In queueing theory, a discipline within the mathematical theory of probability, a Gnetwork (generalized queueing network,^{[1]}^{[2]} often called a Gelenbe network^{[3]}) is an open network of Gqueues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic rerouting or traffic destruction, as well as a model for neural networks.^{[4]}^{[5]} A Gqueue is a network of queues with several types of novel and useful customers:
A productform solution superficially similar in form to Jackson's theorem, but which requires the solution of a system of nonlinear equations for the traffic flows, exists for the stationary distribution of Gnetworks while the traffic equations of a Gnetwork are in fact surprisingly nonlinear, and the model does not obey partial balance. This broke previous assumptions that partial balance was a necessary condition for a productform solution. A powerful property of Gnetworks is that they are universal approximators for continuous and bounded functions, so that they can be used to approximate quite general inputoutput behaviours.^{[8]}
A network of m interconnected queues is a Gnetwork if
A queue in such a network is known as a Gqueue.
Define the utilization at each node,
where the for satisfy

(1) 

(2) 
Then writing (n_{1}, ... ,n_{m}) for the state of the network (with queue length n_{i} at node i), if a unique nonnegative solution exists to the above equations (1) and (2) such that ρ_{i} for all i then the stationary probability distribution π exists and is given by
It is sufficient to show satisfies the global balance equations which, quite differently from Jackson networks are nonlinear. We note that the model also allows for multiple classes.
Gnetworks have been used in a wide range of applications, including to represent Gene Regulatory Networks, the mix of control and payload in packet networks, neural networks, and the representation of colour images and medical images such as Magnetic Resonance Images.
The response time is the length of time a customer spends in the system. The response time distribution for a single Gqueue is known^{[9]} where customers are served using a FCFS discipline at rate μ, with positive arrivals at rate λ^{+} and negative arrivals at rate λ^{−} which kill customers from the end of the queue. The Laplace transform of response time distribution in this situation is^{[9]}^{[10]}
where λ = λ^{+} + λ^{−} and ρ = λ^{+}/(λ^{−} + μ), requiring ρ < 1 for stability.
The response time for a tandem pair of Gqueues (where customers who finish service at the first node immediately move to the second, then leave the network) is also known, and it is thought extensions to larger networks will be intractable.^{[10]}