Closed G-networks with Resets: product form solution

We consider a closed queueing network of generalized queues with customers and signals. Each queue has an infinite capacity and one server The service time is exponential. After its service completion a customer moves to another queue and may become a signal. When the signal enters a non empty queue it vanishes while it resets the queue when it enters an empty queue. We prove that the steady state distribution for such a closed network of queues has a product form solution. To the best of our knowledge it is the first closed network of generalized queues with product form solution. We also consider a more complex system where the reset acts upon a set of queues rather than a single one. We also prove that the steady-state distribution exists and has a product form.