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Factorization Identities for Reflected Processes, with Applications

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Brian H. Fralix ,
Johan S. H. van Leeuwaarden  and
Onno J. Boxma
Brian H. Fralix*
Affiliation:
Clemson University
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
Onno J. Boxma*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA. Email address: bfralix@clemson.edu
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
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Abstract

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We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

Keywords

Lévy processPalm distributionrandom walktime-dependent behaviorWiener‒Hopf factorization

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes 60G55: Point processes 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 632 - 653
Copyright
© Applied Probability Trust 

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