JP3774119B2 - Rate function measuring device and method, program, and recording medium - Google Patents

Description

【００２６】
例えば、確率変数Ａ_tとして時間ｔ内に転送されるデータ量（例えば、セル数やパケット数）を扱う場合、時間ｔを十分大きくとると、確率量Ａ_t／ｔの分布は、レートファンクションで表される所定の関数形で表現される。
したがって、確率量Ａ_t／ｔが任意の値γより大きい確率分布Ｐ｛Ａ_t≧γｔ｝は、次のように近似できる。
【００２７】
【数１８】

【００２８】
また、パラメータｔがｋ個の離散的な値をとる場合の確率変数Ａ_tを、パラメータｔの各離散値ごとの確率変数ｘ_kで定義した場合、レートファンクションＩ（γ）は、次のように表すことができる。
【００２９】
【数１９】

【００３０】
これを物理的に解釈すると、上記式は、確率変数Ａ_tのサンプル平均Ａ_t／ｔがγ以上である確率が、ｔを無限大としたとき、ｔの増加に対してどのような割合でその確率が減少していくかを示していることになる。なお、確率変数Ａ_tのサンプル平均Ａ_t／ｔがγ以下の場合についても、確率分布Ｐ｛Ａ_t≦γｔ｝として同様な議論を行うことができる。
【００３１】
本発明では、このような確率分布Ｐ｛Ａ_t≦γｔ｝のキュムラント（対数）を所定の近似式、例えば直交多項式またはべき関数の線形和などの近似式で近似することにより、所望のレートファンクションを算出するものとし、その近似式で用いられる直交多項式またはべき関数の各次数に対応するパラメータａ_t,kを、その確率変数Ａ_tの実現値から求めたキュムラント値を用いて回帰分析などにより算出するようにしたものである。
【００３２】
次に、演算処理部１の構成および動作について詳細に説明する。図２に演算処理部の詳細な構成を示す。
以下では、解析対象システムとしてＡＴＭ通信システムを想定し、確率変数Ａ_tとしてそのシステムの観測点において転送されるセル数をとり、そのパラメータｔとして確率変数Ａ_tの実現値（ここではセル数）を計数する計測期間を用いる場合を例として説明する。
【００３３】
計測手段１０には、実現値計測手段１０Ａと移動平均算出手段１０Ｂとが設けられている。
実現値計測手段１０Ａでは、入力Ｉ／Ｆ部２で取り込まれたＡＴＭ通信システムからの状態信号Ｄ＿ＩＮ（例えば、セル転送タイミングを示すパルス信号など）から、パラメータｔ₁を有する確率変数Ａ_t1の実現値として、計測期間ｔ₁ごとに転送されたセル数を計数し、その計数値を確率変数Ａ_t1のデータ群Ａ_t1（）として出力する。
【００３４】
図３に実現値計測手段の処理手順を示す。
実現値計測手段１０Ａでは、まず、操作入力部５から計測期間としてｔ₁を設定するとともに（ステップ１００）、レートファンクション算出の周期としてＴ（Ｔ≫ｔ₁）を設定し（ステップ１０１）、記憶部４（または演算処理部１内）に設けられた制御カウンタｎをクリアする（ステップ１０２）。
そして、Ｄ＿ＩＮを元にした転送セル数の計数を開始し（ステップ１０３）、計数開始から計測期間ｔ₁だけ経過するまで計数を継続する（ステップ１０４：ＮＯ）。
【００３５】
続いて、計測期間ｔ₁経過後（ステップ１０４：ＹＥＳ）、記憶部４に設けられたデータ群を構成する配列Ａ_t1（ｎ）へその計数値を格納し（ステップ１０５）、制御カウンタｎを１だけインクリメントする（ステップ１０６）。
このようなステップ１０３〜１０６までの計数処理を、周期Ｔが経過するまで繰り返し実施し（ステップ１０７：ＮＯ）、周期Ｔ経過後（ステップ１０７：ＹＥＳ）、得られたデータ群Ａ_t1（）を記憶部４に出力して（ステップ１０８）、実現値計測にかかる一連の処理を終了する。
【００３６】
移動平均算出手段１０Ｂでは、ｔ₁とは異なる計測期間ｔ₂，ｔ₃ごとに転送されたセル数を計数し、その移動平均値を算出出力する。このとき、ｔ₂，ｔ₃としてそれぞれｔ₁の整数倍の値ｎ₂ｔ₁，ｎ₃ｔ₁（但し、ｎ₂，ｎ₃は２以上の整数で、ｎ₂＜ｎ₃）を選択する。
図４に移動平均算出手段での処理手順を示す。
移動平均処理手順１０Ｂでは、まず、操作入力部５から計測期間としてｔ₂を設定するとともに（ステップ１１０）、データ群Ａ_t1（）に対する平均算出データ数ｍとしてｎ₂を設定し（ステップ１１１）、制御カウンタｎをクリアする（ステップ１０２）。
【００３７】
続いて、データ群Ａ_t1（）のうちｎ番目からｎ＋ｍ−１番目までのｍ個の実現値を読み込み（ステップ１１３）、これらｍ個の実現値の平均値を算出する（ステップ１１４）。そして、その平均値をデータ群を構成する配列Ａ_t2（ｎ）へ格納し（ステップ１１５）、制御カウンタｎを１だけインクリメントする（ステップ１１６）。
このようなステップ１１３〜１１６までの計数処理を、Ｔ／ｔ₁−ｍ＋１回だけ繰り返し実施し（ステップ１１７）、得られたデータ群Ａ_t2（）を記憶部４に出力する（ステップ１１８）。
さらに、上記ステップ１１０〜１１８を計測期間ｔ₃について繰り返し実施することにより、データ群Ａ_t3（）を算出して記憶部４に出力し（ステップ１１９）、一連の処理を終了する。
【００３８】
確率分布推定手段１１には、キュムラント算出手段１１Ａと近似式算出手段１１Ｂとが設けられている。
キュムラント算出手段１１Ａでは、計測手段１０で計測されたデータ群Ａ_t1（），Ａ_t2（），Ａ_t3（）に基づき、確率変数Ａ_t1，Ａ_t2，Ａ_t3の確率補分布の対数値を示すキュムラント群Ｃ_t1（），Ｃ_t2（），Ｃ_t3（）を算出出力する。
【００３９】
図５にキュムラント算出手段の処理手順を示す。
キュムラント算出手段１１Ａでは、まず、記憶部４に記憶されているデータ群Ａ_t1（）のうちから、いずれかの未処理データ（γｔ₁）を選択し（ステップ１２０）、以下の式に基づき、データ群Ａ_t1（）のうちγｔ₁以上となるデータに関する確率分布のキュムラントを算出する（ステップ１２１）。
【００４０】
【数２０】

【００４１】
これにより算出された値をその未処理データに対応するキュムラント値として、キュムラント群を構成する配列Ｃｔ₁（）へ格納する（ステップ１２２）。
このようなステップ１２０〜１２２までのステップを、データ群Ａ_t（）のすべてのデータについて繰り返し実施し（ステップ１２３）、得られたキュムラント群Ｃ_t1（）を記憶部４に出力する（ステップ１２４）。
さらに、上記ステップ１２０〜１２４を計測期間ｔ₂，ｔ₃についてそれぞれ繰り返し実施することにより、キュムラント群Ｃ_t2（），Ｃ_t3（）を算出して出力し（ステップ１２５）、一連の処理を終了する。
【００４２】
近似式算出手段１１Ｂでは、確率変数Ａ_t1，Ａ_t2，Ａ_t3で共通の直交多項式やべき関数の線形和を用いた近似式で、各確率変数Ａ_t1，Ａ_t2，Ａ_t3ごとにそれぞれの確率分布のキュムラントを近似し、それぞれのキュムラント群Ｃ_t1（），Ｃ_t2（），Ｃ_t3（）を元にして各近似式に含まれるパラメータを回帰分析などにより算出することにより、各近似式を求める。
図６に近似式算出手段１１Ｂの処理手順を示す。
近似式算出手段１１Ｂでは、まず、確率変数Ａ_t1について、その確率補分布のキュムラントを複数の関数ｆ₁（γ），ｆ₂（γ），…，ｆ_k（γ）（ｋは２以上の整数）とパラメータａ₁₁，ａ₁₂，…，ａ_1kの積和演算式からなる次の近似式Ｆ₁（γ）で近似する。
【００４３】
【数２１】

