WO2023132066A1 - Variance calculation method, variance calculation device, and program - Google Patents

Variance calculation method, variance calculation device, and program Download PDF

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WO2023132066A1
WO2023132066A1 PCT/JP2022/000401 JP2022000401W WO2023132066A1 WO 2023132066 A1 WO2023132066 A1 WO 2023132066A1 JP 2022000401 W JP2022000401 W JP 2022000401W WO 2023132066 A1 WO2023132066 A1 WO 2023132066A1
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vertex
variance
cov
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availability
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健吾 中村
武 井上
正彬 西野
宜仁 安田
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日本電信電話株式会社
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    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10Complex mathematical operations
    • G06F17/18Complex mathematical operations for evaluating statistical data, e.g. average values, frequency distributions, probability functions, regression analysis

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  • the present invention relates to a distributed computing method, a distributed computing device, and a program.
  • k-terminal network reliability (hereinafter abbreviated as k-NR), and the probability is network reliability ( Or simply called reliability).
  • Non-Patent Documents 1 and 2 propose a method of quickly obtaining network reliability using a Binary Decision Diagram (BDD) that can compactly represent a set of combinations. At present, many studies can obtain accurate network reliability for networks with hundreds of links.
  • BDD Binary Decision Diagram
  • network reliability is calculated under the assumption that the availability of each link (that is, the probability that each link will operate correctly) is given accurately.
  • link availability is given with some uncertainty, or is estimated with error, for example, by machine learning. If the availability of each link has uncertainty in the form of variance, then the network reliability value should also have uncertainty in the form of variance. Calculating such network reliability uncertainties is important in a realistic situation.
  • a network system with a network reliability of 99% but with a standard deviation (uncertainty) of 1% and a network system with a network reliability of 98.5% but with a standard deviation (uncertainty) of 0.1% Considering a certain network system, although the former has higher network reliability, it is quite conceivable that the latter is preferred.
  • An embodiment of the present invention has been made in view of the above points, and aims to calculate the distribution of network reliability.
  • FIG. 4 is a flowchart showing an example of the flow of processing executed by the distributed computing device according to the embodiment; 3 is a diagram showing an example of a graph G and an end point set K; FIG. FIG. 2 is an example of a BDD representing a reliability logic function; FIG. 2 illustrates an example of an algorithm for calculating network reliability variance values by BDD;
  • the state of each link (ie, up or down) is assumed to be probabilistically independent of the states of other links. Also, the set of states of all links included in E is called "the state of E".
  • ⁇ K is the set of all states X of E such that all nodes in K are connected in G X .
  • ⁇ K is the set X of all states X of E such that all nodes in K are all connected by the set E X of active links.
  • E[ ⁇ ], Var[ ⁇ ], and Cov[ ⁇ , ⁇ ] be expected values, variances, and covariances of P 1 , . . . , P m , respectively.
  • the “variance of network reliability” to be obtained is the value of the variance Var[R G (K)] of R G (K), and the average p i and variance ⁇ i 2 of the graph G, the endpoint set K, and each link e i
  • the problem to be solved is to find the value of this variance Var[R G (K)] given .
  • Non-Patent Documents 1 and 2 there is no need to enumerate all the states of E such that all the nodes included in K are connected, and the above BDD can be directly constructed. Significant reduction in computation time can be realized compared to enumeration.
  • the idea based on the method described in Non-Patent Document 1 was proposed, and the method described in Non-Patent Document 2 is a part of the method described in Non-Patent Document 1. Improved and faster.
  • the covariance Cov[P(X), P (Y)] can be calculated in a time proportional to the number of links. value can be calculated.
  • the number of states included in ⁇ K generally increases exponentially with the number of links m, it is difficult in terms of calculation time to calculate the variance using this method even in a network with several tens of links. is.
  • there is no method for calculating the value of network reliability variance in a realistic time and a method that can calculate the value of network reliability variance even in a network with more than 100 links is required.
  • BDD is used to quickly calculate the variance value of network reliability.
  • the BDD used in Non-Patent Documents 1 and 2 is a data structure that expresses a set of combinations with a loop-free graph (called a directed acyclic graph) made up of a collection of vertices and oriented edges.
  • a loop-free graph called a directed acyclic graph
  • the nodes and links of the directed acyclic graph are called “vertices" and "edges", respectively.
  • Non-Patent Documents 1 and 2 after constructing a BDD, a network reliability value is calculated by dynamic programming with each vertex of the BDD as one state.
  • a new dynamic programming method in which a pair of vertices of the BDD is one state is used to calculate distribution of network reliability. Calculate a value. This allows computing the value of the network reliability variance in a time proportional to the polynomial of the size of the BDD.
  • the size of BDD for calculating network reliability is often very small compared to the number of states included in ⁇ K , so it becomes possible to calculate the variance even in a network with about 200 links. .
  • FIG. 1 shows a hardware configuration example of the distributed computing device 10 according to this embodiment.
  • the distributed computing device 10 according to the present embodiment is realized by the hardware configuration of a general computer or computer system, and includes an input device 101, a display device 102, an external I/F 103, and a communication I/F. /F 104 , processor 105 and memory device 106 . Each of these pieces of hardware is communicably connected via a bus 107 .
  • the input device 101 is, for example, a keyboard, mouse, touch panel, various physical buttons, and the like.
  • the display device 102 is, for example, a display, a display panel, or the like. Note that the distributed computing device 10 may not have at least one of the input device 101 and the display device 102, for example.
  • the external I/F 103 is an interface with an external device such as the recording medium 103a.
  • the distributed computing device 10 can read from and write to the recording medium 103 a via the external I/F 103 .
  • Examples of the recording medium 103a include CD (Compact Disc), DVD (Digital Versatile Disk), SD memory card (Secure Digital memory card), USB (Universal Serial Bus) memory card, and the like.
  • the communication I/F 104 is an interface for connecting the distributed computing device 10 to a communication network.
  • the processor 105 is, for example, various arithmetic units such as a CPU (Central Processing Unit) and a GPU (Graphics Processing Unit).
  • the memory device 106 is, for example, various storage devices such as HDD (Hard Disk Drive), SSD (Solid State Drive), flash memory, RAM (Random Access Memory), and ROM (Read Only Memory).
  • the distributed computing device 10 has the hardware configuration shown in FIG. 1, so that it can implement various processes described later.
  • the hardware configuration shown in FIG. 1 is an example, and the distributed computing device 10 may have, for example, a plurality of processors 105, a plurality of memory devices 106, Various hardware other than the illustrated hardware may be included.
  • FIG. 2 shows an example of the functional configuration of the distributed computing device 10 according to this embodiment.
  • the distributed computing device 10 according to this embodiment has an input unit 201 , a construction unit 202 , a calculation unit 203 and an output unit 204 .
  • Each of these units is implemented by, for example, processing that one or more programs installed in the distributed computing device 10 cause the processor 105 to execute.
  • G (V, E)
  • K ⁇ V a set of endpoints
  • ⁇ i 2 the availability of each link e i ⁇ E.
  • the input unit 201 inputs the given connected undirected graph G, the end point set K, and the availability mean p i and variance ⁇ i 2 of each link e i ⁇ E.
  • the constructing unit 202 Based on the connected undirected graph G and the endpoint set K input by the input unit 201, the constructing unit 202 generates (constructs) a BDD having all the states in which all the nodes included in the endpoint set K are connected. .
  • the calculation unit 203 calculates the value of the variance of network reliability by dynamic programming. to calculate
  • the output unit 204 outputs the network reliability variance value calculated by the calculation unit 203 to a predetermined arbitrary output destination.
  • the output destination includes, for example, the display device 102 such as a display, the memory device 106, and other devices or devices connected via a communication network.
  • the input unit 201 inputs the given connected undirected graph G, the end point set K, and the mean p i and variance ⁇ i 2 of the availability of each link e i ⁇ E (step S101).
  • the constructing unit 202 generates a BDD having all states in which all nodes included in the endpoint set K are connected based on the connected undirected graph G and the endpoint set K input in step S101. (Build) (step S102). Details of this step will be described later.
  • step S103 the calculation unit 203 dynamically Calculate the value of the variance of the network reliability by the planning method. Details of this step will be described later.
  • the output unit 204 outputs the network reliability variance value calculated in step S103 to a predetermined arbitrary output destination (step S104).
  • step S102> Since the processing executed by the construction unit 202 when constructing the BDD in this step is the same as the method proposed in Non-Patent Documents 1 and 2, the following mainly describes the structure of the BDD constructed in this step. will be explained.
