Energy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and migration algorithms

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1 Energy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and igration algoriths Chaia Ghribi, Makhlouf Hadji and Djaal Zeghlache Institut Mines-Téléco, Téléco SudParis UMR CNRS , Rue Charles Fourier 91011, Evry, France Eail: {chaia.ghribi, akhlouf.hadji, Abstract This paper presents two exact algoriths for energy efficient scheduling of virtual achines (VMs) in cloud data centers. Modeling of energy aware allocation and consolidation to iniize overall energy consuption leads us to the cobination of an optial allocation algorith with a consolidation algorith relying on igration of VMs at service departures. The optial allocation algorith is solved as a bin packing proble with a iniu power consuption objective. It is copared with an energy aware best fit algorith. The exact igration algorith results fro a linear and integer forulation of VM igration to adapt placeent when resources are released. The proposed igration is general and goes beyond the current state of the art by iniizing both the nuber of igrations needed for consolidation and energy consuption in a single algorith with a set of valid inequalities and conditions. Experiental results show the benefits of cobining the allocation and igration algoriths and deonstrate their ability to achieve significant energy savings while aintaining feasible convergence ties when copared with the best fit heuristic. Index Ters Energy efficiency, VM placeent, VM igration, Cloud data center, Linear integer prograing. I. INTRODUCTION Energy efficiency is becoing increasingly iportant for data centers and clouds. The wider adoption of cloud coputing and virtualization technologies has led to cluster sizes ranging fro hundreds to thousands of nodes for ini and large data centers respectively. This evolution induces a treendous rise of electricity consuption, escalating data center ownership costs and increasing carbon footprints. For these reasons, data centers now ebed onitoring capabilities and probes such as sart power distribution units (PDUs) to achieve energy efficiency and reduce overall cost. The fact that electricity consuption is set to rise 76% fro 2007 to 2030 [1] with data centers contributing an iportant portion of this increase ephasizes the iportance of reducing energy consuption in clouds. According to the Gartner Report [2], the average data center is estiated to consue as uch energy as households, and according to McKinsey report [3], The total estiated energy bill for data centers in 2010 is 11.5 billion and energy costs in a typical data center double every five years. Reducing the overall energy bill requires first and foreost energy and ecology conscious designs of data centers hardware and software, the use of solar and photovoltaic energy, relying on renewable energy sources and efficient cooling systes. The second and apparently ore odest contribution to energy efficient clouds is to introduce energy aware scheduling and placeent algoriths and enhanced resource anageent. In fact, cloud data centers are electricity guzzlers especially if resources are peranently switched on even if they are not used. An idle server consues about 70% of its peak power [4]. This waste of idle power is considered as a ajor cause of energy inefficiency. This paper is a contribution to the reduction of such excessive energy consuption using energy aware allocation and igration algoriths to have a axiu nuber of idle servers to put into sleep ode. Intel s Cloud Coputing 2015 Vision [5] stresses also the need for such dynaic resource scheduling approaches to iprove power efficiency of data centers by shutting down and putting to sleep idles servers. Our work proposes an exact energy aware allocation algorith using the classical forulation of the Bin-Packing proble. The ai of this algorith is to reduce the nuber of used servers or equivalently axiize the nuber of idle servers to put in sleep ode. To take into account workloads and service ties a linear integer prograing algorith is used to optiize constantly the nuber of used servers after service departures. This igration algorith is cobined with the exact allocation algorith to reduce overall energy consuption in the data centers. The proposed algoriths act as an energy consuption aware VM scheduler and can be used to enhance current infrastructure anagers and schedulers such as OpenNebula [6] and OpenStack [7]. The power consuption indicators can be provided by energy consuption estiation tools such as jouleeter [8]. A dedicated siulator is used to assess perforance and crosscheck with the perforance results produced by the exact algoriths. Evaluation results show that the exact allocation algorith cobined with igration reduces considerably the nuber of required servers to serve a given load and can thus iniize power consuption in data centers. Section II of this paper describes a related work addressing consolidation using different igration techniques. The syste odel and the derived algoriths are addressed in Section III. Perforance evaluation is the object of section IV. II. RELATED WORK Authors in [9] consolidate applications or tasks on reduced nuber of physical achines to switch off achines in

