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doi: 10.3934/jimo.2021106
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## Performance analysis and system optimization of an energy-saving mechanism in cloud computing with correlated traffic

 1 School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China 2 Key Laboratory for Computer Virtual Technology and System Integration of Hebei Province, Yanshan University, Qinhuangdao 066004, China 3 Langfang Yanjing Vocational Technical College, Langfang 065200, China 4 Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan 5 Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

* Corresponding author: Shunfu Jin

Received  September 2020 Revised  April 2021 Early access June 2021

Energy consumption is becoming a significant part of overall operational cost in cloud data centers. For the purpose of satisfying the Service Level Agreement (SLA) of cloud users while enhancing the energy efficiency in cloud computing systems, in this paper we propose an energy-saving mechanism with a sleep mode. Taking into consideration the traffic's correlation and the stochastical behavior of data arrival requests in a random cloud environment with the proposed energy-saving mechanism, we model the system as a MAP/M/$N$/$N$+$K$ queue with a synchronous multi-vacation. Then, we present a theoretical basis for analyzing and evaluating the system performance by taking a state transition rate matrix in the steady state. Next, we investigate the change trends for the energy saving rate of the system and the average latency of tasks by carrying out numerical experiments. Moreover, we give a

Citation: Xuena Yan, Shunfu Jin, Wuyi Yue, Yutaka Takahashi. Performance analysis and system optimization of an energy-saving mechanism in cloud computing with correlated traffic. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021106
##### References:
 [1] P. Bertoldi, M. Avgerinou and L. Castellazzi, Trends in Data Centre Energy Consumption under the European Code of Conduct for Data Centre Energy Efficiency, Technical Report, Publications Office of the European Union, Luxembourg, 2017. Google Scholar [2] C. Cheng, J. Li and Y. Wang, An energy-saving task scheduling strategy based on vacation queuing theory in cloud computing, Tsinghua Science and Technology, 20 (2015), 28-39.  doi: 10.1109/TST.2015.7040511.  Google Scholar [3] B. D. Choi, B. Kim and D. Zhu, MAP/M/$c$ queue with constant impatient time, Mathematics of Operations Research, 29 (2004), 309-325.  doi: 10.1287/moor.1030.0081.  Google Scholar [4] D. Ding, X. Fan, Y. Zhao, K. Kang, Q. Yin and J. Zeng, Q-learning based dynamic task scheduling for energy-efficient cloud computing, Future Generation Computer Systems, 108 (2020), 361-371.  doi: 10.1016/j.future.2020.02.018.  Google Scholar [5] S. A. Dudin and O. S. Dudina, Call center operation model as a MAP/PH/$N$/$R-N$ system with impatient customers, Problems of Information Transmission, 47 (2011), 364-377.  doi: 10.1134/S0032946011040053.  Google Scholar [6] O. Dudina and S. Dudin, Queueing system MAP/M/$N$/$N$+$K$ operating in random environment as a model of call center, in BWWQT, Minsk, Belarus, 2013, 83–92. doi: 10.1007/978-3-642-35980-4_10.  Google Scholar [7] R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, in MHS'95, Nagoya, Japan, 1995, 39–43. doi: 10.1109/MHS.1995.494215.  Google Scholar [8] A. H. Gandomi, X.-S. Yang, S. Talatahari and A. H. Alavi, Firefly algorithm with chaos, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 89-98.  doi: 10.1016/j.cnsns.2012.06.009.  Google Scholar [9] Q.-M. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.  