübel*
Affiliation:
Universität Hannover
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize Banach's matchbox problem: demands of random size are made on one of two containers, both initially with content t, where the container is selected at random in the successive steps. Let Zt be the content of the other container at the moment when the selected container is found to be insufficient. We obtain the asymptotic distribution of Zt as t → ∞ under quite general conditions. The case of exponentially distributed demands is considered in more detail.

Keywords

Asymptotic normalityBessel functionsconvergence in distributionexponential distributionKnuth's old sumrenewal theory

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60F05: Central limit and other weak theorems 60K05: Renewal theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 650 - 656
Copyright
Copyright © Applied Probability Trust 2002 

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

Abramowitz, M., and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.Google Scholar
Berghahn, H.-H. (1966). Eine Verallgemeinerung des Banachschen ‘Streichholzproblems’. Diploma Thesis, Universität Freiburg.Google Scholar
Cacoullos, T. (1967). Asymptotic distribution for a generalized Banach match box problem. J. Amer. Statist. Assoc. 62, 1252–1257.Google Scholar
Chung, K. L. (1974). A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Holst, L. (1989). A note on Banach's match box problem. Statist. Prob. Lett. 8, 441–443.CrossRefGoogle Scholar
Knuth, D. E. (1984). The toilet paper problem. Amer. Math. Monthly 91, 465–470.Google Scholar
Prodinger, H. (1994). Knuth's old sum—a survey. Bull. EATCS 54, 232–245.Google Scholar
Stirzaker, D. (1988). A generalization of the matchbox problem. Math. Scientist 13, 104–114.Google Scholar
Uppuluri, V. R. R., and Blot, W. J. (1974). Asymptotic properties of the number of replications of a paired comparison. J. Appl. Prob. 11, 43–52.Google Scholar