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Exact Closed-Form M/D/c Queueing Delay Formula
Version 1
: Received: 12 January 2024 / Approved: 15 January 2024 / Online: 15 January 2024 (08:50:49 CET)
How to cite: Wong, D.T.C. Exact Closed-Form M/D/c Queueing Delay Formula. Preprints 2024, 2024011070. https://doi.org/10.20944/preprints202401.1070.v1 Wong, D.T.C. Exact Closed-Form M/D/c Queueing Delay Formula. Preprints 2024, 2024011070. https://doi.org/10.20944/preprints202401.1070.v1
Abstract
It has been more than a century since Erlang published his paper in 1909. The classical queueing system for the M/D/c queue has been investigated by many authors. The key is to find the roots of an equation using explicit or numerical methods. We present an alternate form of explicit solution(s) of this/these root(s) using the Lambert-W function in this paper. In addition, the unsaturated probabilities of the number of customers in the system is derived explicitly in terms of this/these root(s). Finally, the mean number of customers in the system and the mean queueing delay are obtained from these probabilities. The exact queueing delay formula for the M/D/c queue is expressed in closed-form in terms of the root(s). Numerical results show that they are in excellent agreement with the numerical results published by Seelen, Tijms and Van Hoorn.
Keywords
M/D/c Queue; Poisson Arrivals; Deterministic (Constant) Service Time; c Multiple Servers; root(s) of an equation; mean number of customers; mean queueing delay
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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