Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek |
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Authors: | A J E M Janssen J S H van Leeuwaarden‡ |
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Institution: | Philips Research Digital Signal Processing Group, HTC-36, 5656 AE Eindhoven, The Netherlands; Eindhoven University of Technology and EURANDOM, PO Box 513 –5600 MB Eindhoven, The Netherlands |
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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. |
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Keywords: | M/D/s queue Erlang D formula queueing theory roots square-root staffing |
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