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Mathematics
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Applications of Mathematical Analysis in Telecommunications
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Applications of Mathematical Analysis in Telecommunications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Network Science".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 18824

Special Issue Editor

Prof. Dr. Konstantin Samouylov

E-Mail Website
Guest Editor
Applied Mathematics and Communications Technology Institute, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia
Interests: 5G; mobile communication; unmanned aerial vehicles; QoS; сomputer networks; wireless networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical analysis plays an indisputable role in almost all spheres of human activity. The global public telecommunications network is the largest technical object that humans have ever built. This network has been created since about the end of the 19th century and has survived all the stages of the development of industrial society. As of 2020, five generations of the telecommunication network have already been created, and the scientific community is already discussing the future 6th generation of the network. At all stages, starting from the tasks set by the outstanding mathematicians Agner Erlang and Aleksandr Khinchin, engineers turned to various mathematical theories and methods both to evaluate the performance of existing networks and to build models of future systems—even those that have not yet passed the laboratory research stage . Almost no scientific article devoted to the R&D of telecommunication systems and networks is complete without the construction of mathematical models and their mathematical analysis. The created methods in some cases served the development of purely mathematical disciplines such as queuing theory, coding theory, and some others.

We devote this Special Issue of Mathematics to the problems of the application of mathematical analysis in telecommunications and expect contributions from applied mathematicians from various profiles in the form of scientific articles showing their achievements and confirming the relevance of current and future research.

Prof. Dr. Konstantin Samouylov
Guest Editor

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Published Papers (6 papers)

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Research

13 pages, 452 KiB  
Article
Mathematical Model of Call Center in the Form of Multi-Server Queueing System
by Anatoly Nazarov, Alexander Moiseev and Svetlana Moiseeva
Mathematics 2021, 9(22), 2877; https://doi.org/10.3390/math9222877 - 12 Nov 2021
Cited by 4 | Viewed by 2319
Abstract
The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and [...] Read more.
The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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24 pages, 1312 KiB  
Article
Estimation of the Performance Measures of a Group of Servers Taking into Account Blocking and Call Repetition before and after Server Occupation
by Sergey Stepanov and Mikhail Stepanov
Mathematics 2021, 9(21), 2811; https://doi.org/10.3390/math9212811 - 5 Nov 2021
Cited by 13 | Viewed by 2116
Abstract
The model of a fully available group of servers with a Poisson flow of primary calls and the possibility of losses before and after occupying a free server is considered. Additionally, a call can leave the system because of the aging of transmitted [...] Read more.
The model of a fully available group of servers with a Poisson flow of primary calls and the possibility of losses before and after occupying a free server is considered. Additionally, a call can leave the system because of the aging of transmitted information. After each loss, there is some probability that a customer repeats the call. Such models are seen in the modeling of various telecommunication systems such as emergency information services, call and contact centers, access nodes, etc., functioning in overloading situations. The stationary behavior of the system is described by the infinite-state Markov process. It is shown that stationary characteristics of the model can be calculated with the help of an auxiliary model of the same class but without call repetitions due to losses occurring before and after the occupation of a free server and the aging of transmitted information. The performance measurements of the auxiliary model are calculated by solving a system of state equations using a recursive algorithm based on the concept of the truncation of the used state space. This approach allows significant savings of computer resources to be made by ignoring highly unlikely states in the process of calculation. The error caused by truncation is estimated. The presented numerical examples illustrate the use of the model for the elimination of the negative effects of emergency information service overload based on the filtering of the input flow of calls. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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17 pages, 369 KiB  
Article
Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals
by Freek Verdonck, Herwig Bruneel and Sabine Wittevrongel
Mathematics 2021, 9(14), 1709; https://doi.org/10.3390/math9141709 - 20 Jul 2021
Viewed by 2010
Abstract
In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and [...] Read more.
In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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15 pages, 318 KiB  
Article
Vacation Queueing Model for Performance Evaluation of Multiple Access Information Transmission Systems without Transmission Interruption
by Alexander Dudin, Sergei Dudin, Valentina Klimenok and Yuliya Gaidamaka
Mathematics 2021, 9(13), 1508; https://doi.org/10.3390/math9131508 - 28 Jun 2021
Cited by 3 | Viewed by 2064
Abstract
We consider a MAP/PH/1-type queueing system with server vacations as a model that is useful for the analysis of multiple access systems with polling discipline without transmission interruption. Vacation of the server corresponds to the [...] Read more.
We consider a MAP/PH/1-type queueing system with server vacations as a model that is useful for the analysis of multiple access systems with polling discipline without transmission interruption. Vacation of the server corresponds to the service providing competitive information flows to the polling system. In this paper, we consider a vacation queueing model under pretty general assumptions about the probabilistic distributions describing the behavior of the system and the realistic assumption, in many real-world systems, that ongoing service cannot be terminated ahead of schedule. We derive the criterion of the stable operation of the system and the stationary distributions of the system states and the waiting time. An illustrative numerical example is presented. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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17 pages, 629 KiB  
Article
Analysis of Single-Server Multi-Class Queue with Unreliable Service, Batch Correlated Arrivals, Customers Impatience, and Dynamical Change of Priorities
by Alexander Dudin, Olga Dudina, Sergei Dudin and Konstantin Samouylov
Mathematics 2021, 9(11), 1257; https://doi.org/10.3390/math9111257 - 31 May 2021
Cited by 13 | Viewed by 3570
Abstract
A single-server non-pre-emptive priority queueing system of a finite capacity with many types of customers is analyzed. Inter-arrival times can be correlated and batch arrivals are allowed. Possible unreliability of the server, implying the loss of a customer or the necessity of its [...] Read more.
A single-server non-pre-emptive priority queueing system of a finite capacity with many types of customers is analyzed. Inter-arrival times can be correlated and batch arrivals are allowed. Possible unreliability of the server, implying the loss of a customer or the necessity of its service from the early beginning or some phase of the service, is taken into account. Initial priorities provided to various types of customers at the arrival moment can be varied (increased or decreased) after the random amount of time during the customer stay in the buffer. Such a type of queues arises in the modeling operation of various emergency care systems, information, and perishable goods delivering systems, etc. The stationary behavior of the system is described by the finite state multi-dimensional continuous-time Markov chain with the upper-Hessenberg block structure of the generator. The stationary distribution of the system states and some important characteristics of the system are calculated. The presented numerical examples illustrate opportunities to quantitatively evaluate the impact of the buffer capacity and customers’ mean arrival rate on the most important characteristics of the system. The possibility of solving optimization problems is briefly shown. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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16 pages, 1029 KiB  
Article
Priority Multi-Server Queueing System with Heterogeneous Customers
by Valentina Klimenok, Alexander Dudin and Vladimir Vishnevsky
Mathematics 2020, 8(9), 1501; https://doi.org/10.3390/math8091501 - 4 Sep 2020
Cited by 21 | Viewed by 3699
Abstract
In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can [...] Read more.
In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory. Full article
(This article belongs to the Special Issue Applications of Mathematical Analysis in Telecommunications)
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