The Concept of Building a Network of Digital Twins to Increase the Efficiency of Complex Telecommunication Systems

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Introduction
Using the functionality of the cybernetic model [1], called the digital twin, allows us to solve a number of complex problems of various kinds.e difference between the digital twin and the conventional mathematical models [2] is that it allows us to very accurately display the current state of the investigated object or process.At the same time, the digital twin can not only operate with the reality that is supposed to exist objectively but also work with augmented reality [3] and with an altered reality.e concept "altered reality" was proposed in [4].
is feature of the digital twin is illustrated in Figure 1. e dotted line under the number 3 indicates the relationship between augmented and altered reality.e functionality of this connection is the subject of a separate study.Features of the digital twin, as applied to one object, are well described in the technical literature [5][6][7].
In 2003, Michael Grieves first proposed the digital twin terminology as "digital equivalent to a physical product" [8].However, up till now, no formal or widely accepted definition has been achieved, and no unified creating and deploying process, too.Lots of digital twin conceptual models or reference models have been proposed.And different understandings about digital twin and digital twin conceptual models are reviewed by Liu et al. [9].For example, Grieves and Vickers [10] defined two types of digital twins: digital twin prototype (DTP) and digital twin instance (DTI), and DTs were in a digital twin environment (DTE).Bao et al. [11] defined three types of digital twin models from the perspective of the manufacturing process on the shop floor, which were Product Digital Twins, Process Digital Twins, and Operation Digital Twins.Ullah [12] proposed three types of digital twins: object twins, process twins, phenomena twins.
e digital twin model must contradic between the simplified virtual model and the complex behavior of the physical object [9].A variety of frameworks and reference models of digital twin were proposed, but none of them become industry consensus.Some researches adopt an information management-oriented view, focusing on semantic connections and seamless information flow [12].A compromised approach is to implement flexible modeling in a modular way.Negri et al. [13] proposed adding black-box modules to the main simulation model.Black-box modeling using data or grey-box modeling using a combination of physics and data is also feasible.
Digital twin is a base concept of Industry 4.0 manufacturing.In complex cyber-physical production systems with many digital twins, information must be aggregated.is leads to the need for digital twins to interact.erefore, the network models for twin-to-twin communication are needed.e purpose of this paper is to develop a concept that extends the principles of applying digital twins to a set of objects or processes.e main idea is to create a network that unites various digital twins.A complex telecommunication system that provides support for a wide range of services was chosen as an example of such an aggregate.e proposed concept is aimed at ensuring the sustainable functioning of complex telecommunication systems.e developed solutions are illustrated by two main examples.
e first example is related to the organization of a feedback chain for the main participants of the telecommunications market engaged in the life cycle, which is represented by the following set of operations: (i) Formation of principles for the operating telecommunication network modernization.(ii) Development of equipment for the transmission, switching, and processing of information.(iii) Preparation of design documentation for the work, allowing us to begin the modernization of the operating telecommunication network.(iv) Carrying out construction work, as a result of which the telecommunication system as a whole is updated.
(v) Organization of technical operation processes taking into account new properties of the telecommunication system.
e second example is related to monitoring of served traffic.For this example, we consider a special case concerning a radical change in traffic due to emergency situations.Monitoring traffic in such situations has shown that a sharp increase in the activity of users of the telecommunication system begins.If it is possible to find rational solutions for traffic service in such cases, then the relevant experience must be used by all telecommunication operators.e idea of creating a digital twin network, as far as the authors know, is new.For this reason, real projects aimed at building a network of digital twins are not presented in the paper.
e rest of this paper is structured as follows.Section 2 analyzes the features of mathematical models of a digital twin.e next section contains proposed conceptual provisions for creating a network of digital twins.
e two examples of the application of this concept mentioned above are described in the fourth and fifth sections, respectively.Section 6 presents considerations regarding the possibility of using a network of digital twins to predict the behavior of complex systems in emergency situations of various natures.Analysis of the nature of information processes in a network of digital twins is the subject of the seventh section.Conclusion and future research are presented in Section 8.

