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A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

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Abstract

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GIX/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GIX/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GIX/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

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Acknowledgements

This research was supported (in part) by the Department of National Defense Applied Research Program grant GRC0000B1638.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, Ont, K7K 7B4, Canada

    J. J. Kim & M. L. Chaudhry

  2. School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar, 751024, India

    V. Goswami

  3. School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Permanent Campus Argul, Khurda, Jatni, Odisha, 725 050, India

    A. D. Banik

Authors
  1. J. J. Kim
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  2. M. L. Chaudhry
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  3. V. Goswami
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  4. A. D. Banik
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Contributions

J.J. Kim: as the first author, wrote the paper and conducted the numerical analysis. Chaudhry, M.L.: collaborated with the first author and provided guidance with the literature review. Goswami, V.: assisted first author with numerical analysis. Banik, A.D.: assisted with the literature review of the manuscript.

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Correspondence to J. J. Kim.

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Kim, J.J., Chaudhry, M.L., Goswami, V. et al. A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots. Methodol Comput Appl Probab 23, 273–289 (2021). https://doi.org/10.1007/s11009-020-09836-4

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  • DOI: https://doi.org/10.1007/s11009-020-09836-4

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