Graph-Based Product Form
Céline Comte, Isaac Grosof
TL;DR
This work addresses when Markov chains admit a product-form stationary distribution determined by the transition-graph structure rather than specific rates. It introduces graph-based product form and proves two equivalent conditions for a product-form relationship between states $i$ and $j$: an $i,j$-sourced cut and the absence of a joint ancestor, complemented by a graph-traversal algorithm with complexity $O(|V|^2|E|)$ for finite graphs, and extended to higher-level product form. The framework further develops the cut graph concept and explores recursive, higher-level product-form relationships, with queueing-theory examples illustrating the approach. Practically, the framework provides a structural, symbolic route to identify product-form distributions and offers insights beyond traditional rate-based methods, enabling efficient discovery of product-form relations in finite Markov chains.
Abstract
Product-form stationary distributions in Markov chains have been a foundational advance and driving force in our understanding of stochastic systems. In this paper, we introduce a new product-form relationship that we call "graph-based product form". As our first main contribution, we prove that two states of the Markov chain are in graph-based product form if and only if the following two equivalent conditions are satisfied: (i) a cut-based condition, reminiscent of classical results on product-form queueing systems, and (ii) a novel characterization that we call joint-ancestor freeness. The latter characterization allows us in particular to introduce a graph-traversal algorithm that checks product-form relationships for all pairs of states, with time complexity $O(|V|^2 |E|)$, if the Markov chain has a finite transition graph $G = (V, E)$. We then generalize graph-based product form to encompass more complex relationships, which we call "higher-level product form", and we again show these can be identified via a graph-traversal algorithm when the Markov chain has a finite state space. Lastly, we identify several examples from queueing theory that satisfy this product-form relationship.