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Extreme values of phase-type and mixed random variables with parallel-processing examples

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Sungyeol Kang  and
Richard F. Serfozo
Sungyeol Kang*
Affiliation:
Electronics and Telecommunications Research Institute
Richard F. Serfozo*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: Electronics and Telecommunications Research Institute, Yusong – P.O. Box 106, Taejon, South Korea.
∗∗Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email address: rserfozo@isye.gatech.edu.
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Abstract

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

Keywords

Extreme valuesphase-type distributionsmixed random variablesparallel-processingPERTtask-graphsrandom environments

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 194 - 210
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This research was supported in part by the Air Force Office of Scientific Research under contract 91–0013 and NSF grant DDM-9224520.

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