Features of Mathematical Models
One of the commonly used mathematical models is a black-, grey-, or white-box [1,2].A black-box is considered an object, the essence of which we do not know anything except for a set of functions of the form A(t), B(t), C(t), and D(t).
ese functions describe the external action and operations performed by the black-box, control actions, and the result, respectively.For the grey-box, an example of which is shown in Figure 2; in addition to knowing the functions A(t), B(t), C(t), and D(t), incomplete information on the structure and functions of the investigated object is available.A white-box model is used in cases where information about the structure and functions of the studied object is considered complete and reliable.
An important feature of this model is that it implements a feedback loop.It often allows us to significantly increase the value of the model.Unfortunately, in practice, it is not always possible to create a model in which there is a feedback loop.One of the tasks of the digital twin is to build an effective feedback loop that can change its functions, taking into account the features of the processes occurring in the telecommunication system.e effectiveness of using mathematical models is determined by a number of factors, among which it is appropriate to single out three important points: 2 Complexity (c) e correct interpretation of mathematical results for practical decisions Figure 3 shows three phases: the development of the model, its study, and interpretation of the results.At each phase, processes that lead to the accumulation of errors are realized.As a result, a certain dependence F 1 (x 1 , x 2 , x m ), which can conditionally be considered correct, is gradually transformed into the function F 2 (y 1 , y 2 , y n ). is function contains a number of errors that can significantly distort the necessary results.
It should be noted that the achievements of modern mathematics allow us to state that the minimal errors in the study of the model correspond to point (b).Minimization of errors determined by point (a) is ensured by using the model in the form of a digital twin.e degree of adequacy may approach the maximum possible level.To a large extent, the same conclusion applies to point (c).However, the validity of this statement is associated with the use of a model in the form of a network of digital twins.
is hypothesis is considered in the next section.

Digital Twin Network Model
At first glance, the proposed model is formed as a simple collection of digital twins.In fact, the digital twin network is a more complex design.Figure 4 shows a model consisting of seven digital twins.Each digital twin serves as a model of one object.A typical example of such an object is a switching node.Interfaces between model elements are designated as NNI J , Network Node Interface of the jth type.e given illustration contains three types of NNI J interfaces, A, B, and C.
e difference in interfaces is determined mainly by the stack of protocols used [13].At the Network Access Layer, Ethernet technology is typically used.At the Internet and Transport layers, the TCP/IP protocol stack is usually dominant.At the Application layer, most likely, the development of a new protocol or even a set of protocols is required.It can be argued that Big Data [14], Data Mining [15], Neural Networks [16], and the like will be widely used in a network of digital twins.Obviously, in a network of digital twins, both cloud and fog computing will be widely used [17,18].
Information accumulated by a network of digital twins will become a good base for scenarios for the development of telecommunication networks [19].Processing this information will reduce the risks [20], inevitable for the development of complex systems of any kind due to the uncertainty of the driving forces of evolutionary processes.
An assertion of this kind should be illustrated by several examples.Two such examples are discussed in the following sections of the paper.

Digital Twin Network and Feedback Loops
e lifecycle of a telecommunication system can be represented in the form of five successive stages, shown in Figure 5.It is assumed that the end of the life cycle of a telecommunication system is a qualitatively new stage in its modernization. is circumstance and the limitation of the number of stages explain the use of the word "simplified" in the name of the proposed model.
Set W 1 includes a set of theoretical research results that form the concept of creating or upgrading a telecommunication system.e values of the remaining sets W k (k � 2, 3, 4, 5) are determined by the names of the corresponding rectangles.
Some of the results are used by developers of the transmission, switching, and processing equipment.At the same time, some theoretical provisions are being revised.
e measure of the difference in solutions for steps i and j (j � i + 1) is denoted as P ij .e Q ij value is considered a measure of the effectiveness of feedback between solution developers for steps i and j (j � i − 1).
An analysis of the main stages of the lifecycle carried out for a number of telecommunication systems showed that the Description of the subject of study Entity model

Mathematical model Model elaboration
Interpretation of the results Feedback loop Complexity differences between P ij are very significant, and the feedback efficiency between solution developers is close to zero.A radical change in the current situation can be ensured by creating a network of digital twins. is network will allow the solution developer to track all changes made in the subsequent stages and intervene in the development of the telecommunication system.In other words, the values of P ij can be minimized.Automation of information exchange provides maximization of Q ij values.ere are no quantitative estimates of such an analysis yet.Apparently, a network of digital twins will help get the needed results.

