Characterization of the stationary distribution of the batch-transfer networks as a product of biased geometric distributions

Consideration was given to the class of exponential queuing networks with batch arrivals and batch servicing. The messages arriving to the network nodes form random-size batches of customers associated with these nodes. Upon completion of servicing a current batch, it discharges the network and sends a message, if any, to the rest of the nodes according to some routing matrix. The necessary and sufficient conditions under which the stationary distribution of the network states is multiplicative were established in terms of the isolated nodes placed in a dummy random medium. These conditions underlie characterization of the stationary distribution in the form of biased geometric distributions.