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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Matrix analytic method ： ウィキペディア英語版 Matrix analytic method In probability theory, the matrix analytic method is a technique to compute the stationary probability distribution of a Markov chain which has a repeating structure (after some point) and a state space which grows unboundedly in no more than one dimension. Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. ==Method description== An M/G/1-type stochastic matrix is one of the form〔 ::$P\; =\; \backslash begin$ B_0 & B_1 & B_2 & B_3 & \cdots \\ A_0 & A_1 & A_2 & A_3 & \cdots \\ & A_0 & A_1 & A_2 & \cdots \\ & & A_0 & A_1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end where ''B''''i'' and ''A''''i'' are ''k'' × ''k'' matrices. (Note that unmarked matrix entries represent zeroes.) Such a matrix describes the embedded Markov chain in an M/G/1 queue. If ''P'' is irreducible and positive recurrent then the stationary distribution is given by the solution to the equations〔 ::$P\; \backslash pi\; =\; \backslash pi\; \backslash quad\; \backslash text\; \backslash quad\; \backslash mathbf\; e^\backslash text\backslash pi\; =\; 1$ where e represents a vector of suitable dimension with all values equal to 1. Matching the structure of ''P'', ''π'' is partitioned to ''π''1, ''π''2, ''π''3, …. To compute these probabilities the column stochastic matrix ''G'' is computed such that〔 ::$G\; =\; \backslash sum\_^\backslash infty\; G^i\; A\_i.$ ''G'' is called the auxiliary matrix. Matrices are defined〔 ::$\backslash begin$ \overline_ &= \sum_^\infty G^A_j \\ \overline_i &= \sum_^\infty G^B_j \end then ''π''0 is found by solving〔 ::$\backslash begin$ \overline_0 \pi_0 &= \pi_0\\ \quad \left(\mathbf e^}\left(I - \sum_^\infty \overline_i\right)^\sum_^\infty \overline_i\right) \pi_0 &= 1 \end and the ''π''''i'' are given by Ramaswami's formula,〔 a numerically stable relationship first published by Vaidyanathan Ramaswami in 1988. ::$\backslash pi\_i\; =\; (I-\backslash overline\_1)^\; \backslash left(\backslash overline\_\; \backslash pi\_0\; +\; \backslash sum\_^\; \backslash overline\_\backslash pi\_j\; \backslash right),\; i\; \backslash geq\; 1.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Matrix analytic method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.