Conditions for a product-form stationary distribution of one queueing system with batch transfers and a disaster flow

We consider an open exponential network with two types of arrival flows at the network nodes: a message flow and a disaster flow. Messages arriving at the nodes form batches of customers of a random size. A disaster arrival at a node completely empties the queue at the node if it is nonempty and has no effect otherwise. Customers are served in batches of a random size. After a batch is served at a node, the batch quits the network and, according to a routing matrix, either sends a message or a disaster to another node or does not send anything. We find conditions for the stationary distribution of the network state probabilities to be represented as a product of shifted geometric distributions.