Steady-state exponential networks with bypassing of communications and with generation of arriving and servicing random size customer batches

The exponential queueing networks class with Jackson traffic of communications and with possibility of instant bypassing of nodes by communications with probabilities depending on their states is considered. The communications taking over by thenodes generate customers’ batches of random sizes and will be joined to these nodes. Customers' batch of random size is served simultaneously, the size being determinated at the moment of completing the service due to the assemble-transfer batch service discipline. When customers batch service is finished it leaves the network and sends communication to other nodes or leaves the network with according to some routing matrix. Sufficient conditions are established for the product form stationary network distribution in terms of insulating nodes in fictitious random environment. As an example we considered the case when the sizes of customers' batches taking over by nodes have a geometric distribution and the network state stationary distribution has the product form of factors which have shift quasigeometric distribution.