雲端服務在最近幾年中，已經成為在日常生活中隨處可見的服務方式。使用雲端服務可以帶來許多益處，包括易用性、線上處理資料的彈性、更簡易之管理，以及有成本效益的建置。以一個雲端計算中心來說，不一樣的伺服器可能有著不一樣之計算能力，也就是說，雲端計算中心可能具有異質之伺服器。儘管雲端計算帶來許多的益處，另一方面，管理一個雲端計算中心其最主要的問題在於：雲端計算消耗偌大的能量，不僅僅會導致高昂的運作成本，同時也是對地球不環保的表現。在這篇研究中，我們研究如何在一個有著異質伺服器的雲端計算中心，其中每一個伺服器可能有著不同之平均服務速率，能夠降低成本同時又能維持可接受的服務品質(也就是平均系統延遲)。為了要能反映真實的電力市場之情況，我們考慮時變電價。明確地說，我們假設有兩種狀態或區間，也就是高電價區間以及低電價區間。我們提出將考慮的系統塑模成一具有多伺服器門檻之MMPP/Mi/C/K模型。我們推導所考慮系統之解析模型。我們利用疊代演算法以求得穩態機率分布並計算感興趣之效能指標。感興趣的效能指標包括平均系統數目、平均佇列長度、平均系統延遲、平均佇列延遲、遺失機率、成功送達率、以及總成本。各種系統參數對於不同效能之影響也在隨後被研究，例如高電價平均抵達速率、低電價平均抵達速率、以及高至低轉換速率。我們呈現了對於三種有著不同高低伺服器門檻設定之方法之效能以供比較，其中第一種方法有著設定為低的較高門檻與設定為低的較低門檻、第二種(我們所提出)方法為有著設定為高的較高門檻與設定為低的較低門檻、以及第三種方法為有著設定為高的較高門檻與設定為高的較低門檻。我們發現，相對於第一種方法，我們所提出之方法能夠達成較少之總成本；相對於第三種方法，則能達成較好之平均系統延遲。最後，我們以C語言撰寫電腦模擬程式來驗證解析結果之準確性。

Cloud computing services have infiltrated into our daily life in recent years. Using cloud computing service brings several benefits, such as ease of use, flexibility in manipulating data online, simpler management and cost-efficient implementation. Within a cloud computing center, the computing power may differ from server to server, that is, the cloud computing centers may have heterogeneous servers. Despite the benefits of cloud computing, on the other hand, the main problem of managing a cloud computing center lies on one factor: it consumes significant amount of energy, and thus not only contributes to high operating cost but also is non-green to the earth. In this work, we study how to achieve cost reduction while maintaining an acceptable QoS (i.e., average system delay) in cloud computing center with heterogeneous servers, where each server may have a different average service rate. To reflect the real electricity market, time-varying electricity prices are considered. Specifically, we assume that there are two states or periods, i.e., high electricity price periods and low electricity price periods. We propose to model the considered system as an MMPP/Mi/C/K with server thresholds. The analytical models are derived for the system considered. We develop an iterative algorithm to find the steady state probability distribution and the performance measures of interest are computed. The performance measures of interest are average number in system, average number in queue, average system delay, average queueing delay, loss probability, throughput, and total cost. The effect of various system parameters on different performance measures are studied, e.g., the high electricity price arrival rate, the low electricity price arrival rate, and the high-to-low switching rate. For comparison, we present the performance of three considered schemes with different upper and/or lower thresholds, where the first scheme with a low upper threshold and a low lower threshold, the second (proposed) scheme with a high upper threshold and a low lower threshold, and the third scheme with a high upper threshold and a high lower threshold. It is shown that our proposed scheme outperforms the first scheme in terms of the total cost, and outperforms the third scheme in terms of the average system delay. Finally, the computer simulation is written in C to verify the accuracy of the analytical results.

[1] L. Wang, G. Laszewski, A. Younge, X. He, M. Kunze, J. Tao, and C. Fu, “Cloud Computing: a Perspective Study,” New Generation Computing,vol. 28, no. 2, pp. 137–146, Apr. 2010.
[2] A. Ghazizadeh, “Cloud Computing Benefits And Architecture In E-Learning,” 2012 IEEE Seventh International Conference on Wireless, Mobile and Ubiquitous Technology in Education (WMUTE), pp. 199-201, Mar. 27-30 2012.
[3] J. Gibson, R. Rondeau, D. Eveleigh, Q.Tan, “Benefits and Challenges of Three Cloud Computing Service Models,” 2012 Fourth International Conference on Computational Aspects of Social Networks (CASoN), pp. 199-205, Nov. 2012.
[4] H. Khazaei, J. Misic, V. Misic, “Modelling of Cloud Computing Centers Using M/G/m Queues,” 2011 31st International Conference on Distributed Computing Systems Workshops(ICDCSW), pp. 87-92, Jun. 20-24 2011.
