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

Josef Zuk () and David Kirszenblat ()
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Josef Zuk: Defence Science and Technology Group
David Kirszenblat: Defence Science and Technology Group

Queueing Systems: Theory and Applications, 2024, vol. 107, issue 1, No 5, 159-198

Abstract: 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.

Keywords: Queueing theory; Non-pre-emptive priority; Markovian queue; Queue-length distribution; 90B22; 60K25; 60J74 (search for similar items in EconPapers)
JEL-codes: C44 C63 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11134-024-09912-3

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