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Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is aMarkov 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 $Y_t$ isdriven by $\varphi_t$ as well as by $X_t$. That paper derived a closed-formexpression for the joint stationary distribution, given in terms of operatorsacting on measures, which does not lend itself easily to numericalcomputations.  Here, we construct a discontinuous Galerkin method for approximating thisstationary distribution, and illustrate the methodology using an on-offbandwidth sharing system, which is a special case of a stochastic fluid-fluidprocess.

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