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Stability Analysis of a Multi-server Model with Simultaneous Service and a Regenerative Input Flow

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  • Larisa Afanaseva

    (Lomonosov Moscow State University)

  • Elena Bashtova

    (Lomonosov Moscow State University)

  • Svetlana Grishunina

    (Lomonosov Moscow State University
    National Research University Higher School of Economics)

Abstract

We study the stability conditions of a multi-server queueing system in which each customer requires a random number of servers simultaneously. The input flow is assumed to be a regenerative one and random service times are identical for all occupied servers. The service time has a hypoexponential distribution which belongs to the class of phase-type distributions. We introduce an auxiliary queueing system in which there are always customers in the queue and define an auxiliary service process as the number of served customers in this system. Then we construct the sequence of common regeneration points for the regenerative input flow and the auxiliary service process. Based on the relationship between the real and the auxiliary service processes we obtain upper and lower estimates for the mean of the number of actually served customers during the common regeneration period. It allows us to deduce the stability criterion of the model under consideration. It turns out that the stability condition does not depend on the structure of the input flow. It only depends on the rate of this process.

Suggested Citation

  • Larisa Afanaseva & Elena Bashtova & Svetlana Grishunina, 2020. "Stability Analysis of a Multi-server Model with Simultaneous Service and a Regenerative Input Flow," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1439-1455, December.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:4:d:10.1007_s11009-019-09721-9
    DOI: 10.1007/s11009-019-09721-9
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    References listed on IDEAS

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    1. Tkachenko Andrey, 2013. "Multichannel queuing systems with balking and regenerative input fl ow," HSE Working papers WP BRP 14/STI/2013, National Research University Higher School of Economics.
    2. P. H. Brill & M. J. M. Posner, 1981. "The System Point Method in Exponential Queues: A Level Crossing Approach," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 31-49, February.
    3. Percy H. Brill, 2008. "Level Crossing Methods in Stochastic Models," International Series in Operations Research and Management Science, Springer, number 978-0-387-09421-2, September.
    4. Percy H. Brill & Linda Green, 1984. "Queues in Which Customers Receive Simultaneous Service from a Random Number of Servers: A System Point Approach," Management Science, INFORMS, vol. 30(1), pages 51-68, January.
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    Cited by:

    1. Mor Harchol-Balter, 2021. "Open problems in queueing theory inspired by datacenter computing," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 3-37, February.

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