Resource Allocation for Real-Time Tasks using Cloud Computing

Reource Allocation for Real-Time Tak uing Cloud Computing Karthik Kumar, Jing Feng, Yamini Nimmagadda, and Yung-Hiang Lu School of Electrical and Computer Engineering, Purdue Univerity, Wet Lafayette, IN, 47907. Abtract Thi paper preent a method to allocate reource for real-time tak uing the Infratructure a a Service model offered by cloud computing. Real-time tak have to be completed before deadline, and cloud computing offer election of reource with different peed and cot. In cloud computing, reource allocation can be caled up baed on the requirement; thi i called elaticity and i the key difference from exiting multiproceor tak allocation. Scalable reource make economical allocation of reource an important problem. We analyze the problem of allocating reource for a et of realtime tak uch that the economic cot i minimized and all the deadline are met. We formulate the problem a a contrained optimization problem and propoe a polynomial-time olution to allocate reource efficiently. We compare the economic cot and performance provided by our olution with the optimal olution and an EDF (earliet deadline firt) method. We how how the cot varie baed on the ditribution of the tak. Index Term reource allocation; cloud computing; cheduling; I. INTRODUCTION Cloud computing offer a uer the ervice (called Infratructure a a Service - IaaS) of renting computing reource over the Internet. The uer can elect from different type of computing reource baed on the requirement. For example, Amazon EC2 Cloud ervice provide different option for electing reource a hown in Table I. The uer can rent an arbitrarily large number of reource: thi i called calable computing or elaticity, ince the number can be caled dynamically to meet the requirement. One et of application that can benefit from calable computing are mixed-parallel application [5], [12], [16], [18], [20]. Thee application exhibit high tak and data parallelim. Many of thee application are real-time [5], [12], [20] and require their workload to be completed before deadline; example include voice and object recognition, image and video retrieval, navigation ytem, financial ytem, and miion-critical ytem. For example, an object recognition engine [20] may be hoted on the cloud. Each tak i an object query; the object mut be recognized within a pecified time duration to be of value to the uer. Since the cloud offer calable computing, reource allocation can be caled baed on the arrival time, workload, and deadline of the tak. In cloud computing, a reource i a virtual machine that guarantee a certain level of performance to the uer. For example, Amazon EC2 cloud ervice define virtual machine with peed in compute unit ; thi provide a hardwareindependent definition of peed for the virtual machine by abtracting away variation in the underlying phyical hardware. The type and amount of virtual machine allocated determine the cot paid by the uer. Amazon cot for different virtual machine are hown in Table I. There are three different type tandard, high memory, and high CPU, including different amount of proceor, memory, and torage at different cot. Each virtual machine i rented multiple hour, and the uer Virtual Machine Compute Memory Storage Cot/ type Unit hour High Memory 6.5 17.1 GB 420 GB $0.69 Standard 8.0 15.0 GB 1690 GB $1.04 High CPU 20.0 7.0 GB 1690 GB $1.24 TABLE I VIRTUAL MACHINE TYPES AVAILABLE IN AMAZON S EC2 CLOUD SERVICE (SIZE EXTRA LARGE, US-N CALIFORNIA). HTTP://AWS.AMAZON.COM/EC2/PRICING/ i charged a fixed cot irrepective of the virtual machine utilization within the hour. Thi motivate the need to find a cot-efficient allocation for a given et of tak. In the ret of thi paper, we ue the term reource, virtual machine (VM), and proceor interchangeably. Several reearcher have developed efficient allocation for real-time tak on multi-proceor ytem [7], [23], [13]. However, previou tudie chedule tak on a fixed number of proceor. For calable computing, the virtual machine (VM) are rented, and can be caled up to any number. Thi create ome fundamental change in the problem. Firt, it implie that if a tak i ufficiently parallelizable, it deadline can alway be met ince more VM can be allocated to complete the tak before it deadline. Thi i different from previou tudie that examine