Virtual Machine Allocation in Cloud Computing for Minimizing Total Execution Time on Each Machine

Virtual Machine Allocation in Cloud Computing for Minimizing Total Execution Time on Each Machine Quyet Thang NGUYEN Nguyen QUANG-HUNG Nguyen HUYNH TUONG Van Hoai TRAN Nam THOAI Faculty of Computer Science & Engineering Ho Chi Minh city University of Technology Vietnam 268 Ly Thuong Kiet Ho Chi Minh Vietnam Email: nqthang;hungnq2;htnguyen;hoai;nam}@cse.hcmut.edu.vn Abstract This paper considers a virtual machine allocation problem. Each physical machine in cloud has a lot of virtual machines. Each job needs to use a number of virtual machines during a given and fixed period. The objective aims to minimize the cost induced by total execution time on each physical machine. This allocation problem is proved to be N P -hard. Additionally three mixed integer linear mathematical models are constructed to represent and solve the problem. The performance comparison of the three proposed models is analyzed through some empirical results. Keywords: virtual machine allocation; cloud computing; MILP. I. INTRODUCTION Cloud computing is driven by economies of scale. A cloud system uses virtualization technology to provide cloud resources (e.g. CPU memory) to users in form of virtual machines. Virtual machine which is a sandbox for user application fits well in the education environment to provide computational resources for the needs in teaching and research. According to resource owner s view they want to reduce operation costs which induces electronic bill of large-scale data center system. Virtual Machine (VM) allocation problem in virtualized data centers is a challenging topic. The VM allocation problem can be seen in static and dynamic mapping. A VM allocation aims to map each VM to physical machines (PMs) to optimize a given objective function. The objective function can be maximizing performance minimizing power/energy consumption or maximize provider s profit etc. Static VM allocation can be seen as a d-dimensional Vector Bin Packing problem (V BP d ) [5] [6] in which VMs are items and physical machines are bins. The V BP d (d 1) bin packing problem is NP- Hard. A dynamic VM allocation differs from static VM allocation is that in dynamic VM allocation a system event (e.g. low CPU utilization system temperature hardware/software failure etc.) can trigger a re-mapping of the set of VMs and the set of PMs. Virtualization software (e.g. XEN KVM VMWare server ESXi) currently supports to execute more than one operating systems (OS) on the same physical machine i.e. some VMs can share same physical hardware. In this study we consider VM allocation problem in a cloud composed by m physical machines. Each of these machines can supply a number of operations demanded at any instant. These operations could be executed independently and in parallel though virtual machine generation mechanism. Depending to physical material each machine has an upper-bound on the maximum number of virtual machines allocated at the same time. Each user/task in system could demands a service which needs a number of virtual machines in a particular period. The objective aims to find a feasible assignment satisfying the demand of all users in order to reduce the total energy consumption of the whole system. With assumption that power consumption on a physical machine does not increase so much when cloud system needs to use more virtual machines the objective could be transformed as minimizing the total execution time on each physical machine. Our main contribution is not only about determining some polynomial cases and N P -Hard proof but also about proposing some Mixed Integer Linear Programming (MILP) model. Some experimental results show the performance of these proposed models on a well-known free solver COIN-OR CBC. The paper is organized as follows. In Section II some related works are discussed. Section III some notation used in this paper is introduced and the considered problem is addressed in detail. NP-completeness of problem will be presented in section IV. Then several linear mathematical models describing the considered problem are proposed in Section V. Section VI presents the computational experiments and analysis. Section VII concludes this study and future works are discussed. II. RELATED WORK A. Background Virtual machine is a software entity which is executed and managed by a Virtual Machine Monitor (VMM) or hypervisor such as XEN KVM VMware ESXi server etc. There are three generic problems of multi-dimensional packing multiprocessor