Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T17:40:31.121Z Has data issue: false hasContentIssue false
  • English
  • Français

On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

Part of: Stochastic processes Mathematical economics Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
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. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.

Keywords

Growth-collapse processrisk processshot-noise process

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G10: Stationary processes 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 363 - 371
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported by grant number 964/06 from the Israel Science Foundation and the Vigevani Chair in Statistics.

References

[1] Altman, E., Avrachenkov, K. and Barakat, C. (2005). A stochastic model of TCP/IP with stationary random losses. IEEE/ACM Trans. on Networking 13, 356–369.Google Scholar
[2] Altman, E., Avrachenkov, K., Barakat, C. and Nùñez-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] Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221–243.Google Scholar
[4] Brandt, A. (1986). The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients. J. Appl. Prob. 18, 211–220.Google Scholar
[5] 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
[6] Eliazar, I. and Klafter, J. (2006). Growth–collapse and decay–surge evolutions, and geometric Langevin equations. Physica A 367, 106–128.CrossRefGoogle Scholar
[7] 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
[8] Löpker, A. H. and van Leeuwaarden, J. S. H. (2007). Hitting times for multiplicative growth-collapse processes. EURANDOM Report 2007-049. Available at http://www.eurandom.tue.nl/EURANDOMreports.htm.Google Scholar
[9] Löpker, A. H. and van Leeuwaarden, J. S. H. (2007). Transient moments of the window size in TCP. EURANDOM Report 2007-019. Available at http://www.eurandom.tue.nl/EURANDOMreports.htm.Google Scholar
[10] Sigman, K. (1995). Stationary Marked Point Processes. Chapman and Hall, New York.Google Scholar