  • the constructing unit 202 Based on the graph G and the endpoint set K, the constructing unit 202 expresses a set ⁇ K that collects all the states X of E such that all the nodes included in the endpoint set K are connected in GX . Construct a Binary Decision Diagram (BDD). Since the BDD is originally a data structure that expresses a logic function, first consider a logic function fK corresponding to ⁇ K .
  • BDD Binary Decision Diagram
  • N is a vertex set and A is an edge set.
  • a root is a vertex that has no edges toward itself, and is hereinafter expressed as r ⁇ N.
  • the vertices included in the vertex set N are roughly divided into terminal vertices, which are special vertices with no outgoing edges, and other internal vertices. Below, the terminal vertex is
  • first terminal vertex first terminal vertex
  • second terminal vertex second terminal vertex
  • LO and HI edges have two outgoing edges, called LO and HI edges, and integer values called labels.
  • the vertices pointed to by the LO edge and HI edge of internal vertex v are called LO child vertices and HI child vertices, respectively, and are denoted by lo(v) and hi(v), respectively.
  • the label of v is expressed as lb(v) ⁇ 1, . . . , c ⁇ . where c is the number of binary variables that the logic function represented by the BDD takes as input.
  • the value of the BDD label needs to increase as it goes from the root to the edges. That is, lb(v) ⁇ lb(lo(v)) and lb(v) ⁇ lb(hi(v)) for each interior vertex v.
  • the size of the BDD is defined as the number of vertices
  • FIG. 5 shows a BDD expressing the reliability logic function f K corresponding to ⁇ K given the graph G and the endpoint set K shown in FIG.
  • circles represent nodes of graph G
  • black circles represent endpoints (that is, nodes included in endpoint set K among nodes of graph G).
  • a 1 to a 8 are internal vertices
  • a 9 and a 10 are terminal vertices
  • the solid line represents the HI edge
  • the broken line represents the LO edge. represents a label.
  • a BDD representing a reliability logic function f K is defined from its root r to the first terminal vertex
  • each path from the root r of the BDD to the first terminal vertex corresponds to one or more states included in ⁇ K .
  • Non-Patent Documents 1 and 2 are for efficiently constructing a BDD that expresses the reliability logic function fK . Therefore, please refer to Non-Patent Documents 1 and 2, etc. for a specific method of constructing a BDD that expresses the reliability logic function fK .
  • Step S103> the calculation unit 203 calculates the value of the network reliability variance by a new dynamic programming method in which the pair of vertices of the BDD is one state.
  • the idea for calculating the network reliability variance value will be described, and then the algorithm representing the specific procedure for performing the calculation will be described.
  • R r corresponding to the root r is the network reliability R G (K).
  • the fourth line calls Cov(r, r), which is a recursive procedure (function).
  • the procedure Cov(u, v) is a recursive procedure for calculating the covariance Cov[R u , R v ], and its content is described on lines 5-17.
  • 0 is returned as an answer if at least one of u and v is a terminal vertex.
  • lines 8 and 9 if the covariance Cov[R u , R v ] has already been calculated, the calculated value is returned as an answer.
  • an associative array c is prepared as a cache for storing the calculated Cov[R u , R v ], and the value of Cov[R u , R v ] is stored in c[(u, v)]. .
  • Cov[R u , R v ] is calculated by recursively calling the procedure Cov(u, v) using equations (5) and (6). That is, after the smaller of the label of u and the label of v is stored in i on the 10th line, if the label of u is smaller than the label of v, Cov[R u , R v ] is calculated using equation (6).
  • the distributed computing device 10 can quickly calculate the distributed value of network reliability, which could not be calculated until now except by a naive method.
  • the distributed value of network reliability can be calculated even for a real network with a link scale of about 100 to 200. This makes it possible to quantitatively evaluate the impact of the uncertainty of availability of each link on the uncertainty of network reliability. Therefore, for example, when a link is added to the network with no track record of operation and whose exact availability is unknown, it is possible to quantitatively analyze how the link affects the uncertainty of network reliability.
  • the distributed computing device 10 may perform various network controls using the calculated network reliability variance value. For example, if the network reliability is the same, the network is controlled to have a low variance value, or even if the network reliability is low, if it is within a certain reference range, the network is controlled to have a low variance value. You may perform control such as performing.

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Abstract

In a variance calculation method according to an embodiment, a computer performs: an input procedure for inputting a connected undirected graph G=(V, E), an endpoint set K⊂V, and the mean pi and variance σi 2 of the availability of links ei∈E constituting the undirected graph G; a construction procedure for constructing a binary decision diagram having all states in which all nodes included in the endpoint set K are connected, on the basis of the undirected graph G and the endpoint set K; and a calculation procedure for calculating the variance value of a network reliability representing the probability that all nodes included in the endpoint set K are connected, on the basis of the binary decision diagram and the mean pi and the variance σi 2 of the availability of the links ei∈E.

Description

分散計算方法、分散計算装置及びプログラムDistributed computing method, distributed computing device and program
 本発明は、分散計算方法、分散計算装置及びプログラムに関する。 The present invention relates to a distributed computing method, a distributed computing device, and a program.
 通信ネットワークや電力網、交通網等のネットワークシステムでは、リンクが故障しそのリンクが使えなくなってしまうことが時折発生する。そのようなリンクの故障を確率的な事象と捉えると、或る指定した複数のノード間が接続している確率(つまり、或る指定した複数のノードのうちの任意の2つのノード間を繋ぐ道が存在する確率)を計算することができる。指定したk個のノード間が接続している確率を求める問題はk-ターミナル信頼性問題(k-terminal network reliability、以下、k-NRと略す。)と呼ばれ、その確率はネットワーク信頼性(又は、単に信頼性)とも呼ばれる。 In network systems such as communication networks, power networks, and transportation networks, it sometimes happens that a link fails and becomes unusable. If such a link failure is regarded as a probabilistic event, the probability that a plurality of specified nodes are connected (that is, the probability that any two nodes the probability that a road exists) can be calculated. The problem of obtaining the probability that k specified nodes are connected is called k-terminal network reliability (hereinafter abbreviated as k-NR), and the probability is network reliability ( Or simply called reliability).
 k-NRは#P-完全という計算が困難な問題に属することが知られており、愚直な方法では数十リンク程度のネットワークでもネットワーク信頼性の計算に多くの時間を要してしまう。これに対して、例えば、非特許文献1や2では、組合せ集合をコンパクトに表現できる二分決定グラフ(BDD:Binary Decision Diagram)を用いて高速にネットワーク信頼性を求める手法が提案されている。現在では、数多くの研究により数百リンク程度のネットワークに対して正確なネットワーク信頼性を求めることができる。 It is known that k-NR belongs to the problem of #P-completeness, which is difficult to calculate, and a simple method would take a lot of time to calculate network reliability even in a network with several tens of links. On the other hand, for example, Non-Patent Documents 1 and 2 propose a method of quickly obtaining network reliability using a Binary Decision Diagram (BDD) that can compactly represent a set of combinations. At present, many studies can obtain accurate network reliability for networks with hundreds of links.
 一方で、既存研究では、各リンクの可用性(すなわち、各リンクが正しく動作する確率)は正確に与えられるという仮定の下でネットワーク信頼性が計算されている。しかしながら、現実的にはリンクの可用性はいくらかの不確かさを持って与えられたり、例えば機械学習により誤差を持った形で推定されたりもする。もし各リンクの可用性が不確かさを分散(variance)の形で持っているならば、ネットワーク信頼性の値も不確かさを分散の形で持っているはずである。このようなネットワーク信頼性の不確かさを計算することは現実的な状況では重要である。例えば、ネットワーク信頼性は99%であるがその標準偏差(不確かさ)が1%あるネットワークシステムと、ネットワーク信頼性は98.5%であるがその標準偏差(不確かさ)は0.1%であるネットワークシステムとを考えた場合、前者の方がネットワーク信頼性自体は高いものの、後者の方が好まれる状況も十分に考えられる。 On the other hand, in existing research, network reliability is calculated under the assumption that the availability of each link (that is, the probability that each link will operate correctly) is given accurately. However, in reality, link availability is given with some uncertainty, or is estimated with error, for example, by machine learning. If the availability of each link has uncertainty in the form of variance, then the network reliability value should also have uncertainty in the form of variance. Calculating such network reliability uncertainties is important in a realistic situation. For example, a network system with a network reliability of 99% but with a standard deviation (uncertainty) of 1% and a network system with a network reliability of 98.5% but with a standard deviation (uncertainty) of 0.1% Considering a certain network system, although the former has higher network reliability, it is quite conceivable that the latter is preferred.
