Peter G. Harrison, P. M. Lonsdale
Fluid models have for some time been used to approximate stochastic networks with discrete state. These range from traditional 'heavy traffic' approximations to the recent advances in bio-chemical system models. Here we obtain an exact solution for a pair of two queues linked in tandem with on-off rarivals at the first and departures from the second. The solution method of the resulting vector differential equation is via Laplace transforms, which yields joint moments (e.g. covariance) directly and can be inverted to give the steady-state joint probability distribution of the fluid levels at the two queues. The exactness of this result can be used to validate simulations, which in turn can be used to validate more complex models, and also suggests a route to analytical solution of larger networks.
The full solution to this problem follows in section 3 of "An Approximate Compositional Approach to the Analysis of Fluid Queue Networks" A. J. Field, P. G. Harrison, Performance Evaluation 64 (9-12) pp.1137-1152 October, 2007 http://aesop.doc.ic.ac.uk/pubs/fluidsPERF07/
Information from pubs.doc.ic.ac.uk/tandem-fluid-queue.