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Dynamic rate Erlang-A queues

Author

Listed:
  • William A. Massey

    (Princeton University)

  • Jamol Pender

    (Cornell University)

Abstract

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large-scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al. (Queueing Syst 30(1):149–201, 1998) for Markovian service networks. These asymptotics were inspired by the Halfin and Whitt (Oper Res 29(3):567–588, 1981) scaling for multi-server queues. In this paper, we provide a review and an in-depth analysis of the Erlang-A queueing model. We prove new results about cumulant moments of the Erlang-A queue, the transient behavior of the Erlang-A limit cycle, new fluid limits for the delay time of a virtual customer, and optimal static staffing policies for healthcare systems. We combine tools from queueing theory, ordinary differential equations, complex analysis, cumulant moments, orthogonal polynomials, and dynamic optimization to obtain new insights about this fundamental queueing model.

Suggested Citation

  • William A. Massey & Jamol Pender, 2018. "Dynamic rate Erlang-A queues," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 127-164, June.
    • Handle: RePEc:spr:queues:v:89:y:2018:i:1:d:10.1007_s11134-018-9581-2
    DOI: 10.1007/s11134-018-9581-2
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    References listed on IDEAS

    as
    1. Stephen G. Eick & William A. Massey & Ward Whitt, 1993. "The Physics of the Mt/G/∞ Queue," Operations Research, INFORMS, vol. 41(4), pages 731-742, August.
    2. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
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    4. Shlomo Halfin & Ward Whitt, 1981. "Heavy-Traffic Limits for Queues with Many Exponential Servers," Operations Research, INFORMS, vol. 29(3), pages 567-588, June.
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    6. Otis B. Jennings & Avishai Mandelbaum & William A. Massey & Ward Whitt, 1996. "Server Staffing to Meet Time-Varying Demand," Management Science, INFORMS, vol. 42(10), pages 1383-1394, October.
    7. Liu, Yunan & Whitt, Ward, 2017. "Stabilizing performance in a service system with time-varying arrivals and customer feedback," European Journal of Operational Research, Elsevier, vol. 256(2), pages 473-486.
    8. Armann Ingolfsson & Elvira Akhmetshina & Susan Budge & Yongyue Li & Xudong Wu, 2007. "A Survey and Experimental Comparison of Service-Level-Approximation Methods for Nonstationary M(t)/M/s(t) Queueing Systems with Exhaustive Discipline," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 201-214, May.
    9. Jamol Pender & Young Myoung Ko, 2017. "Approximations for the Queue Length Distributions of Time-Varying Many-Server Queues," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 688-704, November.
    10. Zohar Feldman & Avishai Mandelbaum & William A. Massey & Ward Whitt, 2008. "Staffing of Time-Varying Queues to Achieve Time-Stable Performance," Management Science, INFORMS, vol. 54(2), pages 324-338, February.
    11. Young Myoung Ko & Natarajan Gautam, 2013. "Critically Loaded Time-Varying Multiserver Queues: Computational Challenges and Approximations," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 285-301, May.
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    Cited by:

    1. Eugene Furman & Adam Diamant & Murat Kristal, 2021. "Customer Acquisition and Retention: A Fluid Approach for Staffing," Production and Operations Management, Production and Operations Management Society, vol. 30(11), pages 4236-4257, November.
    2. Zhang, Xin & Huang, Ning & Sun, Lina & Zheng, Xiangyu & Guo, Ziyue, 2022. "Modeling congestion considering sequential coupling applications: A network-cell-based method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    3. Ward Whitt, 2018. "A broad view of queueing theory through one issue," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 3-14, June.
    4. Gregor Selinka & Raik Stolletz & Thomas I. Maindl, 2022. "Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 20-38, January.

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