 しかしながら、上述したように、ネットワーク信頼性の不確かさを考慮した既存研究は存在せず、従ってネットワーク信頼性の分散を計算する手法も存在しない。 However, as mentioned above, there is no existing research that considers the uncertainty of network reliability, and therefore there is no method for calculating the variance of network reliability.
 本発明の一実施形態は、上記の点に鑑みてなされたもので、ネットワーク信頼性の分散を計算することを目的とする。 An embodiment of the present invention has been made in view of the above points, and aims to calculate the distribution of network reliability.
 上記目的を達成するため、一実施形態に係る分散計算方法は、連結な無向グラフG=(V,E)と、端点集合K⊂Vと、前記無向グラフGを構成するリンクe∈Eの可用性の平均p及び分散σ とを入力する入力手順と、前記無向グラフGと、前記端点集合Kとに基づいて、前記端点集合Kに含まれる全てのノードが接続する状態を全て持つ二分決定グラフを構築する構築手順と、前記二分決定グラフと、前記リンクe∈Eの可用性の平均p及び分散σ とに基づいて、前記端点集合Kに含まれる全てのノードが接続する確率を表すネットワーク信頼性の分散値を計算する計算手順と、をコンピュータが実行する。 To achieve the above object, a distributed computation method according to an embodiment includes a connected undirected graph G=(V, E), a set of endpoints K⊂V, and links e i ε forming the undirected graph G A state in which all nodes included in the endpoint set K are connected based on an input procedure for inputting the mean p i and the variance σ i 2 of the availability of E, the undirected graph G, and the endpoint set K and based on the BDD and the mean p i and variance σ i 2 of the availability of the links e i ∈E, all and a computing procedure for calculating a network reliability variance representing the probability that a node will connect.
 ネットワーク信頼性の分散を計算することができる。 It is possible to calculate the distribution of network reliability.
本実施形態に係る分散計算装置のハードウェア構成の一例を示す図である。It is a figure which shows an example of the hardware constitutions of the distributed computing device which concerns on this embodiment. 本実施形態に係る分散計算装置の機能構成の一例を示す図である。It is a figure showing an example of functional composition of a distributed computing device concerning this embodiment. 本実施形態に係る分散計算装置が実行する処理の流れの一例を示すフローチャートである。4 is a flowchart showing an example of the flow of processing executed by the distributed computing device according to the embodiment; グラフG及び端点集合Kの一例を示す図である。3 is a diagram showing an example of a graph G and an end point set K; FIG. 或る信頼性論理関数を表現するBDDの一例を示す図である。FIG. 2 is an example of a BDD representing a reliability logic function; BDDによりネットワーク信頼性の分散値を計算するアルゴリズムの一例を示す図である。FIG. 2 illustrates an example of an algorithm for calculating network reliability variance values by BDD;
 以下、本発明の一実施形態について説明する。 An embodiment of the present invention will be described below.
 <問題設定>
 「ネットワーク信頼性の分散」を数学的に定式化すると以下のようになる。
<Problem setting>
A mathematical formulation of "distribution of network reliability" is as follows.
 ネットワークは、連結な無向グラフG=(V,E)としてモデル化する。ここで、n=|V|をノード数、m=|E|をリンク数とする。各リンクe∈Eには可用性P∈[0,1]が与えられる。これは、リンクeは確率Pで正しく動作し、確率1-Pで故障する、ということを意味する。各リンクの状態(つまり、動作しているか又は故障しているか)は他のリンクの状態と確率的に独立であるものとする。また、Eに含まれる全てのリンクの状態の集まりを「Eの状態」と呼ぶことにする。このとき、Eの状態Xが与えられると、E⊆Eを動作しているリンクの集合として、動作しているリンクのみからなるGの部分グラフG=(V,E)を考えることができる。この状態Xが起こる確率は以下の式(1)で計算できる。 The network is modeled as a connected undirected graph G=(V,E). Let n=│V│ be the number of nodes and m=│E│ be the number of links. Each link e i εE is given availability P i ε[0,1]. This means that link e i will work correctly with probability P i and will fail with probability 1−P i . The state of each link (ie, up or down) is assumed to be probabilistically independent of the states of other links. Also, the set of states of all links included in E is called "the state of E". Then, given the state X of E, let E X ⊆E be the set of active links, and consider a subgraph G X =(V, E X ) of G consisting of only active links. can be done. The probability of this state X occurring can be calculated by the following equation (1).
Figure JPOXMLDOC01-appb-M000001
 ノードの集合K⊆Vが与えられたとき、Kに含まれるノードが全て接続する確率(ネットワーク信頼性)R(K)は以下の式(2)で計算できる。
Figure JPOXMLDOC01-appb-M000001
When a set of nodes K⊆V is given, the probability (network reliability) R G (K) that all the nodes included in K are connected can be calculated by the following equation (2).
Figure JPOXMLDOC01-appb-M000002
 ここで、ΓはKに含まれる全てのノードがG中で接続しているようなEの状態Xを全て集めた集合である。言い換えれば、Γは、Kに含まれる全てのノードが、動作しているリンクの集合Eにより全て接続しているようなEの状態Xを全て集めた集合である。
Figure JPOXMLDOC01-appb-M000002
where Γ K is the set of all states X of E such that all nodes in K are connected in G X . In other words, Γ K is the set X of all states X of E such that all nodes in K are all connected by the set E X of active links.
 いま、不確かさを反映するために、各リンクの可用性Pは平均p、分散σ の確率分布に従うものと仮定する。つまり、各リンクの可用性を確率変数とみなすことにする。このとき、各リンクの状態の独立性を担保するため、i≠jに対してPとPは統計的に独立であるものとする。すると、Eの状態Xを一つ固定したときの式(1)のP(X)や式(2)のR(K)は、P,・・・,Pに従属する確率変数となる。 Now, to reflect uncertainty, it is assumed that the availability P i of each link follows a probability distribution with mean p i and variance σ i 2 . In other words, let us consider the availability of each link as a random variable. At this time, in order to ensure the independence of the state of each link, P i and P j are assumed to be statistically independent for i≠j. Then, when one state X of E is fixed, P(X) in Equation (1) and R G (K) in Equation (2) are random variables dependent on P 1 , . . . , P m Become.
 以降、E[・]、Var[・]、Cov[・,・]をそれぞれP,・・・,Pに関する期待値、分散、共分散とする。各リンクの可用性がP=pと固定されたときの従来のネットワーク信頼性の値を考えると、これはR(K)の期待値E[R(K)]に一致する。求める「ネットワーク信頼性の分散」は、R(K)の分散Var[R(K)]の値であり、グラフGと端点集合Kと各リンクeの平均p及び分散σ とが与えられたときにこの分散Var[R(K)]の値を求めるのが解くべき問題である。 Henceforth, let E[·], Var[·], and Cov[·,·] be expected values, variances, and covariances of P 1 , . . . , P m , respectively. Considering the conventional network reliability value when the availability of each link is fixed as P i =p i , this agrees with the expected value E[R G (K)] of R G (K). The “variance of network reliability” to be obtained is the value of the variance Var[R G (K)] of R G (K), and the average p i and variance σ i 2 of the graph G, the endpoint set K, and each link e i The problem to be solved is to find the value of this variance Var[R G (K)] given .
 <従来技術>
 k-NRで求めるネットワーク信頼性の値は上記の式(2)で表されるため、Γ、すなわちKに含まれる全てのノードがG中で接続しているようなEの状態Xを全て列挙することができれば計算できる。しかし、Eの状態はリンクの数に対して指数的に多く存在するため、素直な列挙では数十リンクのネットワークでも計算時間が膨大になってしまう。そこで、Kに含まれる全てのノードが接続するようなリンクの組合せEを全て持つBDDを構築し、そのBDDを確率計算に用いることで、計算時間を削減することができる。特に、非特許文献1や2では、Kに含まれる全てのノードが接続するようなEの状態を全て列挙することなく、上記のようなBDDを直接構築できる手法となっているため、素直な列挙と比べて大幅な計算時間の削減を実現できる。なお、非特許文献1に記載されている手法がもととなるアイデアを提案したものであり、非特許文献2に記載されている手法は非特許文献1に記載されている手法の一部を改良しより高速にしたものである。
<Conventional technology>
Since the network reliability value obtained by k-NR is represented by the above equation (2), the state X of E such that Γ K , that is, all nodes included in K are connected in G X is If you can enumerate all of them, you can calculate. However, since the number of states of E is exponentially greater than the number of links, simple enumeration would result in a huge calculation time even for a network with several tens of links. Therefore, the calculation time can be reduced by constructing a BDD that has all combinations of links EX that connect all the nodes included in K, and using the BDD for probability calculation. In particular, in Non-Patent Documents 1 and 2, there is no need to enumerate all the states of E such that all the nodes included in K are connected, and the above BDD can be directly constructed. Significant reduction in computation time can be realized compared to enumeration. In addition, the idea based on the method described in Non-Patent Document 1 was proposed, and the method described in Non-Patent Document 2 is a part of the method described in Non-Patent Document 1. Improved and faster.
