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Diffusion approximation of the stationary distribution of a two-level single server queue

Part of: Special processes Stochastic processes Markov processes

Published online by Cambridge University Press:  27 May 2025

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
*
*Postal address: Yamazaki 2641, Noda, Chiba, Japan, 278-8051. Email address: miyazawa@rs.tus.ac.jp
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Abstract

We consider a single server queue that has a threshold to change its arrival process and service speed by its queue length, which is referred to as a two-level GI/G/1 queue. This model is motivated by an energy saving problem for a single server queue whose arrival process and service speed are controlled. To obtain its performance in tractable form, we study the limit of the stationary distribution of the queue length in this two-level queue under scaling in heavy traffic. Except for a special case, this limit corresponds to its diffusion approximation. It is shown that this limiting distribution is truncated exponential (or uniform if the drift is null) below the threshold level and exponential above it under suitably chosen system parameters and generally distributed interarrival times and workloads brought by customers. This result is proved under a mild limitation on arrival parameters using the so-called basic adjoint relationship (BAR) approach studied in Braverman, Dai, and Miyazawa (2017, 2024) and Miyazawa (2017, 2024). We also intuitively discuss about a diffusion process corresponding to the limit of the stationary distribution under scaling.

Keywords

single server queuelevel dependent arrival and servicestationary distributionheavy trafficdiffusion approximationPalm distributionstochastic integral equationmulti-level SRBM

MSC classification

Primary: 60K25: Queueing theory 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Atar, R., Castiel, E. and Reiman, M. (2024). Asymptotic optimality of switched control policies in a simple parallel server system under an extended heavy traffic condition. Stochastic Systems (to appear).10.1287/stsy.2022.0022CrossRefGoogle Scholar
Atar, R. and Wolansky, G. (2024). Invariance principle and McKean-Vlasov limit for randomized load balancing in heavy traffic. Preprint. Technion - Israel Institute of Technology.Google Scholar
Braverman, A., Dai, J. and Miyazawa, M. (2017). Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach. Stochastic Systems 7, 143–196.10.1287/15-SSY199CrossRefGoogle Scholar
Braverman, A., Dai, J. and Miyazawa, M. (2024). The BAR-approach for multiclass queueing networks with SBP service policies. Stochastic Systems. Published online in Articles in Advance.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman & Hall.10.1007/978-1-4899-4483-2CrossRefGoogle Scholar
Harrison, J. M. (2013). Brownian Models of Performance and Control. Cambridge University Press, New York.10.1017/CBO9781139087698CrossRefGoogle Scholar
Kingman, J. F. C. (1961). The single server queue in heavy traffic. Mathematical Proceedings of the Cambridge Philosophical Society 57, 902–904.10.1017/S0305004100036094CrossRefGoogle Scholar
Mandelbaum, A. and Pats, G. (1998). State-dependent stochastic networks. Part I: approximations and applications with continuous diffusion limits. Annals of Applied Probability 8, 569–646.10.1214/aoap/1028903539CrossRefGoogle Scholar
Miyazawa, M. (1994). Rate conservation laws: a survey. Queueing Systems 15, 1–58.10.1007/BF01189231CrossRefGoogle Scholar
Miyazawa, M. (2017). A unified approach for large queue asymptotics in a heterogeneous multiserver queue. Adv. in Appl. Probab. 49, 182–220.10.1017/apr.2016.84CrossRefGoogle Scholar
Miyazawa, M. (2024). Multi-level reflecting Brownian motion on the half line and its stationary distribution. Journal of the Indian Society for Probability and Statistics 25, 543–574.10.1007/s41096-024-00205-9CrossRefGoogle Scholar
Miyazawa, M. (2024). Palm problems arising in bar approach and its applications. Queueing Systems 108, 253–273.10.1007/s11134-024-09915-0CrossRefGoogle Scholar
Miyazawa, M. (2024). The stationary distributions of state-dependent diffusions reflected at one and two sides. Technical report.Google Scholar
Rudin, W. (1987). Real complex analysis 3rd ed. International Series in Pure and Applied Mathematics. McGraw-Hill.Google Scholar
Yamada, K. (1995). Diffusion approximation for open state-dependent queueing networks in the heavy traffic situation. Annals of Applied Probability 5, 958–982.10.1214/aoap/1177004602CrossRefGoogle Scholar