the chedulability on a given number of proceor. Second, ince the number of available VM i nearly infinite, at every time intant, there are option uing different number and type of VM, baed on their computing peed and cot. For example, to finih a tak, a uer can elect a larger number of lower, cheaper VM or a maller number of fater, more expenive VM, or a combination between them. Third, acquiring VM by rent implie a fixed charge for a given rental period. Suppoe a uer rent a VM for an hour and the tak complete before the end of the hour, the VM become available for running other tak that arrive within the hour. Thu the allocation of reource for one tak at the preent may affect the election of reource for future tak. Recent tudie on allocating cloud VM for real time tak [19], [12], [14], [3] focu on different apect like the infratructure to enable real-time tak on VM, election of VM for power management in the data center, etc. Unfortunately, none of thee tudie conider how the uer can make a cot-efficient election from a et of different VM for real-time tak. In thi paper, we develop an algorithm for allocating VM to application with real-time tak. The allocation i formulated a a contrained optimization problem. Since an exhautive earch for olution ha exponential complexity, we propoe a polynomial-time heuritic to olve the problem. We compare the cot obtained by our heuritic with the optimal olution, and an Earliet Deadline Firt (EDF-) trategy. We how the condition when our heuritic outperform the EDF trategy. We alo perform a enitivity analyi for the 978-1-4577-0638-7 /11/$26.00 2011 IEEE

Paper Heterogeneou Workload Number of Proceor Proceor [9] Ye Fixed Fixed [22], [4] No Probabilitic Fixed [7], [23], [13] No Fixed Fixed Thi paper Ye Fixed Scalable TABLE II COMPARISON OF DIFFERENT STUDIES ON REAL-TIME SCHEDULING. parameter of the problem. The remainder of thi paper i organized a follow: Section II decribe the type of application that can be ued for our problem, and highlight our contribution by comparing it with related work. Section III decribe our problem formulation, and the propoed olution. Section IV decribe the evaluation of the propoed olution, and Section V conclude the paper. II. BACKGROUND AND RELATED WORK A. Scheduling Parallel Application Scheduling mixed-parallel application [16], [18] on multiproceor ytem i a known NP-Complete problem [16]. Many of thee parallel application are real-time [12], [5], [20]. The tak in thee application have two important characteritic: (1) highly parallelizable [11], [2], [16] and (2) real-time contraint (or deadline) [15], [17]. Each tak can be partitioned into maller and parallel unit. We ue p a the mallet unit of computation. For example, image retrieval may compare image with a query to find a match, and the mallet unit of computation i comparing a ingle image with the query. The tak find a match for a query image, called img, from a collection of one million image. Thi tak mut be completed within d econd. We may ue one million VM and each compare only one image. If all VM can execute imultaneouly and the torage ytem can provide a ufficient bandwidth, thi would be the fatet approach but it would alo ue the larget number of VM. Since we only need to find a match for img (ye/no), the reult can be merged very quickly in logarithmic time. Another option ue 1000 VM and each VM compare 1000 image. It i alo poible to ue a ingle VM for the one million image; thi will take much longer. The actual election of VM may be contrained by the deadline d, the peed and cot of different type of VM. In thi paper, we conider cot-efficient VM election for uch application. B. Scheduling for Multiproceor Previou tudie have conidered energy-efficient cheduling. Uniproceor cheduling cheme like Earliet-Deadline Firt (EDF) [8] and Rate Monotonic (RM) cheduling [1] are adapted to multiproceor ytem. Many reearcher [7], [4], [23], [13] conider real-time tak allocation on multiproceor ytem. Several paper have tudied Dynamic Voltage and Frequency Scaling (DVFS). We do not conider DVFS in thi paper a it i a well tudied problem. Table II how how our work differ from the current real-time multiproceor cheduling technique. C. Virtual Machine Allocation for Cloud Computing Thi include two different problem: (1) electing virtual reource by the uer and (2) mapping virtual reource to phyical reource by the ervice provider. Thi paper focue on the firt problem. Several tudie [10], [21], [6] conider cot-effective reource election for cloud ytem. The common focu of all thee work i cot-efficient VM allocation; the key difference from the above work i that we conider real-time tak. Recent tudie have been performed on allocation of VM for real time tak. Aymerich et al. [3] develop an infratructure for deploying real-time financial tak on cloud ytem. Tai et al. [19] dicu real-time partitioning of databae tak on cloud infratructure. Liu et al. [14] how how to chedule real-time tak with different utility function; however they do not conider different type of VM. The cloet work to our i that of Kim et al. [12]; they conider cheduling real-time tak on cloud ytem. However, in their work the real-time contraint i pecified in a ervice level agreement (SLA). In uch cloud model, the uer doe not elect individual VM and VM allocation i left to the ervice provider. Their work examine power management while allocating VM to meet the SLA. In our work, we conider the IaaS model where the uer elect and pay the cot for the VM (imilar to Amazon model), and we propoe a cheme to reduce cot for the uer. Thu the work of Kim et al. [12] benefit the ervice provider and our work benefit the uer. Thi paper ha the following contribution: (1) Thi i the firt paper to conider cot-efficient reource allocation in heterogenou cloud for real-time tak. (2) We formulate reource allocation a a contrained optimization problem. (3) We propoe a polynomial-time algorithm to allocate VM while meeting tak real-time contraint and we how how the cot varie baed on the ditribution of the tak et. III. REAL-TIME CLOUD SCHEDULING Thi ection decribe the problem of cheduling real-time tak. Section III-A and III-B define the problem, and how a imple example. Section III-C formulate a contrained optimization problem. Section III-D decribe a olution and Section III-E preent our algorithm to olve the problem. A. Problem Definition The application ha a et of tak T. Each tak α i T ha an arrival time a i and a deadline d i (pecified in minute). We ue cycle to quantify the tak workload w i ; the uage of cycle in thi context i a general meaure of the amount of computation required for the workload. The workload of each tak i parallelizable into maller unit (ubtak); the ize of the mallet poible ubtak i p. R i the et of available VM. Each VM v j R ha a computation peed j and the correponding cot c j. The peed j i the number of cycle the VM can complete per minute. The uer i charged the cot c j for renting v j for D minute continuouly, regardle of the utilization within the interval. D i the minimum time unit for renting. We aume that tak α i can alway meet it deadline d i if enough VM are allocated. The condition p i i, j, d i a i max( j) ; in other word, the fatet VM max( j ) can compute the mallet ubtak p before the deadline. The problem can be tated a follow: find an offline mapping from T onto a ubet of R to minimize the overall cot while meeting the deadline of all tak. B. Motivating Example We decribe a imple motivating example. Table III how the peed and the cot of the reource available for election. The cot are in the ame ratio a table I. The value of D i 60 minute and p i 10. We conider a ingle tak α 1 with an arrival time a 1 of 0 min, workload w 1 of 600 cycle,

(a) (b) (c) Fig. 1. Allocation of three reource v 1, v 2,andv 3 for two tak: α 1 i hown in olid rectangle, and α 2 in empty rectangle. (a) Auming v 1 i allocated to α 1 from 0 to 60 min, allocating two of v 2 at 40 min to α 2. The cot i 1+2 1.5 =4.(b)Ifv 2 i allocated to α 1, v 2 can be alo ued for α 2 from 40 min to 60 min. Thi along with one of v 3 at 40 min can complete α 2. The cot i 1.5+2=3.5. (c)ifv 3 i allocated to α 1, v 3 can alo be ued for α 2 from 40 min to 80 min. Thi along with one of v 1 at 40 min can complete α 2 with the lowet total cot 2+1=3. and a deadline d 1 of 70 min. Tak α 1 take 60 (= w1 1 ), 30 (= w1 2 ), and 20 (= w1 3 ) minute on v 1, v 2, and v 3 repectively. The objective i to reduce the cot while meeting the deadline. A chedule elect the reource with the lowet cot while meeting α 1 deadline. Renting v 1 from 0 min to 60 min can finih α 1 before d 1 with the lowet cot. For each tak