Extraction of exact symbolic stationary probability formulas for Markov chains with finite space with application to production lines. Part I: description of methodology - Boulas, Konstantinos S.; Dounias, Georgios D.; Papadopoulos, Chrissoleon T. (Frontiers in Manufacturing Technology, 2025)
IntroductionMarkov 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.MethodThe 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.ResultsA 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.DiscussionThe 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.