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

Michel Mandjes; Peter Taylor

Journal article

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

Michel Mandjes, Peter Taylor

Probability in the Engineering and Informational Sciences | Cambridge University Press (CUP) | Published : 2016

DOI: 10.1017/S026996481500039X

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.

University of Melbourne Researchers

Peter Taylor Author

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Grants

Awarded by NWO Gravitation Project NETWORKS

Awarded by Australian Research Council (ARC) Laureate Fellowship

Funding Acknowledgements

The authors wish to thank an anonymous referee for some insightful comments and for pointing out references [2,3,10]. Michel Mandjes' research is partly funded by the NWO Gravitation Project NETWORKS-grant number 024.002.003. Peter Taylor's research is supported by the Australian Research Council (ARC) Laureate Fellowship FL130100039 and the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).