α i,weuec αi to denote the cot for the tak to meet it deadline and c α1 =1. VM v j j = cycle c min j = cot hour v 1 10 1.0 v 2 20 1.5 v 3 30 2.0 TABLE III RESOURCES AVAILABLE FOR SELECTION Tak Arrival Deadline Workload α i a i d i w i α 1 0 70 600 α 2 40 80 1500 TABLE IV TASKS TO BE COMPLETED Next, we conider tak α 2 with arrival time a 2 =40min, workload w 2 = 1500 cycle, and deadline d 2 =80min.We ue the election v 1 made earlier for α 1, and examine how to allocate reource to execute α 2. We need to complete α 2 in d 2 a 2 = 40 minute; thi can be accomplihed with four v 1 ( 1500 150 10 =150 minute, 40 =4v 1), two v 2 or, two v 3.The correponding cot for both tak are 1+4=5, 1+3=4, and 1+4=5 repectively. If we conider uing more than one type of VM for α 2, uing one v 3 and one v 2 reduce the cot to 1+(2+1.5) = 4.5. Uing one v 3 and one v 1 further reduce the cot to 1+(2+1) = 4 and thi i the lowet total cot. Figure 1(a) how the example with c α1 +c α2 =4, uing v 1 for α 1 and two of v 2 for α 2. If we conider both tak imultaneouly, a better olution allocate v 2 to α 1, and v 3 to α 2, hown in Figure 1(b). Thi reult in c α1 + c α2 =1.5 + 2 = 3.5, while meeting both deadline. The cot i lower becaue α 1 utilize v 2 from 0 minto30min,andα 2 can be run on v 2 from 40 min to 60 min, thu completing 400 cycle of α 2.Onev 3 can be rented at 40 min, and the remaining 1100 cycle of α 2 can be run on v 3 before the deadline d 2. The example how that a trategy may not give the lowet total cot. The tak mut be conidered imultaneouly to minimize the total cot. In the next ection, we formulate reource allocation a a contrained optimization problem. C. Problem Formulation We formulate the problem decribed in Section III-A a a contrained optimization problem. Each type of VM v j R ha peed j and cot c j, j =1,2,...x. We may elect any number of VM of a given type and we ue indicator variable δ(j, k) to repreent the election of VM. More pecifically, δ(j, k) = 1 if the k VM of type v j are ued, where k = 1, 2, 3... For example, δ(2, 3) = 1 indicate that three v 2 are ued. The objective function i to minimize the total cot. Thi may be given by umming the cot of all the elected VM a hown in equation (1): x y min c j δ(j, k) (1) j=1 k=1 The outer loop in equation (1) correpond to the x different type of VM, and the inner loop ummed to y correpond to the number of available VM of each type. Since cloud computing offer the election of an arbitrary number of VM, the value of y =. In reality, we can derive an upper bound for y for a given et of tak T. The total computation for T i given by C = T i=1 w i. The computation C i partitioned among the VM. If we conider the extreme cae that each VM perform the mallet unit p, then the maximum number of VM that can be ued for T i C p.the( C p +1)th VM cannot be ued a each of the previou C p VM i already performing the mallet unit of computation; there i no more computation for any additional VM. Thu we have an upper bound for y to be y C T p = 1 p w i. (2) i=1 Equation (1) repreent the total cot to be minimized and i the objective function. We now examine the contraint: all tak need to meet their deadline. We need to elect ufficient VM in equation (1) uch that each tak α i can be completed after it arrival time a i and before it deadline d i.theδ in equation (1) only indicate the type and the number of VM allocated; they do not indicate when the VM are allocated. The time of allocation i important to enure that the VM allocated for α i are available after a i and before d i. In order to conider the time of allocation, we make a retriction (for now) that each allocated VM i available for 1 minute (D=1), and we introduce a new et of indicator variable θ that include

