🤖 AI Summary
Existing characterizations of product-form steady-state distributions in Markov chains lack structural, graph-theoretic foundations.
Method: We introduce the novel concept of *graph-based product-form*, wherein the steady-state probability decomposes as a product of functions depending only on the ancestral sets induced by a directed graph structure. Leveraging graph cuts and joint ancestor independence, we establish multiple equivalent characterizations and design an $O(|V|^2|E|)$ graph-traversal algorithm to automatically verify the condition for any state pair. We further generalize the theory to higher-order product-forms and unify the framework under a graph-structural decision procedure.
Results: The approach is validated on classical models—including Jackson networks and BCMP queues—demonstrating both effectiveness and broad applicability. This work provides the first systematic, computationally tractable, graph-theoretic methodology for determining the existence of product-form steady-state distributions.
📝 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.