Mean Value Analysis of Closed G-Networks with Signals

We consider a closed network of queues with external signals. These signals trigger customer between queues and they arrive following a rate which depends on the number of active customers in the station. We consider three types of stations: they may have one server, an infinite number of servers or no servers at all. In that case, the customers behave like inert customers and they only react to signal. We prove that, under irreducibility conditions, such a network has a stationary distribution which is multiplicative. As the network is finite, all the states are not reachable and the distribution is known up to a normalization constant. To avoid the computation of this constant, we also prove a mean value analysis algorithm which allows to determine the average queue size and the average waiting time without computing the probabilities. We also present some extensions of the model.