IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v38y1990i3p499-505.html
   My bibliography  Save this article

Convexity Properties of the Erlang Loss Formula

Author

Listed:
  • Arie Harel

    (Rutgers University, Newark, New Jersey)

Abstract

We prove that the throughput of the M/G/x/x system is jointly concave in the arrival and service rates. We also show that the fraction of customers lost in the M/G/x/x system is convex in the arrival rate, if the traffic intensity is below some Ρ * and concave if the traffic intensity is greater than Ρ * . For 18 or less servers, Ρ * is less than one. For 19 or more servers, Ρ * is between 1 and 1.5. Also, the fraction lost is convex in the service rate, but not jointly convex in the two rates. These results are useful in the optimal design of queueing systems.

Suggested Citation

  • Arie Harel, 1990. "Convexity Properties of the Erlang Loss Formula," Operations Research, INFORMS, vol. 38(3), pages 499-505, June.
    • Handle: RePEc:inm:oropre:v:38:y:1990:i:3:p:499-505
    DOI: 10.1287/opre.38.3.499
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.38.3.499
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.38.3.499?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ward Whitt, 2006. "Sensitivity of Performance in the Erlang-A Queueing Model to Changes in the Model Parameters," Operations Research, INFORMS, vol. 54(2), pages 247-260, April.
    2. Öner, K.B. & Kiesmüller, G.P. & van Houtum, G.J., 2010. "Optimization of component reliability in the design phase of capital goods," European Journal of Operational Research, Elsevier, vol. 205(3), pages 615-624, September.
    3. Zhang, Xuelu & Wang, Jinting & Do, Tien Van, 2015. "Threshold properties of the M/M/1 queue under T-policy with applications," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 284-301.
    4. Felipe Caro & David Simchi-Levi, 2012. "Optimal static pricing for a tree network," Annals of Operations Research, Springer, vol. 196(1), pages 137-152, July.
    5. Opher Baron & Oded Berman & Yael Deutsch, 2018. "On the optimality of the sequential approach for network design problems of service operations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(5), pages 363-377, August.
    6. Fernanda Campello & Armann Ingolfsson & Robert A. Shumsky, 2017. "Queueing Models of Case Managers," Management Science, INFORMS, vol. 63(3), pages 882-900, March.
    7. Jingui Xie & Yiming Fan & Mabel C. Chou, 2017. "Flexibility design in loss and queueing systems: efficiency of k-chain configuration," Flexible Services and Manufacturing Journal, Springer, vol. 29(2), pages 286-308, June.
    8. Erhun Özkan & Geert-Jan Houtum & Yasemin Serin, 2015. "A new approximate evaluation method for two-echelon inventory systems with emergency shipments," Annals of Operations Research, Springer, vol. 224(1), pages 147-169, January.
    9. E. Carrizosa & E. Conde & M. Muñoz-Márquez, 1998. "Admission Policies in Loss Queueing Models with Heterogeneous Arrivals," Management Science, INFORMS, vol. 44(3), pages 311-320, March.
    10. Fernanda Campello & Armann Ingolfsson & Robert A. Shumsky, 2018. "Queueing Models of Case Managers," Management Science, INFORMS, vol. 64(1), pages 7-26, January.
    11. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.
    12. Doǧan A. Serel & Erdal Erel, 2008. "Coordination of staffing and pricing decisions in a service firm," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(4), pages 307-323, July.
    13. R. Bekker & A. Bruin, 2010. "Time-dependent analysis for refused admissions in clinical wards," Annals of Operations Research, Springer, vol. 178(1), pages 45-65, July.
    14. Westerweel, Bram & Basten, Rob J.I. & van Houtum, Geert-Jan, 2018. "Traditional or Additive Manufacturing? Assessing Component Design Options through Lifecycle Cost Analysis," European Journal of Operational Research, Elsevier, vol. 270(2), pages 570-585.
    15. Sleptchenko, Andrei & Al Hanbali, Ahmad & Zijm, Henk, 2018. "Joint planning of service engineers and spare parts," European Journal of Operational Research, Elsevier, vol. 271(1), pages 97-108.
    16. Endre Boros & Lei Lei & Yao Zhao & Hua Zhong, 2008. "Scheduling vessels and container-yard operations with conflicting objectives," Annals of Operations Research, Springer, vol. 161(1), pages 149-170, July.
    17. Arie Harel, 2011. "Convexity Results for the Erlang Delay and Loss Formulae When the Server Utilization Is Held Constant," Operations Research, INFORMS, vol. 59(6), pages 1420-1426, December.
    18. Legros, Benjamin & Fransoo, Jan C., 2023. "Admission and pricing optimization of on-street parking with delivery bays," Other publications TiSEM 6d41ee5c-27dc-4d34-aff1-4, Tilburg University, School of Economics and Management.

    More about this item

    Keywords

    queues: loss system; design of queues;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:38:y:1990:i:3:p:499-505. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.