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Heavy-traffic fluid limits for periodic infinite-server queues

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Abstract

To better understand what stochastic model might be appropriate in applications with system data, we study the consequences of fitting a stationary birth-and-death (BD) process to the sample path of a periodic \(M_t/GI/\infty \) model. The fitted BD process will necessarily have the correct steady-state distribution (appropriately defined), but will not have the correct transient behavior. Nevertheless, the fitted birth-rate and death-rate functions have structure determined by the \(M_t/GI/\infty \) model that should be seen with data if the \(M_t/GI/\infty \) model is appropriate. In this paper, we establish heavy-traffic fluid limits that yield explicit approximation formulas for the fitted birth-rate and death-rate functions that can help evaluate whether a periodic \(M_t/GI/\infty \) model is appropriate. We also establish many-server heavy-traffic fluid limits for the steady-state distribution in the periodic \(M_t/GI/\infty \) model. For the special case of sinusoidal arrival rates, the limiting steady-state distribution has an arcsine law.

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Acknowledgments

The author thanks James Dong for his contribution to this grey-box modeling project, including extensive simulations that improved the author’s understanding of this subject. The author acknowledges support from NSF Grant CMMI 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. Heavy-traffic fluid limits for periodic infinite-server queues. Queueing Syst 84, 111–143 (2016). https://doi.org/10.1007/s11134-016-9494-x

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  • DOI: https://doi.org/10.1007/s11134-016-9494-x

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