  Markov-modulated fluids have a long history. They form a simple class of
Markov additive processes, and were initially developed in the 1950s as models
for dams and reservoirs, before gaining much popularity in the 1980s as models
for buffers in telecommunication systems, when they became known as fluid
queues. More recent applications are in risk theory and in environmental
studies. In telecommunication systems modelling, the attention focuses on
determining the stationary distribution of the buffer content. Early ODE
resolution techniques have progressively given way to approaches grounded in
the analysis of the physical evolution of the system, and one only needs now to
solve a Riccati equation in order to obtain several quantities of interest. To
the early algorithms proposed in the Applied Probability literature, numerical
analysts have added new algorithms, improved in terms of convergence speed,
numerical accuracy, and domain of applicability. We give here a high-level
presentation of the matrix-analytic approach to the analysis of fluid queues,
briefly address computational issues, and conclude by indicating how this has
been extended to more general processes.