[5] N. Gharbi, L. Charabi, and L. Mokdad, “Performance Evaluation of Heterogeneous Servers Allocation Disciplines in Networks with Retrials,” 2015 IEEE 17th International Conference on High Performance Computing and Communications (HPCC), 2015 IEEE 7th International Symposium on Cyberspace Safety, and Security (CSS), and 2015 IEEE 12th International Conf on Embedded Software and Systems (ICESS), pp. 904-907, Aug. 24-26, 2015.
[6] I. S. Moreno, J. Xu, “Customer-Aware Resource Overallocation to Improve Energy Efficiency in RealTime Cloud Computing Data Centers,” 2011 IEEE International Conference on Service-Oriented Computing and Applications (SOCA), Dec. 12-14 2011.
[7] C. Wen, X. Long, Y. Yang, F. Ni, Y. Mu, “System Power Model and Virtual Machine Power Metering for Cloud Computing Pricing,” 2013 Third International Conference on Intelligent System Design and Engineering Applications (ISDEA), pp. 1379-1382, Jan. 16-18 2013.
[8] M. Lin, A. Wierman, L. Andrew, E. Thereska, “Dynamic Right-Sizing for Power-Proportional Data Centers,” IEEE/ACM Transactions on Networking, vol. 21, issue 5, pp. 1378-1391, Nov. 22 2012.
[9] H. Dou, Y. Qi, P. Wang, “Hybrid Power Control and Electricity Cost Management for Distributed Internet Data Centers in Cloud Computing,” 2013 10th Web Information System and Application Conference (WISA), pp. 394-399, Nov. 10-15 2013.
[10] R. Hans, U. Lampe, R. Steinmetz, “QoS-Aware, Cost-Efficient Selection of Cloud Data Centers,” 2013 IEEE Sixth International Conference on Cloud Computing, pp946-947, June 28 - July 03, 2013.
[11] S. Subbiah, V. Perumal, “Power Aware Resource Optimization in Cloud,” 2013 International Conference on Recent Trends in Information Technology (ICRTIT), pp. 318-322, Jul. 25-27 2013.
[12] G. Faraci, G. Schembra, “An Analytical Model for Electricity-Price-Aware Resource Allocation in Virtualized Data Centers,” 2015 IEEE International Conference on Communications (ICC), pp. 5839-5845, Jun. 8-12 2015.
[13] X. Lu, F. Kong, J. Yin, X. Liu, H. Yu, and G. Fan, “Geographical Job Scheduling in Data Centers with Heterogeneous Demands and Servers,” 2015 IEEE 8th International Conference on Cloud Computing, pp. 413-420, June 27-July 2, 2015.
[14] F. S. Q. Alves, H. C. Yehia, L. A. C. Pedrosa, F. R. B. Cruz, and L. Kerbache, “Upper Bounds on Performance Measures of Heterogeneous M/M/c Queues,” Mathematical Problems in Engineering, vol. 2011, May 2011.
[15] W. Lin and P.R. Kumar, “Optimal Control of a Queueing System with Two Heterogeneous Servers,” IEEE Transactions on Automatic Control, vol. 29, pp. 696–703, Aug. 1984.
[16] X. Jin, G. Min, “Performance Analysis of A Hybrid Scheduling Scheme Under Heterogeneous MMPP and Self-Similar Traffic,” GLOBECOM '07. IEEE Global Telecommunications Conference, 2007, pp. 2714-2718, Nov. 26-30 2007.
[17] A.T. Andersen, B.F. Nielsen, “A Markovian Approach for Modeling Packet Traffic with Long-Range Dependence,” IEEE Journal on Selected Areas in Communications, vol. 16, issue 5, pp. 719-732, Jun. 1998.
[18] H. Heffes, D. Lucantoni, “A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer performance,” IEEE Journal on Selected Areas in Communications, vol. 4, issue 6, pp.856-868, Sep. 1986.
[19] S. Vakilinia, M. Cheriet, J. Rajkumar, “Dynamic Resource Allocation of Smart HomeWorkloads in the Cloud,” 2016 12th International Conference on Network and Service Management (CNSM), pp. 367-370, Oct. 31 – Nov. 4, 2016.
[20] C. Negru, V. Cristea, “Cost models – pillars for efficient cloud computing: position paper,” International Journal of Intelligent Systems Technologies and Applications, vol. 12, issue 1, pp. 28-38, Jul. 2013.
[21] Z. Rui, T. Bingyong, “The Pricing of Cloud Computing with Preferential Policies,” 2016 IEEE 13th International Conference on e-Business Engineering (ICEBE), pp. 232-237, Nov. 4-6 2016.