In this paper we consider the first passage process of a spectrally negative
Markov additive process (MAP). The law of this process is uniquely
characterized by a certain matrix function, which plays a crucial role in
fluctuation theory. We show how to identify this matrix using the theory of
Jordan chains associated with analytic matrix functions. Importantly, our
result also provides us with a technique, which can be used to derive various
further identities. We then proceed to show how to compute the stationary
distribution associated with a one-sided reflected (at zero) MAP for both the
spectrally positive and spectrally negative cases as well as for the two sided
reflected Markov-modulated Brownian motion; these results can be interpreted in
terms of queues with MAP input.Comment: 16 page