【００４４】
そして、キュムラント算出手段１１Ａで算出した確率変数Ａ_t1に関するキュムラント群Ｃ_t1（）を元にして、任意のγに対するキュムラント値をその近似式にそれぞれ当てはめて回帰分析などを行うことにより、近似式のパラメータａ₁₁，ａ₁₂，…，ａ_1kを算出し、確率変数Ａ_t1における近似式を求める。（ステップ１３０）。このとき、例えば最小二乗法により任意のγに対するキュムラント値とその近似式による近似値との誤差の二乗の総和が最小となるように、パラメータａ₁₁，ａ₁₂，…，ａ_1kを選択すればよい。
【００４５】
同様にして、確率変数Ａ_t2，Ａ_t3について、その確率分布のキュムラントを次の近似式でそれぞれ近似し、キュムラント群Ｃ_t2（），Ｃ_t3（）を元にして回帰分析などを行う。このとき、近似式としては、確率変数Ａ_t1で用いたものと同じ関数ｆ₁（γ），ｆ₂（γ），…，ｆ_k（γ）から構成され、これら関数ｆに乗算されるパラメータａ_i1，ａ_i2，…，ａ_ikのみが異なる近似式Ｆ_i（γ）を用いる。
【００４６】
【数２２】

【００４７】
これにより、近似式のパラメータａ₂₁，ａ₂₂，…，ａ_2kおよびパラメータａ₃₁，ａ₃₂，…，ａ_3kをそれぞれ算出し、確率変数Ａ_t2，Ａ_t3における近似式を求め（ステップ１３１，１３２）、算出したこれらパラメータａ_i1，ａ_i2，…，ａ_ik（但し、ｉ＝１，２，３）を記憶部４に記憶して、一連の処理を終了する。
【００４８】
レートファンクション算出手段１２には、仮レートファンクション算出手段１２Ａ、平均算出手段１２Ｂ、出力値補正手段１２Ｃ、信頼区間算出手段１２Ｄが設けられている。
仮レートファンクション算出手段１２Ａでは、近似式算出手段１１Ｂで算出された、確率変数Ａ_t1，Ａ_t2，Ａ_t3における各近似式Ｆ₁（γ），Ｆ₂（γ），Ｆ₃（γ）から、仮レートファンクションＩ'₂（γ），Ｉ'₃（γ）を算出する。
【００４９】
図７に仮レートファンクション算出手段１２Ａの処理手順を示す。
仮レートファンクション算出手段１２Ａでは、まず、記憶部４に記憶されたパラメータａ_i1，ａ_i2，…，ａ_ikを読み出し、近似式算出手段１２で算出された各近似式を読み込む（ステップ１４０）。そして、次の式に基づき仮レートファンクションＩ'₂（γ），Ｉ'₃（γ）を求めて（ステップ１４１）、一連の処理を終了する。
【００５０】
【数２３】

【００５１】
図８に平均値算出手段１２Ｂの処理手順を示す。
平均算出手段１２Ｂでは、計測手段１０の実現値計測手段１０Ａで計測された確率変数Ａ_t1に関するデータ群Ａ_t1（）から、その標本平均値γ_AVGを算出し（ステップ１５０）、一連の処理を終了する。
【００５２】
出力値補正手段１２Ｃでは、レートファンクションＩ（γ）がγ＝標本平均値γ_AVGのとき、Ｉ（γ）＝０となる性質に基づき、平均算出手段１２Ｂで算出された標本平均値γ_AVGを用いて、仮レートファンクション算出手段１２Ａで算出された仮レートファンクションＩ'₂（γ），Ｉ'₃（γ）を補正することにより所望のレートファンクションＩを算出する。
【００５３】
図９に出力値補正手段１２Ｃの処理手順を示す。
出力値補正手段１２Ｃでは、まず、γに対して平均算出手段１２Ｂで算出された標本平均値γ_AVGを設定する（ステップ１６０）。次に、仮レートファンクション算出手段１２Ａで求められた仮レートファンクションＩ'₂（γ）を元にして、γを補正するγ₂を用いて表現した、次のようなレートファンクションＩ₂（γ）を定義する。
【００５４】
【数２４】

【００５５】
そして、Ｉ₂（γ_AVG）＝０となるようにγ₂を選択して上記式を補正し、所望のレートファンクションＩ₂（γ）の式を求める（ステップ１６１）。
同様にして、仮レートファンクション算出手段１２Ａで求められた仮レートファンクションＩ'₃（γ）を元にして、γを補正するγ₃を用いて表現した、次のようなレートファンクションＩ₃（γ）を定義する。
【００５６】
【数２５】

【００５７】
そして、Ｉ₃（γ_AVG）＝０となるようにγ₃を選択して上記式を補正し、所望のレートファンクションＩ₃（γ）の式を求め（ステップ１６２）、一連の処理を終了する。
【００５８】
信頼区間算出手段１２Ｄでは、出力値補正手段１２Ｃで求められたレートファンクションＩ₂（γ），Ｉ₃（γ）の値の差が、所定の許容誤差ε内に収まるγの範囲γ_min，γ_maxを求め、レートファンクションＩ₂（γ）またはＩ₃（γ）とともに記憶部４または表示部６に出力する。理論的にレートファンクションＩ₂（γ），Ｉ₃（γ）はいずれも同じ関数となるが、有限の測定結果に基づき近似したレートファンクションには誤差が含まれる。
したがって、異なるパラメータｔ₂，ｔ₃について得られたレートファンクションＩ₂（γ）とレートファンクションＩ₃（γ）との差が、所望の許容誤差ε内に収まるγの範囲を提示することにより、所望の信頼性をもってレートファンクションを利用できる。
【００５９】
図１０に信頼区間算出手段の処理手順を示す。
まず、信頼区間算出手段１２Ｄでは、レートファンクションに対する所望の許容誤差をεに設定し（ステップ１７０）、次の式を満足するγの範囲γ_min，γｍａｘを求める（ステッ１７１）。
【００６０】
【数２６】

