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The lifetime of a random set

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Peter C. Kiessler  and
Kanoktip Nimitkiatklai
Peter C. Kiessler*
Affiliation:
Clemson University
Kanoktip Nimitkiatklai*
Affiliation:
Clemson University
*
∗Postal address: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA.
∗Postal address: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA.
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Abstract

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We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝn. Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

Keywords

Weibull distributionextreme value theoryinteracting particle system

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 566 - 580
Copyright
© Applied Probability Trust 2005 

References

Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Griebel, M., Jager, L. and Voigt, A. (2004). Computing diffusion coefficients of intrinsic point defects in crystalline silicon. In Proc. 10th Internat. Conf. Composites Eng., ed. Hui, D., International Community for Composites Engineering, New Orleans, pp. 487–488.Google Scholar
Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press, New York.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley, New York.Google Scholar
Meeker, W. O. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. John Wiley, New York.Google Scholar
Neudeck, P. G. and Powell, J. A. (2004). Performance limiting micropipe defects in silicon carbide wafers. Res. Rept., NASA Lewis Research Center. Available at http://www.grc.nasa.gov/WWW/SiC/publications/micropipeEDLPaper.html.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Révész, P. (1994). Random Walks on Infinitely Many Particles. World Scientific, River Edge, NJ.Google Scholar