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On Arrivals That See Time Averages

Author

Listed:
  • Benjamin Melamed

    (NEC Research Institute, Princeton, New Jersey)

  • Ward Whitt

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

Abstract

We investigate when Arrivals See Time Averages (ASTA) in a stochastic model; i.e., when the stationary distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the stationary distribution of the observed process. We also characterize the relation between the two distributions when ASTA does not hold. We introduce a Lack of Bias Assumption (LBA) which stipulates that, at any time, the conditional intensity of the point process, given the present state of the observed process, be independent of the state of the observed process. We show that LBA, without the Poisson assumption, is necessary and sufficient for ASTA in a stationary process framework. Consequently, LBA covers known examples of non-Poisson ASTA, such as certain flows in open Jackson queueing networks, as well as the familiar Poisson case (PASTA). We also establish results to cover the case in which the process is observed just after the points, e.g., when departures see time averages. Finally, we obtain a new proof of the Arrival Theorem for product-form queueing networks.

Suggested Citation

  • Benjamin Melamed & Ward Whitt, 1990. "On Arrivals That See Time Averages," Operations Research, INFORMS, vol. 38(1), pages 156-172, February.
    • Handle: RePEc:inm:oropre:v:38:y:1990:i:1:p:156-172
    DOI: 10.1287/opre.38.1.156
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    Citations

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    Cited by:

    1. (Yew Sing) Lee, Thomas, 1997. "A simple approach for analyzing feedback vacation queues with levy input process," European Journal of Operational Research, Elsevier, vol. 96(2), pages 299-316, January.
    2. Tetsuji Hirayama, 2003. "Mean sojourn times in multiclass feedback queues with gated disciplines," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 719-741, October.
    3. David L. Jagerman & Benjamin Melamed, 2003. "Models and Approximations for Call Center Design," Methodology and Computing in Applied Probability, Springer, vol. 5(2), pages 159-181, June.
    4. Weiwei Chen & Benjamin Melamed & Oleg Sokolinskiy & Ben Sopranzetti, 2017. "Cash Conversion Systems in Corporate Subsidiaries," Manufacturing & Service Operations Management, INFORMS, vol. 19(4), pages 604-619, October.
    5. Edieal J. Pinker & Robert A. Shumsky, 2000. "The Efficiency-Quality Trade-Off of Cross-Trained Workers," Manufacturing & Service Operations Management, INFORMS, vol. 2(1), pages 32-48, July.
    6. Weina Wang & Mor Harchol-Balter & Haotian Jiang & Alan Scheller-Wolf & R. Srikant, 2019. "Delay asymptotics and bounds for multitask parallel jobs," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 207-239, April.
    7. Glynn, Peter & Sigman, Karl, 1998. "Independent sampling of a stochastic process," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 151-164, June.
    8. Hassin, Refael & Haviv, Moshe & Oz, Binyamin, 2023. "Strategic behavior in queues with arrival rate uncertainty," European Journal of Operational Research, Elsevier, vol. 309(1), pages 217-224.

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