G-networks: a unifying model for neural and queueing networks

Abstract

We survey results concerning a new stochastic network we have developed [1–7], which was initially motivated by neural network modelling [1], or — as we called it — by queueing networks with positive and negative customers [2, 3]. Indeed, it is well known that signals in neural networks are formed by impulses or action potentials, traveling much like customers in a queueing network. We call this model a G-network because it serves as a unifying basis for diverse areas of stochastic modelling in queueing networks, computer networks, computer system performance and neural networks. In its simplest version, “negative” and “positive” signals or customers circulate among a finite set of units, modelling inhibitory and excitatory signals of a neural network, or “negative and positive customers” of a queueing network. Signals can arrive either from other units or from the outside world. Positive signals are accumulated at the input of each unit, and constitute its signal potential. The state of each unit or neuron is its signal potential (which is equivalent to the queue length), while the network state is the vector of signal potentials at each neuron. If its potential is positive, a unit or neuron fires, and sends out signals to the other neurons or to the outside world. As it does so, its signal potential is depleted. In the Markovian case, this model has product form, i.e. the steady-state probability distribution of its potential vector is the product of the marginal probabilities of the potential at each neuron. The signal flow equations of the network, which describe the rate at which positive or negative signals arrive to each neuron, are non-linear. We discuss the relationship between this model and the usual connectionist (formal) model of neural networks, and present applications to combinatorial optimization and to image texture processing. Extensions of the model to the case of “multiple signal classes”, and to “networks with triggered customer motion” are presented. We also examine the general stability conditions which guarantee that the network has a well-defined steady-state behaviour.