the temporal information: θ(j, k, m) = 1, ifthek VMoftype j are available at the m th minute. Reconider the example in Section III-B. For tak α 1, a 1 = 0, d 1 =70, w 1 = 600, and p =10. To enure thi tak i completed before it deadline, VM mut be allocated uch that the number of cycle completed in the interval between a 1 (0 min) and d 1 (70 min) i at leat w 1 (600 cycle). The number of cycle completed at each minute in thi interval depend on the peed of the VM allocated at each minute within the interval; formulating thi uing θ give x y j θ(j, k, m) (3) j=1 k=1 cycle at the m th minute. In thi example, x i 3 becaue there are three different type of VM. From equation (2), the value of y cannot exceed 1 10 600 =60. Thi mean that the maximum number of VM we can ue i 60, with each computing 10 cycle. In equation (3), if θ(2, 1, 1) = 1 and θ(1,k,1) = θ(3,k,1) = 0 for 1 k 60, the peed at the 1 t minute i 2 =20cycle/min. The amount of work that can be done during thi minute i 20 cycle. To enure that at leat 600 cycle are completed for the tak, we need to make ure the ummation of work done at each minute from 0 to 70 i at leat 600. To generalize thi acro the duration of tak α i T, we obtain the following requirement, for each α i : d i x y j θ(j, k, m) w i (4) m=a i j=1 k=l i The lower limit of k for the i th tak equal l i becaue ome θ are already allocated for previou i-1 tak (up to tak α i 1 ); thi enure that θ for the current tak α i doe not with the θ allocated for tak α 0, α 1... α i 1. Thi prevent haring VM among tak; however, our aumption that D = 1 enure that each VM i already fully utilized at the minute it i available. Equation (4) i baed on the aumption that D=1. In order to generalize our formulation uch that a VM i available for D minute, we need to define a function to repreent a time interval of D minute. We ue the window function u(m) u(m D), where u(m) i the unit tep function. We now need to find the number of D minute window to perform the ame amount of computation a the allocation for θ in equation (4). In order to do thi, we define equation (5) and (6) for each VM type j. At each time intant m, θ(j, k, m) hould be le than or equal to n (m z)j ; n (m z)j i the mallet number of D minute window required to do theameworkaθ(j, k, m) at time m. Weueaubcript of m z becaue a D-minute window that begin z minute before m, can be ued at m, a long a z D. Thevalueof m i between the arrival of the firt tak a 1 and the deadline of the lat tak d N. f(m, z) =u(m z) u(m z + D) d N y D ( θ(j, k, m) n (m z)j f(m, z)) 0 m=a 1 k=1 z=0 To find the total number of VM of each type j that are required, we define t j = d N m=a 1 D (5) n mj (6) Speed Cot VM ued 10 1.0 v 1 20 1.5 v 2 30 2.0 v 3 40 3.0 v 1 + v 3 50 3.5 v 2 + v 3 60 4.0 v 3 + v 3 70 5.0 v 1 +2 v 3 80 5.5 v 2 +2 v 3 90 6.0 v 3 +2 v 3... TABLE V LOOK-UP TABLE FOR DIFFERENT SPEED REQUIREMENTS AND THE CORRESPONDING COSTS AND VMS. Equation (6) um all the n mj at every poible m, ranging from D minute before the arrival of the firt tak, to deadline of the lat tak. Thi um t j give the total number of VM of type j that are required. In order to make thi election a minimum cot election in equation (1), we et δ(j, k) =1 k t j. (7) The olution to thi ILP formulation i intractable. In Section III-D and III-E, we decribe a algorithm and our polynomial time heuritic to olve thi problem. D. EDF-Greedy Algorithm The firt olution in Section III-B can be decribed a a trategy baed on EDF (earliet deadline firt). The tak are conidered in the order of their deadline. The trategy firt trie to allocate a tak to VM available from the allocation for previou tak. If thee VM are inufficient to complete the tak before it deadline, the trategy elect the cheapet et of VM that can complete the tak before the deadline. To find the cheapet et of VM for each tak, we contruct a lookup table baed on the peed and cot in Table III. The lookup table contain a range of poible computing peed contructed from the different type of VM. In Table III, the VM have peed {10, 20, 30}. Uing thee peed a the bae, we can obtain the combinatorial et of peed S = {10, 20, 30, 40, 50, 60,...}. For each peed in S, thevm et to give that peed i identified and tored with that peed in Table V. For example, to achieve peed of 50, we can ue one v 2 and one v 3 with cot = 2+1.5 =3.5, ortwov 2 and one v 1 with cot = 2 1.5+1=4. The former ha a lower cot and it i a better allocation. The entrie in the lookup table are orted in order of their peed. For a given tak workload, the trategy earche the lookup table to find the lowet peed that can finih the workload before the deadline. For tak α 1 in Section III-B, the lowet peed needed to finih the tak of w 1 = 600 cycle in d 1 a 1 =70minute i 10 cycle/min. If d 1 i changed to 10 min, then the lowet peed needed to compute 600 cycle in 10 min in the table i 60 cycle/min. If thi lowet peed in the lookup table i αi, the correponding cot i c αi.the lookup table i finite becaue the number of row i bounded by the highet peed needed to atify any tak. Thi bound i computed by w i max( ), α i T. (8) d i a i Thi guarantee that the trategy can find a VM allocation for all the tak. The complexity of computing the

