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

Doubling Algorithms for Stationary Distributions of Fluid Queues: A Probabilistic Interpretation