We consider a stochastic fluid network with inputs which are independent subordinators. We show that under some natural conditions the distribution of the fluid content process converges in total variation to a proper limit and that the limiting distribution has a positive mass at the origin. As a consequence of the methodology used, we obtain upper and lower bounds for the expected values of this limiting distribution. For the two-dimensional case, under certain conditions, explicit formulas for the means, variances and covariance of the steady-state fluid content are given. Further, for the two-dimensional case, it is shown that, other than for trivial setups, the limiting distribution cannot have product form.
"Stability and nonproduct form of stochastic fluid networks with Lévy inputs." Ann. Appl. Probab. 6 (1) 186 - 199, February 1996. https://doi.org/10.1214/aoap/1034968070