lookup table depend on the number of type of VM x. In the above example, x =3and it i oberved that the VM have peed-cot ratio of 10, 13.33, and 15 for v 1, v 2, and v 3 repectively. We contruct the lookup table by trying to allocate VM in decreaing order of their peed-cot ratio, i.e., we try to allocate a many of v 3 before we allocate v 2, and o on. To get a peed x, we can compute the required VM to be x = t 2 of v 2, and 3 = t 3 of v 3, x (t3 3) 2 x (t3 3+t2 2) 1 of v 1. Thu a peed of 80 i obtained by uing 80 30 =2ofv 3 (t 3 =2), and 80 2 30 20 =1ofv 2, and 80 (2 30+1 20) 10 =0ofv 1,givingv 2 +2 v 3, a een in Table V. Similar formulation may be obtained for different type of VM with different peed and cot. E. Allocation Conidering Temporal Overlap The optimization problem formulated in Section III-C may be olved by exhautive combination of allocating different type of reource to all tak at variou time intant. Thi approach unfortunately i intractible. The algorithm in Section III-D allocate VM for each tak eparately. When the tak in time, thi trategy may produce under-utilized and higher-cot allocation. Thi ection decribe our polynomial-time algorithm that conider all tak together in the order of their deadline. For each tak α i, the algorithm (1) identifie ping tak in the future and (2) allocate reource conidering thee tak. (1) Identifying Overlapping Tak: For a given tak α i, we conider the tak in the future whoe deadline are after d i and examine whether thee tak temporally with α i.takα j temporally with α i if a j <d i + D. Thi mean that a VM can be allocated uch that it i hared by α i and α j.weuet i to denote the et of tak that with α i : T i {α j d j d i, and a j <d i + D} (9) (2) Allocating Reource: Firt, we ue Table V to obtain the lowet peed αi required to compute tak α i (Section III-D). Next, we examine if allocating a VM with peed greater than αi for tak α i can benefit the ping tak in T i. We conider each VM peed αi in the lookup table and compute the VM time t ij for each tak α j T i. We define t ij a the amount of time the VM allocated for α i i till available for running α j.thevalueoft ij depend on the arrival time, workload, and deadline of α i and α j, and the peed of the VM. It i calculated by: min(d D ij,d j a j ) if a j a i wi t ij = D D ij, if a j a i < wi,d j a i D d j (a i + D ij ), if a j a i < wi,d j a i <D where D ij = w i + max(a j d i, 0). (10) In the above equation, the time to complete the α i i wi min. The firt cae how the condition when the arrival time of the two tak are ufficiently apart uch that α i i completed before the arrival time of α j, i.e. (a j a i wi ). Tak α j can ue the VM for t j min. The value of t j i given by the leer of two term, D D ij, and d j a j. The firt term D D ij denote remaining time for the VM for α j,after taking into account the time it i ued by α i,= wi, and a poible gap a j d i between the deadline of the firt tak and the arrival of the econd tak; during thi time, the VM cannot be ued by both tak. For example, aume the firt tak ue w 1 =10min of the VM, mut be completed by d 1 =20min, and the econd tak arrive at time a 2 =40min, and mut be completed by d 2 =60min. Then the value of D 12 from equation (10) i 10 + max((40 20), 0)=30, and D D 12 i 60 30=30. However, d 2 a 2 =60 40=20. Thu, the VM that i available for 30 min at a 2 i ued only for min(30,20) minute, and thi i t 12 for the firt cae in the above equation. The other two cae are imilarly contructed. Note that the VM may not be ued for the entire t ij if tak α j doe not have ufficient workload w j to utilize the VM for t ij min. In uch a cenario, the temporal by α j i reduced from t ij to wj where i the peed of the VM being conidered, αi.weuetheo to repreent the actual. O min( w j,t ij), for αi (11) The maller of wj and t ij i the amount of time a VM allocated for a previou tak can be actually ued for a future tak. Baed on the O, we compute a revied cot for the current tak α i. The revied cot c α i i not the actual cot paid for the reource, and i ued only for