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Mt/G/\infty Queues with Sinusoidal Arrival Rates

Author

Listed:
  • Stephen G. Eick

    (AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636)

  • William A. Massey

    (AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636)

  • Ward Whitt

    (AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636)

Abstract

In this paper we describe the mean number of busy servers as a function of time in an M t /G/\infty queue (having a nonhomogeneous Poisson arrival process) with a sinusoidal arrival rate function. For an M t /G/\infty model with appropriate initial conditions, it is known that the number of busy servers at time t has a Poisson distribution for each t, so that the full distribution is characterized by its mean. Our formulas show how the peak congestion lags behind the peak arrival rate and how much less is the range of congestion than the range of offered load. The simple formulas can also be regarded as consequences of linear system theory, because the mean function can be regarded as the image of a linear operator applied to the arrival rate function. We also investigate the quality of various approximations for the mean number of busy servers such as the pointwise stationary approximation and several polynomial approximations. Finally, we apply the results for sinusoidal arrival rate functions to treat general periodic arrival rate functions using Fourier series. These results are intended to provide a better understanding of the behavior of the M t /G/\infty model and related M t /G/s/r models where some customers are lost or delayed.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormnsc:v:39:y:1993:i:2:p:241-252
    DOI: 10.1287/mnsc.39.2.241
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    Citations

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    Cited by:

    1. 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.
    2. Song-Hee Kim & Ward Whitt, 2013. "Statistical Analysis with Little's Law," Operations Research, INFORMS, vol. 61(4), pages 1030-1045, August.
    3. Rouba Ibrahim & Mor Armony & Achal Bassamboo, 2017. "Does the Past Predict the Future? The Case of Delay Announcements in Service Systems," Management Science, INFORMS, vol. 63(6), pages 1762-1780, June.
    4. Yang, Feng & Liu, Jingang, 2012. "Simulation-based transfer function modeling for transient analysis of general queueing systems," European Journal of Operational Research, Elsevier, vol. 223(1), pages 150-166.
    5. Yiran Liu & Harsha Honnappa & Samy Tindel & Nung Kwan Yip, 2021. "Infinite server queues in a random fast oscillatory environment," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 145-179, June.
    6. Wenwen Li & Alexander Goldenshluger, 2024. "Adaptive minimax estimation of service time distribution in the $$M_t/G/\infty $$ M t / G / ∞ queue from departure data," Queueing Systems: Theory and Applications, Springer, vol. 108(1), pages 81-123, October.
    7. Eugene Furman & Alex Cressman & Saeha Shin & Alexey Kuznetsov & Fahad Razak & Amol Verma & Adam Diamant, 2021. "Prediction of personal protective equipment use in hospitals during COVID-19," Health Care Management Science, Springer, vol. 24(2), pages 439-453, June.
    8. Linda V. Green & Peter J. Kolesar, 1998. "A Note on Approximating Peak Congestion in Mt/G/\infty Queues with Sinusoidal Arrivals," Management Science, INFORMS, vol. 44(11-Part-2), pages 137-144, November.
    9. 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.
    10. 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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