Google Scholar [10] T. Hirai, H. Masuyama, S. Kasahara and Y. Takahashi, Performance optimization of parallel-distributed processing with checkpointing for cloud environment, Journal of Industrial and Management Optimization, 14 (2018), 1423–-1442. doi: 10.3934/jimo.2018014.  Google Scholar [11] X. Huang, D. Wu and N. Zhao, Study of performance measures and energy consumption for cloud computing centers based on queueing theory, Journal of Physics: Conference Series, 1631 (2020), 25-26.  doi: 10.1088/1742-6596/1631/1/012155.  Google Scholar [12] S. Jin, H. Wu and W. Yue, Pricing policy for a cloud registration service with a novel cloud architecture, Cluster Computing, 22 (2019), 271-283.  doi: 10.1007/s10586-018-2854-z.  Google Scholar [13] S. Jin, S. Hao, X. Qie and W. Yue, A virtual machine scheduling strategy with a speed switch and a multi-sleep mode in cloud data centers, Journal of Systems Science and Systems Engineering, 28 (2019), 194-210.  doi: 10.1007/s11518-018-5401-9.  Google Scholar [14] S. Jing, S. Ali, K. She and Y. Zhong, State-of-the-art research study for green cloud computing, The Journal of Supercomputing, 65 (2013), 445-468.  doi: 10.1007/s11227-011-0722-1.  Google Scholar [15] H. Khazaei, J. Mišić and V. B. Mišić, Performance analysis of cloud computing centers, in QShine, Houston, USA, 2010,251–264. doi: 10.1007/978-3-642-29222-4_18.  Google Scholar [16] Q.-L. Li and Y. Q. Zhao, A MAP/G/1 queue with negative customers, Queueing Systems, 47 (2004), 5-43.  doi: 10.1023/B:QUES.0000032798.65858.19.  Google Scholar [17] Y. Liu, L. Wang, X. Wang, X. Xu and P. Jiang, Cloud manufacturing: Key issues and future perspectives, International Journal of Computer Integrated Manufacturing, 32 (2019), 858-874.  doi: 10.1080/0951192X.2019.1639217.  Google Scholar [18] L. Luo, W. Wu and F. Zhang, Energy modeling based on cloud data center, Journal of Software, 25 (2014), 1371-1387.   Google Scholar [19] A. Manzoor, Cloud Security: Concepts, Methodologies, Tools, and Applications, IGI Global, Hershey, PA, 2019. Google Scholar [20] S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications, 27 (2016), 1053-1073.  doi: 10.1007/s00521-015-1920-1.  Google Scholar [21] B. K. Patle, A. Pandey, A. Jagadeesh and D. R. Parhi, Path planning in uncertain environment by using firefly algorithms, Defence Technology, 14 (2018), 691-701.  doi: 10.1016/j.dt.2018.06.004.  Google Scholar [22] T. Phung-Duc and K. Kawanishi, Multiserver retrial queue with setup time and its application to data centers, Journal of Industrial and Management Optimization, 15 (2019), 15-35.  doi: 10.3934/jimo.2018030.  Google Scholar [23] QYResearch, Global Cloud Accounting Software Market Size, Status and Forecast 2025, Technical Report, Albany, NY, 2018. Google Scholar [24] J. Shaler Stidham, Optimal Design of Queueing Systems, Chapman and Hall, New York, 2009. doi: 10.1201/9781420010008.  Google Scholar [25] G. Shao and J. Chen, A load balancing strategy based on data correlation in cloud computing, in UCC, Shanghai, China, 2016,364–368. doi: 10.1145/2996890.3007852.  Google Scholar [26] N. Sharma and R. Guddeti, Multi-objective energy efficient virtual machines allocation at the cloud data center, IEEE Transactions on Services Computing, 12 (2019), 158-171.  doi: 10.1109/TSC.2016.2596289.  Google Scholar [27] M. J. Usman, A. S. Ismail, G. Abdul-Salaam, H. Chizari, O. Kaiwartya, A. Y. Gital, M. Abdullahi, A. Aliyu and S. I. Dishing, Energy-efficient nature-inspired techniques in cloud computing datacenters, Telecommunication Systems, 71 (2019), 275-302.  doi: 10.1007/s11235-019-00549-9.  Google Scholar [28] J. Vilaplana, F. Solsona, I. Teixidó, J. Mateo, F. Abella and J. Rius, A queuing theory model for cloud computing, The Journal of Supercomputing, 69 (2014), 492-507.  