the purpoe of deciion making in our algorithm. The cot c α i i calculated c α i min{c αi (1 O D )}, α j T i, αi. (12) In the above equation, the cot c α i i obtained by reducing c αi baed on how much the VM with tak in the future (1 O D ). Since we elect the reource correponding to c α i for each tak α i, thi encourage election of VM that can be ued by other tak in the future. Our algorithm retrict the et of peed earched to the ize of the lookup table. Further, our algorithm conider the et T i (all tak that with the current tak) while making an allocation for α i. A more exhautive analyi could further conider that each tak α j T i ha a correponding et of tak T j that i alo affected by the allocation for α i, and o on. Conidering thi would make our algorithm cloer to exhautive earch. To illutrate our algorithm, we conider the tak α 1 and α 2 from Section III-B and the value of D =60. Originally, we had aigned v 1 to α 1 ince it ha the lowet cot (c α1 =1), and finihe before the deadline. We now examine the cot for VM allocation to α 1 for the temporal- algorithm. The between α 1 and α 2 begin at 40 min. If we allocate v 1 for α 1, the cot remain the ame (t 12 =0,O =0 and c α 1 =c α1 =1) ince α 1 ue v 1 for the entire 60 min, and v 1 cannot be made available for the ping tak α 2. Allocating v 2 for α 1 earlier reulted in c α1 = c 2 =1.5. Ifwe allocate v 2 for α 1 in the temporal- cheme, allocating it from 10 min to 70 min reult in v 2 being available for α 2 from 40 min to 70 min (t 12 = 30). The temporal O = min(30, 1500 20 ) = 30. The cot for thi allocation i c α 1 = c 2 (1 O 30 D ) = 1.5 (1 60 ) = 0.75, which i le than the previou option. Allocating v 3 for α 1 earlier reulted in c α1 = c 3 =2.For the temporal- cheme, if we allocate v 3 from 20 min to 80 min, α 1 i completed from 20 min to 40 min, and v 3 i

available for α 2 from 40 min to 80 min. Thi reult in a cot of c α 1 = c 3 c 3 ( 40 40 60 )=2 2 ( 60 )=0.67, thu making the elector chooe v 3 for α 1 over v 1 and v 2. Aigning v 3 to α 1 a hown in Figure 1(c) reult in 1200 cycle of α 2 being completed from 40 min to 80 min; allocating v 1 for α 2 at 40 min can complete the tak with c α2 =1. The actual cot for α 1 and α 2 uing v 1 and v 3 i c α1 + c α2 = 2+1=3, lower than the total cot obtained in Section III-B. IV. EVALUATION In thi ection, we decribe how we evaluate the different cheduling algorithm. Section IV-A decribe the evaluation etup. Section IV-B compare the cot obtained by the algorithm. Section IV-C and IV-D decribe our enitivity analyi, and Section IV-E compare the complexity. A. Setup We conider three algorithm for our evaluation: (1) EDF (called ) decribed in Section III-D, (2) temporal- (called ) in Section III-E, and (3) exhautive earch, conidering all poible combination to obtain a minimum-cot allocation. It ue neted loop of the following, (1) each tak, (2) each type of VM, (3) any number of reource of a given type, and (4) every poible temporal allocation of each VM. The input to the cheme are the peed and cot of the reource available, and the arrival time, workload, deadline of the tak. We ue the peed and cot hown in Table III for our evaluation. We chooe a Poion ditribution with parameter λ to model the arrival time of the tak ince it i a natural way to model a equence of event randomly paced in time. We conider up to 500 tak for our evaluation and λ ranging from 0.5 to 500. We ue randomly generated deadline of the tak between (d l,d u ) minute and different value of d l and d u, ranging from 10 to 1500. The workload for the tak are randomly generated within a range of (p, max(w)) cycle, where the lower limit i p=10 cycle. We ue a range of upper limit for max(w), from 1500 to 4500. A decribed in Section III-D, the value of q and d l determine the number of entrie in the lookup table. Accordingly, we contruct the lookup table for the reource. B. Cot Analyi We oberve the cot returned by the cheme. The exhautive earch conider all poible allocation of reource, and return the lowet cot. However, ince exhautive earch take everal hour, we compare the cot returned by exhautive earch for only up to ten tak. The and cheme perform within 200% of the reult from exhautive earch for upto ten tak. For a greater number of tak, we only compare the and the cheme Figure 2 (a) how the cot for the and cheme, for