# Elasticity-Aware Virtual Machine Placement in K-ary Cloud Data Centers

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3 have: at most, can support is: VMs. Therefore, the number of VMs that one PM Fig.3. Tree-based Topology According to Eqs. 3 and 4, the combinational elasticity is: Fig.4. One-layer cluster The combinational elasticity in Eq. 5 is our objective to maximize. Notably, maximizing the elasticity can be viewed as minimizing the utilization of the PM and link resources. So we transfer the objective functions Eqs. 3 and 4 into: Given K identical PMs, K min C} in Eq. 9 is the maximal number of VMs that a one-level cluster can support. Hence, given N VM requests, the number of VMs that can be accepted is: Elasticity Maximization in One-level Clusters According to Eqs. 6 and 7, the combinational utilization is: Again, maximizing the elasticity is equivalent to minimizing the utilization. In the next section, we focus on minimizing the utilization in Eq. 8. For simplicity, let each VM's machine resource demand as one VM slot [7]. That is, Let < N, B> denote that there are N VM requests, and each VM's bandwidth is B. III. A ONE-LAYER CLUSTER STUDY We start from a simple case of a one-layer cluster. As shown in Fig. 4, we have N VMs requests, each with a demand of B. Each of the K PMs has a machine capacity of C, and each of the K links' capacity is L. At first, we need to determine how many VM requests can be accepted into this one-layer cluster. First of all, we consider the machine resources of the cluster. For each PM, the sum of the VMs' machine resources should not exceed the capacity limit. Therefore, we have N C. Second of all, we will take care of the link capacity, which will play another role in limiting the total number of supported VMs. On the other hand, constrained by link capacity, each PM under that link can support VM slots. If L BC, one PM can support C VMs. However, if L BC, one PM can support, There are two main factors that determine the elasticity. The first one is the input size. With limited resources, large inputs will consume more resources, leading to worse elasticity. The second is the VM placement, which is our major concern. Given N VM requests, there are VMs accepted into the cluster (Eq. 10). Assume that the PM is allocated to VMs. Due to the symmetry of this K-ary tree, without of loss of generality, we assume that < <,...,<. Therefore, for the machine utilization, Eq. 6 can be transferred as. For the bandwidth utilization, Eq. 7 can also be transferred as: min{, N- }*. Then, considering the linear elasticity relationship between bandwidth and machine resource, the combinational utilization for PMs and links in Eq. 8 is: (11) Since we focus on worst-case, we therefore minimize the worst-case combinational utilization in Eq. 11. In that case, we only pay attention to the allotment of. We can obtain the optimal result to minimize Eq. 11, as shown in Appendix A. We also give the detailed analysis for binary data center in Appendix B. A. Discussion Suppose that c=1, we can see from the result in Appendix A, when BC L, the optimal solution for x is load-balancing. 24

5 PMs have the same machine capacity, therefore, for abstraction nodes with the same accumulative capacity, the inner structure of the original sub-trees are the same. We can see in Fig.5, a cluster consisting of four machines with two lower-layer links are abstracted into a single node of accumulative capacity of 16. Both the abstraction nodes with a capacity of 16 share the same structure of the original abstracted one-layer sub-tree. B. Hierarchical VM Placement for Multi-layer Clusters With the abstraction of a K layer multi-layer cluster into a one-layer cluster, we can use the optimal solution for the discussion in Section III. Based on that result, we propose our hierarchical VM placement algorithm for a multi-layer cluster. THEOREM Under the semi-homogeneous data center configuration, if BC, the proposed hierarchical VM placement algorithm is optimal to minimize the combinational utilization. The proof of optimality is shown in Appendix C. As a matter of fact, the semi-homogeneous configuration with BC is widely adopted in most data centers. Our algorithm can be divided into two steps. Firstly, for each switch from bottom to top, the accumulative capacity of the abstraction node rooted at that switch is calculated. Upon