Computational algorithms for the stationary distribution of Markov chains based on the system of inequalities and its application to queueing models

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K11841
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60020:Mathematical informatics-related
• Research Institution: Osaka University
• Principal Investigator: Takine Tetsuya 大阪大学, 大学院工学研究科, 教授 (00216821)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: マルコフ連鎖 / 条件付き定常分布 / 不等式系 / 数値計算法 / 待ち行列モデル

Outline of Final Research Achievements

We have developed a new computational method for the conditional stationary distribution in continuous-time Markov chains. In previous studies, the (conditional) stationary distribution is characterized as the solution of a system of linear equations, and a numerical calculation method has been developed based on this. In this study, on the other hand, the conditional stationary distribution in a general Markov chain is characterized by a system of linear inequalities obtained from the information contained in the northwest corner of the transition rate matrix, and a new computational method with guaranteed accuracy for the conditional stationary distribution was developed based on this. Furthermore, based on the developed numerical calculation method, we have established a numerical solution method with guaranteed accuracy for a queueing model in which the arrival rate and disaster rate are level-dependent, which could not be handled by conventional matrix analysis methods.

Free Research Field

待ち行列理論

Academic Significance and Societal Importance of the Research Achievements

マルコフ連鎖の定常分布は平衡方程式と呼ばれる等式により特徴付けられる。それ故、従来の研究では等式を元にした解法が議論されてきた。本研究では、遷移確率行列の北西角が持つ定常分布に関する情報を不等式で表現し、解が存在する領域（解空間）を明示的に与えた。さらに、北西角に含まれている状態の内、北西角に含まれない状態から１ステップで到達可能な態の集合が与えられれば、解空間が多面体の相対的内部で与えられることを示した。特別な構造をもたないマルコフ連鎖の定常分布の性質はほとんど議論されておらず、本研究の成果は、今後、さらなる理論の進化に貢献すると思われる。