 一方で、非特許文献1や2に記載されている手法を含む既存研究はいずれも各リンクの可用性が正確に与えられるという仮定の下で行われたものであり、可用性が分散という不確かさを持って与えられた下でネットワーク信頼性の不確かさを議論した研究は存在しない。なお、ネットワーク信頼性の文脈で「分散」について触れているものとしては、式(2)のネットワーク信頼性の値をモンテカルロ法により近似的に計算する場合の誤差を考察した研究が存在するが、この研究はあくまでも近似計算手法に起因する不確かさを議論したものであり、各リンクの可用性の不確かさを考慮したものではない。 On the other hand, all existing research, including the methods described in Non-Patent Documents 1 and 2, were conducted under the assumption that the availability of each link is given accurately, and the uncertainty of availability is distributed. Given that there are no studies that discuss the uncertainty of network reliability. In addition, as a study that touches on "variance" in the context of network reliability, there is a study that considers the error when the value of network reliability in Equation (2) is approximately calculated by the Monte Carlo method. This study only discusses the uncertainty caused by the approximate calculation method, and does not consider the uncertainty of the availability of each link.
 <提案手法>
 上述した通り、ネットワーク信頼性の不確かさについて触れた既存研究は存在せず、従ってネットワーク信頼性の分散を計算する手法も存在しない。一方で、上記の式(2)を用いて、分散Var[R(K)]を以下のように分解することは可能である。
<Proposed method>
As mentioned above, there is no existing research that addresses network reliability uncertainty, and thus no method for calculating network reliability variance. On the other hand, using equation (2) above, it is possible to decompose the variance Var[R G (K)] as follows.
Figure JPOXMLDOC01-appb-M000003
 ここで、共分散Cov[P(X),P(Y)]はリンクの数に比例する時間で計算できるため、Γに含まれる状態を全て列挙できれば上記の式によりネットワーク信頼性の分散の値を計算することができる。しかし、一般に、Γに含まれる状態の個数はリンクの本数mに対して指数的に増加するため、数十リンク程度のネットワークでもこの方法で分散の値を計算するのは計算時間的に困難である。このように、ネットワーク信頼性の分散の値を現実的な時間で計算する手法は存在せず、100リンク以上のネットワークでもネットワーク信頼性の分散の値を計算できるような手法が必要である。
Figure JPOXMLDOC01-appb-M000003
Here, the covariance Cov[P(X), P (Y)] can be calculated in a time proportional to the number of links. value can be calculated. However, since the number of states included in ΓK generally increases exponentially with the number of links m, it is difficult in terms of calculation time to calculate the variance using this method even in a network with several tens of links. is. As described above, there is no method for calculating the value of network reliability variance in a realistic time, and a method that can calculate the value of network reliability variance even in a network with more than 100 links is required.
 そこで、本実施形態では、ネットワーク信頼性の分散の値を高速に計算することができる手法を提案し、この提案手法を実行する分散計算装置10について説明する。 Therefore, in the present embodiment, a technique that can quickly calculate the value of variance of network reliability is proposed, and the distributed computing device 10 that executes the proposed technique will be described.
 本提案手法では、非特許文献1に記載されている手法をベースとして、BDDを用いてネットワーク信頼性の分散値を高速に計算する。非特許文献1や2で用いられているBDDは、頂点と向きのある辺の集まりでできたループの無いグラフ(有向非巡回グラフと呼ぶ)で組合せ集合を表現するデータ構造である。なお、元のネットワークの「ノード」、「リンク」と区別するため、有向非巡回グラフのノード、リンクはそれぞれ「頂点」、「辺」と呼ぶことにする。 In this proposed method, based on the method described in Non-Patent Document 1, BDD is used to quickly calculate the variance value of network reliability. The BDD used in Non-Patent Documents 1 and 2 is a data structure that expresses a set of combinations with a loop-free graph (called a directed acyclic graph) made up of a collection of vertices and oriented edges. In order to distinguish from the "nodes" and "links" of the original network, the nodes and links of the directed acyclic graph are called "vertices" and "edges", respectively.
 非特許文献1や2に記載されている手法では、BDDを構築した後、BDDの各頂点を一状態とする動的計画法によりネットワーク信頼性の値を計算している。一方で、本実施形態に係る分散計算装置10では、非特許文献1と同様にBDDを構築した後、BDDの頂点のペアを一状態とする新たな動的計画法によりネットワーク信頼性の分散の値を計算する。これにより、BDDのサイズの多項式に比例する時間でネットワーク信頼性の分散の値を計算することができる。一般に、ネットワーク信頼性を計算するためのBDDのサイズは、Γに含まれる状態の個数と比べて非常に小さくなることが多いため、200リンク程度のネットワークでも分散の値を計算できるようになる。 In the methods described in Non-Patent Documents 1 and 2, after constructing a BDD, a network reliability value is calculated by dynamic programming with each vertex of the BDD as one state. On the other hand, in the distributed computing device 10 according to the present embodiment, after constructing a BDD in the same manner as in Non-Patent Document 1, a new dynamic programming method in which a pair of vertices of the BDD is one state is used to calculate distribution of network reliability. Calculate a value. This allows computing the value of the network reliability variance in a time proportional to the polynomial of the size of the BDD. In general, the size of BDD for calculating network reliability is often very small compared to the number of states included in ΓK , so it becomes possible to calculate the variance even in a network with about 200 links. .
 <分散計算装置10のハードウェア構成例>
 本実施形態に係る分散計算装置10のハードウェア構成例を図1に示す。図1に示すように、本実施形態に係る分散計算装置10は一般的なコンピュータ又はコンピュータシステムのハードウェア構成で実現され、入力装置101と、表示装置102と、外部I/F103と、通信I/F104と、プロセッサ105と、メモリ装置106とを有する。これらの各ハードウェアは、それぞれがバス107により通信可能に接続される。
<Hardware Configuration Example of Distributed Computing Device 10>
FIG. 1 shows a hardware configuration example of the distributed computing device 10 according to this embodiment. As shown in FIG. 1, the distributed computing device 10 according to the present embodiment is realized by the hardware configuration of a general computer or computer system, and includes an input device 101, a display device 102, an external I/F 103, and a communication I/F. /F 104 , processor 105 and memory device 106 . Each of these pieces of hardware is communicably connected via a bus 107 .
 入力装置101は、例えば、キーボードやマウス、タッチパネル、各種物理ボタン等である。表示装置102は、例えば、ディスプレイや表示パネル等である。なお、分散計算装置10は、例えば、入力装置101及び表示装置102のうちの少なくとも一方を有していなくてもよい。 The input device 101 is, for example, a keyboard, mouse, touch panel, various physical buttons, and the like. The display device 102 is, for example, a display, a display panel, or the like. Note that the distributed computing device 10 may not have at least one of the input device 101 and the display device 102, for example.
 外部I/F103は、記録媒体103a等の外部装置とのインタフェースである。分散計算装置10は、外部I/F103を介して、記録媒体103aの読み取りや書き込み等を行うことができる。なお、記録媒体103aとしては、例えば、CD(Compact Disc)、DVD(Digital Versatile Disk)、SDメモリカード(Secure Digital memory card)、USB(Universal Serial Bus)メモリカード等が挙げられる。 The external I/F 103 is an interface with an external device such as the recording medium 103a. The distributed computing device 10 can read from and write to the recording medium 103 a via the external I/F 103 . Examples of the recording medium 103a include CD (Compact Disc), DVD (Digital Versatile Disk), SD memory card (Secure Digital memory card), USB (Universal Serial Bus) memory card, and the like.
 通信I/F104は、分散計算装置10を通信ネットワークに接続するためのインタフェースである。プロセッサ105は、例えば、CPU(Central Processing Unit)やGPU(Graphics Processing Unit)等の各種演算装置である。メモリ装置106は、例えば、HDD(Hard Disk Drive)やSSD(Solid State Drive)、フラッシュメモリ、RAM(Random Access Memory)、ROM(Read Only Memory)等の各種記憶装置である。 The communication I/F 104 is an interface for connecting the distributed computing device 10 to a communication network. The processor 105 is, for example, various arithmetic units such as a CPU (Central Processing Unit) and a GPU (Graphics Processing Unit). The memory device 106 is, for example, various storage devices such as HDD (Hard Disk Drive), SSD (Solid State Drive), flash memory, RAM (Random Access Memory), and ROM (Read Only Memory).