【００６１】
そして、得られたγの範囲γ_min，γ_maxとレートファンクションＩ₂（γ）またはＩ₃（γ）とを出力し（ステップ１７２）、一連の処理を終了する。
【００６２】
このように、演算処理部１において、解析対象となる確率システムからパラメータｔを有する確率変数Ａ_tの実現値を計測手段１０で計測し、この計測結果に基づき、確率変数Ａ_tをパラメータｔで除算して得られる確率量Ａ_t／ｔが任意の値γ以上となる確率変数Ａ_tの確率補分布Ｐ｛Ａ_t≧γｔ｝を確率分布推定手段１１で推定し、これにより得られた確率補分布Ｐ｛Ａ_t≧γｔ｝の分布状態を示すレートファンクションＩ（γ）をレートファンクション算出手段１２で算出出力するようにしたので、対象となるシステムからレートファンクションを直接計測することが可能となる。
したがって、極限定理の１つであ大偏差原理によって得られる結果（理論）を、実際に利用することが可能になり、大規模化、大容量化したシステムなど極限定理が成立する領域における確率システムの特性を把握でき、性能評価、目標品質を満足する設備設計などへの応用が可能になる。
【００６３】
また、確率分布推定手段１１のキュムラント算出手段１１Ａで、計測手段１０により計測された確率変数Ａ_tの実現値から、確率補分布Ｐ｛Ａ_t≧γｔ｝の対数値logＰ｛Ａ_t≧γｔ｝をキュムラントとして複数算出し、近似式算出手段１１Ｂで、対数値logＰ｛Ａ_t≧γｔ｝をγに関する所定の関数ｆ_i（γ）とパラメータａ_iとの積和演算式からなる近似式Ｆ（γ）を用いて近似し、キュムラント算出手段１１Ａにより算出された各キュムラントを用いてパラメータａ_iを求めることにより近似式を求めるようにしたので、確率変数Ａ_tの実現値を用いて正確に確率補分布Ｐ｛Ａ_t≧γｔ｝を推定できる。
【００６４】
また、確率分布推定手段１１で、異なる２つのパラメータｔ₁，ｔ₂における確率補分布Ｐ｛Ａ_t1≧γｔ₁｝およびＰ｛Ａ_t2≧γｔ₂｝の対数値logＰ｛Ａ_t1≧γｔ₁｝およびlogＰ｛Ａ_t2≧γｔ₂｝をそれぞれ推定し、レートファンクション算出手段１２の仮レートファンクション算出手段１２Ａで、推定手段で得られた対数値logＰ｛Ａ_t1≧γｔ₁｝およびlogＰ｛Ａ_t2≧γｔ₂｝を用いて、仮レートファンクションＩ'₂（γ）を求め、出力値補正手段１２Ｃで、標本平均値γで０（ゼロ）となるように仮レートファンクションＩ'₂（γ）を補正しレートファンクションＩ₂（γ）として出力するようにしたので、γ＝標本平均値の場合にＩ（γ）＝０となるレートファンクションＩ（γ）の性質に則した誤差の少ないレートファンクションを算出できる。
【００６５】
また、確率分布推定手段１１で、上記と同様にしてパラメータｔ₁，ｔ₂とは異なるパラメータｔ₃における確率補分布Ｐ｛Ａ_t3≧γｔ₃｝の対数値logＰ｛Ａ_t3≧γｔ₃｝を推定するとともに、レートファンクション算出手段１２で、その対数値logＰ｛Ａ_t3≧γｔ₃｝を用いて得られた仮レートファンクションＩ'₃（γ）を補正しレートファンクションＩ₃（γ）として出力し、さらに信頼区間算出手段１２Ｄで、パラメータｔ₂，ｔ₃について得られたレートファンクションＩ₂（γ）およびＩ₃（γ）の差が所定の許容絶対誤差ε以下となるγの範囲を信頼区間として求めるようにしたので、有限の測定結果に基づき近似したレートファンクションに含まれる誤差が、所望の許容絶対誤差ε内に収まるγの範囲を提示することができ、得られたレートファンクションを所望の信頼性をもって利用できる。
【００６６】
【発明の効果】
以上説明したように、本発明は、計測手段において、解析対象となる確率システムからパラメータｔを有する確率変数Ａ_tの実現値を計測し、確率分布推定手段において、この計測結果に基づき、確率変数Ａ_tをパラメータｔで除算して得られる確率量Ａ_t／ｔが任意の値γ以上となる確率変数Ａ_tの確率補分布Ｐ｛Ａ_t≧γｔ｝を推定し、レートファンクション算出手段において、確率分布推定手段により得られた確率補分布Ｐ｛Ａ_t≧γｔ｝の分布状態を示すレートファンクションＩ（γ）を算出するようにしたものである。
そして、確率分布推定手段により、計測された確率変数Ａ _t の実現値から、確率補分布Ｐ｛Ａ _t ≧γｔ｝の対数値 log Ｐ｛Ａ _t ≧γｔ｝をキュムラントとして複数算出し、確率補分布Ｐ｛Ａ _t ≧γｔ｝の対数値 log Ｐ｛Ａ _t ≧γｔ｝を、γに関する所定の関数ｆ _i （γ）とパラメータａ _t,i との積和演算式からなる近似式Ｆ _t （γ）を用いて近似し、各キュムラントを用いてパラメータａ _t,i を求めることにより近似式を求めるようにしたものである。
さらに、確率分布推定手段により、異なる２つのパラメータｔ ₁ ，ｔ ₂ における確率補分布Ｐ｛Ａ _t1 ≧γｔ ₁ ｝およびＰ｛Ａ _t2 ≧γｔ ₂ ｝の対数値 log Ｐ｛Ａ _t1 ≧γｔ ₁ ｝および log Ｐ｛Ａ _t2 ≧γｔ ₂ ｝をそれぞれ推定し、レートファンクション算出手段により、確率分布推定で得られた対数値 log Ｐ｛Ａ _t1 ≧γｔ ₁ ｝および log Ｐ｛Ａ _t2 ≧γｔ ₂ ｝を用いて、仮レートファンクションＩ ' ₂ （γ）を求め、確率変数Ａ _t1 の実現値から得られた標本平均値γ _AVG で０（ゼロ）となるように仮レートファンクションＩ ' ₂ （γ）を補正しレートファンクションＩ ₂ （γ）として出力するようにしたものである。
これにより、対象となるシステムからレートファンクションを直接計測することが可能となる。
したがって、極限定理の１つである大偏差原理によって得られる結果（理論）を、実際に利用することが可能になり、大規模化したシステムや大容量化したシステムなど、極限定理が成立する領域における確率システムの特性を把握でき、対象となるシステムの性能評価、目標品質を満足する設備設計などへの応用が可能になる。
【図面の簡単な説明】
【図１】 本発明の一実施の形態によるレートファンクション測定装置を示すブロック図である。
【図２】 演算処理部の詳細を示すブロック図である。
【図３】 演算処理部の実現値計測手段における処理手順を示すフローチャートである。
【図４】 演算処理部の移動平均算出手段における処理手順を示すフローチャートである。
【図５】 演算処理部のキュムラント算出手段におけるの処理手順を示すフローチャートである。
【図６】 演算処理部の近似式算出手段における処理手順を示すフローチャートである。
【図７】 演算処理部の平均算出手段における処理手順を示すフローチャートである。
【図８】 演算処理部の仮レートファンクション算出手段における処理手順を示すフローチャートである。
【図９】 演算処理部の出力値補正手段における処理手順を示すフローチャートである。
【図１０】 演算処理部の信頼区間算出手段における処理手順を示すフローチャートである。
【符号の説明】
１…演算処理部、１０…計測手段、１０Ａ…実現値計測手段、１０Ｂ…移動平均算出手段、１１…キュムラント算出手段、１１Ｂ…近似式算出手段、１２…レートファンクション算出手段、１２Ａ…仮レートファンクション算出手段、１２Ｂ…平均算出手段、１２Ｃ…出力値補正手段、１２Ｄ…信頼区間算出手段、２…入力Ｉ／Ｆ部、３…記憶媒体、３１…プログラム、４…記憶部、５…操作入力部、６…表示部。[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a rate function measuring apparatus and method, a program, and a recording medium, and more particularly to a rate function measuring apparatus related to a rate function that plays a central role in the limit theorem used in performance evaluation of various systems including a communication system, and The present invention relates to a method, a program, and a recording medium.
[0002]
[Prior art]
Various systems such as computer systems and communication systems have been regarded as stochastic systems and have been analyzed. In order to analyze, first, a target system is modeled, and the model is analyzed. As is well known, in a telephone communication system, the arrival of a telephone call is modeled with high accuracy by the Poisson process, and the call hold time is modeled with high accuracy by an exponential distribution.
Conventionally, when a certain system is analyzed as described above, the system has been modeled and performance evaluation is performed by analyzing the model.
[0003]
In recent years, various systems such as computer systems and communication systems have been increased in scale and capacity. However, it is not easy to model such a large-scale or large-capacity system, and it is often very difficult to analyze even if it is modeled. As described above, it is very difficult to evaluate the performance of a large-scale system or a large-capacity system in the same manner using conventional techniques.
In such large-scale and large-capacity systems, it is known that various limit theorems in probability theory give good approximations in system performance evaluation. In the area where the limit theorem holds, the characteristics of the stochastic system are expressed by the rate function. The rate function is a parameter that represents the shape of the tail of the distribution of the target probability amount.
[0004]
[Problems to be solved by the invention]
In modeling and analysis of a stochastic system using such limit theorems, it is important to derive a rate function. For example, `` Duffield et al: '' Entropy of ATM Traffic Streams: A Tool for Estimating QoS Parameters, IEEE JSACVol13, No.6,1995 function of the ATM traffic stream ”, and it is known that the rate function plays a central role in the limit theorem, such as the large deviation principle.
[0005]
However, conventionally, there has been no idea of directly measuring such a rate function from the target system. This is because the above-mentioned various systems such as computer systems and communication systems handle a small amount of probability, and thus cannot be directly measured from the target system. It seems that it was thought that it could not be used for modeling and analysis of stochastic systems using the theorem.
An object of the present invention is to provide a rate function measuring apparatus and method, a program, and a recording medium capable of directly measuring a rate function from a target system.
[0006]
[Means for Solving the Problems]
In order to achieve such an object, a rate function measuring apparatus according to the present invention, when analyzing a stochastic system based on the limit theorem, measures a rate function that expresses a distribution of random variables of the stochastic system. In a measuring device, a random variable A having a parameter t from a stochastic system to be analyzed_tAnd a random variable A based on the measurement result of the measurement means and the measurement result of the measurement means._tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_tProbability distribution estimation means for estimating ≧ γt} and the probability complementary distribution P {A obtained by this estimation means_tRate function calculating means for obtaining a rate function I (γ) indicating a shape of ≧ γt}.
[0007]
  In addition to this,In the probability distribution estimating means, the random variable A measured by the measuring means in the cumulant calculating means._tFrom the actual value of the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is calculated as a cumulant, and the approximate expression calculation means calculates the probability complement distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is a predetermined function f related to γ_i(Γ) and parameter a_{t, i}Approximation formula F consisting of product-sum operation formula with_tThe parameter a is approximated using (γ) and each cumulant calculated by the cumulant calculating means is used._{t, i}The approximate expression is obtained by obtaining.
[0008]
  further,Two different parameters t in the probability distribution estimation means₁, T₂Probability distribution P {A in_t1≧ γt₁} And P {A_t2≧ γt₂} Logarithm logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂} And rate function calculation meansThenTemporary rate function calculation meansBy means of probability distribution estimationThe obtained logarithmic value logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂}, The provisional rate function I ′₂(Γ) is obtained, and the output value correction means calculates the random variable A_t1Sample mean γ obtained from realization of_AVGTemporary rate function I 'to be 0 (zero) at₂Rate function I after correcting (γ)₂This is output as (γ).
[0009]
  On this occasion,Probability distribution estimation means, parameter t₁, T₂Parameter t different from_ThreeProbability distribution P {A in_t3≧ γt_Three} Logarithm logP {A_t3≧ γt_Three}, And rate function calculation meansIsTemporary rate function calculation meansBy means of probability distribution estimationThe obtained logarithmic value logP {A_t3≧ γt_Three}, The provisional rate function I ′_Three(Γ) is obtained, and the sample average value γ is obtained by the output value correcting means._AVGTemporary rate function I 'to be 0 (zero) at_ThreeRate function I after correcting (γ)_Three(Γ), and the parameter t is calculated by the confidence interval calculation means of the rate function calculation means.₂, T_ThreeRate function I obtained for₂(Γ) and I_ThreeA range of γ where the difference of (γ) is equal to or smaller than a predetermined allowable absolute error ε may be obtained as the confidence interval.
[0010]
  The rate function measurement method according to the present invention is an analysis target in a rate function measurement method that measures a rate function that expresses the distribution of random variables of the probability system when analyzing the probability system based on the limit theorem. Random variable A with parameter t from stochastic system_tMeasure actual value ofMeasuring steps toBased on the measurement result, the random variable A_tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_t≧ γt} is estimatedA probability distribution estimation step, Probability distribution P {A obtained by estimation_tObtain rate function I (γ) indicating the shape of ≧ γt}A rate function calculating step..
