Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)1837010.22363/2312-9735-2018-26-2-167-175Research ArticleAnalysis of Queueing Systems with an Infinite Number of Servers and a Small ParameterVasilyevS ACandidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)vasilyev_sa@rudn.universityTsarevaG OPhD student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)gotsareva@gmail.comPeoples’ Friendship University of Russia (RUDN University)1512201826216717521042018Copyright © 2018, Vasilyev S.A., Tsareva G.O.2018In this paper we consider the dynamics of large-scale queueing systems with an infinite number of servers. We assume that there is a Poisson input flow of requests with intensity . We suppose that each incoming request selects two any servers randomly and at the next step of an algorithm is sending this request to the server with the shorter queue instantly. A share () of the servers that have the queues lengths with not less than can be described using a system of ordinary differential equations of infinite order. We investigate this system of ordinary differential equations of infinite order with a small real parameter. A small real parameter allows us to describe the processes of rapid changes in large-scale queueing systems. We use the simulation methods for this large-scale queueing systems analysis. The numerical simulation show that the solution of the singularly perturbed systems of differential equations have an area of rapid change of the solutions, which is usually located in the initial point of the problem. This area of rapid function change is called the area of the mathematical boundary layer. The thickness of the boundary layer depends on the value of a small parameter, and when the small parameter decreases, the thickness of the boundary layer decreases. The paper presents the numerical examples of the existence of steady state conditions for evolutions () and quasi-periodic conditions with boundary layers for evolutions ().countable Markov chainslarge-scale queueing systemssingular perturbed systems of differential equationsdifferential equations of infinite ordersmall parameterсчётные марковские цепикрупномасштабные системы массового обслуживаниясингулярные возмущённые системы дифференциальных уравненийдифференциальные уравнения бесконечного порядкамалый параметрВведенская Н. Д., Добрушин Р. Л., Карпелевич Ф. И. Система обслуживания с выбором наименьшей из двух очередей – асимптотический подход // Проблемы передачи информации. — 1996. — Т. 32, № 1. — С. 20–34.Vvedenskaya N. D., Suhov Yu. M. Dobrushin’s Mean-Field Approximation for a Queue with Dynamic Routing. 3. — 1997. — Pp. 493–526.Введенская Н. Д. Большая система обслуживания с передачей сообщения по нескольким путям. — 1998. — Т. 34, С. 98–108.Afanassieva L. G., Fayolle G., Popov S. Yu. Models for Transportation Networks // Journal of Mathematical Sciences. — 1997. — Vol. 84, No 3. — Pp. 1092–1103.Khmelev D. V., Oseledets V. I. Mean-Field Approximation for Stochastic Transportation Network and Stability of Dynamical System (Preprint No. 434). — Bremen: University of Bremen, 1999.Khmelev D. V. Limit Theorems for Nonsymmetric Transportation Networks. — 2001. — Vol. 7, Pp. 1259–1266.Scherbakov V. V. Time Scales Hierarchy in Large Closed Jackson Networks (Preprint No. 4). — Moscow: French–Russian A. M. Liapunov Institute of Moscow State University, 1997.Oseledets V. I., Khmelev D. V. Global Stability of Infinite Systems of Nonlinear Differential Equations, and Nonhomogeneous Countable Markov Chains. — 2000. — Vol. 36, Pp. 60–76.Chernavskaya E. A. Limit Theorems for an Infinite-Server Queuing System. — 2015. — Vol. 99.Chernavskaya E. A. Limit Theorems for Queueing Systems with Infinite Number of Servers and Group Arrival of Requests. — 2016. — Vol. 71, Pp. 257–260.Gaidamaka Yu., Sopin E., Talanova M. Approach to the Analysis of Probability Measures of Cloud Computing Systems with Dynamic Scaling. — 2016. — Vol. 601, Pp. 121—131.Stochastization of One-Step Processes in the Occupations Number Representation / V. Korolkova, E. G. Eferina, E. B. Laneev et al. — 2016. — Pp. 698–704.Sojourn Time Analysis for Processor Sharing Loss System with Unreliable Server / K. Samouylov, V. Naumov, E. Sopin et al. — Springer Verlag, 2016. — Vol. 9845, Pp. 284—297.Bolotova G. O., Vasilyev S. A., Udin D. N. Systems of Differential Equations of Infinite Order with Small Parameter and Countable Markov Chains. — Springer Verlag, 2016. — Vol. 678, Pp. 565–576.