scheduling bin packing and the knapsack problem [5]. In [5] they considered on vector scheduling problem or namely vector bin packing problem. The vector scheduling problem is how to schedule n dimensional tasks on m parallel machines so as to minimize the maximum load over dimensions of these all parallel machines. In [6] they studied First Fit and Best Fit Decrease (FFD and BFD) heuristics of d-dimensional vector bin packing problem and apply these heuristics to VM allocation problem. B. Virtual machine allocation Sotomayor et al. [7] proposed lease and virtual machine to provide computational resource for short-term resource needs in teaching and researching. They presented First-Come-First-Serve (FCFS) and backfilling scheduling algorithms for schedule user leases and using a greedy algorithm to map all identical VMs which belongs to same user lease onto same physical machine. Disadvantage of the greedy VM allocation algorithm is that two different leases could not be mapped to same physical machines. In [9] the authors proposed two power-aware VM allocation algorithms that represent some combinations of First Fit Decrease (FFD) and shortest duration time heuristics. Although the VM allocation algorithms can reduce total energy consumption for computing physical machines these VM allocation algorithms in [9] however do not lead to optimal solution. Mathematical programming approach has been applied in traditional scheduling problems on non-virtualized systems (e.g. high performance computing cluster) for many years in order to find an optimal schedule for performance [11] [10] or minimize energy consumption of heterogeneous computer clusters [12]. These works did not use virtualization and were not suitable for virtual machine 978-1-4673-2088-7/13/$31.00 2013 IEEE 241

scheduling. Recently there are some interesting works using mathematical programming for scheduling problems on virtualized systems (we focused on virtualized datacenters) such as [4]. Speitkamp and Bichler [4] proposed mathematical programming model for server consolidation in which each service was implemented in a VM to reduce number of used physical machine. Some Integer Liner Programming (ILP) models were proposed for static server allocation problem (SSAP) and extended version of the SSAP. They also claimed that the SSAP is strongly NP-Hard. There were about 600 virtual machines tested on their ILP models. For solving these ILP models they also used both first-fit and best-fit heuristics on the allocation problem. However the ILP models did not address arrival time of user requests or advanced reservation lease. The VM allocation which is mapping of set of various VMs onto set of heterogeneous physical machines with objective to minimizing total energy consumption of physical machines in a single virtualized datacenter was studied [3] [1]. Beloglazov et al. [1] considered a dynamic VM allocation (with migration) where each VM was concerned on two dimensions: CPU usage and power consumption (Watt). Beloglazov et al. claimed that the VM allocation problem is NP-Hard and they proposed Modified Best Fit Decreasing (MBFD) algorithm that maps each VM to heterogeneous physical machines (these machines are not equals in power consumption) such that to minimize increasing power consumption on each placement of VM and some other algorithms on migration of VMs. The MBFD is however a best-fit heuristic. Therefore the mapping of the MBFD is not optimal. Í. Goiri et al. [3] considered an energy-aware scheduling problem with similar VM allocation. In [3] they calculated score of each assignment of each VM onto a physical machine concerns hardware and software requirements power consumption system temperature; their scheduling algorithm was called scorebased scheduler. The score-based algorithm is of hill-climbing search on (N +1)xM matrix each cell is a score on each assignment of each VM to a physical machine with time complexity of O(kxNxM) (k is number of iterations). The score-based algorithm can be optimal cost-based VM allocation. Remark that VM allocation in [3] and in [1] did not consider on starting time of requests. This is an essential difference with our study - VM allocation to jobs with starting constraint for minimizing total execution time on each machine. III. PROBLEM STATEMENT We first define terms that will be used in this paper. A. Terminology notation Notation used in this paper is given below. n: number of jobs (J 1 J 2... J n) s i: starting time of job J i p i: processing time of job J i R i: number of virtual machines needed