  Fluid queues are mathematical models frequently used in stochastic modelling.
Their stationary distributions involve a key matrix recording the conditional
probabilities of returning to an initial level from above, often known in the
literature as the matrix $\Psi$. Here, we present a probabilistic
interpretation of the family of algorithms known as \emph{doubling}, which are
currently the most effective algorithms for computing the return probability
matrix $\Psi$.
  To this end, we first revisit the links described in \cite{ram99, soares02}
between fluid queues and Quasi-Birth-Death processes; in particular, we give
new probabilistic interpretations for these connections. We generalize this
framework to give a probabilistic meaning for the initial step of doubling
algorithms, and include also an interpretation for the iterative step of these
algorithms. Our work is the first probabilistic interpretation available for
doubling algorithms.
