Mathematical Models and Methods of Investigation of Hybrid Communication Networks Based on Laser and Radio Technologies

AbstractNew Radio Access Technology 3GPP New Radio has become the fundamental wireless technology in the fifth-generation networks, which allows us to achieve high data rates due to the ability to work in the millimeter-wave band. But the key feature and the main problem of 5G New Radio networks is that people themselves, cars, buildings, etc. are signal blockers, while the base stations of the fourth generation networks have widescreen broadcasting and such small obstacles do not cause loss of connection. Service providers and mobile operators are already testing the proposed technology. In this connection, the scientific community has the task of analyzing the performance of these systems and increasing it in the future. Currently, there are known studies of “basic” mathematical models of such networks. By this term, we mean models built in the simplest possible assumptions. However, due to the justified necessity of introducing new technology into the daily lives of subscribers, service providers pose the scientific community with the task of analyzing the effectiveness of the most appropriate mathematical models. For example, a technology of splitting transmitted data into two streams using as 5G and both 4G transmission technologies is considered now by 3GPP Project Coordination Group. The paper is devoted to such a problem. We consider a mathematical model of the message transmitting with the implementation of the splitting function in the communication networks of New Radio technology in the form of a resource queueing system with a renewal arrival process and non-exponential service. For this problem, an approximation of a stationary two-dimensional probability distribution of the number of occupied resources in parallel service units is obtained. It is shown that this approximation coincides with the Gaussian distribution, and its area of applicability is shown.

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New classes of graphs with edge $ \; \delta- $ graceful labeling

AIMS Mathematics ◽

10.3934/math.2022197 ◽

2022 ◽

Vol 7 (3) ◽

pp. 3554-3589

Author(s):

Mohamed R. Zeen El Deen ◽

◽

Ghada Elmahdy ◽

Keyword(s):

Positive Integer ◽

Mathematical Models ◽

Communication Networks ◽

Data Security ◽

Coding Theory ◽

Graph Labeling ◽

Graceful Labeling ◽

Extensive Range ◽

Fan Graph ◽

Positive Integers

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

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