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A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems

Author

Listed:
  • Hongyuan Lu

    (Pennsylvania State University)

  • Guodong Pang

    (Pennsylvania State University)

  • Michel Mandjes

    (University of Amsterdam
    CWI)

Abstract

We prove a functional central limit theorem for Markov additive arrival processes where the modulating Markov process has the transition rate matrix scaled up by $$n^{\alpha }$$ n α ( $$\alpha >0$$ α > 0 ) and the mean and variance of the arrival process are scaled up by n. It is applied to an infinite-server queue and a fork–join network with a non-exchangeable synchronization constraint, where in both systems both the arrival and service processes are modulated by a Markov process. We prove functional central limit theorems for the queue length processes in these systems joint with the arrival and departure processes, and characterize the transient and stationary distributions of the limit processes. We also observe that the limit processes possess a stochastic decomposition property.

Suggested Citation

  • Hongyuan Lu & Guodong Pang & Michel Mandjes, 2016. "A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 84(3), pages 381-406, December.
    • Handle: RePEc:spr:queues:v:84:y:2016:i:3:d:10.1007_s11134-016-9496-8
    DOI: 10.1007/s11134-016-9496-8
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    References listed on IDEAS

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    1. D. Anderson & J. Blom & M. Mandjes & H. Thorsdottir & K. Turck, 2016. "A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 153-168, March.
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    Cited by:

    1. Xuefeng Gao & Lingjiong Zhu, 2018. "Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(1), pages 161-206, October.
    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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