In queueing theory, a discipline within the mathematical theory of probability, a G-network (generalized queueing network, often called a Gelenbe network) is an open network of G-queues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic re-routing or traffic destruction, as well as a model for neural networks. A G-queue is a network of queues with several types of novel and useful customers:

A product-form solution superficially similar in form to Jackson's theorem, but which requires the solution of a system of non-linear equations for the traffic flows, exists for the stationary distribution of G-networks while the traffic equations of a G-network are in fact surprisingly non-linear, and the model does not obey partial balance. This broke previous assumptions that partial balance was a necessary condition for a product-form solution. A powerful property of G-networks is that they are universal approximators for continuous and bounded functions, so that they can be used to approximate quite general input-output behaviours.

A network of m interconnected queues is a G-network if

A queue in such a network is known as a G-queue.

Define the utilization at each node,

where the ${\displaystyle \scriptstyle {\lambda _{i}^{+},\lambda _{i}^{-}}}$ for ${\displaystyle \scriptstyle {i=1,\ldots ,m}}$ satisfy

Then writing (n₁, ... ,n_m) for the state of the network (with queue length n_i at node i), if a unique non-negative solution ${\displaystyle \scriptstyle {(\lambda _{i}^{+},\lambda _{i}^{-})}}$ 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 ${\displaystyle \pi }$ satisfies the global balance equations which, quite differently from Jackson networks are non-linear. We note that the model also allows for multiple classes.