2023/12/07 by Josef Zuk, Zuk, Josef, David Kirszenblat +1
Business, Management and Accounting · Decision Sciences · Mathematics · #60J74 (Secondary) #90B22 (Primary) 60K25 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications
paper · pdf · doi:10.48550/arxiv.2312.03992
openalex publication_date 2023/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Explicit and exact results are obtained for the joint queue-length distribution for the two-level non-preemptive Markovian priority queue. 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, that 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.