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- Title
Back to the roots of the M/D/ s queue and the works of Erlang, Crommelin and Pollaczek.
- Authors
Janseen, A. J. E. M.; van Leeuwaarden, J. S. H.
- Abstract
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.
- Subjects
QUEUING theory; PROBABILITY theory; TELECOMMUNICATION traffic; SQUARE root; INFINITE series (Mathematics); FOURIER analysis; COMPARATIVE studies; STOCHASTIC processes; BULK queues
- Publication
Statistica Neerlandica, 2008, Vol 62, Issue 3, p299
- ISSN
0039-0402
- Publication type
Article
- DOI
10.1111/j.1467-9574.2008.00395.x