G-Networks: Development of the Theory of Multiplicative Networks

This is a review on G-networks, which are the generalization of the Jackson and BCMP networks, for which the multi-dimensional stationary distribution of the network state probabilities is also represented in product form. The G-networks primarily differ from the Jackson and BCMP networks in that they additionally contain a flow of the so-called negative customers and/or triggers. Negative customers and triggers are not served. When a negative customer arrives at a network node, one or a batch of positive (ordinary) customers is killed (annihilated, displaced), whereas a trigger displaces a positive customer from the node to some other node. For applied mathematicians, G-networks are of great interest for extending the multiplicative theory of queueing networks and for practical specialists in modeling computing systems and networks and biophysical neural networks for solving pattern recognition and other problems.