Project/Area Number 
11640116

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Osaka University 
Principal Investigator 
AKI Sigeo Osaka University, Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (90132696)

CoInvestigator(Kenkyūbuntansha) 
TANIGUCHI Masanobu Osaka University Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (00116625)
INAGAKI Nobuo Osaka University Graduate School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (10000184)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)

Keywords  discrete patterns / discrete distribution theory / directed tree / probability generating function / Markov chain / system reliability / binomial distribution / waiting time problems / 離散分布 / パターン / 連 / 工学的システム / 信頼性 / 経験分布関数 
Research Abstract 
We have studied waiting time distributions of runs and patterns in random structures such as sequences of random variables and random directed trees. The following results are derived. 1. We have obtained the exact distribution of the number of "1" runs of a specified length on {0, 1} valued Markov trees. This result can be applied to calculate the reliability of a consecutive system on a directed tree. We also obtained the exact distribution of the life time of the consecutive system on the directed tree. 2. We have studied exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We have given systems of linear equations with respect to conditional probability generating functions of the waiting times and have solved them. This result can be applied to calculate the reliability of the linear connected (r, s) outof (r+1, n) : F lattice system. 3. We have introduced a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variables of returnable type and have discussed some of their properties. By using the result we have derived the distribution of numbers of occurrences of runs of specified lengths in a sequence of multistate trials. 4. We have introduced a unified counting scheme for runs called 1overlapping counting. We have given exact probability generating function of the number of 1overlapping 1runs of a specified length in some dependent random sequences such as a Markov chain and a heigher order Markov chain. 5. We have introduced a new type of dependent sequence called a binary sequence of order (k, r) and have derived the exact distributions of sooner and later waiting times for success and failure runs in the sequence.
