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Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times

Author

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  • Lun Yu

    (Department of Industrial Engineering, Tsinghua University, Beijing 100084, P. R. China)

  • Ohad Perry

    (Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208)

Abstract

We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the M / M / n queue when the traffic intensity approaches its critical value 1 from below, and is otherwise a transient diffusion, despite the fact that the prelimit is positive recurrent. To obtain a refined measure of the congestion due to the correlation, we characterize a lower-order fluid (LOF) limit for the case in which the diffusion limit is transient, demonstrating that the queue in this case scales like n 3 / 4 . Under both the diffusion and LOF scalings, we show that the stationary distributions converge weakly to the time-limiting behavior of the corresponding process limit.

Suggested Citation

  • Lun Yu & Ohad Perry, 2023. "Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 1119-1157, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:1119-1157
    DOI: 10.1287/moor.2022.1300
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