 本実施形態に係る分散計算装置10は、図1に示すハードウェア構成を有することにより、後述する各種処理を実現することができる。なお、図1に示すハードウェア構成は一例であって、分散計算装置10は、例えば、複数のプロセッサ105を有していてもよいし、複数のメモリ装置106を有していてもよいし、図示したハードウェア以外の様々なハードウェアを有していてもよい。 The distributed computing device 10 according to the present embodiment has the hardware configuration shown in FIG. 1, so that it can implement various processes described later. Note that the hardware configuration shown in FIG. 1 is an example, and the distributed computing device 10 may have, for example, a plurality of processors 105, a plurality of memory devices 106, Various hardware other than the illustrated hardware may be included.
 <分散計算装置10の機能構成例>
 本実施形態に係る分散計算装置10の機能構成例を図2に示す。図2に示すように、本実施形態に係る分散計算装置10は、入力部201と、構築部202と、計算部203と、出力部204とを有する。これら各部は、例えば、分散計算装置10にインストールされた1以上のプログラムが、プロセッサ105に実行させる処理により実現される。ここで、分散計算装置10には、連結な無向グラフG=(V,E)と、端点集合K⊆Vと、各リンクe∈Eの可用性の平均p及び分散σ とが与えられるものとする。なお、n=|V|をノード数、m=|E|をリンク数とする。
<Functional Configuration Example of Distributed Computing Device 10>
FIG. 2 shows an example of the functional configuration of the distributed computing device 10 according to this embodiment. As shown in FIG. 2 , the distributed computing device 10 according to this embodiment has an input unit 201 , a construction unit 202 , a calculation unit 203 and an output unit 204 . Each of these units is implemented by, for example, processing that one or more programs installed in the distributed computing device 10 cause the processor 105 to execute. Here, the distributed computing device 10 has a connected undirected graph G=(V, E), a set of endpoints K⊆V, and the mean p i and the variance σ i 2 of the availability of each link e i εE. shall be given. Let n=|V| be the number of nodes and m=|E| be the number of links.
 入力部201は、与えられた連結な無向グラフGと端点集合Kと各リンクe∈Eの可用性の平均p及び分散σ とを入力する。 The input unit 201 inputs the given connected undirected graph G, the end point set K, and the availability mean p i and variance σ i 2 of each link e i ∈E.
 構築部202は、入力部201によって入力された連結な無向グラフGと端点集合Kとに基づいて、端点集合Kに含まれる全てのノードが接続する状態を全て持つBDDを生成(構築)する。 Based on the connected undirected graph G and the endpoint set K input by the input unit 201, the constructing unit 202 generates (constructs) a BDD having all the states in which all the nodes included in the endpoint set K are connected. .
 計算部203は、構築部202によって生成されたBDDと各リンクe∈Eの可用性の平均p及び分散σ の値とに基づいて、動的計画法によりネットワーク信頼性の分散の値を計算する。 Based on the BDD generated by the construction unit 202 and the values of the mean p i and the variance σ i 2 of the availability of each link e i εE, the calculation unit 203 calculates the value of the variance of network reliability by dynamic programming. to calculate
 出力部204は、計算部203によって計算されたネットワーク信頼性の分散の値を予め決められた任意の出力先に出力する。なお、当該出力先としては、例えば、ディスプレイ等の表示装置102、メモリ装置106、通信ネットワークを介して接続される他の装置又は機器等が挙げられる。 The output unit 204 outputs the network reliability variance value calculated by the calculation unit 203 to a predetermined arbitrary output destination. Note that the output destination includes, for example, the display device 102 such as a display, the memory device 106, and other devices or devices connected via a communication network.
 <分散計算装置10が実行する処理の流れ>
 本実施形態に係る分散計算装置10が実行する処理の流れについて、図3を参照しながら説明する。
<Flow of Processing Executed by Distributed Computing Device 10>
The flow of processing executed by the distributed computing device 10 according to this embodiment will be described with reference to FIG.
 まず、入力部201は、与えられた連結な無向グラフGと端点集合Kと各リンクe∈Eの可用性の平均p及び分散σ とを入力する(ステップS101)。 First, the input unit 201 inputs the given connected undirected graph G, the end point set K, and the mean p i and variance σ i 2 of the availability of each link e i εE (step S101).
 次に、構築部202は、上記のステップS101で入力された連結な無向グラフGと端点集合Kとに基づいて、端点集合Kに含まれる全てのノードが接続する状態を全て持つBDDを生成(構築)する(ステップS102)。なお、本ステップの詳細については後述する。 Next, the constructing unit 202 generates a BDD having all states in which all nodes included in the endpoint set K are connected based on the connected undirected graph G and the endpoint set K input in step S101. (Build) (step S102). Details of this step will be described later.
 次に、計算部203は、上記のステップS102で生成されたBDDと上記のステップS101で入力された各リンクe∈Eの可用性の平均p及び分散σ とに基づいて、動的計画法によりネットワーク信頼性の分散の値を計算する(ステップS103)。なお、本ステップの詳細については後述する。 Next , the calculation unit 203 dynamically Calculate the value of the variance of the network reliability by the planning method (step S103). Details of this step will be described later.
 そして、出力部204は、上記のステップS103で計算されたネットワーク信頼性の分散の値を予め決められた任意の出力先に出力する(ステップS104)。 Then, the output unit 204 outputs the network reliability variance value calculated in step S103 to a predetermined arbitrary output destination (step S104).
 <ステップS102の詳細>
 本ステップでBDDを構築する際に構築部202が実行する処理は非特許文献1や2で提案されている手法と同一であるため、以下では、主に、本ステップで構築されるBDDの構造について説明する。
<Details of step S102>
Since the processing executed by the construction unit 202 when constructing the BDD in this step is the same as the method proposed in Non-Patent Documents 1 and 2, the following mainly describes the structure of the BDD constructed in this step. will be explained.
 構築部202は、グラフGと端点集合Kとに基づいて、端点集合Kに含まれる全てのノードがG中で接続しているようなEの状態Xを全て集めた集合Γを表現する二分決定グラフ(BDD)を構築する。BDDは、本来、論理関数を表現するデータ構造であるため、まずはΓに対応する論理関数fを考える。 Based on the graph G and the endpoint set K, the constructing unit 202 expresses a set ΓK that collects all the states X of E such that all the nodes included in the endpoint set K are connected in GX . Construct a Binary Decision Diagram (BDD). Since the BDD is originally a data structure that expresses a logic function, first consider a logic function fK corresponding to ΓK .
 二値変数をリンクの本数と同じ個数用意し、x,・・・,xとする。そして、Eの状態Xに対して、リンクeが動作していればx=true、リンクeが故障していればx=falseとして、x,・・・,xへの値の割り当てを対応させる。以上の対応の下で、x,・・・,xに対応するEの状態XがΓに含まれていればtrue、そうでなければfalseの値を取る論理関数をf(x,・・・,x)とする。このfを以下では信頼性論理関数と呼ぶことにする。 The same number of binary variables as the number of links are prepared, and x 1 , . . . , x m . Then, for the state X of E, if the link ei is working, x i =true, if the link ei is broken, x i =false, and x 1 , . . . , x m Match the value assignments. Under the above correspondence, a logical function f K ( x 1 , . . . , x m ). This fK is hereinafter referred to as the reliability logic function.
 以降、BDDの構造について説明する。BDDは、論理関数を根付きの有向非巡回グラフB=(N,A)として表現するデータ構造である。ここで、Nは頂点集合、Aは辺集合とする。根とは、自身に向かう辺が一つも無いような頂点のことであり、以降、r∈Nと表す。 The structure of the BDD will be explained below. A BDD is a data structure that represents a logical function as a directed acyclic graph with roots B=(N,A). Here, N is a vertex set and A is an edge set. A root is a vertex that has no edges toward itself, and is hereinafter expressed as rεN.
 頂点集合Nに含まれる頂点は、出る辺が無い特別な頂点である終端頂点と、それ以外の内部頂点とに大別される。以下、終端頂点を The vertices included in the vertex set N are roughly divided into terminal vertices, which are special vertices with no outgoing edges, and other internal vertices. Below, the terminal vertex is
Figure JPOXMLDOC01-appb-M000004
と表す。以下、これらの終端頂点をそれぞれ「第1の終端頂点」、「第2の終端頂点」ともいう。
Figure JPOXMLDOC01-appb-M000004
is represented as Hereinafter, these terminal vertices are also referred to as "first terminal vertex" and "second terminal vertex", respectively.
