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The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Offer Kella
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
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Abstract

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The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

Keywords

Generalized Pollaczek-Khinchine formulaLévy process with no negative jumpsspectrally positive Lévy processreflected Lévy processsupremum of a Lévy process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 883 - 887
Copyright
© Applied Probability Trust 

Footnotes

Supported in part by grant 434/09 from the Israel Science Foundation and the Vigevani Chair in Statistics.

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