Two-sided reflection of Markov-modulated brownian motion

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.