Queueing networks are used to model the performance of the Internet, of manufacturing and job-shop systems, supply chains, and other networked systems in transportation or emergency management. Composed of service stations where customers receive service, and then move to another service station till they leave the network, queueing networks are based on probabilistic assumptions concerning service times and customer movement that represent the variability of system workloads. Subject to restrictive assumptions regarding external arrivals, Markovian movement of customers, and service time distributions, such networks can be solved efficiently with “product form solutions” that reduce the need for software simulators requiring lengthy computations. G-networks generalise these models to include the effect of “signals” that re-route customer traffic, or negative customers that reject service requests, and also have a convenient product form solution. This paper extends G-networks by including a new type of signal, that we call an “Adder”, which probabilistically changes the queue length at the service center that it visits, acting as a load regulator. We show that this generalisation of G-networks has a product form solution.