2 2 surplus. This approach differs fro ours because consolidation is achieved at the task level, rather than the virtual achine (VM) level (or IaaS level)in our case, and hence fits better the Platfor or Software as a Service (PaaS, SaaS) levels. Allocation or placeent is also static as opposed to our dynaic placeent according to workload where we apply igration to allocate and reallocate VMs. Reference [10] addresses policies for dynaic VMs reallocation using VMs igration according to CPU perforance requireents. Their ost effective policy, a double threshold policy, is based on the idea of setting upper and lower utilization thresholds for hosts and keeping the total utilization of the CPU of all the VMs between these thresholds. If the CPU utilization of a host exceeds the upper threshold, soe VMs are igrated and if it falls below the lower threshold, all the hosted VMs should be igrated. One of the closest work to our paper is found in [11]. Authors treat the proble of consolidating VMs in a server by igrating VMs with steady and stable capacity needs. They proposed an exact forulation based on a linear progra described by a too sall nuber of valid inequalities. Indeed, this description does not allow solving, in reasonable tie and in an optial way, probles involving allocation of a large nuber of ites (or VMs) to any bins (or Servers). In order to scale and find solutions for large sizes, the authors resorted to a heuristic using a static and a dynaic consolidation of VMs to reduce energy consuption of the hosting nodes or servers. In [12], authors presented a server consolidation (Sercon) algorith which consists of iniizing the nuber of used nodes in a data center and iniizing the nuber of igrations at the sae tie. They copared their algorith with the well known placeent heuristic FFD (First-Fit Decreasing) [13], to solve the Bin-Packing proble, and have shown the efficiency of Sercon to consolidate VMs and iniize igrations. However, Sercon is a heuristic that can not always reach or find the optial solution. Our proposed algorith is based on an exact forulation of the consolidation proble with igrations. We anage to optially consolidate VMs in servers while iniizing the energy cost of igrations. In [14], authors presented an approach EnaCloud for dynaic live placeent taking into account energy efficiency in a cloud platfor. They proposed an energy-aware heuristic algorith in order to save energy by iniizing the nuber of running servers. Another study relying on dynaic resource allocation is presented in [15]. The authors presented a natureinspired VM consolidation algorith inspired fro an Ant Colony Optiization. This algorith ais at reducing the nuber of used physical achines and thus saves energy. III. THE SYSTEM MODEL The odel considers infrastructure providers allocating physical resources instances to host users and tenants applications or equivalently for this paper VMs. The physical resources are seen as servers. It is assued that applications are packaged into virtual achines to be hosted by the infrastructure providers. The cloud providers save energy and reduce power consuption by packing and consolidating through igration of VMS to axiize the nuber of idle servers to put to sleep ode. Figure 1 depicts the syste odel coposed of the proposed energy efficient allocation and igration algoriths contributing to scheduling, an energy consuption estiator and cloud anagers that handle infrastructure resource instantiation and anageent. Each odule is briefly described to set the stage for the analytical odeling of the energy efficient resource allocation proble in clouds. Cloud IaaS anager (e.g. OpenStack [7], OpenNebula [6] and Eucalyptus [16]) control and anage cloud resources and handle clients requests, VM scheduling and fetch and store iages in storage spaces. Energy estiation odule is an interediate odule between the cloud infrastructure anager and the energyaware scheduler. The odule can rely for exaple on an energy estiation tool as in Jouleeter [8] that uses power odels to infer power consuption of VMs or servers fro resource usage. Energy-aware VM scheduler responsible for the energy aware VM placeent in the data center is the focus of our energy consuption optiization odel. This green scheduler is basically coposed of two odules. An allocation odule and a igration odule. The role of the allocation odule is to perfor the initial VM