Doubling algorithms for stationary distributions of fluid queues: A probabilistic interpretation

Fluid queues are mathematical models frequently used in stochastic modeling. 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 Ψ. Here, we present a probabilistic interpretation of the family of algorithms known as doubling, which are currently the most effective algorithms for computing the return probability matrix Ψ. To this end, we first revisit the links described in Ramaswami (1999) and da Silva Soares and Latouche (2002) 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.