[0011]
  In addition to this, in the probability distribution estimation step,Measured random variable A_tFrom the actual value of the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is calculated as a cumulant, and the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is a predetermined function f related to γ_i(Γ) and parameter a_{t, i}Approximation formula F consisting of product-sum operation formula with_tApproximation using (γ) and parameter a using each cumulant_{t, i}To find an approximate expression byIt is a thing.
[0012]
  In the probability distribution estimation step,Two different parameters t₁, T₂Probability distribution P {A in_t1≧ γt₁} And P {A_t2≧ γt₂} Logarithm logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂} Respectively,In the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂}, The provisional rate function I ′₂(Γ) is obtained, and the random variable A_t1Sample mean γ obtained from realization of_AVGTemporary rate function I 'to be 0 (zero) at₂Rate function I after correcting (γ)₂Output as (γ)It is a thing.
[0013]
  At this time, in the probability distribution estimation step,Parameter t₁, T₂Parameter t different from_ThreeProbability distribution P {A in_t3≧ γt_Three} Logarithm logP {A_t3≧ γt_Three}In the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t3≧ γt_Three}, The provisional rate function I ′_Three(Γ) is obtained and the sample average value γ_AVGTemporary rate function I 'to be 0 (zero) at_ThreeRate function I after correcting (γ)_ThreeOutput as (γ), parameter t₂, T_ThreeRate function I obtained for₂(Γ) and I_ThreeA range of γ where the difference of (γ) is equal to or smaller than a predetermined allowable absolute error ε may be obtained as the confidence interval.
[0014]
  The program according to the present invention, when analyzing a stochastic system based on the limit theorem, causes a computer to execute a process of measuring a rate function expressing the distribution of random variables of the stochastic system.programIn,
To be analyzedprobabilityRandom variable A with parameter t from the system_tMeasure the real value ofmeasurementBased on the step and the measurement result, the random variable A_tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_tEstimate ≧ γt}Probability distribution estimationStep and probability complement distribution P {A obtained by estimation_tObtain rate function I (γ) indicating the shape of ≧ γt}Rate function calculationStep andHave.
[0015]
  In addition to this, in the probability distribution estimation step,Measured random variable A_tFrom the actual value of the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_tA step of calculating a plurality of ≧ γt} as a cumulant, and a probability complementary distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is a predetermined function f related to γ_i(Γ) and parameter a_{t, i}Approximation formula F consisting of product-sum operation formula with_tApproximation using (γ) and parameter a using each cumulant_{t, i}Obtaining an approximate expression by obtainingIt is something to be executed.
[0016]
  In the probability distribution estimation step,Two different parameters t₁, T₂Probability distribution P {A in_t1≧ γt₁} And P {A_t2≧ γt₂} Logarithm logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂} For each estimationAnd at the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂}, The rate function I ′₂A step of obtaining (γ) and a random variable A_t1Sample mean γ obtained from realization of_AVGTemporary rate function I 'to be 0 (zero) at₂Rate function I after correcting (γ)₂Outputting as (γ)It is something to be executed.
[0017]
  At this time, in the probability distribution estimation step,Parameter t₁, T₂Parameter t different from_ThreeProbability distribution P {A in_t3≧ γt_Three} Logarithm logP {A_t3≧ γt_Three} Step of estimatingAnd at the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t3≧ γt_Three}, The provisional rate function I ′_ThreeThe step of obtaining (γ) and the sample mean value γ_AVGTemporary rate function I 'to be 0 (zero) at_ThreeRate function I after correcting (γ)_ThreeA step of outputting as (γ) and a parameter t₂, T_ThreeRate function I obtained for₂(Γ) and I_ThreeObtaining a range of γ in which the difference of (γ) is less than or equal to a predetermined allowable absolute error ε as a confidence interval;ExecuteYou may do it.
[0018]
  The recording medium according to the present invention, when analyzing a probability system based on the limit theorem, causes a computer to execute a process of measuring a rate function that expresses a distribution of random variables of the probability system.It is a recording of the program., Subject to analysisprobabilityRandom variable A with parameter t from the system_tMeasure the real value ofmeasurementBased on the step and the measurement result, the random variable A_tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_tEstimate ≧ γt}Probability distribution estimationStep and probability complement distribution P {A obtained by estimation_tObtain rate function I (γ) indicating the shape of ≧ γt}Rate function calculationStep andHave.
[0019]
  In addition to this, in the probability distribution estimation step,Measured random variable A_tFrom the actual value of the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_tA step of calculating a plurality of ≧ γt} as cumulants, and a probability complementary distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is a predetermined function f related to γ_i(Γ) and parameter a_{t, i}Approximation formula F consisting of product-sum operation formula with_tApproximation using (γ) and parameter a using each cumulant_{t, i}Obtaining an approximate expression by obtainingIt is something to be executed.
[0020]
  Also,In the probability distribution estimation step,Two different parameters t₁, T₂Probability distribution P {A in_t1≧ γt₁} And P {A_t2≧ γt₂} Logarithm logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂} For each estimationAnd at the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂}, The rate function I ′₂A step of obtaining (γ) and a random variable A_t1Sample mean γ obtained from realization of_AVGTemporary rate function I 'to be 0 (zero) at₂Rate function I after correcting (γ)₂Outputting as (γ)It is something to be executed.
[0021]
  At this time, in the probability distribution estimation step,Parameter t₁, T₂Parameter t different from_ThreeProbability distribution P {A in_t3≧ γt_Three} Logarithm logP {A_t3≧ γt_Three} Step of estimatingAnd at the rate function calculation step, the probability distribution estimation stepThe obtained logarithmic value logP {A_t3≧ γt_Three}, The provisional rate function I ′_ThreeThe step of obtaining (γ) and the sample mean value γ_AVGTemporary rate function I 'to be 0 (zero) at_ThreeRate function I after correcting (γ)_ThreeA step of outputting as (γ) and a parameter t₂, T_ThreeRate function I obtained for₂(Γ) and I_ThreeObtaining a range of γ in which the difference of (γ) is less than or equal to a predetermined allowable absolute error ε as a confidence interval;ExecuteYou may do it.
[0022]
DETAILED DESCRIPTION OF THE INVENTION
Next, embodiments of the present invention will be described with reference to the drawings.
FIG. 1 is a block diagram showing a rate function measuring apparatus according to an embodiment of the present invention. This rate function measuring apparatus includes an input I / F unit 2 that takes in a state signal (or state data) indicating an operation state of the system from various systems such as a computer system or a communication system to be analyzed, and this input I / F. An arithmetic processing unit 1 that measures an actual value of a random variable indicating the operating state of the system from the state signal D_IN obtained via the unit 2, calculates and outputs a desired rate function based on the measurement result, and the arithmetic processing unit 1 A recording medium 3 in which a program 31 describing various processes is recorded, a storage unit 4 that stores information used for various processes in the arithmetic processing unit 1, a rate function measurement start for the arithmetic processing unit 1, etc. Processing input unit 5 for inputting various instructions and processing results such as rate function calculated by the arithmetic processing unit 1 The display unit 6 for displaying is provided.
[0023]
The arithmetic processing unit 1 includes a measuring unit 10 that measures an actual value of a random variable from D_IN fetched by the input I / F unit 2, and a predetermined probability distribution for the random variable based on the measurement result of the measuring unit 10. Probability distribution estimation means 11 for estimation and rate function calculation means 12 for calculating a desired rate function based on the probability distribution estimated by the probability distribution estimation means 11 are provided.
The rate function measuring apparatus is composed of a computer as a whole, and the arithmetic processing unit 1 is composed of a microprocessor that operates by reading a program 31 recorded in the recording medium 3. Therefore, each function means of the arithmetic processing unit 1 is realized by the cooperation of hardware resources including the microprocessor and the storage unit 4 and software called the program 31.
[0024]
The rate function in the present invention will be described.
The rate function is a random variable A having a parameter t such as time._tProbability amount A_tThis is a parameter (function) indicating the shape of the tail of the / t distribution. When an arbitrary value γ is used, the rate function I (γ) can be defined as the following function using γ.
[0025]
[Expression 17]

