The article considers an approach to assessing the importance metrics and reliability of networks. The Monte Carlo method is used to estimate Birnbaum metrics and failure probability with determination of the confidence interval. To conduct a computational experiment, the R software package is used. A description is given of the representation of the control system reliability model in the iGraph package, which provides visualization of the results. The model of a three-level network structure is considered as an example.
Currently, there is a rapid development of information technologies and their implementation
in various areas of human activity [
In addition to such technical characteristics of computer networks as: performance, latency,
security, scalability, extremely important characteristics are complex reliability indicators:
availability factor, average unavailability time per year [
Issues of reliability of systems with a network structure are still relevant [4, 5].
For instance, the reliability of network nodes, termed as the terminal reliability, is the probability that a set of operational edges provides communication paths between every pair of nodes. Another closely related concept with reliability is availability, which can be defined as the probability that a component will be available when demanded [3].
Importance measures (IMs) are used to evaluate the efect of component reliability on system reliability. IMs are useful tools in reliability engineering [14], risk analysis [15, 17], and system reliability optimization. These measures can help reliability engineers to find a better solution rapidly because they can identify the weakest links of the system, which are the premise and foundation of system design, maintenance, and resource configurations. During the system design period, component importance can help designers determine cost efective design ideas with relatively high system reliability and low cost rapidly.
Note that the significance of a system element according to Birnbaum reflects the degree of influence of changes in the element’s readiness coeficient on changes in the system’s readiness coeficient. The significance of a system element according to Barlow-Proshan reflects the probability that a system failure that occurred at a certain point was caused by this element. The significance of a Vesely-Fassel system element reflects the probability that this element is one of the failed elements, provided that the system failed. Vesely-Fassel significance characterizes those elements that are most often involved in system failures.
The cost of increasing the risk for a system element reflects the importance of maintaining the current level of reliability of this element. The cost of reducing the risk for a system element reflects the degree to which the system’s availability coeficient increases if this element is replaced with a flawless element. The critical significance of a system element reflects the probability that this element is critical for the system at a given time. The potential for improvement of a system element reflects the gain in system reliability if this element is replaced with a completely reliable one [16].
The most widely used importance measures were acquired in the component allocation problem. The optimal component allocation problem arises when a pool of available components exists (in the market or in the company store) for each subsystem and a designer chooses a set of components for each subsystem such that the system reliability is maximized subject to constraints such as weight and cost.
The easiest way to estimate is to use Crude Monte-Carlo simulation [19]. Let (1), ..., () be independent identically distributor random vectors with the same distribution as . Then 1 = · ∑︁ ℎ(()) =1 ( ) = 1 · ( − 2) is an unbiased estimator for , where ℎ() is reliability function. Its sample variance is given by (1) (2)
An important measure for the "eficiency" of any estimator is its relative error. Relative error for is given by √︂ 1 − ( ) = (3)
To achieve acceptable accuracy, we assume an error less 0.01, and the number of iterations is 5143.
It is natural to assume that diferent elements afect the system’s behavior in terms of reliability in diferent ways. The ability of the researcher to quantify the nature of the elements influence on the behavior of the system is of particular importance in the analysis of systems. This makes it possible to identify system weaknesses, select optimal redundancy, and make a rational impact on the reliability of the system as a whole.
The importance of the element in the system is defined as a private derivative of the availability factor (the probability of) the system availability (the probability of) the element for which an analysis of its significance: (, ) = ℎ()
This characteristic is called Birnbaum significance (BIM-significance) [ 6]. The significance is estimated by the number of times the system availability coeficient increases when the element availability coeficient increases. BIM-significance does not depend on the readiness coeficient of the element , but depends only on for all = in satisfies the inequalities 0 ≤ () ≤ 1.
Birnbaum importance considers the relationships between the system performance when component i is perfect, the system performance when component i fails, and the current system performance.
For the BIM-significance indicator, you can get an expression in the following form: (, ) = ℎ(1, ) − ℎ(0, ) where ℎ(, ) is reliability function, ℎ(1, ) for absolutely reliable component, ℎ(0, ) for absolutely unreliable component.
Other metrics are also used to analyze the significance of elements. The list is presented in table 1.
Birnbaum () reflects the degree of influence of the change in the availability factor of the element on the change in the availability factor of the system. Risk Decrease () reflects the importance of maintaining the current level of reliability of this element. Fussell-Vesely () reflects the probability that this element is one of the failed elements, provided that the system failed, the Vesely-Fassel significance characterizes those elements that are most often involved in system failures. Risk Increase () reflects the importance of maintaining the current level of reliability of this element. Criticality Importance () reflects the probability that this element is critical for the system at a given time. (4) (5)
Many researches have been devoted to computational aspects of significance estimation, including [7, 8].
R is a programming language for statistical data processing and working with graphics, as well as a free open-source computing environment for the GNU project. The R language contains tools that allow you to create multiple parallel threads of calculations (due to simultaneous loading of several processor cores) and reduce the time spent on modeling several times. To assess the accuracy of the results obtained, the bootstrap method is proposed. The essence of the method in this case is that on the basis of one available sample (obtained using the graph traversal algorithm), a series of pseudo-samples of the same size is formed, consisting of random combinations of the original set of elements. In this case, the "random selection with return" algorithm is used, i.e. the extracted element is returned to the original set and has a chance to be selected again. For each random sample to estimate the probability of failure (or probability of failure) and thus formed the sample probabilities of system failure (or probability of failurefree operation), which further evaluated the necessary statistical data (standard deviation or confidence limits). To calculate the number of iterations and estimate the confidence interval, a standard approach is used in accordance with [3]. Various techniques can also be used to improve accuracy, with the most widespread sampling by significance [10].
