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Exact results for the distribution of the partial busy period for a multi-server queue

Author

Listed:
  • Josef Zuk

    (Defence Science and Technology Group)

  • David Kirszenblat

    (Defence Science and Technology Group)

Abstract

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two.

Suggested Citation

  • Josef Zuk & David Kirszenblat, 2024. "Exact results for the distribution of the partial busy period for a multi-server queue," Queueing Systems: Theory and Applications, Springer, vol. 107(1), pages 63-108, June.
    • Handle: RePEc:spr:queues:v:107:y:2024:i:1:d:10.1007_s11134-024-09906-1
    DOI: 10.1007/s11134-024-09906-1
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    More about this item

    Keywords

    Partial busy period; Multi-server queues; Markovian queue; Spectral decomposition;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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