Impatient service in a G-network

An open exponential queueing network with signals and impatient service is considered. Upon completion of service at a node, a positive customer passes to another node with fixed probabilities either as a positive customer or as a signal, or quits the network. Every signal is activated during a random exponentially distributed amount of time. Activated signals with fixed probabilities either move a customer from the node they arrive to another node or kill a positive customer. Each customer can be served in a node at most a random time (”patient” time) distributed exponentially. When the patient service is finished, the customer with fixed robabilities either goes to another node or quits the network. The stationary state probabilities for such a G-network in which positive customers are processed in each node by a single server is derived in product form. The solution for an analogous symmetrical G-network in which service rate of a positive customer at each node depends on the number of positive customers in this node is expressed in product form too.