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A single-server queue with batch arrivals and semi-Markov services

Author

Listed:
  • Abhishek

    (University of Amsterdam)

  • Marko A. A. Boon

    (Eindhoven University of Technology)

  • Onno J. Boxma

    (Eindhoven University of Technology)

  • Rudesindo Núñez-Queija

    (University of Amsterdam)

Abstract

We investigate the transient and stationary queue length distributions of a class of service systems with correlated service times. The classical $$M^X/G/1$$ M X / G / 1 queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process J(t). The state of $$J(\cdot )$$ J ( · ) at a service initiation time determines the joint distribution of the subsequent service duration and the state of $$J(\cdot )$$ J ( · ) at the next service initiation. Several earlier works have imposed technical conditions, on the zeros of a matrix determinant arising in the analysis, that are required in the computation of the stationary queue length probabilities. The imposed conditions in several of these articles are difficult or impossible to verify. Without such assumptions, we determine both the transient and the steady-state joint distribution of the number of customers immediately after a departure and the state of the process J(t) at the start of the next service. We numerically investigate how the mean queue length is affected by variability in the number of customers that arrive during a single service time. Our main observations here are that increasing variability may reduce the mean queue length, and that the Markovian dependence of service times can lead to large queue lengths, even if the system is not in heavy traffic.

Suggested Citation

  • Abhishek & Marko A. A. Boon & Onno J. Boxma & Rudesindo Núñez-Queija, 2017. "A single-server queue with batch arrivals and semi-Markov services," Queueing Systems: Theory and Applications, Springer, vol. 86(3), pages 217-240, August.
    • Handle: RePEc:spr:queues:v:86:y:2017:i:3:d:10.1007_s11134-017-9531-4
    DOI: 10.1007/s11134-017-9531-4
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    References listed on IDEAS

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    1. D. P. Gaver, 1963. "A comparison of queue disciplines when service orientation times occur," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 10(1), pages 219-235, March.
    2. de Smit, J. H. A., 1986. "The single server semi-markov queue," Stochastic Processes and their Applications, Elsevier, vol. 22(1), pages 37-50, May.
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    Cited by:

    1. Chris Blondia, 2021. "A queueing model for a wireless sensor node using energy harvesting," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 77(2), pages 335-349, June.
    2. Abhishek & Marko A. A. Boon & Michel Mandjes, 2019. "Generalized gap acceptance models for unsignalized intersections," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 385-409, June.

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    1. Abhishek & Marko A. A. Boon & Michel Mandjes, 2019. "Generalized gap acceptance models for unsignalized intersections," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 385-409, June.

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