# Cost Minimization for Big Data Processing in Geo-Distributed Data Centers

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6 6 summarizing all constraints discussed above, we can formulate this cost minimization as a mixed-integer nonlinear programming (MINLP) problem as: MINLP: min : (26), s.t. : (3) (5), (7) (14), (23), (25), x j, y, z, u {0, 1}, j J, k K Note that the above formulation is based on the setting that the number of replicas for each data chunk is a predetermined constant. If it is a part of the optimization, denoted by an integer variable p, the total cost can be further minimized by the formulation below. MINLP-2: min : (26), s.t. : y = p, k K, j J p 1, 5 LINEARIZATION (4), (5), (7) (14), (23), (25), x j, y, z, u {0, 1}, j J, k K. We observe that the constraints (8) and () are nonlinear due to the products of two variables. To linearize these constraints, we define a new variable δ as follows: δ = y λ, j J, k K, (27) which can be equivalently replaced by the following linear constraints: 0 δ λ, j J, k K, (28) λ + y 1 δ y, j J, k K. (29) The constraints (8) and () can be written in a linear form as: δ uk ϕ k, (u, v) E, u, j J, k K, () (u,j) E f (u,j) = (λ δ ) ϕ k, j J, k K. (31) We then consider the remaining nonlinear constraints (14) and (23) that can be equivalently written as: γ (u,v) µ z (u,v) + 1 µ z (u,v) λ u D, (u, v) E, j J, k K. In a similar way, we define a new variable ϵ as: ϵ (u,v) (32) = µ z (u,v), (33) such that constraint (32) can be written as: γ (u,v) ϵ (u,v) + µ ϵ (u,v) λ u D, (u, v) E, j J, k K. (34) The constraint (33) can be linearized by: 0 ϵ (u,v) µ, (u, v) E, j J, k K, () µ + z (u,v) 1 ϵ (u,v) z (u,v), j J, k K. (36) Now, we can linearize the MINLP problem into a mixed-integer linear programming (MILP) as MILP: min : (26), s.t. : (3) (5), (7), (9), (11) (13), (25), (28) (31), (34) (36), x j, y, z, u {0, 1}, j J, k K. 6 PERFORMANCE EVALUATION In this section, we present the performance results of our joint-optimization algorithm ( ) using the MILP formulation. We also compare it against a separate optimization scheme algorithm ( Non-joint ), which first finds a minimum number of servers to be activated and the traffic routing scheme using the network flow model as described in Section 4.2. In our experiments, we consider J = 3 data centers, each of which is with the same number of servers. The intra- and inter-data center link communication cost are set as C L = 1 and C R = 4, respectively. The cost P j on each activated server j is set to 1. The data size, storage requirement, and task arrival rate are all randomly generated. To solve the MILP problem, commercial solver Gurobi [26] is used. The default settings in our experiments are as follows: each data center with a size 20, the number of data chunks K =, the task arrival rates λ k [0.01, 5], k K, the number of replicas P = 3, the data chunk size ϕ k [0.01, 1], k K, and D = 0. We investigate how various parameters affect the overall computation, communication and overall cost by varying one parameter in each experiment group. Fig. 3 shows the server cost, communication cost and overall cost under different total server numbers varying from 36 to 60. As shown in Fig. 3(a), we can see that the server cost always keep constant on any data center size. As observed from Fig. 3(b), when the total number of servers increases from 36 to 48, the communication costs of both algorithms decrease significantly. This is because more tasks and data chunks can be placed in the same data center when more servers are provided in each data center. Hence, the communication cost is greatly reduced. However, after the number of server reaching 48, the communication costs of both algorithms converge. The reason is that most tasks and their corresponding data chunks can be placed in the same data center, or even in the same server. Further increasing the number of servers will not affect the distributions of tasks or data chunks any more. Similar results are observed in Fig. 3(c).

7 7 31 Server Cost Communication Cost Operation Cost Number of Servers (a) Server Cost Number of Servers (b) Communication Cost Number of Servers (c) Overall Cost Server Cost Communication Cost Operation Cost 25 Tasks Arrival Rate (a) Server Cost Tasks Arrival Rate (b) Communication Cost Tasks Arrival Rate (c) Overall Cost Fig. 3. On the effect of the number of servers Fig. 4. On the effect of task arrival rate Then, we investigate how the task arrival rate affects the cost via varying its value from 29.2 to The evaluation results are shown in Fig. 4. We first notice that the total cost shows as an increasing function of the task arrival rates in both algorithms. This is because, to process more requests with the guaranteed QoS, more computation resources are needed. This leads to an increasing number of activated servers and hence higher server cost, as shown in Fig. 4(a). An interesting fact noticed from Fig. 4(a) is that algorithm requires sometimes higher server cost than Non-joint. This is because the first phase of the Non-joint algorithm greedily tries to lower the server cost. However, algorithm balances the tradeoff between server cost and communication cost such that it incurs much lower communication cost and thus better results on the overall cost, compared to the Non-joint algorithm, as shown in Fig. 4(b) and Fig. 4(c), respectively. Fig. 5 illustrates the cost as a function of the total data chunk size from 8.4 to 19. Larger chunk size leads to activating more servers with increased server cost as shown in Fig. 5(a). At the same time, more resulting traffic over the links creates higher communication cost as shown in Fig. 5(b). Finally, Fig. 5(c) illustrates the overall cost as an increasing function of the total data size and shows that our proposal outperforms Nonjoint under all settings. Next we show in Fig. 6 the results when the expected maximum response time D increases from 20 to 0. From Fig. 6(a), we can see that the server cost is a nonincreasing function of D. The reason is that when the delay requirement is very small, more servers will be activated to guarantee the QoS. Therefore, the server costs of both algorithms decrease as the delay constraint