To search the graph for paths between certain vertices, use the width traversal algorithm (an implementation of this algorithm in the iGraph library is used). To generate random numbers with an exponential distribution law, the basic functions of the R language are used [9].
Reliability modeling includes iterations. At each iteration, a random operating time before failure is generated for each system element (vertex) (this time is generated based on the specified failure rate of the system element). After that, the elements are sorted in ascending order of uptime to failure, and the element (vertex) with the lowest time is selected. This vertex is removed from the graph and the existence of paths between certain vertices of the graph is checked (between which vertices the presence of a path should be checked is listed in the description of system failure criteria). If all necessary vertexes are found, the current iteration continues and the next element in increasing time to failure is selected and the corresponding vertex is removed from the graph. Next, it checks again whether there are paths between certain vertexes. If no paths are found between the specified vertexes, the system is considered to have failed. The failure time is fixed and a new iteration begins.
For this experiment, we take the dodecahedron graph (Fig. 1), the model that is widely considered for a network reliability example. The dodecahedron graph with 20 vertices, 30 edges, and fault criteria - loss connection of {1, 20}.
In this model, it is assumed that the vertexes are reliable, and the edges (communication lines) may fail. The probability of edge failure is constant over time and is equal to . Using the Monte-Carlo method we estimate the dependence of the probability of failure of the graph on the probability of failure of each edge. The obtained simulation results are in good agreement with the results obtained [11].
The obtained results show a sharp decrease in the probability of connectivity when the probability of edge failure is less than 10− 1, which correlates well with the presence of one path of length 3, and two paths of length 4 (by length is meant the smallest number of edges lying between the vertices {1, 20}).
Automated process control system is a group of technical and software solutions designed to automate technological processes in industrial enterprises. As a rule, automatic process control systems are understood as a complete solution that ensures the automated execution of the basic operations of the technological process of production. Components of automatic process control systems can be separate automatic control systems and automated devices connected in a single complex. Such as Supervisory control and data acquisition systems (SCADA), distributed control systems (DCS), emergency protection systems.
SCADA is a complex of equipment, distributed across three levels of the hierarchy, depending
on the functional purpose: upper level: process operator panels; mid-level: server racks, central
computing server, lower level: remote control terminal [12]. The architecture of the process
control system takes into account the requirements for the implementation of the principle of a
single failure and has structural redundancy [
The system consists of the following units: 1. The Hardware of the main computing resources (S1, S2); 2. Control servers (CS1, CS2) is designed for collecting, processing and storing information about the operation of system equipment, as well as information interaction. 3. Remote terminal unit (B1-B8) is designed for control field equipment. 4. Four independent Ethernet line;
The representation in SCADA of the system in the form of a graph and its representation in the package R is shown in Fig. 4. The standard reliability data (Failure rate: Central control panel main, redundant - 50 · 10− 6ℎ− 1, Central control unit - 30 · 10− 6ℎ− 1, Remote control terminal - 20 · 10− 6ℎ− 1, Commutator - 10 · 10− 6ℎ− 1) are considered as initial data [12]. It is assumed that issues related to the process of ensuring computational reliability are provided by the necessary capacities [13].
A system failure is considered to be the loss of communication between the fictitious vertex F1 (SCADA system operator) and the field equipment control subsystems B1-B8.
The simulation results of failure probability are presented in Fig. 5 Table 2 shows the reference reliability importance metrics estimated for the SCADA system.
Increasing the reliability of the elements with the biggest significance will allow achieving the required failure probability. As a result of the BIM assessment, CPUs of CS unit make the greatest contribution to system reliability. If further reliability improvements are needed, these elements should be considered. Possible ways to improve reliability can be considered: the introduction of continuous monitoring, the choice of more reliable components, using the of redundancy by reserving.
According to the results of the Monte Carlo simulation (fig. 5), it can be argued that the probability of the SCADA functioning in 5000 hours will be no less than 0.992 with a confidence probability of 0.90. To improve accuracy, methods of reducing the variance of a sample estimate, for example, the Cross-Entropy Monte-Carlo method [11, 18], can be used.
The reliability estimated values and importance metrics can be used for system modernization, in particular when using one of the LKA-LKD algorithms-heuristics [16] for example.
The reliability models of three-level networks based on the model systems with independent elements are also considered, a method for assessing reliability is proposed. The Monte Carlo method is used to estimate failure probability and reliability importance metric with determination of the confidence interval. To conduct a computational experiment, the R software package is used. A description is given of the representation of the control system reliability model in the iGraph package, which provides visualization of the results. R language was primarily created and is continuing to evolve as a statistical data processing tool. The value of the BIM metric is determined for further system improvement. Packet Segmentation (2019) 2019 Wave Electronics and its Application in Information and Telecommunication Systems, WECONF 2019, art. no. 8840643. doi: 10.1109/WECONF.2019.8840643 [3] Gertsbakh B., Shpungin Y. Models of Network Reliability: Analysis, Combinatorics, and
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