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Heavy-traffic fluid limits for periodic infinite-server queues

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  • Ward Whitt

    (Columbia University)

Abstract

To better understand what stochastic model might be appropriate in applications with system data, we study the consequences of fitting a stationary birth-and-death (BD) process to the sample path of a periodic $$M_t/GI/\infty $$ M t / G I / ∞ model. The fitted BD process will necessarily have the correct steady-state distribution (appropriately defined), but will not have the correct transient behavior. Nevertheless, the fitted birth-rate and death-rate functions have structure determined by the $$M_t/GI/\infty $$ M t / G I / ∞ model that should be seen with data if the $$M_t/GI/\infty $$ M t / G I / ∞ model is appropriate. In this paper, we establish heavy-traffic fluid limits that yield explicit approximation formulas for the fitted birth-rate and death-rate functions that can help evaluate whether a periodic $$M_t/GI/\infty $$ M t / G I / ∞ model is appropriate. We also establish many-server heavy-traffic fluid limits for the steady-state distribution in the periodic $$M_t/GI/\infty $$ M t / G I / ∞ model. For the special case of sinusoidal arrival rates, the limiting steady-state distribution has an arcsine law.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:queues:v:84:y:2016:i:1:d:10.1007_s11134-016-9494-x
    DOI: 10.1007/s11134-016-9494-x
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    References listed on IDEAS

    as
    1. Ward Whitt, 2005. "Engineering Solution of a Basic Call-Center Model," Management Science, INFORMS, vol. 51(2), pages 221-235, February.
    2. Whitt, Ward, 2012. "Fitting birth-and-death queueing models to data," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 998-1004.
    3. 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.
    4. Ronald W. Wolff, 1965. "Problems of Statistical Inference for Birth and Death Queuing Models," Operations Research, INFORMS, vol. 13(3), pages 343-357, June.
    5. Stephen G. Eick & William A. Massey & Ward Whitt, 1993. "Mt/G/\infty Queues with Sinusoidal Arrival Rates," Management Science, INFORMS, vol. 39(2), pages 241-252, February.
    6. Song-Hee Kim & Ward Whitt, 2013. "Statistical Analysis with Little's Law," Operations Research, INFORMS, vol. 61(4), pages 1030-1045, August.
    7. Glynn, Peter W. & Whitt, Ward, 1993. "Limit theorems for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 299-314, September.
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    Cited by:

    1. Vijayalakshmi Chetlapalli & K. S. S. Iyer & Himanshu Agrawal, 2020. "Modelling time-dependent aggregate traffic in 5G networks," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 73(4), pages 557-575, April.
    2. Pang, Guodong & Zheng, Yi, 2017. "On the functional and local limit theorems for Markov modulated compound Poisson processes," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 131-140.

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