preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202401.1070.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 12 January 2024 / Approved: 15 January 2024 / Online: 15 January 2024 (08:50:49 CET)
Version 2 : Received: 9 September 2024 / Approved: 9 September 2024 / Online: 9 September 2024 (23:41:28 CEST)

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.

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

Computer Science and Mathematics, Mathematics

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