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Heavy traffic limits for queues with non-stationary path-dependent arrival processes

Author

Listed:
  • Kerry Fendick

    (Johns Hopkins University Applied Physics Laboratory)

  • Ward Whitt

    (Columbia University)

Abstract

In this paper, we develop a diffusion approximation for the transient distribution of the workload process in a standard single-server queue with a non-stationary Polya arrival process, which is a path-dependent Markov point process. The path-dependent arrival process model is useful because it has the arrival rate depending on the history of the arrival process, thus capturing a self-reinforcing property that one might expect in some applications. The workload approximation is based on heavy-traffic limits for (i) a sequence of Polya processes, in which the limit is a Gaussian–Markov process, and (ii) a sequence of P/GI/1 queues in which the arrival rate function approaches a constant service rate uniformly over compact intervals.

Suggested Citation

  • Kerry Fendick & Ward Whitt, 2022. "Heavy traffic limits for queues with non-stationary path-dependent arrival processes," Queueing Systems: Theory and Applications, Springer, vol. 101(1), pages 113-135, June.
    • Handle: RePEc:spr:queues:v:101:y:2022:i:1:d:10.1007_s11134-021-09728-5
    DOI: 10.1007/s11134-021-09728-5
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    References listed on IDEAS

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    1. Francisco Germán Badía & Sophie Mercier & Carmen Sangüesa, 2019. "Extensions of the Generalized Pólya Process," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1057-1085, December.
    2. Hidetoshi Konno, 2010. "On the Exact Solution of a Generalized Polya Process," Advances in Mathematical Physics, Hindawi, vol. 2010, pages 1-12, November.
    3. Avi Mandelbaum & William A. Massey, 1995. "Strong Approximations for Time-Dependent Queues," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 33-64, February.
    4. 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.
    5. Ward Whitt & Wei You, 2019. "Time-Varying Robust Queueing," Operations Research, INFORMS, vol. 67(6), pages 1766-1782, November.
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