Product Form Solution - Equilibrium Distributions

Equilibrium Distributions

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 showed this to not be the case. 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" More recently, product-form solutions have been published for Markov process algebras (e.g. RCAT in PEPA) 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.