Data Centers to Offer Ancillary Services

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4 D. Probability Model of q(μ) Consider a new service request that arrives within time slot Λ. Let Q denote the number of service requests waiting in the service queue right before the arrival of the new service request. Since the data center s service rate is μ, it takes Q/μ seconds until all existing requests are removed from the queue. Hence, the new service request can be handled after Q/μ seconds since its arrival. According to the SLA, if Q/μ D, then the request is handled before the deadline D. If Q/μ > D, the request is not handled before the deadline D and it is dropped. Therefore, we can model the SLA-deadline by a finite-size queue with the length μd. A service request can be handled before the SLA-deadline, if and only if it enters the aforementioned finite size queue. We assume that the service request rate has an arbitrary and general probability distribution function. On the other hand, since the service rate μ is fixed over each time interval of length T, q(μ) can be modeled as the loss probability of a G/D/1 queue. Therefore, following the queuing theoretic analysis in [1], we can obtain q(μ) =α(μ) e 1 min m n(μ) n 1, (9) where 1 α(μ) = λ πσ e (μ λ) σ μ (r μ)e (r λ) σ dr (1) and for each n 1 we have (Dμ + n(μ λ)) m n (μ) =. (11) nc λ () + n 1 C λ (l)(n l) Here, σ = C λ () and C λ denotes the auto-covariance of the service request rate [13]. l=1 IV. PROFIT MAXIMIZATION A. The Case without Offering Ancillary Service For the case where the data center does not offer ancillary service, its profit at each time slot Λ is obtained as Profit = Revenue Cost, (1) where revenue is as in (3) and cost is as in (7). We seek to choose the data center s service rate μ to maximize profit. This can be expressed as the following optimization problem: Maximize Tλ[(1 q(μ))δ q(μ)γ] λ μ M max ( ) Tω, (13) where the probability q(μ) is as in (9). We note that the service rate μ is lower bounded by λ. This is necessary to assure stabilizing the service request queue [1], [14]. We also note that problem (15) needs to be solved separately for every time slot Λ {Λ 1,...,Λ N }, i.e., once every T minutes. Dollars / Time Slot Compensation Profit Loss Profitable Power Reduction Range Δ P (Megawatts) Fig. 3. An example to compare profit loss due to load reduction versus the compensation received due to offering ancillary service. The arrow indicates that the optimal amount of load reduction ΔP =.68 Megawatts. B. The Case with Offering Ancillary Service For the case where data center does offer ancillary service, the data center s profit at each time slot Λ is obtained as Profit = Revenue Cost + Compensation, (14) where compensation is as in (8). We seek to choose the data center s service rate μ to maximize profit. This can be expressed as the following optimization problem: Maximize Tλ[(1 q(μ))δ q(μ)γ] λ μ M max ( Tω ( Tτ P ) + ), (15) where the last term is based on the definition of ΔP in Section III-C and the expression for power consumption in (6). C. Geometric Interpretation Let Profit Base denote the optimal objective value of problem (13). That is, the maximum profit that a data center can obtain, by properly selecting its service rate, when the data center does not offer any ancillary service. Clearly, offering ancillary service is beneficial if the profit when offering ancillary service is greater than Profit Base. That is, Tλ[(1 q(μ))δ q(μ)γ] ( ) Tω + (16) ( ) Tτ P Profit Base. From the definition of ΔP,wehave P (μ) =P ΔP. (17) Therefore, we can write μ in terms of ΔP as follows: μ = P 1 (P ΔP ), (18)

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