Abstract
This paper considers an M/M/c/Setup queueing system motivated from data centers under an on–off policy (that is, with idle servers turned off to save energy). We derive and show features of the expected waiting times of customers at every system state. While the system can be modeled using only two variables, it is necessary to adopt one more dimension to derive the expected waiting time at a specific state. Furthermore, we propose a loop algorithm to compute the equilibrium threshold strategy of strategic customers, and calculate performance measures of the system in equilibrium. Finally, we consider optimal system design through several numerical examples, focusing on maximizing social welfare.
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References
Barroso, L.A., Hölzle, U.: The case for energy-proportional computing. Computer 40(12), 33–37 (2007)
Gandhi, A., Doroudi, S., Harchol-Balter, M., Scheller-Wolf, A.: Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward. Queueing Syst. 77(2), 177–209 (2014). https://doi.org/10.1007/s11134-014-9409-7
Hassin, R., Haviv, M.: To Queue or not to Queue: Equilibrium Behavior in Queueing Systems, vol. 59. Springer Science & Business Media, Heidelberg (2003)
Hsieh, C.Y., Phung-Duc, T., Ren, Y., Chen, J.C.: Design and analysis of dynamic block-setup reservation algorithm for 5G network slicing. IEEE Trans. Mobile Comput. (2022)
Keilson, J., Servi, L.D.: A distributional form of Little’s law. Oper. Res. Lett. 7(5), 223–227 (1988)
Mitrani, I.: Service center trade-offs between customer impatience and power consumption. Perform. Eval. 68(11), 1222–1231 (2011)
Mitrani, I.: Managing performance and power consumption in a server farm. Ann. Oper. Res. 202(1), 121–134 (2013). https://doi.org/10.1007/s10479-011-0932-1
Naor, P.: The regulation of queue size by levying tolls. Econometrica 37(1), 15–24 (1969)
Phung-Duc, T.: Exact solutions for M/M/c/setup queues. Telecommun. Syst. 64(2), 309–324 (2017). https://doi.org/10.1007/s11235-016-0177-z
Phung-Duc, T., Kawanishi, K.: Delay performance of data-center queue with setup policy and abandonment. Ann. Oper. Res. 293(1), 269–293 (2020). https://doi.org/10.1007/s10479-019-03268-1
Acknowledgements
The research of Hung Q. Nguyen was supported by JST SPRING, Grant Number JPMJSP2124. The research of Tuan Phung-Duc was supported in part by JSPS KAKENHI, Grant Number 21K11765 and by F-MIRAI: R &D Center for Frontiers of MIRAI in Policy and Technology, the University of Tsukuba and Toyota Motor Corporation collaborative R &D center.
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Nguyen, H.Q., Phung-Duc, T. (2023). M/M/c/Setup Queues: Conditional Mean Waiting Times and a Loop Algorithm to Derive Customer Equilibrium Threshold Strategy. In: Gilly, K., Thomas, N. (eds) Computer Performance Engineering. EPEW 2022. Lecture Notes in Computer Science, vol 13659. Springer, Cham. https://doi.org/10.1007/978-3-031-25049-1_6
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DOI: https://doi.org/10.1007/978-3-031-25049-1_6
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