Abstract
In this paper, we consider an M\({}^X\)/M/1/SET-VARI queue which has batch arrivals, variable service speed and setup time. Our model is motivated by power-aware servers in data centers where dynamic scaling techniques are used. The service speed of the server is proportional to the number of jobs in the system. The contribution of our paper is threefold. First, we obtain the necessary and sufficient condition for the stability of the system. Second, we derive an expression for the probability generating function of the number of jobs in the system. Third, our main contribution is the derivation of the Laplace–Stieltjes transform (LST) of the sojourn time distribution, which is obtained in series form involving infinite-dimensional matrices. In this model, since the service speed varies upon arrivals and departures of jobs, the sojourn time of a tagged job is affected by the batches that arrive after it. This makes the derivation of the LST of the sojourn time complex and challenging. In addition, we present some numerical examples to show the trade-off between the mean sojourn time (response time) and the energy consumption. Using the numerical inverse Laplace–Stieltjes transform, we also obtain the sojourn time distribution, which can be used for setting the service-level agreement in data centers.
Similar content being viewed by others
References
Adan, I., D’Auria, B.: Sojourn time in a single-server queue with threshold service rate control. SIAM J. Appl. Math. 76(1), 197–216 (2016)
Baba, Y.: The M\({}^X\)/M/1 queue with multiple working vacation. Am. J. Oper. Res. 2(2), 217–224 (2012)
Cong, T.D.: On the M\({}^X\)/G/\(\infty \) queue with heterogeneous customers in a batch. J. Appl. Probab. 31(1), 280–286 (1994)
Downton, F.: Waiting time in bulk service queues. J. R. Stat. Soc. 17(2), 256–261 (1955)
Durbin, F.: Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. Comput. J. 17(4), 371–376 (1974)
Fomin, S.V., et al.: Elements of the Theory of Functions and Functional Analysis, vol. 1. Courier Corporation, North Chelmsford (1999)
Gandhi, A., Harchol-Balter, M., Adan, I.: Server farms with setup costs. Perform. Eval. 67(11), 1123–1138 (2010)
Keilson, J., Servi, L.D.: A distributional form of Little’s law. Oper. Res. Lett. 7(5), 223–227 (1988)
Lu, X., Aalto, S., Lassila, P.: Performance-energy trade-off in data centers: Impact of switching delay. In: Proceedings of 22nd IEEE ITC Specialist Seminar on Energy Efficient and Green Networking (SSEEGN), pp. 50–55 (2013)
Maccio, V.J., Down, D.G.: On optimal policies for energy-aware servers. Perform. Eval. 90, 36–52 (2015)
Mittal, S.: A survey of techniques for improving energy efficiency in embedded computing systems. Int. J. Comput. Aided Eng. Technol. 6, 440–459 (2014)
Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1998)
Rohde, U.L.: Digital PLL Frequency Synthesizers: Theory and Design. Prentice-Hall, Englewood Cliffs, NJ (1983)
Shanbhag, D.N.: On infinite server queues with batch arrivals. J. Appl. Probab. 3(1), 274–279 (1966)
Sueur, E.L., Heiser, G.: Dynamic voltage and frequency scaling: the laws of diminishing returns. In: Proceedings of the 2010 International Conference on Power Aware Computing and Systems, pp. 1–8 (2010)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996)
Wierman, A., Andrew, L., Tang, A.: Power-aware speed scaling in processor sharing systems: optimality and robustness. Perform. Eval. 69, 601–622 (2012)
Wolft, R.W.: Poisson arrivals see time average. Oper. Res. 30, 223–231 (1982)
Acknowledgements
We would like to thank the guest editors and two anonymous referees for their constructive comments which significantly improved the presentation of the paper. The research of TP was partially supported by JSPS KAKENHI Grant Number 26730011.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yajima, M., Phung-Duc, T. Batch arrival single-server queue with variable service speed and setup time. Queueing Syst 86, 241–260 (2017). https://doi.org/10.1007/s11134-017-9533-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-017-9533-2