Towards the analysis of the performance measures of heterogeneous networks by means of two-phase queuing systems

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1. Introduction The Fifth Generation (5G) mobile networks are characterized by advanced algorithms for time-frequency resource allocation schemes in a heterogeneous cell between Base Station (BS) and User Equipment (UE) [1], [2]. Due to a multistage nature of transmission processes in the heterogeneous environment, multiphase queuing systems become one of the most suitable ways for the resource allocation algorithms analysis and network investigation. In [3], researchers have proposed to use single-phase queuing systems for modelling local networks, by giving the necessary physical meaning to the stages of service process using a phase-type service distribution. However, in the case © Rykova T.V., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ of the Next Generation Mobile Networks (NGMN) the given assumptions are not able to take into account the complex structure of a network with intermediate storage of transmitted information. A large number of publications [4]-[6] are devoted to the analysis of multiphase queuing systems that consider various variants of structural parameters: capacitance of the buffers at the phases, the number of servers at phases, an ordinary or non-ordinary arrival flow, blocking of service at a phase or loss of a request given that the buffer of the next phase is fully occupied, the possibility of the retransmission at the phase or in the system in general, and various arrival and service distributions of requests. In the given publications, the number of phases is usually limited to two, and they are considered mainly in continuous time. Only a few works approached to investigate heterogeneous networks by means of multiphase (two-phase) queuing systems in discrete time, see, for example, [7], [8]. However, the models in [7], [8] cannot be used because they do not take into account the complex phase structure when modelling transmission processes in a cell and, therefore, do not fully correspond to solving a resource allocation problem in a context of a NGMN cell. It should be noted that most of the foreign publications when using “discrete” and “tandem queue” terms in their papers cover, in fact, mean cyclical service systems in discrete time, but not multiphase systems. In most of the cases, the number of phases in a multiphase queuing system that is taken as a mathematical model for analysis of the performance measures in a NGMN cell should be taken equal to two. This is due to the fact that each phase itself is a structurally complex queuing system with complex rules of functioning, and a further increase of phases severely complicates formalization of the entire system, leads to multidimensional processes that describe its behaviour and a difficult practical use. The analysis in this case becomes extremely bulky with high risks of obtaining inaccurate results. The decomposition of such a system with the analysis of individual phases or groups of phases is most often not applicable due to the significant mutual influence of phases, in contrast to almost completely decomposable systems [9], [10], and can lead in most cases to significant modelling errors. Cases of independence of the functioning of a phase from the previous phase and, accordingly, an admissible decomposition are rare and arise when conditions [11], are met, for example, when using exponential distributions and buffers of unlimited capacity [12], or under assumptions about specific conditions for the functioning of phases [13]. Summarizing all of that mentioned above, in this paper we briefly overview several two-phase systems of finite capacity in discrete time of a special structure that can be used for analysing resource allocation schemes based on the main performance measures obtained for wireless heterogeneous networks. 2. Two-phase model in discrete time for resource allocation analysis in heterogeneous networks Heterogeneous networks with the utilization of lower power levels Relay Nodes (RN) improve the capacity of the system, coverage due to the availability of the alternative paths to users, located in shadow areas, and lower deployments costs. Moreover, relay nodes are characterized by wireless backbone access. However, to achieve its potential, the heterogeneous networks are to utilize an efficient cooperative resource allocation procedure on various paths, e.g. from the base station (gNB, gNodeB) to the RN, and from the RN to the User Equipment (UE), in order to avoid data shortage or overflow of the data at relay nodes. An analytical model of heterogeneous network in terms of a two-phase model in discrete time is further introduced, that presents an efficient tool to study resource allocation procedures by means of the found stationary probability distribution and derived performance measures. 2.1. Model’s description Let us consider downlink transmission in a heterogeneous network with

About the authors

Tatiana V. Rykova

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: tatiana.rykova@hhi.fraunhofer.de
ORCID iD: 0000-0002-8561-7514

Master of Science in Applied Mathematics and Informatics (PFUR), Master of Science in Information Technology (Tampere University of Technology), researcher at Fraunhofer Heinrich Hertz Institute (Berlin, Germany)

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

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