doi: 10.1007/s11227-014-1177-y.  Google Scholar [29] G.-G. Wang, L. Guo, H. Duan and H. Wang, A new improved firefly algorithm for global numerical optimization, Journal of Computational & Theoretical Nanoscience, 11 (2014), 477-485.  doi: 10.1166/jctn.2014.3383.  Google Scholar [30] X. Wang, J. Zhu, S. Jin, W. Yue and Y. Takahashi, Performance evaluation and social optimization of an energy-saving virtual machine allocation scheme within a cloud environment, Journal of the Operations Research Society of China, 8 (2020), 561-580.  doi: 10.1007/s40305-019-00272-x.  Google Scholar [31] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479.  doi: 10.3934/jimo.2009.5.467.  Google Scholar [32] Z.-Q. Wu and X.-B. Zhao, Frequency ${H}_2/{H}_{ \infty}$ optimizing control for isolated microgrid based on IPSO algorithm, Journal of Industrial and Management Optimization, 14 (2018), 1565-1577.  doi: 10.3934/jimo.2018021.  Google Scholar [33] X. Yan, S. Jin, W. Yue and Y. Takahashi, A MAP-based performance analysis on an energy-saving mechanism in cloud computing, in QTNA, Ghent, Belgium, 2019,369–378. doi: 10.1007/978-3-030-27181-7_22.  Google Scholar [34] X.-S. Yang, Firefly algorithms for multimodal optimization, in SAGA, Sapporo, Japan, 2009,169–178. doi: 10.1007/978-3-642-04944-6_14.  Google Scholar [35] X.-S. Yang, Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.  doi: 10.1504/IJBIC.2010.032124.  Google Scholar [36] H. Yeganeh, A. Salahi and M. A. Pourmina, A novel cost optimization method for mobile cloud computing by capacity planning of green data center with dynamic pricing, Canadian Journal of Electrical and Computer Engineering, 42 (2019), 41-51.  doi: 10.1109/CJECE.2019.2890833.  Google Scholar [37] W. Zhao, X. Wang, S. Jin, W. Yue and Y. Takahashi, An energy efficient task scheduling strategy in a cloud computing system and its performance evaluation using a two-dimensional continuous time Markov chain model, Electronics, 8 (2019), 775-790.  doi: 10.3390/electronics8070775.  Google Scholar [38] Z. Zhou and Z. Zhou, A MAP/M/$N$ retrial queueing model with asynchronous single vacations, in ICVRIS, Changsha, China, 2018,245–249. doi: 10.1109/ICVRIS.2018.00067.  Google Scholar

show all references

##### References:
 [1] P. Bertoldi, M. Avgerinou and L. Castellazzi, Trends in Data Centre Energy Consumption under the European Code of Conduct for Data Centre Energy Efficiency, Technical Report, Publications Office of the European Union, Luxembourg, 2017. Google Scholar [2] C. Cheng, J. Li and Y. Wang, An energy-saving task scheduling strategy based on vacation queuing theory in cloud computing, Tsinghua Science and Technology, 20 (2015), 28-39.  doi: 10.1109/TST.2015.7040511.  Google Scholar [3] B. D. Choi, B. Kim and D. Zhu, MAP/M/$c$ queue with constant impatient time, Mathematics of Operations Research, 29 (2004), 309-325.  doi: 10.1287/moor.1030.0081.  Google Scholar [4] D. Ding, X. Fan, Y. Zhao, K. Kang, Q. Yin and J. Zeng, Q-learning based dynamic task scheduling for energy-efficient cloud computing, Future Generation Computer Systems, 108 (2020), 361-371.  doi: 10.1016/j.future.2020.02.018.  Google Scholar [5] S. A. Dudin and O. S. Dudina, Call center operation model as a MAP/PH/$N$/$R-N$ system with impatient customers, Problems of Information Transmission, 47 (2011), 364-377.  doi: 10.1134/S0032946011040053.  Google Scholar [6] O. Dudina and S. Dudin, Queueing system MAP/M/$N$/$N$+$K$ operating in random environment as a model of call center, in BWWQT, Minsk, Belarus, 2013, 83–92. doi: 10.1007/978-3-642-35980-4_10.  