 各内部頂点  Each internal vertex
Figure JPOXMLDOC01-appb-M000005
は、LO辺及びHI辺と呼ばれる2つの出る辺と、ラベルと呼ばれる整数値とを持つ。内部頂点vのLO辺及びHI辺が指す先の頂点をそれぞれLO子頂点及びHI子頂点と呼び、それぞれlo(v)及びhi(v)と表す。また、vのラベルはlb(v)∈{1,・・・,c}と表す。ここで、cはBDDが表現する論理関数が入力として取る二値変数の個数である。BDDのラベルは、根から辺を辿って行くに従い値が大きくなっていく必要がある。すなわち、各内部頂点vに対してlb(v)<lb(lo(v))かつlb(v)<lb(hi(v))である必要がある。なお、終端頂点については、それぞれ
Figure JPOXMLDOC01-appb-M000005
has two outgoing edges, called LO and HI edges, and integer values called labels. The vertices pointed to by the LO edge and HI edge of internal vertex v are called LO child vertices and HI child vertices, respectively, and are denoted by lo(v) and hi(v), respectively. Also, the label of v is expressed as lb(v)ε{1, . . . , c}. where c is the number of binary variables that the logic function represented by the BDD takes as input. The value of the BDD label needs to increase as it goes from the root to the edges. That is, lb(v)<lb(lo(v)) and lb(v)<lb(hi(v)) for each interior vertex v. For terminal vertices,
Figure JPOXMLDOC01-appb-M000006
と仮定する。最後に、BDDのサイズは、頂点の個数|N|として定義される。
Figure JPOXMLDOC01-appb-M000006
Assume that Finally, the size of the BDD is defined as the number of vertices |N|.
 BDDの構造が与えられると、BDDの各頂点vに対して、その頂点に対応する論理関数fが再帰的に定義される。すなわち、終端頂点 Given the structure of the BDD, for each vertex v of the BDD, the logic function fv corresponding to that vertex is recursively defined. i.e. the terminal vertex
Figure JPOXMLDOC01-appb-M000007
に対しては、それぞれf=true、f=falseとする。これらはそれぞれ恒真関数(入力値に関わらずtrueとなる論理関数)、恒偽関数(入力値に関わらずfalseとなる論理関数)である。それ以外の内部頂点vに対しては、
Figure JPOXMLDOC01-appb-M000007
, set f v =true and f v =false, respectively. These are respectively a true function (logical function that is true regardless of the input value) and a constant function (logical function that is false regardless of the input value). For other interior vertices v,
Figure JPOXMLDOC01-appb-M000008
とする。
Figure JPOXMLDOC01-appb-M000008
and
 一例として、図4に示すグラフG及び端点集合Kが与えられたときのΓに対応する信頼性論理関数fを表現するBDDを図5に示す。ここで、図4に示す例では、丸がグラフGのノード、黒丸が端点(つまり、グラフGのノードのうち、端点集合Kに含まれるノード)を表している。また、図5に示す例では、a~aが内部頂点、a及びa10が終端頂点、実線がHI辺、破線がLO辺を表しており、丸の中の数字が内部頂点のラベルを表している。 As an example, FIG. 5 shows a BDD expressing the reliability logic function f K corresponding to Γ K given the graph G and the endpoint set K shown in FIG. Here, in the example shown in FIG. 4, circles represent nodes of graph G, and black circles represent endpoints (that is, nodes included in endpoint set K among nodes of graph G). In the example shown in FIG. 5, a 1 to a 8 are internal vertices, a 9 and a 10 are terminal vertices, the solid line represents the HI edge, and the broken line represents the LO edge. represents a label.
 信頼性論理関数fを表現するBDDは、その根rから第1の終端頂点 A BDD representing a reliability logic function f K is defined from its root r to the first terminal vertex
Figure JPOXMLDOC01-appb-M000009
までの経路が、Γの中の一部の状態に対応している。すなわち、BDDの根rから第1の終端頂点までの各経路のそれぞれは、Γに含まれる1つ以上の状態に対応している。
Figure JPOXMLDOC01-appb-M000009
corresponds to some states in ΓK . That is, each path from the root r of the BDD to the first terminal vertex corresponds to one or more states included in ΓK .
 BDDの根rから第1の終端頂点までの経路が与えられたとき、ラベルがiである頂点からLO辺を辿ることはラベルiに対応するリンクeが故障していることを意味し、HI辺を辿ることはラベルiに対応するリンクeが動作していることを意味する。また、ラベルがiである頂点を通らないことは、ラベルiに対応するリンクeの状態は考慮しないことを意味する。 Given a path from the root r of the BDD to the first terminal vertex, following the LO edge from the vertex labeled i means that the link e i corresponding to label i is broken, and Tracing the HI edge means that the link ei corresponding to label i is active. Also, not passing through the vertex whose label is i means that the state of the link ei corresponding to the label i is not considered.
 例えば、図5において、ラベル1の頂点a→頂点aのHI辺→ラベル2の頂点a→頂点aのHI辺→ラベル4の頂点a→頂点aのLO辺→ラベル5の頂点a→頂点aのHI辺→第1の終端頂点という経路は、(e,e,e,e,e)=(動作,動作,動作,故障,動作)及び(動作,動作,故障,故障,動作)という2つの状態と対応する。 For example, in FIG. 5, vertex a 1 of label 1→HI edge of vertex a 1 →vertex a 2 of label 2→HI edge of vertex a 2 →vertex a 6 of label 4→LO edge of vertex a 6 →label 5 vertex a 8 → HI edge of vertex a 8 → first terminal vertex is (e 1 , e 2 , e 3 , e 4 , e 5 )=(action, action, action, fault, action) and It corresponds to two states (operation, operation, failure, failure, operation).
 信頼性論理関数fを表現するBDDを効率的に構築するのが、非特許文献1や2に記載されている手法である。従って、信頼性論理関数fを表現するBDDを構築する具体的な方法については、非特許文献1や2等を参照されたい。 Techniques described in Non-Patent Documents 1 and 2 are for efficiently constructing a BDD that expresses the reliability logic function fK . Therefore, please refer to Non-Patent Documents 1 and 2, etc. for a specific method of constructing a BDD that expresses the reliability logic function fK .
 <ステップS103の詳細>
 本ステップでは、計算部203は、BDDの頂点のペアを一状態とする新たな動的計画法によりネットワーク信頼性の分散の値を計算する。以下では、まずネットワーク信頼性の分散の値を計算するための考え方について説明した後、その計算を行うための具体的な手順を表すアルゴリズムについて説明する。
<Details of Step S103>
In this step, the calculation unit 203 calculates the value of the network reliability variance by a new dynamic programming method in which the pair of vertices of the BDD is one state. In the following, first, the idea for calculating the network reliability variance value will be described, and then the algorithm representing the specific procedure for performing the calculation will be described.
 まず、信頼性論理関数fを表現するBDDの各頂点vに対して、e,・・・,elb(v)-1の状態を固定した下で、端点(つまり、端点集合Kに含まれるノード)が全て接続する条件付き確率をRとする。すると、各リンクの可用性Pを用いて、以下の式(3)及び(4)に示す等式が成立する。 First, for each vertex v of the BDD that expresses the reliability logic function f K , under fixing the states of e 1 , . Let R v be the conditional probability that all the included nodes) are connected. Then, using the availability P i of each link, the following equations (3) and (4) are established.
Figure JPOXMLDOC01-appb-M000010
 このとき、根rに対応するRがネットワーク信頼性R(K)である。
Figure JPOXMLDOC01-appb-M000010
Then, R r corresponding to the root r is the network reliability R G (K).
 ここで、P,・・・,Pが確率変数ならば、RはP,・・・,Pに従属する確率変数となる。従って、BDDの頂点のペアu,v∈Nに対して、RとRのP,・・・,Pに関する共分散Cov[R,R]を考えることができる。このとき、ネットワーク信頼性の分散Var[R(K)]は、Cov[R,R]と表すことができる。 Here, if P 1 , . . . , P m are random variables, R v is a random variable dependent on P 1 , . Thus, for a pair of vertices u, v ∈ N in the BDD, we can consider the covariance Cov[R u , R v ] of R u and R v with respect to P 1 , . . . , P m . Then, the network reliability variance Var[R G (K)] can be expressed as Cov[R r , R r ].
 共分散に関する等式を用いることにより、共分散Cov[R,R]はその子頂点に関する共分散に分解することができる。例えば、uとvのラベルがlb(u)=lb(v)=iと等しい場合、q=1-pとして以下の式(5)に示す等式が成立する。 Using the covariance equation, the covariance Cov[R u ,R v ] can be decomposed into covariances with respect to its child vertices. For example, when the labels of u and v are equal to lb(u)=lb(v)=i, the following equation (5) holds with q i =1−p i .
