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  • Michel Mandjes (a1) (a2) (a3) (a4) and Peter Taylor (a5)

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.

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1.Abate, J. & Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA Journal on Computing 7: 3643.
2.Artalejo, J.R., Economou, A. & Gómez-Corral, A. (2007). Applications of maximum queue lengths to call center management. Computers and Operations Research 34: 983996.
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: 121126.
4.Artalejo, J.R. & Gómez-Corral, A. (2008). Retrial queueing systems. a computational approach, Berlin: Springer-Verlag, 2008.
5.Asmussen, S. (2003). Applied Probability and Queues, 2nd edn., New York, NY, USA: Springer.
6.Asmussen, S., Avram, F. & Usabel, M. (2002). The Erlang approximation of finite time ruin probabilities. ASTIN Bulletin 32: 267281.
7.Bright, L.W. & Taylor, P.G. (1995). Calculating the equilibrium distribution in level dependent Quasi-Birth-and-Death processes. Stochastic Models 11: 497526.
8.den Iseger, P. (2006). Numerical transform inversion using Gaussian quadrature. Probability in the Engineering and Informational Sciences 20: 144.
9.Ellens, W., Mandjes, M., van den Berg, H., Worm, D. & Błaszczuk, S. (2015). Performance evaluation using periodic system-state measurements. Performance Evaluation 93: 2746.
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: 124136.
11.Kyprianou, A. (2006). Introductory lectures on fluctuations of Lévy processes with applications, Berlin, Germany: Springer.
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.
13.Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models, Baltimore, MD, USA: Johns Hopkins University Press.
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: 143164.
15.Ramaswami, V., Woolford, D. & Stanford, D. (2008). The Erlangization method for Markovian fluid flows. Annals of Operations Research 160: 215225.
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Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
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