  In this paper we provide the analysis of the limiting conditional
distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of
models in the theory of matrix-analytic methods. So far, transient and
stationary analyses of the SFMs have been only considered in the literature.
The limiting conditional distribution gives useful insights into what happens
when the process has been evolving for a long time, given its busy period has
not ended yet. We derive expressions for the Yaglom limit in terms of the
singularity $s^*$ such that the key matrix of the SFM, ${\bf\Psi}(s)$, is
finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the
uniqueness of the Yaglom limit and illustrate the application of the theory
with simple examples.
