This article considers a Markov-modulated Brownian motion with a two-sided reflection. For this doubly-reflected process we compute the Laplace transform of the stationary distribution, as well as the average loss rates at both barriers. Our approach relies on spectral properties of the matrix polynomial associated with the generator of the free (that is, non-reflected) process. This work generalizes previous partial results allowing the spectrum of the generator to be non-semi-simple and also covers the delicate case where the asymptotic drift of the free process is zero.
Bibliographical noteFunding Information:
The first author is partially supported by the Spanish Ministry of Education and Science Grants MTM2007-63140, SEJ2007-64500 and MTM2010-16519. Part of his research was done when he was visiting the Hebrew University of Jerusalem by partial support of Madrid University Carlos III Grant for Young Researchers’ Mobility. The third author is partially supported by grant 964/06 from the Israel Science Foundation and the Vigevani Chair in Statistics.
- Markov additive process
- Markov-modulated Brownian motion
- Skorohod reflection
- Two-sided reflection