Spatially-coherent uniformization of a stochastic fluid model to a Quasi-Birth-and-Death process

We derive a uniformization of a stochastic fluid model (SFM) to a Quasi-Birth-and-Death process (QBD) that is spatially-coherent since the continuous level in the SFM has a natural correspondence to the discrete level in the QBD. As a consequence of this, the QBD can be used as a direct approximation of the original SFM, in those situations where a discrete state space is an advantage. We treat the unbounded as well as the bounded cases and illustrate the theory with a numerical example. The key fluid generator, $Q$, and matrix $\Psi$ for the SFMs emerge from the QBD calculations in the natural limit.