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

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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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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Keywords

  • Queueing
  • multi-server
  • discrete-time
  • bulk-arrivals
  • finite-buffer
  • heavy-tailed
  • and roots