Explicite analitical expressions for stationary characteristics for queueing networks are not existed generally. Even the conditions for existence of stationary mode for the network are absolutely non-trivial. For this purpose the method of fluid limit was developed. This method gives the conditions for existence of stationary mode by investigation of corresponding deterministic fluid dynamics arising in Euler limit.
There is another method to investigate the properties of stationary distributions of the networks. This method use a special passage to thermodynamical limit and the consideration of mean-field models, or non-linear Markov processes describing the evolution of infinite symmetrical networks. This way leads to the question about the validity of Poisson hypothesis for these infinite networks. This hypothesis was proven for some classes of networks. The counterexamples are founded also.
Using both methods together, it is possible to obtane the new surprising results about asymptotical properties of the networks.