Digital Twin Network and Traffic Monitoring
e results of traffic monitoring serve as an important information base for the modernization of the telecommunication system [21].For the implementation of traffic monitoring functions, both software of switching nodes and specialized tools can be used [22].
e quality of service indicators for a telecommunication system is determined by the recommendations of the International Telecommunication Union [23].e task of all the main participants in the telecommunications market is to provide normalized indicators of the quality of service for multiservice traffic [24].
Figure 6 shows a model of a telecommunication system (left fragment) and its digital twin (right fragment).It is assumed that the communication network consists of five routers.Consequently, the network of digital twins will have five elements.A network of digital twins is an undirected graph with vertices a i and edges b ij .e terms used in the proposed construction are borrowed from the concept of dia$par [25]. is system includes the concept of "digital twin" as one of the main components.
e vertices a i and the edges b ij can be associated with various metrics or their combination.Typical examples of metrics are the cost of the hardware, throughput indicators, and reliability characteristics.From the point of view of the considered example, the main interest is associated with the characteristics of the served traffic, but the cost and reliability metrics can also be claimed.
e results accumulated by the network of digital twins are useful for solving many problems.A very important aspect of traffic monitoring is its behavior in emergency situations [26].Such cases lead to avalanche-like growth of traffic. is process can be aggravated by the failure of parts of the equipment that occur due to the destructive effects inherent in emergency situations.In addition, the list of services demanded by users of the telecommunication system may change significantly.
e prototype of the network of digital twins made it possible to solve a number of urgent problems regarding the functioning of the multiservice network in emergency situations [27,28].
e results obtained indicate the advisability of using feedback loops, an example of which was given in the previous section.

Digital Twin Network and Prediction of the Behavior of Complex Systems in Emergency Situations
Emergency traffic monitoring, mentioned in the previous section, is an example of passive use of the functionality that is typical for a digital twin network.Additional properties of a network of digital twins that are not explicitly inherent in complex systems can be used to predict their behavior in the event of atypical situations of various kinds.Besides, a network of digital twins can potentially work with unstructured data. is issue is briefly discussed in the next section.
One of the typical examples of an atypical situation is the outlier of the observed values of a process.In statistics, the observed value is considered an outlier, which differs markedly from the rest of the sample elements [29].Outliers, as a rule, arise randomly.ey can be caused by measurement errors and also testify to the presence of the socalled "heavy-tailed distribution" [30].Also, other reasons for the appearance of outliers are possible.
Figure 7 shows the behavior of a certain function f(x), for which it is appropriate to distinguish between three ranges along the abscissa.e first range includes the values of the function in question on the interval [0, x 1 ].On the interval (x 1 , x 2 ), no values of the function f(x) are observed.Recent changes in the function f(x) are observed on the interval [x 2 , x 3 ].
Similar situations can be found when measuring Internet traffic.ey are similar to the Tukey-Huber model [31], for which, with a very low probability ε, the investigated process f(x) is represented by the function h(x).Moreover, with a high probability equal to (1 − ε), a process is observed that is described by the function g(x):   4 Complexity ( In a network of digital twins, it is possible to conduct experiments that are difficult to carry out, using real objects, without serious problems for ongoing business processes.For example, the task was set to verify the quality of the telecommunications network in conditions of significant traffic growth.If such traffic is artificially created, then there will be a problem in the telecommunication network with the quality of customer service, which can lead to complaints and the need to pay fines to those users who have entered into a Service Level Agreement with a service provider [32].In a network of digital twins, an experiment with a sharp increase in traffic will not lead to negative consequences.e use of the Tukey-Huber model and similar ones will allow us to develop prognostic estimates necessary to choose the path of modernization of complex systems.It should be emphasized that forecasts based on the use of a network of digital twins usually provide an answer to the question: "What will happen?"A reasonable question "Why can this happen?"requires independent research, for which a network of digital twins will not always be an effective tool.An acceptable level of efficiency can be achieved if the initially complex system is built on the basis of a rational solution [33].In this case, a complex system stays stable when changing external and internal factors affecting the stability of its functioning.
It is possible that, in order to solve forecasting problems, some models in the network of digital twins will have to be perfected.e complexity of the refinement will depend on the nature of the complex system for which a network of digital twins is being created.Moreover, in some cases, the collaboration of two or more networks of digital twins will be useful.
To illustrate the last statement, we can give an example, which is sometimes given by specialists in the Big Data and Data Mining technologies [14,15].At one enterprise, from time to time, mass failures occurred in electronic units, which were assembled manually.For a long time, it was not possible to identify the pattern of occurrence of failures.
en one of the specialists drew attention to the fact that the installation of these blocks was carried out mainly by young women.An analysis of the periods of time during which the failures were observed and the late pregnancy of employees engaged in the installation of electronic components made it possible to establish the pattern.Increased sweating during assembly affected the quality of the contacts, which led to block failures.After pregnant employees were transferred in advance to other operations before going on maternity leave, the refusals ceased.
To obtain such results, it should be possible to combine those networks of digital twins that are initially created to solve completely different problems.us, we can say that we should focus on combining networks of digital twins, which is a very difficult task.As a result of such a combination, a kind of super network of digital twins is created.

