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Dynamic rate Erlang-A queues

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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.

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Acknowledgements

William A. Massey was partially supported by National Science Foundation Grant CMMI-1436334.

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA

    William A. Massey

  2. School of Operations Research and Information Engineering, Cornell University, 228 Rhodes Hall, Ithaca, NY, 14853, USA

    Jamol Pender

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  1. William A. Massey
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Correspondence to Jamol Pender.

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This work is dedicated to Ward Whitt, on the occasion of his 75th birthday. We are eternally grateful for his friendship, mentorship, guidance, kindness, and infinite knowledge about stochastic processes.

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Massey, W.A., Pender, J. Dynamic rate Erlang-A queues. Queueing Syst 89, 127–164 (2018). https://doi.org/10.1007/s11134-018-9581-2

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