Characterization of network stationary distribution with batch moving in geometric product form

We consider the exponential network with two independent Poisson enter flows of customer batches: general and additional. The customers of additional flow arrive in empty nodes. The batches required for the service move to other nodes with possible change of size or leave the network according to some irreducible Markov matrix. The batches which don’t achieve the required size are served but after the service is done they leave the network. The criterion of existence of stationary distribution in the form of the product of geometric distributions is established.