reaching the top-layer root switch, we obtain an abstracted one-layer cluster. Secondly, for the input, provided that they can be accepted, for each switch connecting K sub-trees at each layer from top to bottom, recursively allocate the input VMs into its K sub-trees according to our optimal result. Upon finishing the bottom-layer switch (access switch), all the VMs are allocated into the PMs, as shown in Fig. 5. We summarize our algorithm in Algorithm 1. Fig.6. The process of VM placement For each switch at each layer, our algorithm takes a constant time to calculate the optimal solution, For a K layers cluster, layer k has switches. Therefore, the total time of calculation is. Suppose we have M machines at the bottom, then M=. Our algorithm takes two loops, one is the abstraction from bottom to top, and the other is the placement from top to bottom. One loop takes a time of O (M). Two loops is still O (M). Therefore, the total time complexity of our algorithm is O (M), which is very efficient. Fig.7. Conservative schedulability C. Discussion In a link hungry condition, the abstraction process from bottom to top can be conservative. Therefore, the accumulative capacity of the abstraction node can be smaller than the actual capacity. This will lead to some situations where our algorithm cannot schedule a VM request that should be able to be scheduled. Since our abstraction process is layer-by-layer from bottom to top, each step can be conservative calculated. The boundary situation is that: 26

7 Economic interests are one popular topic that shows up in research literature [12, 13]. They tried to find an optimal VM placement that could either maximize the revenue for the cloud provider, or minimize the costs for the customers. In [13], for maximizing the total economic benefit of the cloud provider, the author introduced an SLA-based dynamic resource allocation. The pricing mechanisms are related to the performance of QoS that the cloud provider could guarantee. The better performance the cloud provider could offer, the more revenue the cloud provider could obtain. Since different service requests have different pricing, higher pricing service could get more resource provisioning from the cloud provider. The author also used a convex optimization to present the optimal resource allocation. On the contrary, the author in [12] proposed an optimal VM placement with the objective of minimizing the total renting of the users. In this paper, the author gave another pricing mechanis m, referring to two payment plans: reservation plan and on-demand plan. Since the total resource demand is uncertain and reservation plan has to be decided in advance, it may not meet the future demand of the user. For that reason, the author used the optimal solution of stochastic integer programming to give an optimal VM placement that could minimize the total renting. However, the VM placement problem in [12] and [13] could achieve an optimal solution, which is not a common case. Many VM placement issues are NP-hard, thus we need to find a good heuristic algorithm to solve the problem, such as the first-fit and best-fit greedy algorithm used in [14]. Besides the above VM placement aiming to get an optimal economic interest, network is one important concern in the VM placement problem. The author in proposed a network-aware VM placement. In [15], when performing VM placement, not only the physical resources (like CPU and memory) are considered, but also the traffic demand between different VMs was taken into account. The author gave a heuristic algorithm to allocate placement to satisfy both the communication demands and physical resource restrictions. In [16], the author proposes minimizing the traffic cost through VM placement. Their objective is to place VMs that have large communication requirements close to each other, so as to reduce network capacity needs in the data center. Oktopus [7] uses the hose model to abstract the tenant's bandwidth request, including both virtual cluster and oversubscribed virtual clusters. They propose a VM allocation algorithm to deal with homogeneous bandwidth demands, which is also what we are using in this paper. The virtual cluster provides tenants with guarantees on the network bandwidth they demand, which, according to [17], can be interpreted as the min -guarantee requirements. However, this min-guarantee fails to consider