[0026]
For example, the random variable A_tAssuming that the amount of data transferred within time t (for example, the number of cells and the number of packets) is handled, if the time t is sufficiently large, the probability amount A_tThe distribution of / t is expressed in a predetermined function form represented by a rate function.
Therefore, the probability amount A_tProbability distribution P {A where / t is larger than an arbitrary value γ_t≧ γt} can be approximated as follows.
[0027]
[Formula 18]

[0028]
Further, the random variable A when the parameter t takes k discrete values._tIs a random variable x for each discrete value of the parameter t._kIn this case, the rate function I (γ) can be expressed as follows.
[0029]
[Equation 19]

[0030]
Physically interpreting this, the above equation can be expressed as a random variable A_tSample average A_tThe probability that / t is equal to or greater than γ indicates the rate at which the probability decreases with increasing t when t is infinite. The random variable A_tSample average A_tEven in the case where / t is γ or less, the probability distribution P {A_tA similar argument can be made with ≦ γt}.
[0031]
In the present invention, such a probability distribution P {A_tA desired rate function is calculated by approximating a cumulant (logarithm) of ≦ γt} with a predetermined approximate expression, for example, an approximate expression such as an orthogonal polynomial or a linear sum of power functions, and the orthogonality used in the approximate expression Parameter a corresponding to each degree of the polynomial or power function_{t, k}, The random variable A_tIt is calculated by regression analysis etc. using the cumulant value obtained from the realization value.
[0032]
Next, the configuration and operation of the arithmetic processing unit 1 will be described in detail. FIG. 2 shows a detailed configuration of the arithmetic processing unit.
In the following, an ATM communication system is assumed as an analysis target system, and a random variable A_tIs the number of cells transferred at the observation point of the system, and the parameter t is a random variable A_tAn example will be described in which a measurement period for counting the realization value (here, the number of cells) is used.
[0033]
The measurement means 10 is provided with an actual value measurement means 10A and a moving average calculation means 10B.
In the actual value measuring means 10A, the parameter t is determined from the status signal D_IN (for example, a pulse signal indicating the cell transfer timing) from the ATM communication system taken in by the input I / F unit 2.₁Random variable A with_t1As an actual value of the measurement period t₁The number of transferred cells is counted every time, and the counted value is expressed as a random variable A._t1Data group A_t1Output as ().
[0034]
FIG. 3 shows a processing procedure of the actual value measuring means.
In the actual value measuring means 10A, first, the measurement period from the operation input unit 5 is t.₁(Step 100) and T (T >> t) as the rate function calculation period₁) Is set (step 101), and the control counter n provided in the storage unit 4 (or in the arithmetic processing unit 1) is cleared (step 102).
Then, counting of the number of transfer cells based on D_IN is started (step 103), and the measurement period t from the start of counting is started.₁The counting is continued until only elapses (step 104: NO).
[0035]
Subsequently, the measurement period t₁After elapse (step 104: YES), the array A constituting the data group provided in the storage unit 4_t1The count value is stored in (n) (step 105), and the control counter n is incremented by 1 (step 106).
The counting process from step 103 to step 106 is repeatedly performed until the period T elapses (step 107: NO), and after the period T elapses (step 107: YES), the obtained data group A_t1() Is output to the storage unit 4 (step 108), and a series of processes relating to realization value measurement is terminated.
[0036]
In the moving average calculation means 10B, t₁Measurement period t different from₂, T_ThreeThe number of transferred cells is counted every time, and the moving average value is calculated and output. At this time, t₂, T_ThreeAs t₁An integer multiple of n₂t₁, N_Threet₁(However, n₂, N_ThreeIs an integer greater than or equal to 2, n₂<N_Three) Is selected.
FIG. 4 shows a processing procedure in the moving average calculation means.
In the moving average processing procedure 10B, first, the measurement period from the operation input unit 5 is t.₂(Step 110) and data group A_t1N as the average number of data calculated for ()₂Is set (step 111), and the control counter n is cleared (step 102).
[0037]
Subsequently, data group A_t1The m actual values from the nth to the (n + m−1) th in () are read (step 113), and the average value of these m actual values is calculated (step 114). Then, the average value is converted into an array A constituting the data group._t2(N) is stored (step 115), and the control counter n is incremented by 1 (step 116).
Such a counting process from steps 113 to 116 is performed by T / t.₁-M + 1 repeatedly (step 117), and the obtained data group A_t2() Is output to the storage unit 4 (step 118).
Further, the above steps 110 to 118 are changed to the measurement period t._ThreeBy repeating the above, data group A_t3() Is calculated and output to the storage unit 4 (step 119), and a series of processing ends.
[0038]
The probability distribution estimating means 11 is provided with a cumulant calculating means 11A and an approximate expression calculating means 11B.
In the cumulant calculating unit 11A, the data group A measured by the measuring unit 10 is used._t1(), A_t2(), A_t3Based on (), random variable A_t1, A_t2, A_t3Cumulant group C showing the logarithmic value of the probability complement distribution of_t1(), C_t2(), C_t3() Is calculated and output.
[0039]
FIG. 5 shows the processing procedure of the cumulant calculating means.
In the cumulant calculating unit 11A, first, the data group A stored in the storage unit 4 is stored._t1Any unprocessed data (γt) from ()₁) (Step 120), and based on the following formula, data group A_t1Γt of ()₁A cumulant of the probability distribution related to the above data is calculated (step 121).
[0040]
[Expression 20]

