# Profit Maximization and Power Management of Green Data Centers Supporting Multiple SLAs

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3 seconds handling service requests of class i within the time interval T. Therefore, by dividing the total CPU busy time T ( q i ( ))λ i M by T, the CPU utilization for each server is obtained as U = (6) ( q i ( ))λ i. (7) M Replacing (2) and (7) in (), the power consumption associated with the data center at the time slot of interest is obtained as [ ] aμi + bλ i ( q i ( )) P =, (8) where a = P idle +(E usage )P peak, (9) b = P peak P idle. () Multiplying (8) by the electricity price ω, the total energy cost at the time interval of interest is obtained as [ ] aμi + bλ i ( q i ( )) Cost = Tω. () C. Probability Model of q i ( ) We model the ith SLA-deadline by a finite size queue with the length D i [7], [8]. A service request of class i can be handled before the ith SLA-deadline, if and only if it enters the aforementioned ith finite size queue. We assume that the ith service request rate has mean λ i and an arbitrary general probability distribution. On the other hand, since the service rate is fixed over each time interval of length T, q i ( ) is modeled as the loss probability of a G/D/ queue. Therefore, following the queuing theoretic analysis in [9], we can obtain where α i ( )= q i ( )=α i ( ) e 2 min n m n,i( ), (2) (μ λ λ i ) 2 2σ e i 2 (r λ i )2 2σ (r )e i 2 dr (3) 2πσ i and for each n we have (D i + n( λ i )) 2 m n,i ( )=. (4) nc λi () + 2 n C λi (l)(n l) Here, σ i = C λi () and C λi (l) denotes the auto-covariance with lag time l of the rate of the incoming service request of ith class. The model in (2) has its most accuracy when the ith service request arrival rate is as a Gaussian process. However, it also works well for any general probability distribution [9], as we will confirm through simulations in Section IV. l= D. Profit Maximization without Local Renewable Generation At each time slot Λ, data center s profit is obtained as Profit = Revenue Cost, (5) where revenue is as in (3) and cost is as in (). We seek to choose the data center s service rates to maximize profit. This can be expressed as the following optimization problem: Maximize λ i Subject To T Tω [ q i ( ))δ i λ i q i ( )γ i λ i ] [ ] (6a) aμi + bλ i ( q i ( )) M max, (6b) where q i ( ) is as in (2) and M max denotes the maximum number of servers that are available in the data center. We note that, for each service class i, the service rate is lower bounded by λ i. This is necessary to assure stabilizing the service request queues [7], [9]. We also note that the N- variable optimization problem in (6) can be decomposed into N separate optimization problems whenever M max is too large. Finally, we note that Problem (6) needs to be solved once for every time slot Λ {Λ,...,Λ N }, i.e., every T minutes using a simple exhaustive search. E. Profit Maximization with Local Renewable Generation When a data center is supplied by both power grid and a local behind-the-meter renewable generator, then the optimum choices of service rates for maximizing profit is obtained by solving the following optimization problem Maximize T [ q i ( ))δ i λ i q i ( )γ i λ i ] λ i [ N ] + (7a) a + bλ i ( q i ( )) Tω G Subject To M max, (7b) where [x] + = max(x, ). Note that, in Problem (7), optimization variables are coupled not only in constraint (7b), but also in objective function (7a) due to the term [x] +. IV. PERFORMANCE EVALUATION A. Simulation Setting Consider a data center and assume that the number of switched on servers M is updated periodically at the beginning of each time slot of length T = 5 minutes by solving Problems (6) and (7), for the cases without and with behindthe-meter renewable power generation at the data center,

4 (a) (b) (c) (d) Fig.. Comparing analytical and simulation results to calculate data center profit. (a) Simulation results when parameter M max = 5. (b) Simulation results when parameter M max = 5. (c) Analytical results when parameter M max = 5. (d) Analytical results when parameter M max = 5. respectively. For each switched on server, we have P peak = 2 watts and P idle = watts [6]. We assume that E usage =.2, which is the reported state of the art power usage effectiveness [7]. The electricity price information in our simulations are based on the real-time pricing tariffs currently practiced in Illinois Zone I, on June, 9, 2 []. The prices are updated every hour. We assume that the data center offers N = 2 classes of services, which are identical to the Gold and Silver services explained in the example SLA model in [7]. We also assume that κ =. and κ 2 =.25. To simulate the total workload, we use the publicly available World Cup 98 web hits data on June 4, 998 and June, 9, 998 as the service request trend for service class and service class 2, respectively []. B. Simulation Results for a Single Time Slot To gain insights about the achievable profit, in this section, we focus on a single time slot of length T = 5 minutes starting at 2: PM and we investigate the solution of the profit maximization problem in (6). The simulation versus analytical results for different choices of system parameter M max are shown in Fig.. For the results in Fig. (a) and (c), we have M max = 5. We can see that, since the number of available servers is large, constraint (6b) is not binding and the choices of optimization variables μ and μ 2 can almost arbitrarily increase. However, given the tradeoff between minimizing energy expenditure and maximizing revenue, optimal solution enforces switching on only a limited number of servers. Furthermore, we can see that the simulation curve in Fig. (a) and the analytical curve in Fig. (c) are very similar, suggesting that our proposed analytical profit model is accurate. For the results in Fig. (b) and (d), we have M max = 5. We can see that, since the number of available servers is small, constraint (6b) is binding. As a result, the optimal solution is achieved when μ /κ +μ 2 /κ 2 = M max. That is, at optimality, we need to switch on all servers that are available in the data center. Nevertheless, we can see that although the feasible set for optimization problem (6) is significantly smaller when M max = 5, the profit curve remains concave, again indicating the trade-off between minimizing energy expenditure and maximizing revenue. C. Simulation Results for an Entire Day For the rest of this section we assume that M max = 5. Simulation results over 24 hours are shown in Fig. 2. The time-of-day price of electricity is as in Fig. 2(a) [2]. We can see that the normalized profit gain in Fig. 2(b) is very close to optimal with 97% optimality on average. The normalized profit gain is calculated as (Profit Profit Base ) divided by (Profit Max Profit Base ). Here, Profit Base is the profit obtained when we simply set = λ i [3] and Profit Max is the maximum of the profit curve obtained by simulation. D. Impact of Behind-the-Meter Renewable Generation Next, assume that the data center is equipped with a wind turbines, with.5 Megawatts peak generation. The power

