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.

Definition

A network of m interconnected queues is a G-network if A queue in such a network is known as a G-queue.

Stationary distribution

Define the utilization at each node, where the for satisfy Then writing (n1, ... ,nm) for the state of the network (with queue length ni at node i), if a unique non-negative solution exists to the above equations and such that ρi for all i then the stationary probability distribution π exists and is given by

Proof

It is sufficient to show \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. G-networks 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.

Response time distribution

The response time is the length of time a customer spends in the system. The response time distribution for a single G-queue is known 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 where λ = λ+ + λ− and ρ = λ+/(λ− + μ), requiring ρ < 1 for stability. The response time for a tandem pair of G-queues (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.