[0041]
An array Ct that constitutes the cumulant group, with the calculated value as a cumulant value corresponding to the unprocessed data₁Store in () (step 122).
Such steps 120 to 122 are referred to as data group A._tRepeatedly performed for all data of () (step 123), and obtained cumulant group C_t1() Is output to the storage unit 4 (step 124).
Further, the above steps 120 to 124 are changed to the measurement period t.₂, T_ThreeCumulant group C_t2(), C_t3() Is calculated and output (step 125), and the series of processes is terminated.
[0042]
In the approximate expression calculation means 11B, the random variable A_t1, A_t2, A_t3And an approximate expression using a linear sum of orthogonal polynomials and power functions common to each random variable A_t1, A_t2, A_t3Approximate the cumulant of each probability distribution for each cumulant group C_t1(), C_t2(), C_t3Each approximate expression is obtained by calculating parameters included in each approximate expression by regression analysis based on ().
FIG. 6 shows a processing procedure of the approximate expression calculation unit 11B.
In the approximate expression calculating means 11B, first, a random variable A_t1, The cumulant of its probability distribution is given by a plurality of functions f₁(Γ), f₂(Γ), ..., f_k(Γ) (k is an integer of 2 or more) and parameter a₁₁, A₁₂, ..., a_1kThe following approximate expression F consisting of₁Approximate by (γ).
[0043]
[Expression 21]

[0044]
The random variable A calculated by the cumulant calculating means 11A_t1Cumulant group C_t1Based on (), a regression analysis or the like is performed by applying a cumulant value for an arbitrary γ to the approximate expression, thereby obtaining a parameter a of the approximate expression.₁₁, A₁₂, ..., a_1kAnd the random variable A_t1Find the approximate expression at. (Step 130). At this time, for example, the parameter a is set so that the sum of the squares of errors between the cumulant value for an arbitrary γ and the approximate value by the approximate expression is minimized by the least square method.₁₁, A₁₂, ..., a_1kShould be selected.
[0045]
Similarly, the random variable A_t2, A_t3, Each of the probability distribution cumulants is approximated by the following approximate expression, and the cumulant group C_t2(), C_t3Perform regression analysis based on (). At this time, as an approximate expression, a random variable A_t1The same function f used in₁(Γ), f₂(Γ), ..., f_kParameter a which is composed of (γ) and is multiplied by these functions f_i1, A_i2, ..., a_ikOnly approximate formula F_i(Γ) is used.
[0046]
[Expression 22]

[0047]
As a result, the parameter a of the approximate expression_{twenty one}, A_{twenty two}, ..., a_2kAnd parameter a₃₁, A₃₂, ..., a_3kAre calculated respectively and the random variable A_t2, A_t3Approximation formula is obtained (steps 131 and 132), and these calculated parameters a_i1, A_i2, ..., a_ik(Where i = 1, 2, 3) is stored in the storage unit 4 and the series of processes is terminated.
[0048]
The rate function calculation unit 12 includes a provisional rate function calculation unit 12A, an average calculation unit 12B, an output value correction unit 12C, and a confidence interval calculation unit 12D.
In the provisional rate function calculation means 12A, the random variable A calculated by the approximate expression calculation means 11B._t1, A_t2, A_t3Each approximate expression F in₁(Γ), F₂(Γ), F_ThreeFrom (γ), provisional rate function I ′₂(Γ), I ′_Three(Γ) is calculated.
[0049]
FIG. 7 shows the processing procedure of the provisional rate function calculation means 12A.
In the provisional rate function calculating means 12A, first, the parameter a stored in the storage unit 4 is used._i1, A_i2, ..., a_ikAnd the approximate formulas calculated by the approximate formula calculation means 12 are read (step 140). Then, based on the following formula, the provisional rate function I ′₂(Γ), I ′_Three(Γ) is obtained (step 141), and the series of processing ends.
[0050]
[Expression 23]

[0051]
FIG. 8 shows a processing procedure of the average value calculation means 12B.
In the average calculation means 12B, the random variable A measured by the actual value measurement means 10A of the measurement means 10_t1Data group A_t1From (), the sample mean γ_AVGIs calculated (step 150), and the series of processing ends.
[0052]
In the output value correcting means 12C, the rate function I (γ) is γ = sample average value γ_AVGIn this case, the sample average value γ calculated by the average calculating means 12B based on the property that I (γ) = 0._AVGIs used to calculate the provisional rate function I ′ calculated by the provisional rate function calculation means 12A.₂(Γ), I ′_ThreeA desired rate function I is calculated by correcting (γ).
[0053]
FIG. 9 shows the processing procedure of the output value correcting means 12C.
In the output value correcting means 12C, first, the sample average value γ calculated by the average calculating means 12B with respect to γ._AVGIs set (step 160). Next, the provisional rate function I ′ obtained by the provisional rate function calculation means 12A.₂Γ that corrects γ based on (γ)₂The following rate function I expressed using₂Define (γ).
[0054]
[Expression 24]

[0055]
And I₂(Γ_AVG) = 0 so that = 0₂To correct the above equation and the desired rate function I₂An expression of (γ) is obtained (step 161).
Similarly, the provisional rate function I ′ obtained by the provisional rate function calculation means 12A._ThreeΓ that corrects γ based on (γ)_ThreeThe following rate function I expressed using_ThreeDefine (γ).
[0056]
[Expression 25]

[0057]
And I_Three(Γ_AVG) = 0 so that = 0_ThreeTo correct the above equation and the desired rate function I_ThreeAn expression of (γ) is obtained (step 162), and a series of processing ends.
[0058]
In the confidence interval calculation means 12D, the rate function I obtained by the output value correction means 12C.₂(Γ), I_ThreeThe range of γ where the difference in the values of (γ) falls within the predetermined tolerance ε_min, Γ_maxAnd rate function I₂(Γ) or I_ThreeIt outputs to the memory | storage part 4 or the display part 6 with ((gamma)). Theoretically rate function I₂(Γ), I_Three(Γ) is the same function, but the rate function approximated based on the finite measurement result includes an error.
Therefore, the different parameter t₂, T_ThreeRate function I obtained for₂(Γ) and rate function I_ThreeBy presenting a range of γ within which the difference from (γ) falls within the desired tolerance ε, the rate function can be used with desired reliability.
[0059]
FIG. 10 shows a processing procedure of the confidence interval calculation means.
First, the confidence interval calculation means 12D sets a desired tolerance for the rate function to ε (step 170), and a range γ satisfying the following equation γ_min, Γmax is obtained (step 171).
[0060]
[Equation 26]