5 4 (a) (b) Electricity Price (Cent/kwh) Normalized Profit Gain (c).4 (d).35 Wind Power (Megawatts).5 Additional Profit with Renewable Generators (%) Fig. 2. Experimental data and the simulation results for 24 hours operation of the data center: (a) Time-of-Day Prices [2]. (b) Obtained normalized profit gain without local renewable power generation. (c) Available wind power [4]. (d) Additional profit (in percentage) with local renewable power generation. output for each turbine is assumed to be as in Fig. 2(c), based on the wind speed data available in [4] for June 4, 2. In this case, optimal service rate is obtained by solving Problem (7). The corresponding additional profit gain (in percentage) due to local renewable generation is shown in Fig. 2(d). We can see that local renewable generation can significantly increase the data center s daily profit. V. CONCLUSIONS AND FUTURE WORK In this paper, we proposed a novel analytical model to calculate profit in large data centers without and with behind-themeter renewable power generation. Our proposed model takes into account several factors including the practical servicelevel agreements that currently exist between data centers and their customers, price of electricity, and the amount of renewable power available. We then used the derived profit model to develop an optimization-based profit maximization strategy for data centers. Using various experimental data and via computer simulations, we assess the accuracy of the proposed mathematical model for profit and also the performance of the proposed optimization-based profit maximization strategy. The results in this paper can be extended in several directions. First, the cost model can be extended to include cost elements other than energy cost. Second, the proposed energy and performance management method can be further extended to daily or monthly planning, e.g., to decide whether installing more servers is economical. Finally, the system model can be adjusted to also include potential profit if a data center participates in ancillary services in the power grid. REFERENCES [] S. Steinke, N. Grunwald, L. Wehmeyer, R. Banakar, M. Balakrishnan, and P. Marwedel, Reducing energy consumption by dynamic copying of instructions onto onchip memory, in Proc. of the International Symposium on System Synthesis, Kyoto, Japan, Oct. 22. [2] J. Heo, D. Henriksson, X. Liu, and T. Abdelzaher, Integrating adaptive components: An emerging challenge in performance-adaptive systems and a server farm case-study, in Proc. of the IEEE International Real- Time Systems Symposium, Tucson, AZ, Dec. 27. [3] H. Mohsenian-Rad and A. Leon-Garcia, Coordination of cloud computing and smart power grids, in Proc. of the IEEE Smart Grid Communications Conference, Gaithersburg, MD, oct 2. [4] X. Fan, W. D. Weber, and L. A. Barroso, Power provisioning for a warehouse-sized computer, in Proc. of the ACM International Symposium on Computer Architecture, San Diego, CA, June 27. [5] L. Rao, X. Liu, L. Xie, and W. Liu, Minimizing electricity cost: Optimization of distributed internet data centers in a multi-electricitymarket environment, in Proc. of IEEE INFOCOM, Orlando, FL, 2. [6] M. Ghamkhari and H. Mohsenian-Rad, Optimal integration of renewable energy resources in data centers with behind-the-meter renewable generators, in Proc. of the IEEE International Conference on Communications, Ottawa, Canada, June 22. [7], Energy and performance management of green data centers: A profit maximization approach, submitted to IEEE Trans. on Smart Grid, May 22. [8], Data centers to offer ancillary services, in Proc. of the IEEE SmartGridComm2, Tainan City, Taiwan, 22. [9] H. Kim and N. Shroff, Loss probability calculations and asymptotic analysis for finite buffer multiplexers, Networking, IEEE/ACM Transactions on, vol. 9, no. 6, pp , dec 2. [] [] [2] [3] X. Zheng and Y. Cai, Reducing electricity and network cost for online service providers in geographically located internet data centers, in Green Computing and Communications (GreenCom), 2 IEEE/ACM International Conference on, aug. 2, pp [4]

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