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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 Jackson's theorem (queueing theory) ： ウィキペディア英語版 Jackson network In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes Jacksonian network) is a class of queueing network where the equilibrium distribution is particularly simple to compute as the network has a product-form solution. It was the first significant development in the theory of networks of queues, and generalising and applying the ideas of the theorem to search for similar product-form solutions in other networks has been the subject of much research, including ideas used in the development of the Internet. The networks were first identified by James R. Jackson and his paper was re-printed in the journal Management Science’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’ Jackson was inspired by the work of Burke and Reich, though Jean Walrand notes "product-form results … () a much less immediate result of the output theorem than Jackson himself appeared to believe in his fundamental paper". An earlier product-form solution was found by R. R. P. Jackson for tandem queues (a finite chain of queues where each customer must visit each queue in order) and cyclic networks (a loop of queues where each customer must visit each queue in order). A Jackson network consists of a number of nodes, where each node represents a queue in which the service rate can be both node-dependent and state-dependent. Jobs travel among the nodes following a fixed routing matrix. All jobs at each node belong to a single "class" and jobs follow the same service-time distribution and the same routing mechanism. Consequently, there is no notion of priority in serving the jobs: all jobs at each node are served on a first-come, first-served basis. Jackson networks where a finite population of jobs travel around a closed network also have a product-form solution described by the Gordon–Newell theorem. ==Definition== In an open network, jobs arrive from outside following a Poisson process with rate $\backslash alpha>0$. Each arrival is independently routed to node ''j'' with probability $p\_\backslash ge0$ and $\backslash sum\_^J\; p\_=1$. Upon service completion at node ''i'', a job may go to another node ''j'' with probability $p\_$ or leave the network with probability $p\_=1-\backslash sum\_^J\; p\_$. Hence we have the overall arrival rate to node ''i'', $\backslash lambda\_i$, including both external arrivals and internal transitions: :$\backslash lambda\_i\; =\backslash alpha\; p\_\; +\; \backslash sum\_^J\; \backslash lambda\_j\; p\_,\; i=1,\backslash ldots,J.\; \backslash qquad\; (1)$ Define $a=(\backslash alpha\; p\_)\_^J$, then we can solve $\backslash lambda=(I-P\text{'})^a$. All jobs leave each node also following Poisson process, and define $\backslash mu\_i(x\_i)$ as the service rate of node ''i'' when there are $x\_i$ jobs at node ''i''. Let $X\_i(t)$ denote the number of jobs at node ''i'' at time ''t'', and $\backslash mathbf=(X\_i)\_^J$. Then the equilibrium distribution of $\backslash mathbf$, $\backslash pi(\backslash mathbf)=P(\backslash mathbf=\backslash mathbf)$ is determined by the following system of balance equations: :$\backslash pi(\backslash mathbf)\; \backslash sum\_^J\; (p\_\; +\backslash mu\_i\; (x\_i)\; (1-p\_))$$=\backslash sum\_^J(\backslash alpha\; p\_+\backslash pi(\backslash mathbf+\backslash mathbf\_i)\backslash mu\_i(x\_i+1)p\_)+\backslash sum\_^J\backslash sum\_\backslash pi(\backslash mathbf+\backslash mathbf\_i-\backslash mathbf\_j)\backslash mu\_i(x\_i+1)p\_.\backslash qquad\; (2)$ where $\backslash mathbf\_i$ denote the $i^$ unit vector. 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「Jackson network」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.