Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T21:37:32.332Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov-Modulated Brownian Motion with Two Reflecting Barriers

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Jevgenijs Ivanovs
Jevgenijs Ivanovs*
Affiliation:
EURANDOM, Eindhoven University of Technology and University of Amsterdam
*
∗Postal address: EURANDOM, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. 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 a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.

Keywords

Markov additive processfluid modelfinite buffertwo-sided reflectioninverse local timeoverflow process

MSC classification

Secondary: 60J55: Local time and additive functionals 60K25: Queueing theory 60K37: Processes in random environments
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1034 - 1047
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Ahn, S., Badescu, A. L. and Ramaswami, V. (2007). Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier. Queueing Systems 55, 207–222.CrossRefGoogle Scholar
[2] Anick, D., Mitra, D. and Sondhi, M. M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.CrossRefGoogle Scholar
[3] Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Commun. Statist. Stoch. Models 11, 21–49.CrossRefGoogle Scholar
[4] Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
[5] Asmussen, S. and Pihlsgård, M. (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308–321.Google Scholar
[6] Breuer, L. (2008). First passage times for Markov additive processes with positive Jumps of phase type. J. Appl. Prob. 45, 779–799.Google Scholar
[7] D'Auria, B., Ivanovs, J., Kella, O. and Mandjes, M. (2010). First passage process of a Markov additive process, with applications to reflection problems. Preprint. Available at http://arxiv.org/abs/1006.2965v1.Google Scholar
[8] Ivanovs, J. and Mandjes, M. (2010). First passage of time-reversible spectrally negative Markov additive processes. Operat. Res. Lett. 38, 77–81.CrossRefGoogle Scholar
[9] Karandikar, R. L. and Kulkarni, V. G. (1995). Second-order fluid flow models: reflected Brownian motion in a random environment. Operat. Res. 43, 77–88.CrossRefGoogle Scholar
[10] Kosten, L. (1974/75). Stochastic theory of a multi-entry buffer. {I}. Delft Progress Rep. 1, 10–18.Google Scholar
[11] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0,a]. Ann. Prob. 35, 1740–1768.Google Scholar
[12] Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queueing, ed. Dshalalow, J. H., CRC, Boca Raton, FL, pp. 321–338.Google Scholar
[13] Lindley, D. (1959). Discussion on Mr. Winsten's paper. J. R. Statist. Soc. B 21, 22–23.Google Scholar
[14] Pistorius, M. (2006). On maxima and ladder processes for a dense class of Lévy processes. J. Appl. Prob. 43, 208–220.Google Scholar
[15] Prabhu, N. U. and Zhu, Y. (1989). Markov-modulated queueing systems. Queueing Systems 5, 215–245.CrossRefGoogle Scholar
[16] Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413.Google Scholar
[17] Williams, R. J. (1992). Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval. J. Appl. Prob. 29, 996–1002.CrossRefGoogle Scholar