On the structure of the space of product-form models

This paper deals with Markovian models which are defined on a finite-dimensional discrete state space, and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object, and explore its structure. We focus on models on an orthant ${cal Z_+^n$, which are homogeneous within subsets of ${cal Z_+^n$ called walls, and permit only state transitions whose $| ~ |_{infty$-length is $1$. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. The selection space of distinct models which share a given product-form state distribution is accounted for. In addition, we consider models which are homogeneous throughout a finite-dimensional grid ${cal Z^n$, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.