The first product-form solutions were found for
equilibrium distributions of
Markov chains. Trivially, models composed of two or more
independent sub-components exhibit a product-form solution by the definition of independence. Initially the term was used in
queueing networks where the sub-components would be individual queues. For example,
Jackson's theorem gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues. After numerous extensions, chiefly the
BCMP network it was thought
local balance was a requirement for a product-form solution.
Gelenbe's
G-network model was the first to show that this is not the case. Motivated by the need to model biological neurons which have a point-process like spiking behaviour, he introduced the precursor of G-Networks, calling it the
random neural network. By introducing "negative customers" which can destroy or eliminate other customers, he generalised the family of product form networks. Then this was further extended in several steps, first by Gelenbe's "triggers" which are customers which have the power of moving other customers from some queue to another. Another new form of customer that also led to product form was Gelenbe's "batch removal". This was further extended by Erol Gelenbe and Jean-Michel Fourneau with customer types called "resets" which can model the repair of failures: when a queue hits the empty state, representing (for instance) a failure, the queue length can jump back or be "reset" to its steady-state distribution by an arriving reset customer, representing a repair. All these previous types of customers in G-Networks can exist in the same network, including with multiple classes, and they all together still result in the product form solution, taking us far beyond the reversible networks that had been considered before. Product-form solutions are sometimes described as "stations are independent in equilibrium". Product form solutions also exist in networks of
bulk queues.
J.M. Harrison and
R.J. Williams note that "virtually all of the models that have been successfully analyzed in classical queueing network theory are models having a so-called product-form stationary distribution") and
stochastic petri nets.
Martin Feinberg's deficiency zero theorem gives a sufficient condition for
chemical reaction networks to exhibit a product-form stationary distribution. The work by Gelenbe also shows that product form G-Networks can be used to model spiking
random neural networks, and furthermore that such networks can be used to approximate bounded and continuous real-valued functions. ==Sojourn time distributions==