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Analytic approach to the non-pre-emptive Markovian priority queue

Author

Listed:
  • Josef Zuk

    (Defence Science and Technology Group)

  • David Kirszenblat

    (Defence Science and Technology Group)

Abstract

A new approach is developed for the joint queue-length distribution of the two-level non-pre-emptive M/M/c (i.e. Markovian) priority queue that allows explicit and exact results to be obtained. Marginal distributions are derived for the general multi-level problem. The results are based on a representation of the joint queue-length probability mass function as a single-variable complex contour integral, which reduces to a real integral on a finite interval arising from a cut on the real axis. Both numerical quadrature rules and exact finite sums, involving Legendre polynomials and their generalization, are presented for the joint and marginal distributions. A high level of accuracy is demonstrated across the entire ergodic region. Relationships are established with the waiting-time distributions. Asymptotic behaviour in the large queue-length regime is extracted.

Suggested Citation

  • Josef Zuk & David Kirszenblat, 2024. "Analytic approach to the non-pre-emptive Markovian priority queue," Queueing Systems: Theory and Applications, Springer, vol. 107(1), pages 159-198, June.
    • Handle: RePEc:spr:queues:v:107:y:2024:i:1:d:10.1007_s11134-024-09912-3
    DOI: 10.1007/s11134-024-09912-3
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    More about this item

    Keywords

    Queueing theory; Non-pre-emptive priority; Markovian queue; Queue-length distribution;
    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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