for job J i ɛ(t): set of jobs executed at time t (ɛ(t) = J i s i t < s i + p i}) T : the last execution time (T = max i [1n] (s i + p i)) m: number of physical machines (M 1 M 2... M m). v j: maximum number of virtual machines allocated from physical machine M j In this paper we deal with VM allocation problem of m physical machines. Physical machine M j can allocate at most v j virtual machines at any time. Each user/task J i in system could demand a service which needs R i virtual machines executed in period [s i s i + p i). The objective function is about minimizing the total execution time on each physical machine (if there exists a set of feasible assignments satisfying the requirement of all users). In order to determine status of N P -hardness of this optimization problem we first determine corresponding decision version as follows. B. Decision problem Decision problem VIRMACALLOC is described as follow. Data input: Given n jobs J 1... J n have to be scheduled without preemption on m parallel machines (m 1). Each job is described by a starting time s i processing time p i a number of virtual machines needed for executing R i. All machines are available at time zero and each one can support a number of virtual machines v j. Question: Does exist a resource assignment such that the total execution time on each machine used is not greater than a given value y? This decision problem will be proved to be N P -complete in the next section. IV. COMPLEXITY A. Some special cases Proposition 1: If all jobs are executed on different (disjoint) time windows the decision problem could be answered in polynomial time. Proof: Feasible decision for each job could be determined directly and independently. Hence this special case can be solved in polynomial time. Proposition 2: If nw = 1 R i = R ( i = 1... n) the decision problem could be answered in polynomial time. Proof: Since the requirement is identical for all jobs we could assign the job (in numerical order) to execute on the physical machines which correspond to the order of non-increasing of v j: i.e. on each physical machine unused virtual machines will be assigned to the jobs until the virtual machines are assigned completely and then we consider the next physical machine). This algorithm has complexity of O(n). Proposition 3: If s i = s ( i = 1... n) and v j = v ( j = 1... m) the decision problem could be answered in polynomial time. Proof: It follows the previous proposition (Proposition 2). In the following the NP-completeness of the considered scheduling problem is proved due to the special case where v j = v ( j = 1... m). B. NP-completeness Theorem 1: Problem VIRMACALLOC is N P -complete. Proof: We prove that PART IT ION VIRMACALLOC i.e. decision problem VIRMACALLOC belongs to N P -complete class by a reduction to PART IT ION which is known to be is N P -complete [2].. Recall that problem PART IT ION is described as follow. Data: Finite set A of r elements a 1 a 2... a r with integer sizes s(a k ) k 1 k r r k=1 s(a k) = 2B. Question: Is there a subset A 1 of indices such that k A 1 s(a k ) = k 12...r}\A 1 s(a k ) = B? Given an arbitrary instance of PART IT ION we construct an instance of VIRMACALLOC as follows: n = m = r for each job J i with i 1 2... n}: s i = 0 p i = 1 R i = a i for physical machine M j with j 1 2... m}: v j = B y = 2. ( ) Given a feasible solution to PART IT ION we can define a solution to VIRMACALLOC by assigning a virtual machine of M 1 for a subset of jobs corresponding to A 1 the remaining jobs will be executed on virtual machines allocated from M 2. This assignment satisfies the conditions and the answer to problem VIRMACALLOC is Yes since the exexcution time on M 1 (and on M 2 resp.) is 1. ( ) If there exists a feasible solution to problem VIRMACALLOC then there exists a virtual-machine allocation such that all virtual machines are allocated from two machines at 242