Information Processes in the Digital Twin Networks
A network of digital twins is a specific information system [34].Some types of information may be represented by structured data.Processing such data is a relatively simple task.Part of the information should be considered unstructured data.eir processing is usually more complicated.Apparently, for a network of digital twins, it will be necessary to develop standards that define the rules for creating and processing information.
For those objects and processes that often change due to the introduction of innovative solutions, standards for the creation and processing of information will have to be reviewed periodically.For this reason, it is necessary to develop a methodological approach that will allow us to quickly and cost-effectively update the relevant software.
One of the possible approaches was proposed in [35].It was developed to solve problems that are significantly influenced by singularity processes [36].
ese processes should be considered when building and developing digital twin networks.For this reason, the methodological approach described in [35] is also interesting for minimizing the costs of supporting information processes in digital twin networks.
In a simplified form, the essence of the methodological approach is shown in Figure 8. Equipment for building a network of digital twins can be represented in the form of two components due to the decomposition operation: hardware and software.Further, each of these components is divided into invariable core and variable component.When developing equipment for a network of digital twins, one should strive to minimize the costs of variable components (for both hardware and software).is fact is indicated by an arrow pointing down.It is possible that, at the initial stage of creating a network of digital twins, this will require an increase in costs for the invariable core, which is indicated by the arrow pointing up.
A certain increase in costs at the initial stage of creating a network of digital twins will be compensated by the high speed of introducing new functionalities, aimed primarily at the adequacy of the model to its original.For the practical implementation of the proposed solution, a very important condition is the prediction of further development paths of the studied objects or processes.For this purpose, various forecasting methods can be used [37].Along with prognostic Complexity estimates, various scenarios for the further development of the studied objects or processes should be developed [19].e data exchanged by digital twins include messages of varying degrees of importance.In addition, they can vary markedly in volume.Sometimes, to identify an event, it is enough to transmit one bit, and in some cases, we have to exchange video with a volume that is estimated in megabytes and even gigabytes.Obviously, for the majority of digital twins, the resources of public telecommunication networks will be used.For this reason, additional traffic can noticeably worsen the quality of service indicators established by applicable standards.
e emerging problem can be solved in various ways.One of them is the rational assignment of priorities [38] to different types of traffic in a network of digital twins.As an example, consider Figure 9 [39], which shows the dependences of the average waiting time W (1)   i for demands of different priority in the load range of the teletraffic system ρ from 0 to 0.6.e demand is considered an IP packet.e subscript of the value W (1)   i determines the priority level.If i � 0, then all applications are served without priorities in the order they are received at the entrance of the teletraffic system.
e curves are constructed for the relationship between the intensity values of demands of the first, second, and third priorities, defined as follows: 10, 30, and 60%.
e dashed line shows the dependence of the average duration of waiting for applications when serving them without priorities.All values W (1)   i are normalized to the average time of their service (processing of IP packets in a router or transmission over information exchange paths).e course of the curves clearly demonstrates that the introduction of a priority service discipline can significantly improve the quality of service indicators for some types of services.For example, with the introduction of three priorities, the average waiting time for applications of the first priority is noticeably reduced.It should be emphasized that, for multiservice networks based on packet technologies, the priority discipline of service is recommended by the International Telecommunication Union [40].
ere are several factors to consider when prioritizing IP packets that transmit information in a digital twin network.
ey can be divided into two large groups, internal and external.A typical example of internal factors is the importance of information in terms of decision-making time to resolve a problem.e effect of the additional load on the public telecommunications network is an important external factor that should be considered when choosing the priority level for IP packets related to the work of digital twins.
e actual utilization of the resources of the public telecommunications network should also be taken into account.In some cases, these resources are redundant.is situation is usually typical after the completion of the next stage of the modernization of the telecommunication network.en, some IP packets that transmit information in a digital twin network can be assigned the first or second priority level.is statement is true if the specified quality indicators are provided for those subscribers who have concluded the so-called Service Level Agreements with the Telecommunications Operator [41].
e queuing theory is based on a number of assumptions that are not always correct for real telecommunications networks.A network of digital twins can accomplish two interrelated challenges.First, it does not introduce assumptions that can significantly affect the accuracy of the problem solution.Secondly, it helps to find new approaches to solving complex problems in the field of teletraffic theory.
Over time, the resources of a network built with some redundancy come in line with the served traffic.en the first and second priorities for IP packets that carry information in a network of digital twins can be assigned very rarely.Such a solution is possible for a minimum number of such IP packets.Apparently, priority service is generally acceptable for short messages of high importance.e main traffic generated in the digital twin network will be served with a lower priority level.

