# A discontinuous Galerkin method for approximating the stationary distribution of stochastic fluid-fluid processes

## TL;DR

This paper develops a discontinuous Galerkin numerical method to approximate the stationary distribution of stochastic fluid-fluid processes, enabling practical computation where previous formulas were analytically explicit but numerically intractable.

## Contribution

It introduces a novel discontinuous Galerkin approach for numerical approximation of the stationary distribution in stochastic fluid-fluid processes.

## Key findings

- Successfully applied to an on-off bandwidth sharing system
- Demonstrates accurate approximation of stationary distributions
- Provides a practical computational tool for complex stochastic processes

## Abstract

Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is a Markov processes $\{X_t, Y_t, \varphi_t\}_{t \geq 0}$, where the first fluid $X_t$ is driven by the Markov chain $\varphi_t$, and the second fluid $Y_t$ is driven by $\varphi_t$ as well as by $X_t$. That paper derived a closed-form expression for the joint stationary distribution, given in terms of operators acting on measures, which does not lend itself easily to numerical computations. Here, we construct a discontinuous Galerkin method for approximating this stationary distribution, and illustrate the methodology using an on-off bandwidth sharing system, which is a special case of a stochastic fluid-fluid process.