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Lévy processes with adaptable exponent

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

R. Bekker ,
O. J. Boxma  and
J. A. C. Resing
R. Bekker*
Affiliation:
VU University Amsterdam
O. J. Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
J. A. C. Resing*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: rbekker@few.vu.nl
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

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In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

Keywords

Reflected Lévy processadaptable exponentworkload distributionscale functionLaplace inversionM/G/1 queuestorage process

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 177 - 205
Copyright
Copyright © Applied Probability Trust 2009 

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