Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T11:54:15.544Z Has data issue: false hasContentIssue false
  • English
  • Français

On some tractable growth-collapse processes with renewal collapse epochs

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Onno Boxma ,
Offer Kella  and
David Perry
Onno Boxma
Affiliation:
EURANDOM and Eindhoven University of Technology, EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
Offer Kella
Affiliation:
The Hebrew University of Jerusalem, Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: [email protected]
David Perry
Affiliation:
Haifa University, Department of Statistics, Haifa University, Haifa 31905, Israel. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

Keywords

Growth collapseshot noise processminus-log-phase-type distributionMarkov modulated

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K25: Queueing theory
Type
Part 5. Stochastic Growth and Branching
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 217 - 234
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Abramowitz, M. and Stegun, I. A., (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Altman, A., Avrachenkov, K., Barakat, C. and Núñez-Queija, R., (2002). State-dependent M/G/1 type queueing analysis for congestion control in data networks. Comput. Networks 39, 789–808.Google Scholar
[3] Asmussen, S. and Kella, O., (1996). Rate modulation in dams and ruin problems. J. Appl. Prob. 33, 523–535.Google Scholar
[4] Asmussen, S. and Kella, O., (2000). A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Prob. 32, 376–393.Google Scholar
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L., (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[6] Boxma, O., Perry, D., Stadje, W. and Zacks, S., (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221–243.Google Scholar
[7] Brandt, A., (1986). The stochastic equation Y_{n+1}=A_n Y_n+B_n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[8] Denisov, D. and Zwart, A. P., (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046.CrossRefGoogle Scholar
[9] Eliazar, I. and Klafter, J., (2004). A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein–Uhlenbeck dynamics. Physica A 334, 1–21.Google Scholar
[10] Erdélyi, A., Oberhettinger, F., Magnus, W. and Tricomi, F. G., (1953). Higher Transcendental Functions, Vol. III. McGraw-Hill, New York.Google Scholar
[11] Hale, J. K. and Verduyn Lunel, S. M., (1993). Introduction to Functional Differential Equations. Springer, New York.Google Scholar
[12] Kaspi, H., Kella, O. and Perry, D., (1997). Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates. Queueing Systems 24, 37–57.Google Scholar
[13] Keilson, J. and Mermin, N. D., (1959). The second-order distribution of integrated shot noise. IRE Trans. Inf. Theory 5, 75–77.Google Scholar
[14] Kella, O., (2009). On growth-collapse processes with stationary structure and their shot noise counterparts. J. Appl. Prob. 46, 363–371.Google Scholar
[15] Kella, O. and Löpker, A., (2010). A Markov-modulated growth-collapse model. Prob. Eng. Inf. Sci. 24, 99–107.Google Scholar
[16] Kella, O. and Stadje, W., (2002). Markov-modulated linear fluid networks with Markov additive input. J. Appl. Prob. 39, 413–420.Google Scholar
[17] Kella, O. and Yor, M., (2010). A new formula for some linear stochastic equations with applications. Ann. Appl. Prob. 20, 367–381.Google Scholar
[18] Löpker, A. H. and van Leeuwaarden, J. S. H., (2008). Transient moments of the TCP window size process. J. Appl. Prob. 45, 163–175.CrossRefGoogle Scholar
[19] Löpker, A. H. and van Leeuwaarden, J. S. H., (2007). Hitting times for multiplicative growth-collapse processes. Tech. Rep. 2007-049, EURANDOM. Available at http://www.eurandom.nl/reports/index.htm.Google Scholar
[20] Volkovich, Y. and Litvak, N., (2008). Asymptotic analysis for personalized Web search. Memorandum 1884, University of Twente, Department of Applied Mathematics.Google Scholar
[21] Vervaat, W., (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar