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Stabilizing performance in a single-server queue with time-varying arrival rate

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Abstract

We consider a class of general \(G_t/G_t/1\) single-server queues, including the \(M_t/M_t/1\) queue, with unlimited waiting space, service in order of arrival, and a time-varying arrival rate, where the service rate at each time is subject to control. We study the rate-matching control, where the service rate is made proportional to the arrival rate. We show that the model with the rate-matching control can be regarded as a deterministic time transformation of a stationary G / G / 1 model, so that the queue length distribution is stabilized as time evolves. However, the time-varying virtual waiting time is not stabilized. We show that the time-varying expected virtual waiting time with the rate-matching service-rate control becomes inversely proportional to the arrival rate in a heavy-traffic limit. We also show that no control that stabilizes the queue length asymptotically in heavy traffic can also stabilize the virtual waiting time. Then we consider two square-root service-rate controls and show that one of these stabilizes the waiting time when the arrival rate changes slowly relative to the average service time, so that a pointwise stationary approximation is appropriate.

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Acknowledgments

The author gratefully acknowledges Columbia doctoral student Ni Ma for all reported simulation results and NSF Grant CMMI Grant 1265070.

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  1. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, USA

    Ward Whitt

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  1. Ward Whitt
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Correspondence to Ward Whitt.

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Whitt, W. Stabilizing performance in a single-server queue with time-varying arrival rate. Queueing Syst 81, 341–378 (2015). https://doi.org/10.1007/s11134-015-9462-x

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  • DOI: https://doi.org/10.1007/s11134-015-9462-x

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