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Fitting birth-and-death queueing models to data

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  • Whitt, Ward

Abstract

Given measurements of the number of customers in a queueing system over a finite time interval, it is natural to try to fit a stationary birth-and-death process model, because it is remarkably tractable, even when the birth and death rates depend on the state in an arbitrary way. Natural estimators of the birth (death) rate in each state are the observed number of transitions up (down) from that state divided by the total time spent in that state. It is tempting to validate the model by comparing the steady-state distribution of the model based on those estimated rates to the empirical steady-state distribution recording the proportion of time spent in each state. However, it is inappropriate to draw strong conclusions from a close fit to the same data, because these two distributions are necessarily intimately related, even if the model assumptions are not nearly satisfied. We elaborate by (i) establishing stochastic comparisons between these two fitted distributions using likelihood-ratio stochastic ordering and (ii) quantifying their difference.

Suggested Citation

  • Whitt, Ward, 2012. "Fitting birth-and-death queueing models to data," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 998-1004.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:5:p:998-1004
    DOI: 10.1016/j.spl.2012.02.010
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    References listed on IDEAS

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    1. Ward Whitt, 2005. "Engineering Solution of a Basic Call-Center Model," Management Science, INFORMS, vol. 51(2), pages 221-235, February.
    2. Ward Whitt, 1992. "Asymptotic Formulas for Markov Processes with Applications to Simulation," Operations Research, INFORMS, vol. 40(2), pages 279-291, April.
    3. Ward Whitt, 1999. "Improving Service by Informing Customers About Anticipated Delays," Management Science, INFORMS, vol. 45(2), pages 192-207, February.
    4. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    5. 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. Aleksandrina Goeva & Henry Lam & Huajie Qian & Bo Zhang, 2019. "Optimization-Based Calibration of Simulation Input Models," Operations Research, INFORMS, vol. 67(5), pages 1362-1382, September.
    2. Azam Asanjarani & Yoni Nazarathy & Peter Taylor, 2021. "A survey of parameter and state estimation in queues," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 39-80, February.
    3. Ward Whitt, 2016. "Heavy-traffic fluid limits for periodic infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 111-143, October.

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