Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

Bernardo D'Auria; Offer Kella

doi:10.1016/j.spa.2012.02.001

Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

Bernardo D'Auria^*, Offer Kella

^*Corresponding author for this work

Department of Statistics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.

Original language	English
Pages (from-to)	1566-1581
Number of pages	16
Journal	Stochastic Processes and their Applications
Volume	122
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2012.02.001
State	Published - Apr 2012

Bibliographical note

Funding Information:
The authors thank the two anonymous referees for their valuable suggestions which helped in improving the paper. This research has been partially supported by the Spanish Ministry of Education and Science Grants MTM2007-63140 , MTM2010-16519 , SEJ2007-64500 and RYC-2009-04671 . The second author was supported in part by grant 434/09 from the Israel Science Foundation and the Vigevani Chair in Statistics .

Keywords

Brownian motion
Fluid queues
Markov modulation
Two sided reflection

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10.1016/j.spa.2012.02.001

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title = "Markov modulation of a two-sided reflected Brownian motion with application to fluid queues",

abstract = "In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.",

keywords = "Brownian motion, Fluid queues, Markov modulation, Two sided reflection",

author = "Bernardo D'Auria and Offer Kella",

note = "Funding Information: The authors thank the two anonymous referees for their valuable suggestions which helped in improving the paper. This research has been partially supported by the Spanish Ministry of Education and Science Grants MTM2007-63140 , MTM2010-16519 , SEJ2007-64500 and RYC-2009-04671 . The second author was supported in part by grant 434/09 from the Israel Science Foundation and the Vigevani Chair in Statistics .",

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N2 - In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.

AB - In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.

KW - Brownian motion

KW - Fluid queues

KW - Markov modulation

KW - Two sided reflection

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ER -

Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

Abstract

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Keywords

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