placeent using our exact VM allocation algorith. The dynaic consolidation of virtual achines is handled by the igration odule that iniizes the nuber of used or activated servers thanks to our exact VM igration algorith. The unused servers are shut down or put into sleep ode. All the needed inforation (servers and VMs) to run the algoriths are retrieved via the Cloud IaaS anager that is also used to execute the VM deployent and igration actions. To derive the syste odel, we consider the size n of client requests in ters of the nuber of required VMs and the types of desired VM instances (e.g., sall, ediu, large). Each V M i is characterized by a lifetie t i and a axiu power consuption p i. Each server or hosting node j, fro the data center, has a power consuption liit or power cap noted. This can be fixed by cloud adinistrators. We assue that all servers are hoogeneous; extending the odel to heterogeneous servers is trivial but will increase coplexity and will not necessarily provide additional insight. The approach adopted to achieve energy efficiency in our proposal is to use a bin packing algorith for optial placeent of user requests and to follow with dynaic consolidation once a sufficient nuber of departures have occurred. The dynaic consolidation is handled by the igration algorith which regroups VMs to free as any servers as possible to put the into sleep ode or to shut the down. A. Exact VM Allocation Algorith The proposed exact VM allocation algorith is an extended Bin-Packing approach through the inclusion of valid conditions expressed in the for of constraints or inequalities. The objective is to pack ites (VMs in our case) into a set of bins (servers or nodes hosting the VMs) characterized by their power consuptions. In addition to n, the nuber

3 3 Fig. 1. The syste odel of requested VMs, We define the nuber of servers,, available in the data center. The servers are assued to have the sae power consuption liit:, {j = 1, 2,..., }. At run-tie, each server j hosting a nuber of VMs is characterized by its current power consuption: P j,current. Since the objective is to iniize the energy consuption of the data centers, we define as key decision variable e j for each server j that is set to 1 if server j is selected to host VMs, 0 if it is not selected. In addition, we define the bivalent variable x ij to indicate that V M i has been placed in server j and set x ij to 1; x ij = 0 otherwise. The objective function to place all the deands (or VMs) in a iniu nuber of servers can be expressed using: in Z = e j (1) This optiization is subject to a nuber of linear constraints reflecting the capacity liits of the servers and obvious facts such as a VM can only be assigned to one server or a server can only host VMs according to the aount of reaining resources: 1) Each server has a power liit that cannot be exceeded when serving or hosting VMs and this occurs according to reaining capacity: n p i x ij e j P j,current, j = 1,..., (2) 2) A cloud provider has to fulfil all requests within a prescribed SLA or quota and each requested VM will be assigned to one and only one server: x ij = 1, i = 1,..., n (3) 3) For servers verifying the condition > P j,current and P j,current 0, the total nuber of used servers is lower bounded by Pj,current. This adds he following inequality to the odel: e j P j,current (4) The exact and extended Bin-Packing VM allocation odel can be suarized by luping the objective function with all the constraints and conditions and constraints into the following set of equations: in Z = e j (5) Subject To: n p i x ij e j P j,current, j = 1,..., (6) x ij = 1, i = 1,..., n (7) e j P j,current { 1, if the server j is used; e j = 0, otherwise. { 1, if the V Mi is placed in server j; x ij = 0, otherwise. (8) (9) (10) All the variables and constants used in the odel are listed for easy reference below: n is the size of the request in nuber of requested VMs. is the nuber of servers in the data center. p i represents the power consuption of V M i. x ij is a bivalent variable indicating that V M i is assigned to a server j. e j is a variable used to indicate whether the server j is used or not. represents the axiu power consuption of server j. P j,current represents the current power consuption of server j (P j,current = P j,idle + k p k with VM k hosted by server j). P j,idle represents the power consuption of server j when it is idle. Constraints in server CPU, eory and storage are also added to the odel to confine even further the odel convex hull: n cpu i x ij CP U j e j (11) where cpu i is the requested CPU by V M i. CP U j is the CPU capacity of server j. e i x ij MEM j e j (12) where e i is the requested eory by V M i and MEM j is the eory capacity of server j. sto i x ij ST O j e j (13) where sto i is the requested storage by V M i and ST O j is the storage capacity of server j.