Google Scholar [7] R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, in MHS'95, Nagoya, Japan, 1995, 39–43. doi: 10.1109/MHS.1995.494215.  Google Scholar [8] A. H. Gandomi, X.-S. Yang, S. Talatahari and A. H. Alavi, Firefly algorithm with chaos, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 89-98.  doi: 10.1016/j.cnsns.2012.06.009.  Google Scholar [9] Q.-M. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.  Google Scholar [10] T. Hirai, H. Masuyama, S. Kasahara and Y. Takahashi, Performance optimization of parallel-distributed processing with checkpointing for cloud environment, Journal of Industrial and Management Optimization, 14 (2018), 1423–-1442. doi: 10.3934/jimo.2018014.  Google Scholar [11] X. Huang, D. Wu and N. Zhao, Study of performance measures and energy consumption for cloud computing centers based on queueing theory, Journal of Physics: Conference Series, 1631 (2020), 25-26.  doi: 10.1088/1742-6596/1631/1/012155.  Google Scholar [12] S. Jin, H. Wu and W. Yue, Pricing policy for a cloud registration service with a novel cloud architecture, Cluster Computing, 22 (2019), 271-283.  doi: 10.1007/s10586-018-2854-z.  Google Scholar [13] S. Jin, S. Hao, X. Qie and W. Yue, A virtual machine scheduling strategy with a speed switch and a multi-sleep mode in cloud data centers, Journal of Systems Science and Systems Engineering, 28 (2019), 194-210.  doi: 10.1007/s11518-018-5401-9.  Google Scholar [14] S. Jing, S. Ali, K. She and Y. Zhong, State-of-the-art research study for green cloud computing, The Journal of Supercomputing, 65 (2013), 445-468.  doi: 10.1007/s11227-011-0722-1.  Google Scholar [15] H. Khazaei, J. Mišić and V. B. Mišić, Performance analysis of cloud computing centers, in QShine, Houston, USA, 2010,251–264. doi: 10.1007/978-3-642-29222-4_18.  Google Scholar [16] Q.-L. Li and Y. Q. Zhao, A MAP/G/1 queue with negative customers, Queueing Systems, 47 (2004), 5-43.  doi: 10.1023/B:QUES.0000032798.65858.19.  Google Scholar [17] Y. Liu, L. Wang, X. Wang, X. Xu and P. Jiang, Cloud manufacturing: Key issues and future perspectives, International Journal of Computer Integrated Manufacturing, 32 (2019), 858-874.  doi: 10.1080/0951192X.2019.1639217.  Google Scholar [18] L. Luo, W. Wu and F. Zhang, Energy modeling based on cloud data center, Journal of Software, 25 (2014), 1371-1387.   Google Scholar [19] A. Manzoor, Cloud Security: Concepts, Methodologies, Tools, and Applications, IGI Global, Hershey, PA, 2019. Google Scholar [20] S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications, 27 (2016), 1053-1073.  doi: 10.1007/s00521-015-1920-1.  Google Scholar [21] B. K. Patle, A. Pandey, A. Jagadeesh and D. R. Parhi, Path planning in uncertain environment by using firefly algorithms, Defence Technology, 14 (2018), 691-701.  doi: 10.1016/j.dt.2018.06.004.  Google Scholar [22] T. Phung-Duc and K. Kawanishi, Multiserver retrial queue with setup time and its application to data centers, Journal of Industrial and Management Optimization, 15 (2019), 15-35.  doi: 10.3934/jimo.2018030.  Google Scholar [23] QYResearch, Global Cloud Accounting Software Market Size, Status and Forecast 2025, Technical Report, Albany, NY, 2018. Google Scholar [24] J. Shaler Stidham, Optimal Design of Queueing Systems, Chapman and Hall, New York, 2009. doi: 10.1201/9781420010008.  Google Scholar [25] G. Shao and J. Chen, A load balancing strategy based on data correlation in cloud computing, in UCC, Shanghai, China, 2016,364–368. doi: 10.1145/2996890.3007852.  Google Scholar [26] N. Sharma and R. Guddeti, Multi-objective energy efficient virtual machines allocation at the cloud data center, IEEE Transactions on Services Computing, 12 (2019), 158-171.  doi: 10.1109/TSC.2016.2596289.  