Figure JPOXMLDOC01-appb-M000011
 また、uのラベルがvのラベルより小さい場合、すなわちi=lb(u)<lb(v)の場合、以下の式(6)に示す等式が成立する。
Figure JPOXMLDOC01-appb-M000011
Also, when the label of u is smaller than the label of v, that is, when i=lb(u)<lb(v), the following equation (6) holds.
Figure JPOXMLDOC01-appb-M000012
 lb(u)>lb(v)の場合も同様の等式(つまり、上記の式(6)と対称な式)が成立する。更に、uとvの少なくともいずれか一方が終端頂点である場合、Cov[R,R]=0となる。
Figure JPOXMLDOC01-appb-M000012
A similar equation (that is, an equation that is symmetrical with equation (6) above) holds when lb(u)>lb(v). Furthermore, Cov[R u , R v ]=0 if at least one of u and v is a terminal vertex.
 以上の等式を再帰的に用いることで、Var[R(K)]=Cov[R,R]を再帰的に分割し計算することが可能になる。 By recursively using the above equation, Var[R G (K)]=Cov[R r , R r ] can be recursively divided and calculated.
 ・ネットワーク信頼性の分散の値を計算する手順
 以下では、BDDによりネットワーク信頼性の分散の値を計算する手順を表すアルゴリズムについて、図6を参照しながら説明する。ここで、本アルゴリズムの入力はBDD B=(N,A)とリンクeの可用性の平均p及び分散σ であり、出力はBが表すネットワーク信頼性の分散値である。なお、本アルゴリズムの各行が表す手順は計算部203によって実行される。
Procedure for Calculating Network Reliability Variance Value An algorithm representing a procedure for calculating the network reliability variance value by BDD will be described below with reference to FIG. where the inputs to the algorithm are BDD B=(N,A) and the mean p i and variance σ i 2 of the availability of links e i , and the output is the variance of the network reliability that B represents. Note that the procedure represented by each line of this algorithm is executed by the calculation unit 203 .
 まず、1行目~3行目では、各v∈Nに対して、e[v]=E[R]を計算する。具体的には、まず、1行目で第1の終端頂点に関しては1.0、第2の終端頂点に関しては0.0をそれぞれ代入する。そして、2行目~3行目で、ラベルが大きい頂点を先に走査する順番(bottom-up order)でe[v]←q・e[lo(v)]+p・e[hi(v)]によりe[v]の値を順に計算する。この過程で、ネットワーク信頼性の期待値E[R(K)]=e[r]も求まる。なお、q=1-pである。 First, in the first to third lines, e[v]=E[R v ] is calculated for each v∈N. Specifically, first, in the first line, 1.0 is substituted for the first terminal vertex, and 0.0 is substituted for the second terminal vertex. Then, in the second and third lines, e[v]←q i ·e[lo(v)]+p i ·e[hi( v)] to calculate the value of e[v] in turn. In this process, the expected value E[R G (K)]=e[r] of network reliability is also obtained. Note that q i =1−p i .
 次に、4行目で再帰的手続き(関数)であるCov(r,r)を呼び出す。手続きCov(u,v)は共分散Cov[R,R]を計算する再帰的手続きであり、その中身は5行目~17行目に記述されている。まず、6行目~7行目で、uとvの少なくともいずれか一方が終端頂点であれば0を答えとして返す。次に、8行目~9行目で、すでに共分散Cov[R,R]を計算していれば、計算済みの値を答えとして返す。ここで、計算済みのCov[R,R]を保存するためのキャッシュとして連想配列cを用意し、c[(u,v)]にCov[R,R]の値を格納する。 Next, the fourth line calls Cov(r, r), which is a recursive procedure (function). The procedure Cov(u, v) is a recursive procedure for calculating the covariance Cov[R u , R v ], and its content is described on lines 5-17. First, in the 6th and 7th lines, 0 is returned as an answer if at least one of u and v is a terminal vertex. Next, in lines 8 and 9, if the covariance Cov[R u , R v ] has already been calculated, the calculated value is returned as an answer. Here, an associative array c is prepared as a cache for storing the calculated Cov[R u , R v ], and the value of Cov[R u , R v ] is stored in c[(u, v)]. .
 以降は式(5)や式(6)を用いて、手続きCov(u,v)を再帰的に呼び出すことによりCov[R,R]を計算する。すなわち、10行目でuのラベルとvのラベルの小さい方をiに格納した後、uのラベルがvのラベルより小さい場合は式(6)を用いてCov[R,R]を計算し(11行目~12行目)、vのラベルがuのラベルより小さい場合は式(6)と対称な式を用いてCov[R,R]を計算し(13行目~14行目)、ラベルuとラベルvが等しい場合は式(5)を用いてCov[R,R]を計算する。そして、最後に、計算したCov[R,R]の値を返す(17行目)。これにより、ネットワーク信頼性の分散値が得られる。 Henceforth, Cov[R u , R v ] is calculated by recursively calling the procedure Cov(u, v) using equations (5) and (6). That is, after the smaller of the label of u and the label of v is stored in i on the 10th line, if the label of u is smaller than the label of v, Cov[R u , R v ] is calculated using equation (6). (11th to 12th lines), and if the label of v is smaller than the label of u, calculate Cov[R u , R v ] using a formula symmetrical to formula (6) (13th line to 14th line), if label u and label v are equal, calculate Cov[R u , R v ] using equation (5). Finally, it returns the calculated value of Cov[R u , R v ] (line 17). This gives the network reliability variance.
 <まとめ>
 以上により、本実施形態に係る分散計算装置10は、これまでは素朴な方法以外では計算できなかったネットワーク信頼性の分散値を高速に計算することが可能となる。特に、100~200程度のリンク規模の実ネットワークに対してもネットワーク信頼性の分散値を計算できることが確かめられた。これにより、各リンクの可用性の不確かさがネットワーク信頼性の不確かさに与える影響を定量的に評価することができる。したがって、例えば、稼働実績が無く正確な可用性がわからないリンクがネットワークに追加された場合に、そのリンクがネットワーク信頼性の不確かさにどう影響するのかを定量的に分析することができる。
<Summary>
As described above, the distributed computing device 10 according to the present embodiment can quickly calculate the distributed value of network reliability, which could not be calculated until now except by a naive method. In particular, it was confirmed that the distributed value of network reliability can be calculated even for a real network with a link scale of about 100 to 200. This makes it possible to quantitatively evaluate the impact of the uncertainty of availability of each link on the uncertainty of network reliability. Therefore, for example, when a link is added to the network with no track record of operation and whose exact availability is unknown, it is possible to quantitatively analyze how the link affects the uncertainty of network reliability.
 なお、本実施形態に係る分散計算装置10は、求めたネットワーク信頼性の分散値を用いて種々のネットワーク制御を行ってもよい。例えば、同程度のネットワーク信頼性であれば分散値が低いネットワークとなるように制御したり、仮にネットワーク信頼性が低くても或る基準範囲内であれば分散値が低いネットワークとなるように制御したりする、といった制御を行ってもよい。 Note that the distributed computing device 10 according to the present embodiment may perform various network controls using the calculated network reliability variance value. For example, if the network reliability is the same, the network is controlled to have a low variance value, or even if the network reliability is low, if it is within a certain reference range, the network is controlled to have a low variance value. You may perform control such as performing.
 本発明は、具体的に開示された上記の実施形態に限定されるものではなく、請求の範囲の記載から逸脱することなく、種々の変形や変更、既知の技術との組み合わせ等が可能である。 The present invention is not limited to the specifically disclosed embodiments described above, and various modifications, alterations, combinations with known techniques, etc. are possible without departing from the scope of the claims. .
