Weak convergence of quasi-birth-and-death processes with rational arrival process components to fluid queues

Abstract

In this paper we construct a new approximation to a fluid queue as a quasi-birth-and-death process with rational arrival process components (QBD-RAP) and prove its convergence. Fluid queues are stochastic processes that move linearly at a rate governed by the state of a continuous-time Markov chain (CTMC), and are widely used to model telecommunications, power, risk, and storage systems. A key motivating application is to fluid-fluid queues, whose analysis proceeds via operator-analytic expressions involving the generator of the underlying fluid queue; these expressions are differential operators that are not, in general, readily computable, so approximation is needed. Existing approximations with a probabilistic interpretation guarantee valid probabilities but require a fine discretisation to be accurate, while methods such as the Discontinuous Galerkin approach are more accurate at a given discretisation level but can produce negative mass or probabilities exceeding one. Our (QBD-RAP) approximation addresses this. Because the QBD-RAP is itself a stochastic process, the approximation it produces automatically retains the defining properties of a probability, while promising improved numerical accuracy over existing probabilistic schemes for a given discretisation level. We prove that the generator of the QBD-RAP converges to the generator of the fluid queue, which is, to our knowledge, the first generator-theoretic convergence result for a process with rational arrival process components. The proof introduces a new technique for the analysis of RAP-modulated processes, analysing the generator via bases of conditional residual time distributions rather than the orbit process used in prior RAP analyses, and along the way establishes that the phase process of the QBD-RAP and of the fluid queue share the same distribution.