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Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services

Author

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  • Hongyuan Lu

    (Pennsylvania State University)

  • Guodong Pang

    (Pennsylvania State University)

Abstract

We study an infinite-server fork–join queueing system with dependent services, which experiences alternating renewal service disruptions. Jobs are forked into a fixed number of parallel tasks upon arrival and processed at the corresponding parallel service stations with multiple servers. Synchronization of a job occurs when its parallel tasks are completed, i.e., non-exchangeable. Service times of the parallel tasks of each job can be correlated, having a general continuous joint distribution function, and moreover, the service vectors of consecutive jobs form a stationary dependent sequence satisfying the strong mixing ( $$\alpha $$ α -mixing) condition. The system experiences renewal alternating service disruptions with up and down periods. In each up period, the system operates normally, but in each down period, jobs continue to enter the system, while all the servers will stop working, and services received will be conserved and resume at the beginning of the next up period. We study the impact of both the dependence among service times and these down times upon the service dynamics, the unsynchronized queueing dynamics, and the synchronized process, assuming that the down times are asymptotically negligible. We prove FWLLN and FCLT for these processes, where the limit processes in the FCLT possess a stochastic decomposition property and the convergence requires the Skorohod $$M_1$$ M 1 topology.

Suggested Citation

  • Hongyuan Lu & Guodong Pang, 2017. "Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services," Queueing Systems: Theory and Applications, Springer, vol. 85(1), pages 67-115, February.
    • Handle: RePEc:spr:queues:v:85:y:2017:i:1:d:10.1007_s11134-016-9505-y
    DOI: 10.1007/s11134-016-9505-y
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    References listed on IDEAS

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    1. Hongyuan Lu & Guodong Pang, 2016. "Gaussian Limits for a Fork-Join Network with Nonexchangeable Synchronization in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 560-595, May.
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    Cited by:

    1. Erhun Özkan & Amy R. Ward, 2019. "On the Control of Fork-Join Networks," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 532-564, May.
    2. Dennis Schol & Maria Vlasiou & Bert Zwart, 2025. "Extreme values for the waiting time in large fork-join queues," Queueing Systems: Theory and Applications, Springer, vol. 109(1), pages 1-26, March.
    3. 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.
    4. Schol, Dennis & Vlasiou, Maria & Zwart, Bert, 2023. "Tail asymptotics for the delay in a Brownian fork-join queue," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 99-138.

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    1. Erhun Özkan & Amy R. Ward, 2019. "On the Control of Fork-Join Networks," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 532-564, May.
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