4 4 In this paper we assue that these constraints are et and verified and we hence only need to focus on the energy efficiency constraints through (2). B. Best-Fit forulation and heuristic algorith The exact and extended Bin-Packing is copared to a Best- Fit heuristic adaptation of the well known Best-Fit algorith [13]. The heuristic proposed to achieve energy efficient VM placeent consists of two steps: 1) sorting the requested VMs in decreasing order of power consuption. This builds soehow an ordered stack that is used in the second step for packing VMs in available servers; 2) The sorted VMs are handled starting fro the top of the stack and attepting to place the ost power consuing VMs in the server with the sallest reaining power consuption budget until a VM down the stack fits in this target server. The process repeats or continues until all VMs in the stack are placed and packed as uch as possible in the ost occupied servers. This will tend to free servers for sleep ode or switching off. As this Best-Fit heuristic algorith tries to approxiate the Bin-Packing algorith, it is selected for coparison with our exact VM allocation proposal. The allocation algoriths are cobined with a igration algorith to iniize overall data center power consuption. In our case, the objective is to benchark the exact VM allocation and igration algoriths with at least one well known heuristic algorith. The Best- Fit heuristic was selected since it is known to achieve good suboptial perforance copared with classical Bin-Packing. C. Exact VMs Migration The placed and running VMs in the servers will gradually leave the syste as their related jobs end. These departures are the opportunity to re-optiize the placeent by igrating VMs always in the syste for consolidation in a iniu nuber of fully packed severs. A igration algorith based on an integer linear progra (ILP) is presented to achieve the consolidation. This ILP algorith consists in introducing a nuber of valid inequalities to reduce the span of the convex hull of the igration proble. The atheatical odel for the VM consolidation via igration relies on a linear integer prograing forulation. The objective for the algorith is to igrate VMs fro nodes selected as source nodes (those the algorith ais at eptying so they can be turned off) to other selected destination nodes (those the algorith ais at filling so they serve a axiu nuber of VMs within their capacity liits). Ideally, the algorith should iniize the nuber of active nodes, axiize the overall nuber of VMs handled by the active nodes and hence axiize the nuber of unused, epty or idle nodes. The algorith should also iniize the power consuption caused by igrations. If the power consuption or cost of VM igration is unifor or hoogeneous across hosting nodes or servers, the objective reduces to iniizing the nuber of igrations. The igration concerns the set of non idle servers, <, whose power consuptions are lower than with j in. Despite the slight reduction in size <, the proble reains NP-hard. Hence, we resort to an exact algorith based on linear integer prograing to address optial igration for practical proble sizes or nuber of instances. Fig. 2. Exaple of VMs igration The objective function for the optial VM igration and consolidation can be expressed as the axiization of the nuber of idle servers in the infrastructure: q i ax M = P i,idle y i p kz ijk (14) k=1 where y i = 1 is used to indicate that server i is idle and y i = 0 eans that at least one VM is active in server i. P i,idle is the power consued by idle servers, p k is the cost in ters of consued power when igrating V M k. A fixed power consuption p k is assigned to each VM instance according to instance type to differentiate between sall, ediu and large VMs. Variable z ijk is the bivalent variable expressing igration of V M k fro server i to server j. Variable q i is the total nuber of VMs hosted on server i and that are candidate for igration into destination servers, especially server j in equation (14). The objective function (14) is subject to the igration constraints cited earlier. These conditions are forally expressed through valid inequalities and constraints that have to be respected when iniizing overall energy consuption. 1) When igrating V M k fro a server i to a server j (see figure 3), the algorith ust prevent backward igrations and can only igrate into one specific destination node. Stated in an equivalent way: if a V M k is igrated fro a server i (source) to a server j (destination), it can not be igrated to any other server l (l j). The proposed inequality (15) also ensures that VMs in destination node and VMs igrated to destination nodes are not igrated as we are aiing at filling these nodes instead of eptying the obviously. This is reflected by the inequality : z ijk + z jlk 1; (15) 2) To strengthen further the previous condition, a valid inequality is added to ensure that when a V M k is igrated fro server i to server j (see figure 4), igration to other nodes l (l j) are prevented or forbidden:,j i z ijk 1; (16)

