2023 Fiscal Year Final Research Report
The computation of the stationary distribution in random-walk-type Markov chains: via unraveling the trinity of stability
| Project/Area Number |
21K11770
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 60020:Mathematical informatics-related
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
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| Project Period (FY) |
2021-04-01 – 2024-03-31
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| Keywords | ランダム・ウォーク型マルコフ連鎖 / 定常分布 / 安定性 / 上部ヘッセンベルグ型マルコフ連鎖 / GI/G/1型マルコフ連鎖 / M/G/1型マルコフ連鎖 |
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| Outline of Final Research Achievements |
We have derived several convergence formulas for the last-column block-augmented truncation approximation and the level-increment truncation approximation, as well as computable error upper bounds. We have also developed a quasi-algorithmic solution construction method (a method for generating a sequence of approximate solutions that converge to an exact solution by iterating finite procedures). These have been carried out for M/G/1-type, GI/G/1-type, and upper Block-Hessenberg Markov chains (which are representative random-walk-type Markov chains), considering the 〝Trinity of Stability" in random-walk-type Markov chains. Note that the 〝Trinity of Stability" is the relationship between the three factors related to stability: (i) the tail decay rate of the equilibrium distribution of increments; (ii) the ergodic convergence rate, (iii) the tail decay rate of the stationary distribution.
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| Free Research Field |
応用確率論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は、ランダム・ウォーク(RW)型マルコフ連鎖の安定性に関する新たな視点、「安定性のトリニティ」に注目し、増分平衡分布の裾減衰率、エルゴード収束率、定常分布の裾減衰率という3つの要素の関係性についての知見の活用と深化に取り組んだ。これら3つの要素は、RW型マルコフ連鎖の切断近似の精度と深く関連している。本研究の成果により、RW型マルコフ連鎖の安定性と切断近似の誤差評価についての理論的な理解が深まるとともに、その応用として、待ち行列システムやネットワークの性能評価、リスク評価などの問題に対するより精度の高い解析・数値計算手法の提供につながると期待される。
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