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Concavity and reflected Lévy processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella*
Affiliation:
Yale University
*
Postal address: Department of Operations Research, Yale University, 84 Trumbull Street, New Haven, CT 06511, USA.

Abstract

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

Type
Short Communications
Copyright
Copyright © Applied Probability Trust 1992 

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