Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T09:07:14.807Z Has data issue: false hasContentIssue false
  • English
  • Français

A Markovian growth-collapse model

Part of: Markov processes Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Onno Boxma ,
David Perry ,
Wolfgang Stadje  and
Shelemyahu Zacks
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
David Perry*
Affiliation:
University of Haifa
Wolfgang Stadje*
Affiliation:
University of Osnabrück
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗ Postal address: Department of Statistics, University of Haifa, Haifa, 31909, Israel. Email address: [email protected]
∗∗∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. 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.

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

Keywords

Growth-collapse processpiecewise-deterministic Markov processstationary distributionhitting timeuniform cut-offdualityMarkov modulation

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 221 - 243
Copyright
Copyright © Applied Probability Trust 2006 

References

Albrecher, H. and Boxma, O. J. (2004). A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econom. 35, 245–254.CrossRefGoogle Scholar
Albrecher, H. and Boxma, O. J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom. 37, 650–672.Google Scholar
Altman, E., Avrachenkov, K., Barakat, C. and Núñez-Queija, R. (2001). State-dependent M/G/1 type queueing analysis for congestion control in data networks. In Proc. IEEE INFOCOM 2001 (Anchorage, AK, April 2001), IEEE, Piscataway, NJ, pp. 1350–1359.Google Scholar
Altman, E., Avrachenkov, K., Kherani, A. A. and Prabhu, B. J. (2005). Performance analysis and stochastic stability of congestion control protocols. In Proc. IEEE INFOCOM 2005 (Miami, FL, March 2005), eds Makki, K. and Knightly, E., IEEE, Piscataway, NJ.Google Scholar
Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, Singapore.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus, New York.Google Scholar
Bekker, R., Borst, S. C., Boxma, O. J. and Kella, O. (2003). Queues with workload-dependent arrival and service rates. Queueing Systems 46, 537–556.Google Scholar
Boxma, O. J. and Perry, D. (2001). A queueing model with dependence between service and interarrival times. Europ. J. Operat. Res. 128, 611–624.CrossRefGoogle Scholar
Boxma, O. J., Kella, O. and Perry, D. (2001). An intermittent fluid system with exponential on-times and semi-Markov input rates. Prob. Eng. Inf. Sci. 15, 189–198.Google Scholar
Boxma, O. J., Kaspi, H., Kella, O. and Perry, D. (2005). On/off storage systems with state-dependent input, output and switching rates. Prob. Eng. Inf. Sci. 19, 1–14.Google Scholar
Budhiraja, A., Hernandez-Campos, F., Kulkarni, V. G. and Smith, F. D. (2004). Stochastic differential equation for TCP window size: analysis and experimental validation. Prob. Eng. Inf. Sci. 18, 111–140.Google Scholar
Cohen, J. W. (1982). The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.Google Scholar
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
Keilson, J. and Mermin, N. D. (1959). The second-order distribution of integrated shot noise. IRE Trans. Inf. Theory 5, 75–77.Google Scholar
Kella, O. and Stadje, W. (2001). On hitting times for compound Poisson dams with exponential Jumps and linear release. J. Appl. Prob. 38, 781–786.Google Scholar
Kherani, A. A. and Kumar, A. (2002). Stochastic models for throughput analysis of randomly arriving elastic flows in the Internet. In Proc. IEEE INFOCOM 2002 (New York, June 2002), IEEE, Piscataway, NJ.Google Scholar
Perry, D. and Stadje, W. (2003). Duality of dams via mountain processes. Operat. Res. Lett. 31, 451–458.Google Scholar
Polyanin, A. D. and Zaitsev, V. F. (1995). Handbook of Exact Solutions for Ordinary Differential Equations. CRC, Boca Raton, FL.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.Google Scholar
Scheinhardt, W., van Foreest, N. and Mandjes, M. (2005). Continuous feedback fluid queues. Operat. Res. Lett. 33, 551–559.CrossRefGoogle Scholar