隨者資通產業的發展，和企業對於雲端計算近幾年來的投資，雲端計算有了迅速的進步，和廣泛的運用在各領域。明確地說，基礎設施即服務（IaaS）可以為用戶提供各種不同的硬體資源，如CPU核心，記憶體，硬碟空間。更明確地說，IaaS的雲端供應商以虛擬機器（VMs）的形式來提供資源，其中虛擬機器是部署在實體機器(PMs）裡頭。為了降低各種管理成本，且不違反服務品質（QoS）的要求，對於IaaS來說，一個有效的效能評估方法是必須的。如眾所皆知的，解析模型和電腦模擬是兩種有效的評估方法。在我們的研究中，我們專注於具有異質虛擬機器需求的雲端資料中心的效能評估。明確地說，每位用戶的虛擬機器請求數是介於一和最大虛擬機器需求之間。我們考量兩種情境：均勻和非均勻。對於均勻情境，每位用戶請求一與最大虛擬機器需求之間的任何數目的虛擬機器的機率是相等的。另一方面，對於非均勻情境，每位用戶請求一與最大虛擬機器需求的任何數目的虛擬機器的機率是不相等的。我們假定在占用任何虛擬機器之前，用戶必須先完成配置的階段。只有在佇列是空的，且用戶的虛擬機器需求數目不超過閒置虛擬機器數目之下，一個新抵達的用戶才可以直接進入配置伺服器。此外我們假設即使有用戶離開，未完成配置的用戶仍然無法占用任何虛擬機器。首先，我們推導釋放機率，其中考量到虛擬機器的占用。然後，我們推導所討論系統的解析模型。我們開發一種疊代演算法來得出穩態機率分佈和感興趣的效能評估。感興趣的效能評估包括立即服務機率、遺失機率、系統延遲、成功送達率和系統平均人數。我們研究各種系統參數對於各種效能指標的影響。最後但並非最不重要的，我們自行撰寫電腦模擬來驗證解析結果的準確性。

With the development of ICT industry, and the investment of cloud computing by enterprises in recent years, cloud computing has made rapid progress, and are widely used in various fields. Specifically, an Infrastructure as a service (IaaS) may provide different kinds of hardware resources to customers, e.g., CPU core, memory, disk space. More specifically, IaaS cloud providers offer customers with resources in the form of virtual machines (VMs), which are deployed on Physical Machines (PMs). In order to lower various management costs without violating Quality of Service (QoS) requirements, an effective performance evaluation scheme for IaaS is indispensable. As is well known, the analytical modeling and computer simulation are two effective assessment methods. In our work, we focus on the performance evaluation of cloud data centers with heterogeneous VM requirements. Specifically, the VM requirement of each user is between one and an upper bound. We consider two scenarios: uniform and non-uniform. For uniform scenarios, the probability that each user requests any number of VMs between one and maximum VM requirement is equal. On the other hand, for non-uniform scenarios, the probability that each user requests any number of VMs between one and maximum VM requirement is not equal. It is assumed that before occupying any VM, a user has to finish the provision stage first. A newly arrived user can enter the provision server directly only if the queue is empty and the VM requirement of the user is no greater than the number of idle VMs. Furthermore, it is assumed that a user under provision cannot occupy any VM even if there is a user departure. First, we derive the releasing probability by taking account of the VM occupancy. Then, we derive the analytical models for the system considered. An iterative algorithm is developed to find the steady state probability distribution and the performance measures of interest. The performance measures of interest include immediate service probability, loss probability, system delay, throughput, and average number in system. We study the effect of various system parameters on different performance measures. The performance comparison of uniform and non-uniform scenarios is also conducted. Last but not least, the computer simulation is written to verify the accuracy of the analytical results.

[1] Y. Amanatullah, C. Lim, H. P. Ipung, and A. Juliandri, “Toward cloud computing reference architecture: cloud service management perspective”, The International Conference on ICT for Smart Society (ICISS), Jakarta, pp. 1-4, June 2013.
[2] C. Reiss, A. Tumanov, G. Ganger, R. Katz, and M. Kozuch, “Heterogeneity and dynamicity of clouds at scale: google trace analysis,” in ACM Symp. Cloud Comp., pp. 1–13, 2012.
[3] D. Bruneo, F. Longo, and A. Puliafito, “A stochastic model to investigate data center performance and QoS in iaaS cloud computing systems,” IEEE Trans. Parallel Distrib. Syst., vol. 25, no. 3, pp. 560–569, 2014.
[4] R. Buyya, R. Ranjan, and R. Calheiros, “Modeling and simulation of scalable cloud computing environments and the cloudsim toolkit: challenges and opportunities,” Int. Conf. on High Performance Computing & Simulation, pp. 1-11, June 2009.
[5] L. Wei, B. He, and C. H. Foh, “Towards multi-resource physical machine provisioning for iaas clouds,” IEEE International Conference on Communications (ICC), pp. 3469–3472, 2014.
[6] H. Khazaei, J. V. Misic, V. B. Misic, and N. B. Mohammadi, “Modeling the performance of heterogeneous iaaS cloud centers,” IEEE ICDCS Workshops, pp. 232–237, 2013.
[7] H. Khazaei, J. Misic, and V. B. Misic, “Performance analysis of cloud centers under burst arrivals and total rejection policy,” IEEE GLOBECOM ’11, pp. 1-6, Dec 2011.
[8] R. Ghosh, K. S. Trivedi, V. K. Naik, and D. S. Kim, “End-to-End performability analysis for infrastructure-as-a-service cloud: an interacting stochastic models approach,” IEEE 16th Pacific Rim Int’l Symp. Dependable Computing (PRDC ’10), pp. 125-132, 2010.
[9] R. Ghosh, F. Longo, V. K. Naik, and K. S. Trivedi, “Modeling and performance analysis of large scale iaaS clouds,” Future Gener. Comput. Syst.,vol. 29, no. 5, pp. 1216–1234, Jul 2013.
[10] H. Khazaei, J. Misic, V. Misic, and S. Rashwand, “Analysis of a pool management scheme for cloud computing centers,” IEEE Trans. Parallel Distrib. Syst., vol. 24, no. 5, pp. 849–861, May 2012.
[11] X. Chang, R. Xia, J. K. Muppala, K. S. Trivedi, and J. Liu, “Effective modeling approach for iaaS data center performance analysis under heterogeneous workload,” IEEE Transactions on Cloud Computing, pp.1-1, April 2016.
[12] X. Chang, B. Wang, J. K. Muppala, and J. Q. Liu, “Modeling active virtual machines on iaaS clouds using an m/G/m/m+k queue, ” IEEE Transactions on Services Computing, vol. 9, no. 3, pp. 408–420, May 2016.
[13] B. Wang, X. Chang, and J. Liu, “Modeling heterogeneous virtual machines on iaas data centers,” IEEE Communications Letters, vol. 19, no. 4, pp. 537-540, April 2015.
[14] R. Birke, A. Podzimek, L. Y. Chen, and E. Smirni, “State-of-the-practice in data center virtualization: Toward a better understanding of VM usage,” IEEE International Conference on Dependable Systems and Networks (DSN), pp. 1–12, 2013.