5 5 Fig. 3. A server candidate to a igration should not igrate his own VMs z ijk + z jlk 1 (22) i = 1,...,, j = 1,...,, k = 1,..., q i, k = 1,..., q j, j i, and l = 1,...,, l j, k k.,j i z ijk 1 (23) i = 1,...,, j = 1,...,, k = 1,..., q i, l = 1,...,, l j. q i p k z ijk ( P j,current ) (1 y j ) (24) k=1 j = 1,...,, j i Fig. 4. A V M k can not be igrated to any servers in the sae tie 3) A server j is liited by its power consuption liit. The inequality (17) allows each server j to host VMs without exceeding its power liit: q i p k z ijk ( P j,current ) (1 y j ) (17) k=1 Where P j,current is the current power consuption of server j. 4) If a non-idle server i is a source of VM igration, then it should igrate all of its hosted VMs in order to be put to sleep ode or shut down once copletely eptied: q i z ijk = q i y i, i = 1,...,, j i (18) k=1 5) Another valid inequality is the upper bound in the total nuber of epty servers: y i P j,current (19) 6) Another iportant aspect is to avoid igration of VMs whose lifetie or leftover lifetie t k is shorter than the tie needed to ake igration decisions T 0 : z ijk t k T 0, (20) where t k = t k CurrentT ie, where CurrentT ie represents current or VM igration handling tie. The optial VM consolidation and igration odel and objective function (14) can be suarized for convenience with all the valid conditions as: Subject To: q i ax M = P i,idle y i p kz ijk (21) k=1 q i z ijk = q i y i, i = 1,...,, j i (25) k=1 y i P j,current (26) z ijk t k T 0, (27) { 1, if the V Mk is igrated fro a server i to a server j; z ijk = 0, otherwise. (28) { 1, if the Server i is idle; y i = (29) 0, otherwise. D. Cobination of allocation and igration algoriths Figure 5 suarizes how the allocation and igration algoriths are cobined to achieve inial energy consuption in infrastructure nodes and hence data centers. Both the exact Bin-Packing extension and the Best-Fit heuristic are used to ensure optial and suboptial placeent respectively. Recall that the two algoriths allow us to cross check their relative perforance and benchark the odified Best-Fit with the proposed exact allocation solution since the algoriths exhibit different optiality and convergence tie characteristics. Upon arrival of VM placeent and resource requests these two algoriths select the ost appropriate nodes to host new VMs in available and active nodes. Whenever necessary these algoriths ay resort to turning new nodes on, when the set of active nodes are full and cannot host the new arriving VM instances (sall, ediu or large). Both algoriths will attept serving the requests in the currently active nodes and will of course typically avoid turning any new nodes on. The algoriths are cobined with the igration algorith that is launched if a nuber of VM jobs terinate since their dedicated resources becoe available for opportunistic reuse and for ore efficient resource allocation and distribution. These departures are the opportunity for the consolidation algorith to rearrange allocations by oving VMs into the

6 6 Fig. 5. Cobination of the igration algorith with the two allocation algoriths SP ECcpu2006 [20] workloads (471.onetpp, 470.lb and 445.gobk) were evaluated. Estiated power consuptions were found to be between 25 and 28 watts for these eleents. Without loss of generality and to ease intuitive verification of the results, we refer to these published consuption to associate to each VM type (sall, ediu and large) an energy consuption p i respectively equal to 10 watts (low), 20 watts (ediu) and 30 watts (high) to stay in line with published values. The requests for resources to serve VMs have a constant arrival rate. The requested VM instance types are discrete unifor in [1, 3] (1 for sall, 2 for ediu and 3 for large instances). The VM sizes are arbitrarily drawn as unifor in [1, 3] and classified according to their type. We retained only the rando drawings that fulfill the VM characteristics in size and type. sallest possible set of nodes. All eptied or freed servers (or nodes) are turned off to iniize energy consuption. The consolidation is achieved by the exact igration algorith that oves VMs fro selected source nodes to selected destination nodes. The end result is the activation and use of the sallest set of nodes in the data centers. IV. NUMERICAL RESULTS Our proposed algoriths are evaluated through a Java language ipleentation and the linear solver CPLEX [17]. A dedicated siulator is developed to conduct the perforance assessents and the coparison. The objective of the nuerical evaluation is to quantify the percentage of energy savings or power consuption savings that