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Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek

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  • A. J. E. M. Janssen
  • J. S. H. Van Leeuwaarden

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.

Suggested Citation

  • A. J. E. M. Janssen & J. S. H. Van Leeuwaarden, 2008. "Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 62(3), pages 299-313, August.
    • Handle: RePEc:bla:stanee:v:62:y:2008:i:3:p:299-313
    DOI: 10.1111/j.1467-9574.2008.00395.x
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    References listed on IDEAS

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    1. Sem Borst & Avi Mandelbaum & Martin I. Reiman, 2004. "Dimensioning Large Call Centers," Operations Research, INFORMS, vol. 52(1), pages 17-34, February.
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    Cited by:

    1. Britt W. J. Mathijsen & A. J. E. M. Janssen & Johan S. H. Leeuwaarden & Bert Zwart, 2018. "Robust heavy-traffic approximations for service systems facing overdispersed demand," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 257-289, December.
    2. A. Oblakova & A. Al Hanbali & R. J. Boucherie & J. C. W. Ommeren & W. H. M. Zijm, 2019. "An exact root-free method for the expected queue length for a class of discrete-time queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 92(3), pages 257-292, August.
    3. M. A. A. Boon & A. J. E. M. Janssen & J. S. H. Leeuwaarden & R. W. Timmerman, 2019. "Pollaczek contour integrals for the fixed-cycle traffic-light queue," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 89-111, February.
    4. Alan Scheller-Wolf, 2022. "A multi-server/single-server duality," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 249-251, April.

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