Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-21T21:35:24.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy efficiency of consecutive fragmentation processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Joaquín Fontbona ,
Nathalie Krell  and
Servet Martínez
Joaquín Fontbona*
Affiliation:
Universidad de Chile
Nathalie Krell*
Affiliation:
Université Rennes 1
Servet Martínez*
Affiliation:
Universidad de Chile
*
∗Postal address: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile.
∗∗∗Postal address: Institut de Recherche Mathématique de Rennes (IRMAR), Université Rennes 1, UMR 6625 CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: nathalie.krell@univ-rennes1.fr
∗Postal address: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Motivated by a problem arising in the mining industry, we present a first study of the energy required to reduce a unit mass fragment by consecutively using several devices. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced in Bertoin and Martínez (2005), we compute the average energy required to attain a size η0 with this two-device procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that, for a certain range of parameters of the fragmentation processes and of their energy cost functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or vice versa.

Keywords

Fragmentation processfragmentation energysubordinatorLaplace exponent

MSC classification

Secondary: 60J85: Applications of branching processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 543 - 561
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Berestycki, J. (2002). Ranked fragmentations. ESAIM Prob. Statist. 6, 157–175.CrossRefGoogle Scholar
[2] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
[3] Bertoin, J. (1999). Subordinators: examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997; Lecture Notes Math. 1717), Springer. Berlin, pp. 1–91.CrossRefGoogle Scholar
[4] Bertoin, J. (2001). Homogeneous fragmentation processes. Prob. Theory Relat. Fields 121, 301–318.CrossRefGoogle Scholar
[5] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press.CrossRefGoogle Scholar
[6] Bertoin, J. and Martínez, S. (2005). Fragmentation energy. Adv. Appl. Prob. 37, 553–570.CrossRefGoogle Scholar
[7] Bertoin, J., van Harn, K. and Steutel, F. W. (1999). Renewal theory and level passage by subordinators. Statist. Prob. Lett. 45, 65–69.CrossRefGoogle Scholar
[8] Hoffmann, M. and Krell, N. (2010). {Statistical analysis of self-similar conservative fragmentation chains}. To appear in Bernoulli.Google Scholar
[9] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[10] Walker, W. H., Lewis, W. K., McAdams, W. H. and Gilliland, E. R. (1937). Principles of Chemical Engineering. McGraw-Hill, New York.Google Scholar