[0061]
And the range of γ obtained γ_min, Γ_maxAnd rate function I₂(Γ) or I_Three(Γ) is output (step 172), and the series of processing ends.
[0062]
Thus, in the arithmetic processing unit 1, a random variable A having the parameter t from the probability system to be analyzed._tIs measured by the measuring means 10, and based on the measurement result, the random variable A_tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_t≧ γt} is estimated by the probability distribution estimating means 11, and the probability complementary distribution P {A_tSince the rate function I (γ) indicating the distribution state of ≧ γt} is calculated and output by the rate function calculating means 12, the rate function can be directly measured from the target system.
Therefore, it is possible to actually use the result (theory) obtained by the large deviation principle, which is one of the limit theorems, and the stochastic system in the region where the limit theorem is established, such as a system with large scale and large capacity. This makes it possible to grasp the characteristics of the product and apply it to performance evaluation and facility design that satisfies the target quality.
[0063]
Further, the random variable A measured by the measuring means 10 by the cumulant calculating means 11A of the probability distribution estimating means 11 is shown._tFrom the actual value of the probability distribution P {A_tLogarithmic value logP {A of ≧ γt}_t≧ γt} is calculated as a cumulant, and the approximate expression calculation means 11B uses the logarithmic value logP {A_t≧ γt} is a predetermined function f related to γ_i(Γ) and parameter a_iIs approximated using an approximate expression F (γ) consisting of a product-sum operation expression and a parameter a using each cumulant calculated by the cumulant calculating means 11A._iSince the approximate expression is obtained by obtaining, the random variable A_tAccurately using the realization value of P {A_t≧ γt} can be estimated.
[0064]
Further, the probability distribution estimating means 11 uses two different parameters t₁, T₂Probability distribution P {A in_t1≧ γt₁} And P {A_t2≧ γt₂} Logarithm logP {A_t1≧ γt₁} And logP {A_t2≧ γt₂} Are estimated, and the logarithmic value logP {A obtained by the estimation unit is calculated by the provisional rate function calculation unit 12A of the rate function calculation unit 12._t1≧ γt₁} And logP {A_t2≧ γt₂}, The provisional rate function I ′₂(Γ) is obtained, and the provisional rate function I ′ is obtained by the output value correcting means 12C so that the sample average value γ becomes 0 (zero).₂Rate function I after correcting (γ)₂Since (γ) is output, it is possible to calculate a rate function with little error in accordance with the property of the rate function I (γ) where I (γ) = 0 when γ = sample average value.
[0065]
Further, the probability distribution estimation means 11 performs the parameter t in the same manner as described above.₁, T₂Parameter t different from_ThreeProbability distribution P {A in_t3≧ γt_Three} Logarithm logP {A_t3≧ γt_Three} And the rate function calculation means 12 calculates the logarithmic value logP {A_t3≧ γt_Three}, The provisional rate function I ′ obtained using_ThreeRate function I after correcting (γ)_Three(Γ), and the confidence interval calculation means 12D outputs the parameter t₂, T_ThreeRate function I obtained for₂(Γ) and I_ThreeSince the range of γ in which the difference of (γ) is less than or equal to the predetermined allowable absolute error ε is obtained as the confidence interval, the error included in the rate function approximated based on the finite measurement result is the desired allowable absolute error ε. The range of γ within the range can be presented, and the obtained rate function can be used with desired reliability.
[0066]
【The invention's effect】
  As described above, the present inventionIn the measuring means,Random variable A with parameter t from the stochastic system to be analyzed_tMeasure the real value ofIn the probability distribution estimation means,Based on this measurement result, the random variable A_tThe probability amount A obtained by dividing the parameter by the parameter t_tRandom variable A where / t is greater than or equal to an arbitrary value γ_tProbability distribution P {A_t≧ γt},In the rate function calculation means, probability distribution estimation meansProbability complement distribution P {A obtained by_tRate function I (γ) indicating the distribution state of ≧ γt}It is to be calculated.
  The probability variable A measured by the probability distribution estimation means _t From the actual value of the probability distribution P {A _t Logarithmic value of ≧ γt} log P {A _t ≧ γt} is calculated as a cumulant, and the probability distribution P {A _t Logarithmic value of ≧ γt} log P {A _t ≧ γt} is a predetermined function f related to γ _i (Γ) and parameter a _{t, i} Approximation formula F consisting of product-sum operation formula with _t Approximation using (γ) and parameter a using each cumulant _{t, i} The approximate expression is obtained by obtaining.
  Further, two different parameters t are obtained by the probability distribution estimation means. ₁ , T ₂ Probability distribution P {A in _t1 ≧ γt ₁ } And P {A _t2 ≧ γt ₂ } Logarithmic value log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ }, And the logarithmic value obtained from the probability distribution estimation by the rate function calculation means log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ }, Provisional rate function I ' ₂ (Γ) is obtained, and the random variable A _t1 Sample mean γ obtained from realization of _AVG Temporary rate function I to be 0 (zero) at ' ₂ Rate function I after correcting (γ) ₂ This is output as (γ).
  ThisThe rate function can be measured directly from the target system.
  Therefore, it is possible to actually use the results (theories) obtained by the large deviation principle, which is one of the limit theorems, and the areas where the limit theorem is established, such as large-scale systems and large-capacity systems. This makes it possible to grasp the characteristics of the stochastic system and to apply it to performance evaluation of the target system and facility design that satisfies the target quality.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a rate function measuring apparatus according to an embodiment of the present invention.
FIG. 2 is a block diagram showing details of an arithmetic processing unit.
FIG. 3 is a flowchart showing a processing procedure in an actual value measuring means of the arithmetic processing unit.
FIG. 4 is a flowchart showing a processing procedure in a moving average calculation means of an arithmetic processing unit.
FIG. 5 is a flowchart showing a processing procedure in the cumulant calculating means of the arithmetic processing unit.
FIG. 6 is a flowchart illustrating a processing procedure in an approximate expression calculation unit of the arithmetic processing unit.
FIG. 7 is a flowchart showing a processing procedure in an average calculating means of the arithmetic processing unit.
FIG. 8 is a flowchart showing a processing procedure in provisional rate function calculation means of the arithmetic processing unit.
FIG. 9 is a flowchart showing a processing procedure in the output value correcting means of the arithmetic processing unit.
FIG. 10 is a flowchart illustrating a processing procedure in a confidence interval calculation unit of the arithmetic processing unit.
[Explanation of symbols]
DESCRIPTION OF SYMBOLS 1 ... Arithmetic processing part, 10 ... Measuring means, 10A ... Realized value measuring means, 10B ... Moving average calculating means, 11 ... Cumulant calculating means, 11B ... Approximate expression calculating means, 12 ... Rate function calculating means, 12A ... Temporary rate function Calculation means, 12B ... average calculation means, 12C ... output value correction means, 12D ... confidence interval calculation means, 2 ... input I / F unit, 3 ... storage medium, 31 ... program, 4 ... storage unit, 5 ... operation input unit , 6 ... display section.

Claims (8)

When analyzing a stochastic system based on the limit theorem, in a rate function measuring device that measures the rate function that expresses the distribution of random variables in the stochastic system,
Measuring means for measuring the actual values of the random variable A _t with parameter t from the probability system to be analyzed,
Based on the measurement result in the measuring means, the probability complement distribution P {A _t ≧ random variable A _t the random variable A _t the parameter t in a division with a probability obtained amount A _t / t is an arbitrary value γ or probability distribution estimating means for estimating γt};
Rate function I (γ) indicating the shape of the probability complementary distribution P {A _t ≧ γt} obtained by this estimation means

And a rate function calculating means for calculating a,
The probability distribution estimation means includes
Wherein the realization of the random variable A _t measured by the measuring means, and cumulants calculating means calculates a plurality of logarithmic value log P of the probability auxiliary distribution _{P {A t ≧ γt} {} A t ≧ γt} as cumulants, the The logarithmic value log P {A _t ≧ γt} of the probability complement distribution P {A _t ≧ γt} is expressed by an approximate expression F consisting of a product-sum operation expression of a predetermined function f _i (γ) related to γ and a parameter at _{, i t} (γ)

And approximate expression calculation means for obtaining the approximate expression by obtaining the parameter a _{t, i} using each cumulant calculated by the cumulant calculation means, and having two different parameters t _1, the probability complement distribution P in _{t 2 {a t1 ≧ γt 1} } and P {a _t2 ≧ γt _2} logarithm _{log P {a t1 ≧ γt 1} } and of _{log P {a t2 ≧ γt 2} } to estimate each And
The rate function calculating means uses the logarithmic values log P {A _t1 ≧ γt ₁ } and log P {A _t2 ≧ γt ₂ } obtained by the probability distribution estimating means ,

A temporary rate function calculating means for calculating a temporary rate function I _'2 (gamma) from the random variable A 0 obtained in sample average value gamma _AVG from realization of _t1 (zero) and so as to the temporary rate function I A rate function measuring device comprising: output value correcting means for correcting ' ₂ (γ) and outputting it as rate function I ₂ (γ) . The rate function measuring device according to claim 1,
The probability distribution estimation means estimates a logarithmic value log P {A _t3 ≧ γt ₃ } of a probability complementary distribution P {A _t3 ≧ γt ₃ } in a parameter t ₃ different from the parameters t ₁ and t ₂ ,
The rate function calculating means is a provisional rate function calculating means, using the logarithmic value log P {A _t3 ≧ γt ₃ } obtained by the probability distribution estimating means ,

