ORIGINAL RESEARCH article

Front. Manuf. Technol.

Sec. Sustainable Life Cycle Engineering and Manufacturing

Volume 5 - 2025 | doi: 10.3389/fmtec.2025.1439421

Extraction of Exact Symbolic Stationary Probability Formulas for Markov Chains withFinite Space with Application to Production Lines. Part I: Description of Methodology

Provisionally accepted
  • 1University of the Aegean, Mytilene, North Aegean, Greece
  • 2Aristotle University of Thessaloniki, Thessaloniki, Greece

The final, formatted version of the article will be published soon.

Introduction: Markov chains are a powerful tool for modeling systems in various scientific domains, including queueing theory. These models are characterized by their ability to maintain complexity at a low level due to a property known as the Markov property, which enables the connection between states and transition probabilities. The transition matrices of Markov chains are represented by graphs, which show the properties and characteristics that help analyze the underlying processes.Method: The graph representing the transition matrix of a Markov chain is formed from the transition state diagram, with weights representing the mean transition rates. A probability space is thus created, containing all the spanning trees of the graph that end up in the states of the Markov chain (anti-arborescences). A successive examination of the graph's vertices is initiated to form monomials as products of the weights of the edges forming the symbolic solution.Results: A general algorithm that commences with the Markov chain transition matrix as an input element and forms the state transition diagram. Subsequently, each vertex within the graph is examined, followed by a rearrangement of the vertices according to a depth-first search strategy. In the context of an inverted graph, implementing a suitable algorithm for forming spanning trees, such as the Gabow and Myers algorithm, is imperative. This algorithm is applied sequentially, resulting in the formation of monomials, polynomials for each vertex, and, ultimately, the set of polynomials of the graph. Utilizing these polynomials facilitates the calculation of the stationary probabilities of the Markov chain and the performance metrics.Discussion: The proposed method provides a positive response to the inquiry regarding the feasibility of expressing the performance metrics of a system modeled by a Markov chain through closed-form equations. The study further posits that these specific equations are of considerable magnitude. The intricacy of their formulation enables their implementation in smaller systems, which can serve as building blocks for other methodologies. The correlation between Markov chains and graphs has the potential to catalyze novel research directions in both discrete mathematics and artificial intelligence.

Keywords: Graph theory1, Markov chains2, anti-arborescence3, closed form formulas4, stationary probabilities5, spanning trees6

Received: 27 May 2024; Accepted: 02 Jun 2025.

Copyright: © 2025 Boulas, Dounias and Papadopoulos. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Chrissoleon Papadopoulos, Aristotle University of Thessaloniki, Thessaloniki, Greece

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