 10    分散計算装置
 101   入力装置
 102   表示装置
 103   外部I/F
 103a  記録媒体
 104   通信I/F
 105   プロセッサ
 106   メモリ装置
 107   バス
 201   入力部
 202   構築部
 203   計算部
 204   出力部
10 distributed computing device 101 input device 102 display device 103 external I/F
103a recording medium 104 communication I/F
105 Processor 106 Memory Device 107 Bus 201 Input Unit 202 Construction Unit 203 Calculation Unit 204 Output Unit

Claims (8)

  1.  連結な無向グラフG=(V,E)と、端点集合K⊂Vと、前記無向グラフGを構成するリンクe∈Eの可用性の平均p及び分散σ とを入力する入力手順と、
     前記無向グラフGと、前記端点集合Kとに基づいて、前記端点集合Kに含まれる全てのノードが接続する状態を全て持つ二分決定グラフを構築する構築手順と、
     前記二分決定グラフと、前記リンクe∈Eの可用性の平均p及び分散σ とに基づいて、前記端点集合Kに含まれる全てのノードが接続する確率を表すネットワーク信頼性の分散値を計算する計算手順と、
     をコンピュータが実行する分散計算方法。
    An input that inputs a connected undirected graph G=(V, E), a set of endpoints K⊂V, and the mean p i and variance σ i 2 of the availability of the links e i εE that make up the undirected graph G a procedure;
    a construction procedure for constructing a binary decision graph having all states in which all nodes included in the endpoint set K are connected, based on the undirected graph G and the endpoint set K;
    Variance value of network reliability representing the probability that all nodes included in the end point set K are connected based on the binary decision diagram and the average p i and variance σ i 2 of the availability of the link e i εE a computational procedure for computing
    is a distributed computing method performed by a computer.
  2.  前記計算手順は、
     前記リンクe∈Eの可用性の平均p及び分散σ を用いて、前記二分決定グラフの頂点ペアに対する共分散を再帰的に計算することで、前記ネットワーク信頼性の分散値を計算する、請求項1に記載の分散計算方法。
    The calculation procedure is
    Compute the variance of the network reliability by recursively computing the covariance for the pair of vertices of the BDD using the mean p i and the variance σ i 2 of the availability of the link e i εE. , the variance calculation method of claim 1.
  3.  前記計算手順は、
     前記二分決定グラフの各頂点vに対して、各リンクe,・・・,elb(v)-1(ただし、lb(v)は頂点vのラベル)の各々が動作しているか又は故障しているかのいずれであるかを表す状態を固定した下で、前記端点集合Kに含まれる全てのノードが接続する条件付き確率をR、前記リンクeの可用性をP、前記二分決定グラフの根である頂点をrとしたとき、RとRのP,・・・,P(ただし、m=|E|)に関する共分散の値を、前記ネットワーク信頼性の分散値として計算する、請求項1又は2に記載の分散計算方法。
    The calculation procedure is
    For each vertex v of the BDD, each link e 1 , . R v is the conditional probability that all nodes included in the end point set K are connected, P i is the availability of the link ei , and the binary decision When r is the vertex that is the root of the graph, the covariance value of R r and R r with respect to P 1 , . 3. The variance calculation method according to claim 1 or 2, wherein the calculation is performed as
  4.  前記計算手順は、
     前記二分決定グラフの任意の頂点ペアを(u,v)としたときのRとRのP,・・・,Pに関する共分散をCov[R,R]として、
     頂点uのラベルと頂点vのラベルの等しい場合に成り立つ第1の等式と、頂点uのラベルが頂点vのラベルより小さい場合に成り立つ第2の等式と、頂点vのラベルが頂点uのラベルより小さい場合に成り立つ第3の等式とのいずれかにより前記共分散Cov[R,R]を再帰的に計算することで、前記RとRのP,・・・,Pに関する共分散をCov[R,R]の値を計算する、請求項3に記載の分散計算方法。
    The calculation procedure is
    Let Cov [R u , R v ] be the covariance of R u and R v with respect to P 1 , .
    A first equation that holds if the label of vertex u equals the label of vertex v, a second equation that holds if the label of vertex u is less than the label of vertex v, and a second equation that holds if the label of vertex u is less than the label of vertex u. P 1 , . . . , 4. The variance calculation method of claim 3, wherein the covariance with respect to Pm is calculated as Cov[ Rr , Rr ] values.
  5.  前記計算手順は、
     前記頂点uのラベルをlb(u)、前記頂点vのラベルをlb(v)、前記頂点uのHI辺が指す子頂点をhi(u)、前記頂点uのLO辺が指す子頂点をlo(u)、前記頂点vのHI辺が指す子頂点をhi(v)、前記頂点vのLO辺が指す子頂点をlo(v)、q=1-p、E(・)を期待値として、
     lb(u)=lb(v)=iである場合、前記第1の等式は、Cov[R,R]=(q +σ )Cov[Rlo(u),Rlo(v)]+(p-σ )(Cov[Rlo(u),Rhi(v)]+Cov[Rhi(u),Rlo(v)])+(p -σ )Cov[Rhi(u),Rhi(v)]+σ (E[Rhi(u)]-E[Rlo(u)])(E[Rhi(v)]-E[Rlo(v)])であり、
     i=lb(u)<lb(v)である場合、前記第2の等式は、Cov[R,R]=qCov[Rlo(u),R]+pCov[Rhi(u),R]であり、
     i=lb(v)<lb(u)である場合、前記第3の等式は、Cov[R,R]=qCov[R,Rlo(v)]+pCov[R,Rhi(v)]である、請求項4に記載の分散計算方法。
    The calculation procedure is
    The label of the vertex u is lb(u), the label of the vertex v is lb(v), the child vertex pointed to by the HI edge of the vertex u is hi(u), and the child vertex pointed to by the LO edge of the vertex u is lo Expect (u), hi(v) for the child vertex pointed to by the HI edge of the vertex v, lo(v) for the child vertex pointed to by the LO edge of the vertex v, q i =1−p i , E(・) as the value
    If lb(u)=lb(v)=i, then the first equation is Cov[R u ,R v ]=(q i 2i 2 )Cov[R lo(u) ,R lo (v) ]+(p i q i −σ i 2 )(Cov[R lo(u) ,R hi(v) ]+Cov[R hi(u) ,R lo(v) ])+(p i 2 −σ i 2 )Cov[R hi(u) ,R hi(v) ]+σ i 2 (E[R hi(u) ]−E[R lo(u) ])(E[R hi(v) ] -E[R lo(v) ]);
    If i=lb(u)<lb(v), then the second equation is Cov[R u ,R v ]=q i Cov[R lo(u) ,R v ]+p i Cov[R hi(u) ,R v ],
    If i=lb(v)<lb(u), then the third equation is Cov[R u ,R v ]=q i Cov[R u ,R lo(v) ]+p i Cov[R u , R hi(v) ].
  6.  前記構築手順は、
     リンクe∈Eが動作していればx=true、リンクe∈Eが故障していればx=falseとなる二値変数x,・・・,xを入力として前記端点集合Kに含まれる全てのノードが接続していればtrue、そうでなければfalseを出力とする論理関数を表現する二分決定グラフを構築する、請求項1乃至5の何れか一項に記載の分散計算方法。
    The construction procedure includes:
    , xm with binary variables x 1 , . 6. The binary decision diagram according to any one of claims 1 to 5, which constructs a binary decision graph representing a logic function whose output is true if all nodes included in the set K are connected, and false otherwise. Variance calculation method.
  7.  連結な無向グラフG=(V,E)と、端点集合K⊂Vと、前記無向グラフGを構成するリンクe∈Eの可用性の平均p及び分散σ とを入力するように構成されている入力部と、
     前記無向グラフGと、前記端点集合Kとに基づいて、前記端点集合Kに含まれる全てのノードが接続する状態を全て持つ二分決定グラフを構築するように構成されている構築部と、
     前記二分決定グラフと、前記リンクe∈Eの可用性の平均p及び分散σ とに基づいて、前記端点集合Kに含まれる全てのノードが接続する確率を表すネットワーク信頼性の分散値を計算するように構成されている計算部と、
     を有する分散計算装置。
    Input a connected undirected graph G=(V, E), a set of endpoints K⊂V, and the mean p i and variance σ i 2 of the availability of the links e i εE that make up said undirected graph G. an input section configured in
    a constructing unit configured to construct a binary decision graph having all states in which all nodes included in the endpoint set K are connected, based on the undirected graph G and the endpoint set K;
    Variance value of network reliability representing the probability that all nodes included in the end point set K are connected based on the binary decision diagram and the mean p i and variance σ i 2 of the availability of the link e i εE a computing unit configured to compute
    A distributed computing device with
  8.  コンピュータに、請求項1乃至6の何れか一項に記載の分散計算方法を実行させるプログラム。 A program that causes a computer to execute the distributed computing method according to any one of claims 1 to 6.
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Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
SASAKI YUYA ET AL.: "Approximate computing of network reliability by BDD", IEICE TECHNICAL REPORT, vol. 117, no. 374, 15 December 2017 (2017-12-15), pages 85 - 90, XP009547085 *
SASAKI YUYA: "Approximate computing of efficient network reliability in vague graphs", DBSJ JAPANESE JOURNAL, vol. 18-J, no. 7, 1 March 2020 (2020-03-01), pages 1 - 8, XP093077139 *

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