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A functional law of the iterated logarithm for multi-class queues with batch arrivals

Author

Listed:
  • Yongjiang Guo

    (Beijing University of Posts and Telecommunications)

  • Xiyang Hou

    (Huazhong University of Science and Technology)

  • Yunan Liu

    (North Carolina State University)

Abstract

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.

Suggested Citation

  • Yongjiang Guo & Xiyang Hou & Yunan Liu, 2021. "A functional law of the iterated logarithm for multi-class queues with batch arrivals," Annals of Operations Research, Springer, vol. 300(1), pages 51-77, May.
  • Handle: RePEc:spr:annopr:v:300:y:2021:i:1:d:10.1007_s10479-020-03864-6
    DOI: 10.1007/s10479-020-03864-6
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    References listed on IDEAS

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    1. Hong Chen & Hanqin Zhang, 1997. "Stability of Multiclass Queueing Networks Under FIFO Service Discipline," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 691-725, August.
    2. Sofian De Clercq & Koenraad Laevens & Bart Steyaert & Herwig Bruneel, 2013. "A multi-class discrete-time queueing system under the FCFS service discipline," Annals of Operations Research, Springer, vol. 202(1), pages 59-73, January.
    3. Peter W. Glynn & Ward Whitt, 1988. "An LIL Version of L = (lambda) W," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 693-710, November.
    4. Sakalauskas, L. L. & Minkevicius, S., 2000. "On the law of the iterated logarithm in open queueing networks," European Journal of Operational Research, Elsevier, vol. 120(3), pages 632-640, February.
    5. Yongjiang Guo & Yunan Liu & Renhu Pei, 2018. "Functional law of the iterated logarithm for multi-server queues with batch arrivals and customer feedback," Annals of Operations Research, Springer, vol. 264(1), pages 157-191, May.
    6. Rishi Talreja & Ward Whitt, 2008. "Fluid Models for Overloaded Multiclass Many-Server Queueing Systems with First-Come, First-Served Routing," Management Science, INFORMS, vol. 54(8), pages 1513-1527, August.
    7. Hong Chen & Hanqin Zhang, 2000. "Diffusion Approximations for Some Multiclass Queueing Networks with FIFO Service Disciplines," Mathematics of Operations Research, INFORMS, vol. 25(4), pages 679-707, November.
    8. Tsai, Tsung-Hsi, 2000. "Empirical law of the iterated logarithm for Markov chains with a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 175-191, October.
    9. Avi Mandelbaum & William A. Massey, 1995. "Strong Approximations for Time-Dependent Queues," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 33-64, February.
    10. Caramellino, Lucia, 1998. "Strassen's law of the iterated logarithm for diffusion processes for small time," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 1-19, May.
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