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THE RUNNING MAXIMUM OF A LEVEL-DEPENDENT QUASI-BIRTH-DEATH PROCESS

Published online by Cambridge University Press:  21 December 2015

Michel Mandjes  and
Peter Taylor
Michel Mandjes
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands CWI, P.O. Box 94079, 1090 GB Amsterdam, the Netherlands Eurandom, Eindhoven University of Technology, Eindhoven, the Netherlands IBIS, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands E-mail: m.r.h.mandjes@uva.nl
Peter Taylor
Affiliation:
School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia E-mail: p.taylor@ms.unimelb.edu.au
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Abstract

The objective of this note is to study the distribution of the running maximum of the level in a level-dependent quasi-birth-death process. By considering this running maximum at an exponentially distributed “killing epoch” T, we devise a technique to accomplish this, relying on elementary arguments only; importantly, it yields the distribution of the running maximum jointly with the level and phase at the killing epoch. We also point out how our procedure can be adapted to facilitate the computation of the distribution of the running maximum at a deterministic (rather than an exponential) epoch.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 212 - 223
Copyright
Copyright © Cambridge University Press 2015 

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