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

TL;DR

This paper provides the first probabilistic interpretation of doubling algorithms used for computing stationary distributions in fluid queues, linking them to stochastic processes and enhancing understanding of their effectiveness.

Contribution

It introduces a novel probabilistic interpretation of doubling algorithms, connecting them to fluid queues and Quasi-Birth-Death processes, and generalizes the framework for better insight.

Findings

First probabilistic interpretation of doubling algorithms

New connections between fluid queues and stochastic processes

Enhanced understanding of algorithmic effectiveness

Abstract

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.