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A MARKOV-MODULATED GROWTH COLLAPSE MODEL

Published online by Cambridge University Press:  21 December 2009

Offer Kella
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905; Israel E-mail: offer.kella@huji.ac.il
Andreas Löpker
Affiliation:
EURANDOM and Eindhoven University of Technology, P.O. Box 513; 5600 MB Eindhoven, The Netherlands E-mail: lopker@eurandom.tue.nl

Abstract

We consider a growth collapse model in a random environment for which the inputrates might depend on the state of an underlying irreducible Markov chain and atstate change epochs there is a possible downward jump to a level that is arandom fraction of the level just before the jump. The distributions of thesejumps are allowed to depend on both the originating and target states. Under avery weak assumption we develop an explicit formula for the conditional moments(of all orders) of the time stationary distribution. We then consider specialcases and show how to use this result to study a growth collapse process inwhich the times between collapses have a phase-type distribution.

Information

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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References

1.Altman, E., Avrachenkov, K., Kherani, A.A. & Prabhu, B.J. (2005). Performance analysis and stochastic stability of congestion control protocols. In Proceedings of IEEE Infocom.Google Scholar
2.Boxma, O., Perry, D., Stadje, W. & Zacks, S. (2006). A Markovian growth-collapse model. Advances in Applied Probability 38(1): 221243.CrossRefGoogle Scholar
3.Brandt, A. (1986). The stochastic equation Y n+1=A nY n=B n with stationary coefficients. Advances in Applied Probability 18: 211220.Google Scholar
4.Davis, M.H.A. (1993). Markov models and optimization. Monographs on Statistics and Applied Probability. London: Chapman & Hall.CrossRefGoogle Scholar
5.Dumas, V., Guillemin, F. & Robert, Ph. (2002). A Markovian analysis of additive-increase, multiplicative-decrease (AIMD) algorithms. Advances in Applied Probability 34(1): 85111.CrossRefGoogle Scholar
6.Eliazar, I. & Klafter, K. (2004). A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein-Uhlenbeck dynamics. Physica A 334: 121.CrossRefGoogle Scholar
7.Guillemin, F., Robert, Ph. & Zwart, B. (2004). AIMD algorithms and exponential functionals. Annals of Applied Probability 14(1): 90117.CrossRefGoogle Scholar
8.Kella, O. (2009). On growth collapse processes with stationary structure and their shot-noise counterparts. Journal of Applied Probability 46: 363371.CrossRefGoogle Scholar
9.Löpker, A.H. & van Leeuwaarden, J.S.H. (2008). Transient moments of the TCP window size process. Journal of Applied Probability 45(1): 163175.CrossRefGoogle Scholar
10.Ott, T., Kemperman, J. & Mathis, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Available fromwwww.teunisott.com.Google Scholar
11.Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Advances in Applied Probability 11: 750783.CrossRefGoogle Scholar