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Strong Convexity Results for Queueing Systems

Author

Listed:
  • Arie Harel

    (Rutgers University, Newark, New Jersey)

  • Paul H. Zipkin

    (Columbia University, New York, New York)

Abstract

We prove a strong (and seemingly odd) result about the M / M / c queue: the reciprocal of the average sojourn time is a concave function of the traffic intensity. We use this result to show that the average itself is jointly convex in arrival and service rates. The standard deviation has the same properties. Also, we determine conditions under which these properties are exhibited by a standard approximation for the M / G / c queue. These results are useful in design studies for telecommunications and production systems.

Suggested Citation

  • Arie Harel & Paul H. Zipkin, 1987. "Strong Convexity Results for Queueing Systems," Operations Research, INFORMS, vol. 35(3), pages 405-418, June.
    • Handle: RePEc:inm:oropre:v:35:y:1987:i:3:p:405-418
    DOI: 10.1287/opre.35.3.405
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    Citations

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    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. Shweta Upadhyaya & Richa Sharma & Divya Agarwal & Geetika Malik, 2023. "Convexity analysis and cost optimization of a retrial queue with Bernoulli vacation and delayed phase mending," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 14(5), pages 1671-1690, October.
    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. 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.
    5. M. Eric Johnson & Margaret L. Brandeau, 1999. "Design of an Automated Shop Floor Material Handling System with Inventory Considerations," Operations Research, INFORMS, vol. 47(1), pages 65-80, February.
    6. 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.
    7. Bitran, Gabriel R. & Morabito, Reinaldo., 1994. "Open queueing networks : optimization and performance evaluation models for discrete manufacturing systems," Working papers 3743-94., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    8. J. Smith, 2015. "Optimal workload allocation in closed queueing networks with state dependent queues," Annals of Operations Research, Springer, vol. 231(1), pages 157-183, August.

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