most (due to y = 2 and all jobs start at the same time). Since all machines have the same capacity assume that all jobs are served from the first machine M 1 or two first machines M 1 and M 2. Due to the fact that n i=1 Ri = r i=1 ai = 2B the first machine M1 with capacity B could not support all of jobs. So we need two first machines to serve this job set. Moreover the maximum number of virtual machines could be used from these two machines is v 1 + v 2 = B + B = 2B which corresponds exactly to the number of virtual machines needed. So there is no redundant resource from these two machines. Hence the total virtual machines that serve the jobs executed on M 1 is equal to v 1 = B and the sub-set of jobs defines then A 1. Consequently the answer to PART IT ION is Yes. V. MIXED INTEGER LINEAR MATHEMATICAL MODEL There are many objectives that have been defined in literature such as minimizing the system execution cost minimizing the number of physical machines needed minimizing the number of physical machines executed per time unit... (refers to [1] [4] [7] [9]). Mathematical models proposed in this paper use the below objective: Minimizing the total execution time on each machine. A. Constraints Before building mathematical models for the problem its constraints should be listed out here. Constraint 1: 1) The number of virtual machines assigned to job J i must be exactly its requirement (R i) 2) At any time total number of virtual machines used in physical machine (J i) does not exceed its capacity (v j). 3) No virtual machine can execute 2 jobs at the same time. 4) Once a virtual machine is assigned for a job that job must be executed continuously until complete on that virtual machine. B. Objective function The objective used for the problem is about minimizing the total execution time on each physical machine. To model the objective function intermediate variables are defined as below: 1 machine Mj is used at t j [1 m] t [0 T ] Then the objective function can be formulated as: min m T j=1 t=0 Note that value of the y jt variables will be calculated from decision variables in each following mathematical model. C. Mathematical model 1 The decision variables in this model are defined as below: x ijkt = 1 Job Ji is executed using k th VM on M j at time t y jt i [1 n] j [1 m] k [1 v j] t [0 T ]. Then the y jt variable can be calculated by: max x ijkt j [1 m] t [0 T ] i [1n]k [1v j ] or y jt x ijkt i [1 n] and k [1 v j] since they are in binary and the objective is relational to minimizing each value of y jt for all j [1 m] and for all t [0 T ]. Constraints are formulated as below: 243 Constraint 1: v m j x ijkt = R i i [1 n] t [s i s i + p i) j=1 k=1 Constraint 2: v n j x ijkt v j j [1 m] t [0 T ] i=1 k=1 Constraint 3: n x ijkt 1 j [1 m] k [1 v max] t [0 T ] i=1 Constraint 4: x ijkt = x ijksi i [1 n] j [1 m] k [1 v j] t (s i s i+p i) and x ijkt = 0 i [1 n] j [1 m] k [1 v j] t / [s i s i+p i) D. Mathematical model 2 In this model the t factor is removed from the decision variable. The decision variable is represented as below: 1 Job Ji is executed using k x ijk = th VM on M j i [1 n] j [1 m] k [1 v max]. Then the y jt variables can be calculated by: max x ijk i ɛ(t)k [1v j ] or y jt x ijk i ɛ(t) k [1 v j] as all of them are binary values. Constraints are then represented as below: Constraint 1: v m j x ijk = R i i [1 n] Constraint 2: j=1 k=1 v j x ijk v j j [1 m] t [0 T ] i ɛ(t) k=1 Constraint 3: x ijk 1 j [1 m] k [1 v j] t [0 T ] i ɛ(t) Constraint 4: predetermined implicitly through decision variables definition. E. Mathematical model 3 The decision variable in this model is defined as: x ij = k Job Ji is executed on machine M j using k VM i [1 n] j [1 m]. In this case the y jt variables are calculated by: x ij > 0 1 i [1 n] j [1 m] t [0 T ] or x ij y jt 0 where x ij is integer number and y jt is binary number. And constraints are formulated as:

Constraint 1: x ij = R i i Constraint 2: x ij v j j t j i ɛ(t) Constraint 3: predetermined implicitly Constraint 4: predetermined implicitly VI. EMPIRICAL RESULTS To test the effectiveness of the models we have used COIN-OR CBC (free version) a well-known solver to implement proposed MILP models and compared their computational times. We have measured the performance of models on personal computer with the following configuration: Intel Core i5 2.50GHz 4GB memory run on Ubuntu 12.04 operating system. In this experiment according to each couple (n m) = (number of jobs number of machines) the necessary data test set are generated as below: Processing time p i is generated randomly by integer uniform distribution in [1 3]; Number of virtual machine required by a job R i is generated randomly by integer uniform distribution in [1 5]; Starting time for a job s i is chosen with equal probability for each value in range [0 9] Number of virtual machine for a physical machine v j is generated randomly by integer uniform distribution in [1 5]; For each (n m) couple 10 instances are generated executed using COIN-OR CBC solver and then below information are extracted from the result: computational time that the solver needs to execute the model number of iterations the solver has gone through to archive the optimal solution total number of cuts the