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)3510710.22363/2658-4670-2023-31-2-105-119Research ArticleHeterogeneous queueing system with Markov renewal arrivals and service times dependent on states of arrival processPolinEvgeny P.<p>Assistant of Department of Probability Theory and Mathematical Statistics</p>polin_evgeny@mail.ruhttps://orcid.org/0000-0002-0250-2368MoiseevaSvetlana P.<p>Doctor in Physics and Mathematics, Professor at Department of Probability Theory and Mathematical Statistics</p>smoiseeva@mail.ruhttps://orcid.org/0000-0001-9285-1555MoiseevAlexander N.<p>Doctor in Physics and Mathematics, Head of the Department of Software Engineering</p>moiseev.tsu@gmail.comhttps://orcid.org/0000-0003-2369-452XNational Research Tomsk State UniversityNational Research Tomsk Polytechnic University3006202331210511929062023Copyright © 2023, Polin E.P., Moiseeva S.P., Moiseev A.N.2023<p style="text-align: justify;">In the proposed work, we consider a heterogeneous queueing system with a Markov renewal process and an unlimited number of servers. The service time for requests on the servers is a positive random variable with an exponential probability distribution. The service parameters depend on the state of the Markov chain nested over the renewal moments. It should be noted that these parameters do not change their values until the end of maintenance. Thus, the devices in the system under consideration are heterogeneous. The object of the study is a multidimensional random process - the number of servers of each type being served with different intensities in the stationary regime. The method of asymptotic analysis under the condition of equivalent growing of service times in the units of servers is applied for the study. The method of asymptotic analysis is implemented in the construction of a sequence of asymptotic of increasing order, in which the asymptotic of the first order determines the asymptotic mean value of the number of occupied servers. The second-order asymptotic allows one to construct a Gaussian approximation of the probability distribution of the number of occupied servers in the system. It is shown that this approximation coincides with the Gaussian distribution.</p>queuing systemrandom environmentMarkov renewal processasymptotic analysis methodсистема массового обслуживанияслучайная средапоток марковского восстановленияметод асимптотического анализа[A. Dudin, V. Klimenok, and V. Vishnevsky, The Theory of Queuing Systems with Correlated Flow. Springer Nature, 2020. DOI: 10.1007/9783-030-32072-0.][V. K. Malinovskii, “Asymptotic expansions in the central limit theorem for recurrent Markov renewal processes,” Theory of Probability & Its Applications, vol. 51, no. 3, pp. 523–526, 1987. DOI: 10.1137/1131073.][Y. Lim, S. Hur, and J. 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