Complexity
An alternative option for building a network of digital twins is renting the necessary resources from the Telecommunications Operator or building dedicated information and communication system. is approach will be acceptable for objects of special importance since it is associated with very high investments.

Future Research Directions
It should be emphasized that the creation of digital twins does not negate the relevance of serious scientific research.Digital twins only make it possible to increase the efficiency of scientific research through the use of models that are as adequate as possible to the objects and processes under consideration.
Historically, digital twins are associated by some experts with an approach called the BIM, Building Information Model [42].e similarity between BIM and the digital twin lies mainly in the general principles of mapping real objects and processes to the world of models.
e true value of digital twins, as well as the dia$par concept, will be manifested in conducting interdisciplinary studies [43].ey are considered by the authors of this paper as one of the most important areas of further research.e digital twin network mentioned above is designed to solve problems that require the use of multidisciplinary approaches.Among interdisciplinary works, it is appropriate to note the following publications [44][45][46].
e second line of further research is aimed at implementing the concept of "Society 5.0" [47], also called Super Smart Society.One of the important provisions of "Society 5.0" is the widespread use of promising digital technologies.
is fact allows us to consider the practical application of digital twins as an integral system solution for the concept of "Society 5.0." As the third area of further research, the development of economic and mathematical methods for the study of models presented in the form of digital twins should be mentioned.Such methods seem to be very important and cannot be replaced by other ones of obtaining new knowledge.On the other hand, new methods will be used in conjunction with perspective solutions that expand.
Standard methods of teletraffic models investigate with queues [48,49].In conclusion, it is appropriate to formulate the influence of a network of digital twins on the chain "Data-Information-Knowledge-Wisdom" [50].e corresponding concept is known by the abbreviation DIKW. Figure 10 shows an additional layer, "Understanding," which is used by some DIKW researchers.
Most likely, the network of digital twins will most of all influence three layers in the chain "Information-Knowledge-Understanding." e degree of positive influence will vary.According to the authors of this paper, the most significant influence will fall on the "Understanding" layer.
is assumption is true at the current level of artificial intelligence [51].It is possible that in the future a network of digital twins will also create such results that could be attributed to the Wisdom layer.

Figure 5 :
Figure 5: Simplified lifecycle of the telecommunication network.

Figure 7 :
Figure 7: Example of the behavior of the function f(x).

Figure 9 :Figure 8 :
Figure 9: Mean values of waiting time for demands with different priorities.