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Convexity Properties of the Erlang Loss Formula

Author

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  • 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
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Fernanda Campello & Armann Ingolfsson & Robert A. Shumsky, 2017. "Queueing Models of Case Managers," Management Science, INFORMS, vol. 63(3), pages 882-900, March.
    13. 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.
    14. Fernanda Campello & Armann Ingolfsson & Robert A. Shumsky, 2018. "Queueing Models of Case Managers," Management Science, INFORMS, vol. 64(1), pages 7-26, January.
    15. 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.
    16. 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.
    17. 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.
    18. 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.

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    Keywords

    queues: loss system; design of queues;

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