8 8 Server Cost Communication Cost Operation Cost Data Size (a) Server Cost Data Size (b) Communication Cost Data Size (c) Overall Cost Fig. 5. On the effect of data size Server Cost Communication Cost Operation Cost Delay 80 0 (a) Server Cost Delay (b) Communication Cost Delay 80 0 (c) Overall Cost Fig. 6. On the effect of expected task completion delay increases. A looser QoS requirement also helps find costefficient routing strategies as illustrated in Fig. 6(b). Moreover, the advantage of our over Non-joint can be always observed in Fig. 6(c). Finally, Fig. 7 investigates the effect of the number of replicas for each data chunk, which is set from 1 to 6. An interesting observation from Fig. 7(c) is that the total cost first decreases and then increases with the increasing number of replicas. Initially, when the replica number increases from 1 to 4, a limited number of activated servers are always enough for task processing, as shown in Fig. 7(a). Meanwhile, it improves the possibility that task and its required data chunk are placed on the same server. This will reduce the communication cost, as shown in Fig. 7(b). When the replica number becomes large, no further benefits to communication cost will be obtained while more servers must be activated only for the purpose of providing enough storage resources. In this case, the server and hence the overall costs shall be increased, as shown in Fig. 7(a) and Fig. 7(c), respectively. Our discovery that the optimal number of chunk replicas is equal to 4 under the network setting above is verified one more time by solving the formulation given in Section 4.4 that is to minimize the number of replicas with the minimum total cost. Additional results are given under different settings via varying the task arrival rate and chunk size in the ranges of [0.1, λ U ] and [ϕ L, 1.0], respectively, where a number of combinations of (λ U, ϕ L ) are shown in Fig. 8. We observe that the optimal number of replica a non-decreasing function of the task arrival rate under the same chunk size while a non-increasing function of the data chunk size under the same task arrival rate.

9 9 Server Cost Communication Cost Operation Cost Number of Replica 5 6 (a) Server Cost Number of Replica 5 6 (b) Communication Cost Number of Replica 5 6 (c) Overall Cost Fig. 7. On the effect of the number of replica 7 CONCLUSION In this paper, we jointly study the data placement, task assignment, data center resizing and routing to minimize the overall operational cost in large-scale geo-distributed data centers for big data applications. We first characterize the data processing process using a two-dimensional Markov chain and derive the expected completion time in closed-form, based on which the joint optimization is formulated as an MINLP problem. To tackle the high computational complexity of solving our MINLP, we linearize it into an MILP problem. Through extensive experiments, we show that our joint-optimization solution has substantial advantage over the approach by two-step separate optimization. Several interesting phenomena are also observed from the experimental results. Optimal Number of Replica (3,0.1) (4,0.1) (5,0.1) (3,0.8) (4,0.8) (5,0.8) Arrival Rate & Chunk Size Fig. 8. Optimal number of replica REFERENCES [1] Data Center Locations, centers/inside/locations/index.html. [2] R. Raghavendra, P. Ranganathan, V. Talwar, Z. Wang, and X. Zhu, No Power Struggles: Coordinated Multi-level Power Management for the Data Center, in Proceedings of the 13th International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS). ACM, 2008, pp [3] L. Rao, X. Liu, L. Xie, and W. Liu, Minimizing Electricity Cost: Optimization of Distributed Internet Data Centers in a Multi- Electricity-Market Environment, in Proceedings of the 29th International Conference on Computer Communications (INFOCOM). IEEE, 20, pp [4] Z. Liu, M. Lin, A. Wierman, S. H. Low, and L. L. Andrew, Greening Geographical Load Balancing, in Proceedings of International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS). ACM, 2011, pp [5] R. Urgaonkar, B. Urgaonkar, M. J. Neely, and A. Sivasubramaniam, Optimal Power Cost Management Using Stored Energy in Data Centers, in Proceedings of International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS). ACM, 2011, pp [6] B. L. Hong Xu, Chen Feng, Temperature Aware Workload Management in Geo-distributed Datacenters, in Proceedings of International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS). ACM, 2013, pp [7] J. Dean and S. Ghemawat, Mapreduce: simplified data processing on large clusters, Communications of the ACM, vol. 51, no. 1, pp , [8] S. A. Yazd, S. Venkatesan, and N. Mittal, Boosting energy efficiency with mirrored data block replication policy and energy scheduler, SIGOPS Oper. Syst. Rev., vol. 47, no. 2, pp. 33, [9] I. Marshall and C. Roadknight, Linking cache performance to user behaviour, Computer Networks and ISDN Systems, vol., no. 223, pp , [] H. Jin, T. Cheocherngngarn, D. Levy, A. Smith, D. Pan, J. Liu, and N. Pissinou, Host-Network Optimization for Energy- Efficient Data Center Networking, in Proceedings of the 27th International Symposium on Parallel Distributed Processing (IPDPS), 2013, pp [11] A. Qureshi, R. Weber, H. Balakrishnan, J. Guttag, and B. Maggs, Cutting the Electric Bill for Internet-scale Systems, in Proceedings of the ACM Special Interest Group on Data Communication (SIGCOMM). ACM, 2009, pp [12] X. Fan, W.-D. Weber, and L. A. Barroso, Power Provisioning for A Warehouse-sized Computer, in Proceedings of the 34th Annual International Symposium on Computer Architecture (ISCA). ACM, 2007, pp [13] S. Govindan, A. Sivasubramaniam, and B. Urgaonkar, Benefits and Limitations of Tapping Into Stored Energy for Datacenters, in Proceedings of the 38th Annual International Symposium on Computer Architecture (ISCA). ACM, 2011, pp

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