the potential growth of a tenant's network demand. In that case, the virtual cluster allocation based on this min -guarantee policy will not have enough resources to accommodate tenants' future growth demands, which can lead to the loss of customers. In order to alleviate this problem, in our previous paper [18], we propose the concept of elasticity, which considers existing customers' potential growing bandwidth demand in the future. However, that paper only considers the binary tree data center topology. To obtain a more general result, we study the K-ary tree topology data center in this paper. VIII. CONCLUSIONS In this paper, we study the resource management problem on the cloud data center, which is now suffering from both limited resources and the variety of users' requests. Compared to the previous work, we focus on guaranteeing the on-demand scaling of cloud computing, and we propose the concept of elasticity of the VM requests. To maximize the elasticity of the 28

9 another allocation for the four machines of [a',b',c',d'], that is better than our algorithm in minimizing the combinational utilization in Eq. 30. We assume a' b' c' d'. Due to the symmetry of binary-tree, we could swap each PM's allotment, so as to make [a',b',c',d'] corresponds to [a,b,c,d]. Based on our algorithm, for the first allocation step, we allocate the N requests into the two abstraction nodes of representing the two sub-trees of the root switch. By Eq. 14, the accumulative capacity of each abstraction node is = min {2, 2C} =2C. For the one-layer cluster, our algorithm optimally allocates the N VMs based on the results of Eq. 13. Since a b c d and a' b' c' d', therefore, for the first step, we have: (22) Then, after finishing the allocation for the top layer, according to our algorithm, we are about to do the allocation for the two switches in the second layer. The combinational utilization for the second-layer switch is max {,, }. Assuming our algorithm is not optimal, we have: (23) Since C, the result in Eq. 21 will evenly divide the input of the second-layer switch. Therefore, we have a=b, c=d. Then, min{a+b+c,d}=d, and. Therefore, we have: Since d=c, then, = Combing Eq. 24, we have: (24) (25) Since a' b' c' d', therefore, d' c'. Then,. Therefore, we have: Hence, combing Eqs. 25, 26 (26) generalization of optimality for ones-step. Since all PMs' capacities are the same, and the capacity of the links at the same layer are the same, from the view of physical meaning, our abstraction process misses no information of the lower layer resources for both links and machines. Hence, we conclude that the optimality is secured for one-step generalization. Based on this observation, we can use induction method to prove the optimality of our algorithm in a k layer cluster. When k=1, it is a one-layer cluster, and the optimal solution is given. Suppose that, for the layer, our algorithm is optimal, then, the optimality from the to layer is also secured. Therefore, our algorithm is optimal. APPENDIX D BINARY HETEROGENEOUS CLUSTER ANALYSIS Here, we consider the scenario that c=1. Similarly, we firstly study a one-layer cluster under heterogeneous configuration. However, the calculation of accumulative capacity is different, as shown in Fig. 8. With an input < N, B>, considering the total physical resources of the cluster, we have: N +. Secondly, with the link capacity constraint, we have: N, }+ }. Considering both link and physical capacity limitation, we have: (28) This is the maximal number of VMs that this heterogeneous one-layer cluster can support. We still use as the number of VMs that can be accepted into this cluster, as follows: (29) We try to obtain an optimal result for the one-layer cluster. With accepted into this cluster, we allocate x VMs on the left node, leaving x on the right node. In a heterogeneous scenario, the configuration of a binary cluster is asymmetric; hence, we need to consider all 1 different ways to allocate the VMs. We will discuss the problem separately in the interval of x and x, and then lead to the final result. For x the utilization of the links is max{x, x }, (27) A contradiction with the assumption in Eq. 23, therefore, the proof is complete for C. B. Generalization of Optimality From one-layer to two-layer, the potential factor that can change the optimality of this greedy step is the change of resource capacity, since we use the accumulative capacity to complete the first step. In other words, = can be smaller than max {, }. However, according to the allocation scheme of our algorithm, we have proven that max {, } is equal to, which guarantees the and the utilization of physical resources is max{x, ( -x) }. Then, the combinational utilization of the cluster is: (30) The domain of x is: (31) Mathematically, there is a minimal point for U(x), when x is: (32) 30

