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Back to the roots of the M/D/ s queue and the works of Erlang, Crommelin and Pollaczek
Author(s) -
Janssen A. J. E. M.,
Van Leeuwaarden J. S. H.
Publication year - 2008
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.2008.00395.x
Subject(s) - erlang (programming language) , queue , queueing theory , mathematics , burke's theorem , fourier series , staffing , discrete mathematics , queue management system , computer science , statistics , mathematical analysis , theoretical computer science , computer network , fork–join queue , functional programming , management , economics
A.K. Erlang introduced the M/D/ s queue in 1917, while F. Pollaczek and C.D. Crommelin formalized the theory using complex analysis and transforms. Let D ( s , λ ) denote the stationary probability of experiencing no waiting time in the M/D/ s queue with arrival rate λ and service requirement 1. We use D ( s , λ ) as a vehicle to give an overview of some of the results we obtained over the last years, including explicit characterizations of the roots, the derivation of infinite series from expressions in terms of roots using Fourier sampling and heavy‐traffic limits obtained from square‐root staffing. We propose to call D ( s , λ ) the Erlang D formula, for which several new results are presented and compared with the results of Pollaczek.