can be expected when cobining the exact allocation algorith and the consolidation process using our proposed exact igration algorith. The answers provided by the nuerical analysis concern also the scalability and coplexity of the proposed algoriths in the size of the data centers and the arrival rate of requests for resources to host VMs which is also synonyous to load on the syste. Note, however, that the siulation are conducted for an arrival rate strictly lower than the rate of VM job departures fro the syste; thus siulations correspond to cases where the likelihood of finding an optial or a good solution is high. The assessent scenarios correspond to data centers with 100 servers or nodes for the first five experients. In the last two experients 200 servers are considered. We collect essentially as perforance indicators, the percentage of used servers (which autoatically provides the energy consued or saved by the algoriths) and the tie required for the algoriths to find their best solutions (optial for the exact algoriths). All the servers have a power consuption cap set to 200 watts (the peak power of a typical server is around 250 watts [18]). To perfor per-vm power estiation we referred to a power estiation odel proposed in [19]. Three SP ECcpu2006 [20] workloads (454.calculix, 482.sphinx and 435.groacs) with high, ediu and low power consuption were considered. Their associated power consuption is close to 13 watts, 11 watts and 10 watts respectively. The power estiation odel proposed in [8] provided additional insight. The power consuption of other Fig. 6. Coparison between the exact and heuristic allocation algoriths Figure 6 depicts results of a coparison between the adapted Best-Fit heuristic and our exact extended Bin- Packing allocation algoriths. The siulations correspond to 100 servers and resource requests in nuber of VMs in the [1, 200] range. The lifetie of the VMs are unifor in [30s, 180s]. That is VM jobs last at least 30s and will terinate in less than 180s. The exact allocation algorith as expected outperfors the Best-Fit heuristic for the 1000s tie interval siulated and reported in Figure 6. The Best-Fit heuristic uses ore often all available nodes or servers (100% ordinate value in Figure 6) while the exact algorith anages to use fewer nodes with 10 to 50% ore unused servers that Best-Fit. Figure 7 extends the analysis for the exact and extended Bin- Packing allocation algorith by coparing its perforance with and without consolidation. When the exact algorith is cobined with the igration algorith (that uses igration to epty soe nodes) it can significantly provide additional power savings or energy consuption reduction. The average gain can be estiated to be 10 to 20% ore servers that could be turned off. The average line for the exact algorith is around 80% of servers used while the average for the exact algorith with igration is ore in the order of 60%. Figure 8 pursues the analysis for the exact bin-packing VM allocation algorith by reporting perforance as a function

7 7 Figures 9, 10 and 11 address the perforance of the consolidation algorith via the analysis of the tie needed to achieve igrations of VMs fro source nodes to destination nodes in order to free as any servers as possible and gain the opportunity to shut the down. The assessent is perfored consequently on the active servers, i.e. those currently serving VMs and candidate for consolidation. The perforance as a function of increasing nuber of active nodes to consolidate is reported in the three figures for = 5, = 10 and = 20. For = 5, consolidation after igration fro source to destination nodes can be achieved in the illiseconds tie scales (few to 300 s in Figure 9). The nuber of hosted VMs to consolidate varies fro 5 to 30 for this siulation. Fig. 9. Execution tie of the exact igration algorith ( = 5) Fig. 7. Perforance coparison of the exact allocation algorith with and without igration of data center sizes and VM requests induced load. The tie before convergence to the optial placeent is reported as a function of data center size (fro 100 to 1000 nodes or servers) for request sizes ranging fro 50 to 500 VMs. Clearly, because the proble is NP-Hard, the convergence tie of the exact algorith grows exponentially for requests exceeding 300 VMs; especially for nuber of servers beyond 400. The tie needed to find the optial solutions reains acceptable and reasonable, within 10 s, for data center sizes below 500 receiving requests less than 400 VMs. The tie needed for convergence grows unacceptably high outside of this operating range for the siulated scenarios (tens of seconds to few inutes). This otivated the use of the Best-Fit algorith to find solutions faster even if they are bound to be suboptial as reported in Figure 6. The tie needed for consolidation increases to seconds