10 Obviously, is within the domain Eq. 31. Still, the value of from Eq. 32 may not be an integer. We choose both values of and for future comparison with another interval. Similarly, for x, the objective function for the utilization of the cluster is: To minimize U(x) in Eq. 33, the domain of x is: There is also a minimal point, which is: (34) (35) We will choose the values of,, and for comparison. Therefore, the optimal result of allocation for a one-layer heterogeneous cluster is shown below: REFERENCES [1]. S. Kandula, S. Sengupta, A. Greenberg, P. Patel, and R. Chaiken, The nature of data center traffic: measurements & analysis, in Proceedings of the 9th ACM SIGCOMM conference on Internet measurement conference, IMC 09, (New York, NY, USA), pp ,ACM, [2]. D. Kusic, J. Kephart, J. Hanson, N. Kandasamy, and G. Jiang, Power and performance management of virtualized computing environments via lookahead control, in Proceedings of ICAC 2008, pp. 3 12, june [3]. J. E. Bin, O. Biran, O. Boni, E. Hadad, E. Kolodner, Y. Moatti, and D. Lorenz, Guaranteeing high availability goals for virtual machine placement, in Proceedings of ICDCS 2011, pp , june [4]. E. P. W J. T. Piao and J. Yan, A network-aware virtual machine placement and migration approach in cloud computing, in Proceedings of GCC 2010, pp , nov [5]. K. Li, H. Zheng, J. Wu, Migration-based Virtual Machine Placement in Cloud Systems in Proc. of IEEE CloudNet [6]. J. Zhu, D. Li, J. Wu, H. Liu, Y. Zhang, and J. Zhang, Towards bandwidth guarantee in multi-tenancy cloud computing networks, in Proc. of ICNP 2012, pp [7]. H. Ballani, P. Costa, T. Karagiannis, and A. Rowstron, Towards predictable datacenter networks, in Proc. of the ACM SIGCOMM 2011, pp [8]. J. R. Lange and P. Dinda, Symcall: symbiotic virtualization through vmm-to-guest upcalls, in Proceedings of the 7th ACM SIGPLAN/ SIGOPS international conference on Virtual execution environments, VEE 11, (New York, NY, USA), pp , ACM, [9]. S. T. Jones, A. C. Arpaci-Dusseau, and R. H. Arpaci-Dusseau, Vmmbased hidden process detection and identification using lycosid, in Proceedings of the fourth ACM SIGPLAN/SIGOPS international conference on Virtual execution environments, VEE 08, (New York, NY, USA), pp , ACM, [10]. M. Xu, X. Jiang, R. Sandhu, and X. Zhang, Towards a vmm-based usage control framework for os kernel integrity protection, in Proceedings of the 12th ACM symposium on Access control models and technologies,sacmat 07, (New York, NY, USA), pp , ACM, [11]. A. Ranadive, A. Gavrilovska, and K. Schwan, Ibmon: monitoring vmm-bypass capable infiniband devices using memory introspection, in Proceedings of the 3rd ACM Workshop on System-level Virtualization for High Performance Computing, HPCVirt 09, (New York, NY, USA), pp , ACM, [12]. S. Chaisiri, B.-S. Lee, and D. Niyato, Optimal virtual machine placement across multiple cloud providers, in Proceedings of IEEE Asia-Pacific Services Computing Conference, 2009, pp , dec [13]. G. Feng, S. Garg, R. Buyya, and W. Li, Revenue maximization using adaptive resource provisioning in cloud computing environments, in Proceedings of Grid Computing 2012, pp , sept [14]. J. Xu and J. Fortes, Multi-objective virtual machine placement in virtualized data center environments, in Proceedings of CPSCom 2010, pp , dec [15]. J. T. Piao and J. Yan, A network-aware virtual machine placement and migration approach in cloud computing, in Proceedings of GCC 2010, pp , nov [16]. X. Meng, V. Pappas, and L. Zhang, Improving the scalability of data center networks with traffic-aware virtual machine placement, in Proc. of IEEE INFOCOM 2010, pp. 1 9, [17]. L. Popa, A. Krishnamurthy, S. Ratnasamy, and I. Stoica, Faircloud: sharing the network in cloud computing, in Proc. of ACM HotNets-X 2011, pp. 22:1 22:6. [18]. K. Li, J. Wu, A. Blaisse, Elasticity-aware Virtual Machine Placement for Cloud Datacenters in Proc. of IEEE CloudNet

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