A new solution method for level-dependent Markov chains of M/G/1 type and its application to queueing models

• Project/Area Number: 16K00034
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical informatics
• Research Institution: Osaka University
• Principal Investigator: Takine Tetsuya 大阪大学, 工学研究科, 教授 (00216821)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: M/G/1型マルコフ連鎖 / レベル依存 / 数値計算法 / 再呼のある待ち行列 / 途中退去のある待ち行列 / 条件付き定常状態確率 / 誤差上界

Outline of Final Research Achievements

Markov chains of level-dependent M/G/1 type is a class of bivariate Markov chains with countably infinite levels and finite phases in each level, where the level variable can decreases only by one when the state transition occurs. In this research, we established the computational algorithm for the stationary distribution of Markov chains of level-dependent M/G/1 type. Particular features of this algorithm are as follows. It is based on the theory without any heuristics/approximation, and it is applicable any Markov chains of level-dependent M/G/1 type, in contrast to the existing ones, where additional assumptions on the transition structure are imposed.

Academic Significance and Societal Importance of the Research Achievements

本研究の学術的意義は、マルコフ連鎖の定常分布を不等式系の解として特徴づけたところにある。マルコフ連鎖の定常分布は、大域平衡方程式と呼ばれる連立方程式の解として与えられるが、本研究ではその解をある不等式系の解として特徴づけ、この観察を基に数値計算アルゴリズムを開発した。また、従来の数値計算法で仮定していた付加的な仮定を排除しており、適用範囲が格段に広がった。

Research Products

• [Journal Article] Computing the conditional stationary distribution in Markov chains of level-dependent M/G/1-type (2018)
  • Author(s): Masatoshi Kimura and Tetsuya Takine
  • Journal Title: Stochastic Models Volume: 34 Issue: 2 Pages: 207-238
  • DOI: 10.1080/15326349.2018.1451753
  • Peer Reviewed
• [Journal Article] Multi-Class M/PH/1 Queues with Deterministic Impatience Times (2017)
  • Author(s): Y. Sakuma and T. Takine
  • Journal Title: Stochastic Models Volume: 33 Issue: 1 Pages: 1-29
  • DOI: 10.1080/15326349.2016.1197778
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A Computational Algorithm for the Loss Probability in the M/G/1+PH Queue (2017)
  • Author(s): Y. Inoue and T. Takine
  • Journal Title: Stochastic Models Volume: 33 Issue: 1 Pages: 116-148
  • DOI: 10.1080/15326349.2016.1219955
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Analysis and Computation of the Stationary Distribution in a Special Class of Markov Chains of Level-Dependent M/G/1-Type and Its Application to BMAP/M/∞ and BMAP/M/c+M Queues (2016)
  • Author(s): T Takine
  • Journal Title: Queueing Systems Volume: 84 Issue: 1-2 Pages: 49-77
  • DOI: 10.1007/s11134-016-9482-1
  • Peer Reviewed / Acknowledgement Compliant