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Fluid Limit for Closed Queueing Network with Several Multi-servers

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Analytical and Computational Methods in Probability Theory (ACMPT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10684))

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Abstract

A closed network consists of several multi-servers with n customers. Service requirements of customers at a multi-server have a common cdf. State parameters of the network: for each multi-server empirical measure of the age of customers being serviced and for the queues the numbers of customers in them, all multiplied by \(n^{-1}\).

Our objective: asymptotics of dynamics as \(n\rightarrow \infty \). The asymptotics of dynamics of a single multi-server and its queue with an arrival process as the number of servers \(n\rightarrow \infty \) is currently studied by famous scientists K. Ramanan, W. Whitt et al. Presently there are no universal results for general distributions of service requirements — the results are either for continuous or for discrete time ones; the same for the arrival process. We establish the asymptotics for a network in discrete time, find its equilibrium and prove convergence as \(t\rightarrow \infty \).

Motivation for studying such models: they represent call/contact centers and help to construct them effectively.

S. Anulova—This work was partially supported by RFBR grants No. 16-08-01285 A “Control of stochastic, deterministic, and quantum systems in phases of quick movement.” and No. 17-01-00633 A “Problems of stability and control in stochastic models”.

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Authors and Affiliations

  1. IPU RAN, Profsoyuznaya, 65, 117997, Moscow, Russian Federation

    Svetlana Anulova

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  1. Svetlana Anulova
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Correspondence to Svetlana Anulova .

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Editors and Affiliations

  1. Department of Applied Probability and Informatics, RUDN University, Moscow, Russia

    Vladimir V. Rykov

  2. City University of Hong Kong, Hong Kong, Hong Kong

    Nozer D. Singpurwalla

  3. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Andrey M. Zubkov

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Anulova, S. (2017). Fluid Limit for Closed Queueing Network with Several Multi-servers. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_4

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  • DOI: https://doi.org/10.1007/978-3-319-71504-9_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71503-2

  • Online ISBN: 978-3-319-71504-9

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