solvers has done using its cutting algorithms. Below tables show the average and maximum values accumulated from the solver output: 10 2 25.5 0.342 55.8 79 0.65 141 10 3 1263.3 0.723 547.1 7798 1.62 2395 20 2 160.1 0.688 212.1 520 1.07 611 20 3 10085.3 4.395 1296.8 30653 12.00 2685 30 3 5625.8 4.918 1612.1 13309 8.77 3074 30 5 34564.9 17.274 955.7 291780 84.23 1792 40 5 185068.2 66.622 1353.1 1209247 311.25 4731 Fig. 1. CBC experiment result for model 1 10 2 30.6 0.288 48.2 12 0.69 122 10 3 1653.5 0.703 648.5 7499 1.7 2669 20 2 100 0.567 151.6 296 1.17 321 20 3 4421.3 2.396 1024.3 19254 7.17 2385 30 3 2182.3 2.507 868.8 11599 4.17 2929 30 5 7675.3 7.963 753.4 40446 18.36 1867 40 5 230861.4 78.821 2439.6 1570957 498.47 16635 Fig. 2. CBC experiment result for model 2 We can conclude the following from the experiment result: 10 2 0 0.028 0 0 0.05 0 10 3 126.8 0.107 171.8 1083 0.25 1057 20 2 0 0.042 0 0 0.07 0 20 3 3575.1 0.721 1607.3 8066 1.29 4359 30 3 2967.4 1.024 2041.3 6020 1.69 5030 30 5 5738.3 1.995 1677.4 24696 9.2 4169 40 5 886519.5 180.136 104367.8 6846602 1436.92 798533 244 Fig. 3. CBC experiment result for model 3 When the number of jobs increased the third model is the most effective one to solve. It is because when eliminating parameters from the model several constraints are removed and no new constraint is added. The complexity does not depend only on the number of jobs but it depends on other parameters also. We can see this when seeing the much difference between the average time and maximum time the solver needs to solve models. VII. CONCLUSION In this paper we have considered the resource allocation problem in cloud computing where each physical machine in cloud has a lot of virtual machine and each job needs to use a number of virtual machines during a given and fixed period. The objective aims to minimize the cost induced by total execution time on each physical machine. We prove that this allocation problem is N P -hard. Three mixed integer linear mathematical models are then proposed to represent the considering optimization problem. The performance comparison of the three proposed models is analyzed through some empirical results. Further research can be undertaken to prove whether VIRMACALLOC is weakly or strongly N P -complete. Another research direction should focus on improving solver performance with some optimization technical. Applying relaxation techniques and model-based heuristics for solving will be also an interesting research topic. REFERENCES [1] A. Beloglazov J. Abawajy and R. Buyya Energy-aware resource allocation heuristics for efficient management of data centers for Cloud computing Future Generation Computer Systems vol. 28 no. 5 pp. 755-768 2012. DOI: 10.1016/j.future.2011.04.017. [2] Garey M.R. et Johnson D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Company San Francisco. [3] Í. Goiri F. Julià R. Nou J. Berral J. Guitart and J. Torres Energyaware Scheduling in Virtualized Datacenters IEEE International Conference on Cluster Computing (CLUSTER 2010) pp. 58 67 2010. [4] B. Speitkamp and M. Bichler A Mathematical Programming Approach for Server Consolidation Problems in Virtualized Data Centers IEEE Transactions on Services Computing 3 (4) pp. 266 278 2010. [5] Chandra Chekuri S. Khanna On Multi-Dimensional Packing Problems SODA 1999 pp. 185 194 1999. [6] R. Panigrahy K. Talwar and L. Uyeda Heuristics for Vector Bin Packing research.microsoft.com 2011. [7] B. Sotomayor K. Keahey and I. Foster Combining batch execution and leasing using virtual machines HPDC 08 Proceedings of the 17th international symposium on High performance distributed computing pp. 87 96 2008. DOI: 10.1145/1383422.1383434 [8] B. Sotomayor Provisioning Computational Resources Using Virtual Machines and Leases PhD Thesis submited to The University of Chicago. US 2010.

[9] N. Q. Hung N. Thoai and N. T. Son Performance constraint and power-aware allocation for user requests in virtual computing lab Journal of Science and Technology (Vietnam) vol. 49 4A no. Special on Int. Conf. on Advanced Computing and Applications (ACOMP2011) pp. 383 392 2011. [10] J. R. Correa and M. R. Wagner LP-based online scheduling : from single to parallel machines Math. Program. Ser. A (2009) vol. 119 pp. 109 136 2009. [11] A. Schulz Scheduling unrelated machines by randomized rounding SIAM Journal on Discrete Mathematics vol. 15 no. 4 p. 450 2002. [12] I. Al Azzoni Power-Aware Linear Programming Based Scheduling for Heterogeneous Computer Clusters Future Generation Computer Systems vol. 28 no. 5 pp. 745 754 May 2011. 245