Abstract
We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton–Watson processes, the Laplace–Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.
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Acknowledgements
We are very grateful to a referee for pointing out a problem in our initial proof of the upper bound in Sect. 6.2. We also thank a second referee and an associate editor for many useful suggestions. The first author thanks Dr. Quan Zhou and Dr. Guodong Pang for helpful conversations. The first author is grateful to the Dobelman Family for support in the form of the Dobelman Family Junior Chair. Finally, the first author gratefully acknowledges the support of ARO-YIP-71636-MA, NSF DMS-1811936, and ONR N00014-18-1-2192.
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Ernst, P.A., Asmussen, S. & Hasenbein, J.J. Stability and busy periods in a multiclass queue with state-dependent arrival rates. Queueing Syst 90, 207–224 (2018). https://doi.org/10.1007/s11134-018-9587-9
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DOI: https://doi.org/10.1007/s11134-018-9587-9