From 'with obtaining the ₃ (gamma), with the output value correcting means, the sample mean value gamma a such that 0 (zero) at _AVG temporary rate function I' temporary rate function I corrected ₃ (gamma) Rate Output as function I ₃ (γ)
The rate function calculating means determines a range of γ in which a difference between the rate functions I ₂ (γ) and I ₃ (γ) obtained for the parameters t ₂ and t ₃ is equal to or less than a predetermined allowable absolute error ε. A rate function measuring apparatus , further comprising: a confidence interval calculating means to obtain as follows. When analyzing a stochastic system based on the limit theorem, in a rate function measurement method that measures the rate function that expresses the distribution of random variables in the stochastic system,
Random variable A with parameter t from the stochastic system to be analyzed _t A measurement step to measure the actual value of
Based on this measurement result, the random variable A _t The probability amount A obtained by dividing the parameter by the parameter t _t Random variable A where / t is greater than or equal to an arbitrary value γ _t Probability distribution P {A _t A probability distribution estimation step for estimating ≧ γt};
Probability complement distribution P {A obtained by estimation _t Rate function I (γ) indicating the shape of ≧ γt}

A rate function calculating step for obtaining
In the probability distribution estimating step, the measured random variable A _t From the realized value of the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t A step of calculating a plurality of ≧ γt} as cumulants, and the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t ≧ γt} is a predetermined function f related to γ _i (Γ) and parameter a _{t, i} Approximation formula F consisting of product-sum operation formula with _t (Γ)

And the parameter a using each cumulant. _{t, i} Obtaining the approximate expression by obtaining, and two different parameters t ₁ , T ₂ Probability distribution P {A in _t1 ≧ γt ₁ } And P {A _t2 ≧ γt ₂ } Logarithmic value log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ } Each estimating
In the rate function calculating step, the logarithmic value obtained log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ }Using,

To provisional rate function I ' ₂ (Γ) and a random variable A _t1 Sample mean γ obtained from realization of _AVG The provisional rate function I is set to 0 (zero) at ' ₂ Rate function I after correcting (γ) ₂ And outputting as (γ)
Rate function measurement method characterized by this. In the rate function measuring method according to claim 3,
In the probability distribution estimating step, the parameter t ₁ , T ₂ Parameter t different from _Three Probability distribution P {A in _t3 ≧ γt _Three } Logarithmic value log P {A _t3 ≧ γt _Three } To estimate,
The rate function calculating step includes logarithmic values obtained in the probability distribution estimating step. log P {A _t3 ≧ γt _Three }Using,

To provisional rate function I ' _Three A step of obtaining (γ) and the sample average value γ _AVG The provisional rate function I is set to 0 (zero) at ' _Three Rate function I after correcting (γ) _Three Outputting as (γ) and the parameter t ₂ , T _Three The rate function I obtained for ₂ (Γ) and I _Three Obtaining a range of γ where the difference of (γ) is equal to or less than a predetermined allowable absolute error ε as a confidence interval.
Rate function measurement method characterized by this. When analyzing a probability system based on the limit theorem, in a program for causing a computer to execute a process of measuring a rate function that expresses the distribution of random variables of the probability system,
Random variable A with parameter t from the stochastic system to be analyzed _t A measurement step to measure the actual value of
Based on this measurement result, the random variable A _t The probability amount A obtained by dividing the parameter by the parameter t _t Random variable A where / t is greater than or equal to an arbitrary value γ _t Probability distribution P {A _t A probability distribution estimation step for estimating ≧ γt};
Probability complement distribution P {A obtained by estimation _t Rate function I (γ) indicating the shape of ≧ γt}

A rate function calculating step for obtaining
In the probability distribution estimating step, the measured random variable A _t From the realized value of the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t A step of calculating a plurality of ≧ γt} as cumulants, and the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t ≧ γt} is a predetermined function f related to γ _i (Γ) and parameter a _{t, i} Approximation formula F consisting of product-sum operation formula with _t (Γ)

And the parameter a using each cumulant. _{t, i} Obtaining the approximate expression by obtaining, and two different parameters t ₁ , T ₂ Probability distribution P {A in _t1 ≧ γt ₁ } And P {A _t2 ≧ γt ₂ } Logarithmic value log P {A _t1 ≧ γt ₁ }and log P { A _t2 ≧ γt ₂ } Each estimating
In the rate function calculating step, the logarithmic value obtained log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ }Using,

To provisional rate function I ' ₂ (Γ) and a random variable A _t1 Sample mean γ obtained from realization of _AVG The provisional rate function I is set to 0 (zero) at ' ₂ Rate function I after correcting (γ) ₂ And outputting as (γ)
A program characterized by that. The program according to claim 5,
In the probability distribution estimating step, the parameter t ₁ , T ₂ Parameter t different from _Three Probability distribution P {A in _t3 ≧ γt _Three } Logarithmic value log P {A _t3 ≧ γt _Three } To estimate,
The rate function calculating step includes logarithmic values obtained in the probability distribution estimating step. log P {A _t3 ≧ γt _Three }Using,

To provisional rate function I ' _Three A step of obtaining (γ) and the sample average value γ _AVG The provisional rate function I is set to 0 (zero) at ' _Three Rate function I after correcting (γ) _Three Outputting as (γ) and the parameter t ₂ , T _Three The rate function I obtained for ₂ (Γ) and I _Three Obtaining a range of γ where the difference of (γ) is equal to or less than a predetermined allowable absolute error ε as a confidence interval.
A program characterized by that. When analyzing a stochastic system based on the limit theorem, in a recording medium recording a program for causing a computer to execute a process of measuring a rate function expressing the distribution of random variables of the stochastic system,
Random variable A with parameter t from the stochastic system to be analyzed _t A measurement step to measure the actual value of
Based on this measurement result, the random variable A _t The probability amount A obtained by dividing the parameter by the parameter t _t Random variable A where / t is greater than or equal to an arbitrary value γ _t Probability distribution P {A _t A probability distribution estimation step for estimating ≧ γt};
Probability complement distribution P {A obtained by estimation _t Rate function I (γ) indicating the shape of ≧ γt}

A rate function calculating step for obtaining
In the probability distribution estimating step, the measured random variable A _t From the realized value of the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t A step of calculating a plurality of ≧ γt} as cumulants, and the probability complementary distribution P {A _t Logarithmic value of ≧ γt} log P {A _t ≧ γt} is a predetermined function f related to γ _i (Γ) and parameter a _{t, i} Approximation formula F consisting of product-sum operation formula with _t (Γ)

And the parameter a using each cumulant. _{t, i} Obtaining the approximate expression by obtaining, and two different parameters t ₁ , T ₂ Probability distribution P {A in _t1 ≧ γt ₁ } And P {A _t2 ≧ γt ₂ } Logarithmic value log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ } Each estimating
In the rate function calculating step, the logarithmic value obtained log P {A _t1 ≧ γt ₁ }and log P {A _t2 ≧ γt ₂ }Using,

To provisional rate function I ' ₂ (Γ) and a random variable A _t1 Sample mean γ obtained from realization of _AVG The provisional rate function I is set to 0 (zero) at ' ₂ Rate function I after correcting (γ) ₂ And outputting as (γ)
A recording medium that records the program. The recording medium according to claim 7, wherein
In the probability distribution estimating step, the parameter t ₁ , T ₂ Parameter t different from _Three Probability distribution P {A in _t3 ≧ γt _Three } Logarithmic value log P {A _t3 ≧ γt _Three } To estimate,
The rate function calculating step includes logarithmic values obtained in the probability distribution estimating step. log P {A _t3 ≧ γt _Three }Using,

To provisional rate function I ' _Three A step of obtaining (γ) and the sample average value γ _AVG The provisional rate function I is set to 0 (zero) at ' _Three Rate function I after correcting (γ) _Three Outputting as (γ) and the parameter t ₂ , T _Three The rate function I obtained for ₂ (Γ) and I _Three Obtaining a range of γ where the difference of (γ) is equal to or less than a predetermined allowable absolute error ε as a confidence interval.
A recording medium that records the program.

Images