in Figure 10 for = 10. The curve is reported for up to 60 VMs hosted in the nodes considered or subject to consolidation/igration. When the nuber of servers to consolidate increases further, as shown in Figure 11 for = 20, the convergence ties ove to orders of tens to hundreds of inutes (for the extree case, on the curve upper right corner, this reaches 180 inutes for 120 hosted VMs). These three figures highlight the liits of the exact igration algorith with increasing nuber of servers to consolidate. Fig. 10. Execution tie of the exact igration algorith ( = 10) Fig. 8. Execution tie of the Exact Allocation Algorith Fig. 11. Execution tie of the exact igration algorith ( = 20) The next experients and siulations address the achievable energy savings using different VM requests inter-arrival ties (noted by λ 1 ) and lifeties (represented by µ 1 that also reflects the service rate µ 1 ) in order to assess the perforance for variable syste loads since the ration λ/µ governs perforance. The nuber of servers has been fixed to 200 hosting nodes for the reported results in Table I. One hundred (100) siulation runs are average for each paraeter setting in the table. Table I reports the energy savings with

8 8 the igration algorith copared to the allocation algorith without igration. TABLE I TABLE OF PERCENTAGE OF GAINED ENERGY WHEN MIGRATION IS USED λ µ 1 (s) (s) ,55 36,59 00,00 00,00 00,00 00, ,29 34,00 35,23 38,50 00,00 00, ,48 27,39 35,21 40,32 41,89 36, ,77 18,85 22,02 32,31 39,90 40, ,86 16,17 19,85 22,30 39,20 36, ,63 14,29 18,01 22,13 25,15 30, ,10 14,00 14,86 15,90 22,91 23, ,01 10,20 10,91 15,34 17,02 21, ,80 09,52 10,31 14,70 16,97 19, ,90 07,50 08,40 12,90 16,00 14,97 Energy savings depend evidently on the service rate or the lifetie of VMs or the duration of their jobs relative to the load induced by the VM resource requests. Savings in the siulation can reach as high as 41.89% for inter-arrival ties of 25 seconds and job durations of 30 seconds. For less favorable ratios or loads, the savings for the scenarios tested in the evaluation are less significant but reain respectable (5.90% for the highest loads (inter-arrivals tie is equal to 5seconds) and longer job durations (of 100sec). Fig. 12. Energy savings In order to coplete the analysis, the energy savings that can be achieved by the Best-Fit, the exact allocation and the exact allocation cobined with igration are copared for siilar scenarios with a restricted set of paraeter settings (λ 1 = 10s). All the servers are initially considered OFF, which eans that the energy saving is initialized to 100%. Figure 12 depicts the evolution of the percentage of energy saved by the three algoriths. The obvious dependence on syste load is reflected by the gradual decrease of energy savings for increasing VM lifeties (or increasing job durations). For the exact Bin-Packing allocation algorith the energy savings reain quite high at low load; 90% of energy savings for the exact allocation only and 95% cobined igration. Energy savings achieved by the Best-Fit heuristic stay below the percentages achieved by the exact allocation algorith with or without igration at all different loads. As usual the gap will vanish for saturated systes as the servers will be always busy and peranently turned on, the exact algorith can nevertheless provide gains if intelligent groupings and rearrangeents are applied (this is however out of scope of this paper). V. CONCLUSIONS This paper explored the proble of VM placeent in cloud providers data centers, which is known as NP-hard. Our original contribution consists in using exact algoriths for placeent and consolidation that iniize jointly energy consuption and igration costs in acceptable convergence ties copared slightly faster heuristics such as best fit. We have proposed a linear integer progra corresponding to an exact allocation algorith coupled with an efficient exact VM igration algorith to reduce energy consuption via consolidation. Significant energy savings can be obtained depending on syste loads. At low load the gains can be quite significant. These gains at high loads even if uch lower reain sufficiently valuable using the proposed algoriths. REFERENCES [1] World Energy Outlook 2009 FACT SHEET. [2] Gartner report, financial ties, [3] J. Kaplan, W. Forrest, and N. Kindler, Revolutionizing data center energy efficiency, McKinsey Copany Tech Rep, p. 15, July [4] E. Naone, Conjuring clouds, Technology Review, vol. 112, no. 4, pp , [5] Intel s cloud coputing 2015 vision. [6] Open nebula. [7] OpenStack. [8] A. Kansal, F. Zhao, J. Liu, N. Kothari, and A. A. Bhattacharya, Virtual achine power etering and provisioning, in Proceedings of the 1st ACM syposiu on Cloud coputing, ser. SoCC 10. New York, NY, USA: ACM, 2010, pp [Online]. 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