different number of tak ranging from 100 to 500, with λ =1.5, d l =10, and d u = 1500. The value in the figure are averaged over 100 trial. The figure how that the cheme perform conitently better than the cheme. For 500 tak, the cheme return an average cot of 546, 22% reduction when compared to the value of 698. Thi i becaue the cheme conider haring VM temporally for ping tak. C. Varying Poion parameter λ The value of λ determine the ditribution of the tak. A lower value reult in a more concentrated ditribution, with more tak with a common arrival time. Figure 2 (b) how the hitogram of the arrival time of a mall et of tak from our evaluation, for different value of λ. It i oberved that the et of tak with lower value of λ have more in their arrival time. Thi in the arrival time correpond to an between the duration of the tak. Thu lower value of λ correpond to greater between the tak. We vary λ from 0.5 to 500 and oberve the cot returned by the and cheme in Figure 2 (c). The number of tak i 100. The cot returned by the cheme doe not vary ignificantly with different λ becaue the cheme doe not conider the tak imultaneouly. For the cheme, the cot obtained increae for higher value of λ. For example, for λ = 0.5, the cot returned by the cheme i 27.2% le than the cheme; for λ = 500, the improvement i reduced to 7.2%. Thi i becaue a λ increae, there i le between the tak, and hence conidering the tak imultaneouly ha le benefit. An extreme cae would be a cenario when there i no between the tak. In thi cae, the cheme would return the optimal olution. D. Varying workload Increaing the workload of the tak reult in a proportionate increae in the cot function. We vary the limit of the random generator (p, q) and oberve the variation in the cot. Figure 3(a) how the cot with p =10, for different value of q. Figure 3(b) how the cot with q = 4500 and different value of p. It i oberved that the cot for both cheme increae with the workload. E. Complexity analyi Exhautive earch ha exponential complexity and i not a calable olution. The cheme ha the following tep (1) Contruct the lookup table. The number of entrie i max( wi ) and each entry i linear in the number of different type of VM. Thu the complexity of thi tep i given by O(x max( wi ) ). (2) Viit each tak once and earch the entrie of the lookup table to find the allocation. Since the table i orted, earching can be done in O( max( wi ) ) time. Since there are N tak, thi reult in a complexity of O(N max( wi ) ). The overall complexity i the um of the above tep (1) and (2). If N>>xand max( wi ) i mall, the complexity of the olution i linear in the number of tak, i.e., O(N). For the cheme, we have the ame tep (1) and (2). In addition, for each tak, we conider all the tak with deadline greater than the current tak and examine the haring of VM between tak. Thi reult in each tak allocation having a complexity of O(N 1); ince there are N uch tak allocation, the overall complexity i given by O(N (N 1)) = O(N 2 ). Figure 3(c) how the time taken by the and cheme for different number of tak. We do not how the exhautive earch in the figure becaue it i exponential in the number of tak, number of reource, and type of reource and take everal hour for a et of 20 tak on a 3 GHz ytem with 4 GB RAM. We fit the data point in Figure 3(c), and we verify that the and

Cot 800 600 400 Cot 150 100 200 50 0 100 200 300 400 500 Number of tak (a) (b) 0 0.5 1.5 10 100 500 Value of λ (c) Fig. 2. (a) Cot obtained by the and cheme, for different number of tak (lower cot i better). The value of λ i 1.5. (b) Hitogram of arrival time of the tak for different value of λ. The point in the hitogram are interpolated to how the ditribution of the arrival time. (c) Cot obtained by the cheme and the cheme for different value of λ for 100 tak. A the value of λ increae, the amount of decreae and thu the the cot obtained by the cheme increae. Cot 300 200 1500 2500 3500 4500 Upper bound on workload q (a) Cot 500 400 300 200 10 1500 2500 3500 Lower bound on workload p (b) Time (m) 150 100 50 0 0 200 400 600 800 1000 Number of tak (c) Fig. 3. (a) the cot of the and cheme with lower bound p =10, for different value of upper bound q. (b) the cot of the and cheme with upper bound q = 4500 and different value of lower bound p. 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