Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-07T11:37:28.581Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Abate, J. & Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA Journal on Computing 7: 36–43.Google Scholar
2.Artalejo, J.R., Economou, A. & Gómez-Corral, A. (2007). Applications of maximum queue lengths to call center management. Computers and Operations Research 34: 983–996.CrossRefGoogle Scholar
3.Artalejo, J.R., Economou, A. & Lopez-Herrero, M.J. (2007). Algorithmic analysis of the maximum queue length in a busy period for the M/M/c retrial queue. INFORMS Journal on Computing 19: 121–126.Google Scholar
4.Artalejo, J.R. & Gómez-Corral, A. (2008). Retrial queueing systems. a computational approach, Berlin: Springer-Verlag, 2008.CrossRefGoogle Scholar
5.Asmussen, S. (2003). Applied Probability and Queues, 2nd edn., New York, NY, USA: Springer.Google Scholar
6.Asmussen, S., Avram, F. & Usabel, M. (2002). The Erlang approximation of finite time ruin probabilities. ASTIN Bulletin 32: 267–281.CrossRefGoogle Scholar
7.Bright, L.W. & Taylor, P.G. (1995). Calculating the equilibrium distribution in level dependent Quasi-Birth-and-Death processes. Stochastic Models 11: 497–526.Google Scholar
8.den Iseger, P. (2006). Numerical transform inversion using Gaussian quadrature. Probability in the Engineering and Informational Sciences 20: 1–44.CrossRefGoogle Scholar
9.Ellens, W., Mandjes, M., van den Berg, H., Worm, D. & Błaszczuk, S. (2015). Performance evaluation using periodic system-state measurements. Performance Evaluation 93: 27–46.Google Scholar
10.Gomez-Corral, A. & García, M.L. (2014). Maximum queue lengths during a fixed time interval in the M/M/c retrial queue. Applied Mathematics and Computation 235: 124–136.Google Scholar
11.Kyprianou, A. (2006). Introductory lectures on fluctuations of Lévy processes with applications, Berlin, Germany: Springer.Google Scholar
12.Latouche, G. & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modelling. ASA/SIAM Series on Statistics and Applied Probability. Philadelphia PA, USA.Google Scholar
13.Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models, Baltimore, MD, USA: Johns Hopkins University Press.Google Scholar
14.Ramaswami, V. & Taylor, P.G. (1996). Some properties of the rate matrices in level dependent Quasi-Birth-and-Death processes with a countable number of phases. Stochastic Models 12: 143–164.CrossRefGoogle Scholar
15.Ramaswami, V., Woolford, D. & Stanford, D. (2008). The Erlangization method for Markovian fluid flows. Annals of Operations Research 160: 215–225.Google Scholar