Graph-Based Product Form
C\'eline Comte, Isaac Grosof

TL;DR
This paper introduces a new graph-based product-form relationship for Markov chains, providing a characterization, an efficient algorithm for checking it, and generalizations to higher-level forms, with applications in queueing theory.
Contribution
It presents a novel graph-based product-form characterization, an algorithm for verification, and extends the concept to higher-level forms with practical queueing examples.
Findings
Characterization of graph-based product form via cut-based and joint-ancestor freeness conditions
Development of a graph-traversal algorithm with $O(|V|^2 |E|)$ complexity for checking product form
Identification of queueing theory examples satisfying the new product-form relationship
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 , if the Markov chain has a finite transition graph . We then generalize graph-based product form…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsManufacturing Process and Optimization
