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 Mat. Zametki, 1977, Volume 22, Issue 4, Pages 561–569 (Mi mz8078)

The ergodicity of service systems with an infinite number of servomechanisms

A. Yu. Veretennikov

M. V. Lomonosov Moscow State University

Abstract: Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate $\lambda_k$ depending on the number $k$ of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean $\mu$, under the condition $\mu\varlimsup\limits_{k\to\infty}\frac{\lambda_k}{k+1}<1$. For this system we have, in particular, Erlang's formula
$$p_k(t)\underset{k\to\infty}\longrightarrow p_k=\frac{\lambda_0…\lambda_{k-1}\mu^k}{k!}p_0,\quad k=0,1,…,\quad p_0^{-1}=\sum_{k=0}^\infty\frac{\lambda_0…\lambda_{k-1}\mu^k}{k!},\quad\lambda_{-1}=1.$$

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English version:
Mathematical Notes, 1977, 22:4, 804–808

Bibliographic databases:

UDC: 519.2

Citation: A. Yu. Veretennikov, “The ergodicity of service systems with an infinite number of servomechanisms”, Mat. Zametki, 22:4 (1977), 561–569; Math. Notes, 22:4 (1977), 804–808

Citation in format AMSBIB
\Bibitem{Ver77} \by A.~Yu.~Veretennikov \paper The ergodicity of service systems with an infinite number of servomechanisms \jour Mat. Zametki \yr 1977 \vol 22 \issue 4 \pages 561--569 \mathnet{http://mi.mathnet.ru/mz8078} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=518585} \zmath{https://zbmath.org/?q=an:0408.60095} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 4 \pages 804--808 \crossref{https://doi.org/10.1007/BF01146428} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Yu. Veretennikov, “On the rate of convergence to the stationary distribution in the single-server queuing systems”, Autom. Remote Control, 74:10 (2013), 1620–1629
2. A. Yu. Veretennikov, “On convergence rate for Erlang–Sevastyanov type models with infinitely many servers. In memory and to the 90th anniversary of A.D. Solovyev (06.09.1927–06.04.2001)”, Theory Stoch. Process., 22(38):1 (2017), 89–103
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