Google Scholar [27] M. J. Usman, A. S. Ismail, G. Abdul-Salaam, H. Chizari, O. Kaiwartya, A. Y. Gital, M. Abdullahi, A. Aliyu and S. I. Dishing, Energy-efficient nature-inspired techniques in cloud computing datacenters, Telecommunication Systems, 71 (2019), 275-302.  doi: 10.1007/s11235-019-00549-9.  Google Scholar [28] J. Vilaplana, F. Solsona, I. Teixidó, J. Mateo, F. Abella and J. Rius, A queuing theory model for cloud computing, The Journal of Supercomputing, 69 (2014), 492-507.  doi: 10.1007/s11227-014-1177-y.  Google Scholar [29] G.-G. Wang, L. Guo, H. Duan and H. Wang, A new improved firefly algorithm for global numerical optimization, Journal of Computational & Theoretical Nanoscience, 11 (2014), 477-485.  doi: 10.1166/jctn.2014.3383.  Google Scholar [30] X. Wang, J. Zhu, S. Jin, W. Yue and Y. Takahashi, Performance evaluation and social optimization of an energy-saving virtual machine allocation scheme within a cloud environment, Journal of the Operations Research Society of China, 8 (2020), 561-580.  doi: 10.1007/s40305-019-00272-x.  Google Scholar [31] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479.  doi: 10.3934/jimo.2009.5.467.  Google Scholar [32] Z.-Q. Wu and X.-B. Zhao, Frequency ${H}_2/{H}_{ \infty}$ optimizing control for isolated microgrid based on IPSO algorithm, Journal of Industrial and Management Optimization, 14 (2018), 1565-1577.  doi: 10.3934/jimo.2018021.  Google Scholar [33] X. Yan, S. Jin, W. Yue and Y. Takahashi, A MAP-based performance analysis on an energy-saving mechanism in cloud computing, in QTNA, Ghent, Belgium, 2019,369–378. doi: 10.1007/978-3-030-27181-7_22.  Google Scholar [34] X.-S. Yang, Firefly algorithms for multimodal optimization, in SAGA, Sapporo, Japan, 2009,169–178. doi: 10.1007/978-3-642-04944-6_14.  Google Scholar [35] X.-S. Yang, Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.  doi: 10.1504/IJBIC.2010.032124.  Google Scholar [36] H. Yeganeh, A. Salahi and M. A. Pourmina, A novel cost optimization method for mobile cloud computing by capacity planning of green data center with dynamic pricing, Canadian Journal of Electrical and Computer Engineering, 42 (2019), 41-51.  doi: 10.1109/CJECE.2019.2890833.  Google Scholar [37] W. Zhao, X. Wang, S. Jin, W. Yue and Y. Takahashi, An energy efficient task scheduling strategy in a cloud computing system and its performance evaluation using a two-dimensional continuous time Markov chain model, Electronics, 8 (2019), 775-790.  doi: 10.3390/electronics8070775.  Google Scholar [38] Z. Zhou and Z. Zhou, A MAP/M/$N$ retrial queueing model with asynchronous single vacations, in ICVRIS, Changsha, China, 2018,245–249. doi: 10.1109/ICVRIS.2018.00067.  Google Scholar
State transition of a PM with the energy-saving mechanism
Change trend for the energy saving rate $\omega$ of the system
Change trend for the average latency $\sigma$ of tasks
Change trend for the cost function $\phi \left(\alpha \right)$
Numerical results for the optimization of the energy-saving mechanism
 Buffer size $K$ VM-number $N$ Optimal sleep parameter $\alpha^*$ Minimal system cost $\phi \left( \alpha^* \right)$ 6 0.2860 0.8414 24 7 0.4127 0.6951 8 0.5025 0.6046 6 0.2887 0.8920 27 7 0.4407 0.7224 8 0.5439 0.6203 6 0.2942 0.9401 30 7 0.4746 0.7462 8 0.5886 0.6330 6 0.3026 0.9857 33 7 0.5141 0.7666 8 0.6346 0.6432
 Buffer size $K$ VM-number $N$ Optimal sleep parameter $\alpha^*$ Minimal system cost $\phi \left( \alpha^* \right)$ 6 0.2860 0.8414 24 7 0.4127 0.6951 8 0.5025 0.6046 6 0.2887 0.8920 27 7 0.4407 0.7224 8 0.5439 0.6203 6 0.2942 0.9401 30 7 0.4746 0.7462 8 0.5886 0.6330 6 0.3026 0.9857 33 7 0.5141 0.7666 8 0.6346 0.6432
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