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Using a birth‐and‐death process to estimate the steady‐state distribution of a periodic queue

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  • James Dong
  • Ward Whitt

Abstract

If the number of customers in a queueing system as a function of time has a proper limiting steady‐state distribution, then that steady‐state distribution can be estimated from system data by fitting a general stationary birth‐and‐death (BD) process model to the data and solving for its steady‐state distribution using the familiar local‐balance steady‐state equation for BD processes, even if the actual process is not a BD process. We show that this indirect way to estimate the steady‐state distribution can be effective for periodic queues, because the fitted birth and death rates often have special structure allowing them to be estimated efficiently by fitting parametric functions with only a few parameters, for example, 2. We focus on the multiserver Mt/GI/s queue with a nonhomogeneous Poisson arrival process having a periodic time‐varying rate function. We establish properties of its steady‐state distribution and fitted BD rates. We also show that the fitted BD rates can be a useful diagnostic tool to see if an Mt/GI/s model is appropriate for a complex queueing system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 664–685, 2015

Suggested Citation

  • James Dong & Ward Whitt, 2015. "Using a birth‐and‐death process to estimate the steady‐state distribution of a periodic queue," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(8), pages 664-685, December.
  • Handle: RePEc:wly:navres:v:62:y:2015:i:8:p:664-685
    DOI: 10.1002/nav.21672
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    References listed on IDEAS

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    1. Linda Green & Peter Kolesar, 1991. "The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals," Management Science, INFORMS, vol. 37(1), pages 84-97, January.
    2. William A. Massey & Ward Whitt, 1996. "Stationary-Process Approximations for the Nonstationary Erlang Loss Model," Operations Research, INFORMS, vol. 44(6), pages 976-983, December.
    3. Ward Whitt, 1991. "The Pointwise Stationary Approximation for Mt/Mt/s Queues Is Asymptotically Correct As the Rates Increase," Management Science, INFORMS, vol. 37(3), pages 307-314, March.
    4. Ward Whitt, 1984. "Departures from a Queue with Many Busy Servers," Mathematics of Operations Research, INFORMS, vol. 9(4), pages 534-544, November.
    5. Ronald W. Wolff, 1965. "Problems of Statistical Inference for Birth and Death Queuing Models," Operations Research, INFORMS, vol. 13(3), pages 343-357, June.
    6. Joseph Abate & Ward Whitt, 1999. "Computing Laplace Transforms for Numerical Inversion Via Continued Fractions," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 394-405, November.
    7. Song‐Hee Kim & Ward Whitt, 2014. "Choosing arrival process models for service systems: Tests of a nonhomogeneous Poisson process," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(1), pages 66-90, February.
    8. Song-Hee Kim & Ward Whitt, 2014. "Are Call Center and Hospital Arrivals Well Modeled by Nonhomogeneous Poisson Processes?," Manufacturing & Service Operations Management, INFORMS, vol. 16(3), pages 464-480, July.
    9. Lawrence Brown & Noah Gans & Avishai Mandelbaum & Anat Sakov & Haipeng Shen & Sergey Zeltyn & Linda Zhao, 2005. "Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 36-50, March.
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    Cited by:

    1. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    2. Giorno, Virginia & Nobile, Amelia G., 2020. "On a class of birth-death processes with time-varying intensity functions," Applied Mathematics and Computation, Elsevier, vol. 379(C).

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