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A Computer Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page M S BE aG F 46 COVER FEATURE We calculate the incremental net present value (ΔNPV) as follows: ΔNPV = NPV − NPV P L ⇒ PT − C P T ( 1+ IK ) T N L S P − C L T T T ∑ + − E − + T T T =0 ( 1+ IK ) ( 1+ IK ) ( 1+ IR ) T N N ∑ ∑ T =0 T =0 ⇒ C L P − CT T ( 1+ IK ) T N S L T ∑ + + N T =0 ( 1+ IK ) ( 1+ IR ) T N ∑ − E. T =0 Assuming that profit from storage is equal in both the buy and lease cases—that is, using 1 Tbyte of storage from a purchased disk or from a storage cloud results in the same level of productivity—the above simplification results in the removal of the profit term P T . Capital cost Three components make up the cost of purchasing storage E: The consumer needs a disk controller to house the purchased disks. The consumer purchases disk drives in blocks based on increasing storage needs. The consumer must periodically replace disks due to failure. These components contribute to the future cash flow E as follows: E = COMPUTER DERIVATION OF THE DECISION MODEL ( ( ⎡⎢ V ⎤ − ⎡⎢ V ⎤ T ⎥Ω T −1⎥Ω )∗Ω+R T )∗G T ( ) T E T 1+ I K ⇒ E = + C T ( 1+ IK ) E = V T ( ( ⎡⎢ ⎤ − ⎡⎢ V ⎤ T ⎥Ω T −1⎥Ω )∗Ω+R T )∗G , T where Ω represents the hard disk drive size in Gbytes; V T is the storage requirement in Gbytes at year T; ⎡V T ⎤ Ω is an operator that returns the minimum number of Ω-sized disk drives needed to store V T ; G T is the predicted cost per Gbyte of disk storage at year T; R T is the disk replacement in Gbytes at year T; and C is the disk controller cost. The important insight in this formulation is that the capital cost isn’t all incurred at the start of the project, unlike in traditional NPV models. Rather, it’s a time-varying formula in which users can grow their storage systems as their requirements evolve. We modify ΔNPV to reflect this growth in capital cost: ΔNPV = N ∑ T =0 S + 1+ IK L P C − CT − ET T ( ) ( ) T 1+ I K Operating cost − C LT + N ( ) T N ∑ T =0 1+ I R P We estimate the operating cost of purchased storage, C , by T calculating the electric utility cost associated with running the + C disk controller and the disk units, plus the cost of a human operator to manage the system/data. These cost components P contribute to C as follows: T P C = (365⋅24)∗δ ∗ PC + P T ( ∗⎡⎢ V ⎤ D T ⎥Ω )+α∗H , T where δ is the utility cost ($/kilowatt per hour); P is the control- C ler’s power requirement in kWs; P is the power requirement in D kWs per disk drive; and α is the proportion of the human operator cost, H , required to maintain the system/data at year T. T L The operating cost for leased storage, C , only includes the T cost of a human operator to manage the data. Thus, we calculate L L C as CT = β∗HT , where β is the proportion of the human opera- T tor cost required to maintain the data on the leased storage at year T. Substituting ρ for (α – β), we can modify ΔNPV to reflect the operating costs as follows: ΔNPV = N ∑ T =0 −ρ∗H T − 365⋅24 ( )∗δ ∗( P + P ∗⎡⎢ V ⎤ C D T ⎥Ω )− ET ( ) T L T 1+ I K S − C + + . N ( 1+ IK ) ( 1+ IR ) T ∑ T =0 Approximating the best outcomes N Consumers often find it difficult to estimate their capital cost and the lease financing rates available to them. We therefore suggest an approximate method of evaluating the buy-or-lease decision by deriving the best possible outcome in both the purchase and lease cases. From the well-known economic Law of One Price, 1 we can derive the upper bounds for NPV P and NPV L by substituting I K and I R with the risk-free interest rate I F . The “Corollary for Estimating the Upper-bound Present Value” sidebar explains this in more detail. In turn, we can estimate the risk-free interest rate, I F , from the published return on an instrument such as government treasury bills. We can therefore derive an approximation of the decision criteria using these best NPVs, letting us further simplify the currently derived version of ΔNPV to: ΔNPV = N ∑ T =0 ( )∗δ ∗( P + P ∗⎡⎢ V ⎤ C D T ⎥Ω )− E + L T T ( ) T −ρ∗H T − 365∗24 S − C + . ( 1+ IF ) N Disk price trends 1+ I F We haven’t yet provided a function for the term G , which we need T to calculate E and to estimate the cost of disk storage at year T. T Since April 2003, we’ve been collecting SATA disk price data from Pricewatch.com on a weekly basis. We collected price data for all available disk drive sizes each week—250 Gbytes, 500 Gbytes, and so on—regardless of manufacturer or model. Figure A plots the 10 lowest prices each week for SATA disk drives across all the available drive sizes. Approximating the observed exponential trend line using regression analysis, we obtain the formula ′ − 0.0012 X G = 1.2984e . X A Computer Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page M S BE aG F
A Computer Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page M S BE aG F ′ The function G predicts the cost per Gbyte X of SATA disk storage X days from 20 April 2003, ′ with G = 1.2984 when X = 0. We can therefore X approximate the function G by assuming that T the future disk price trend conforms to the equation K ∗ eC ∗ T , where K represents the lowest storage price per Gbyte available to the consumer at T = 0; and T represents the number of years in the future. We therefore derive G as G T T = K ∗ e –0.0012∗ 365 ∗ T⇒ G = K ∗ e T –0.438∗ T . Disk replacement rates Any realistic cost model for disk storage ownership must estimate the disk replacement cost. A recent large-scale study of disk failures measured the annualized replacement rate (ARR) of disk drives in real data centers. 2 The study observed ARRs in the range of 0.5 to 13.5 percent, with the most commonly observed ARRs in the 3 percent range. In our model, we approximate R T with this empirical approximation of the disk replacement rate by using the formula R T = 0.03 * Ω * ⎡V T ⎤ Ω . In this formula, the constant 0.03 represents the observed 3 percent disk replacement rate. 2 Thus, we can simplify E T to: –0.438∗ T E = (( ⎡V ⎤ – ⎡V ⎤ ) ∗ Ω + 0.03 ∗ Ω ∗⎡V⎤ ) ∗ K ∗ e T T Ω T – 1 Ω T Ω ⇒ E = (1.03 ∗⎡V⎤ – ⎡V ⎤ ) ∗ Ω ∗ K · e T T Ω T – 1 Ω –0.438∗ T . Disk salvage value $/GByte We assume a hard disk drive can be sold in the used market for some salvage value at the end of its life. To predict this salvage value, we leverage the future disk price prediction formula, discounting the predicted price by some depreciation factor, γ, in the We assume this vendor lets users purchase raw disk storage over the Internet. We’re only interested in the storage cloud’s pricing structure for the end user, and we don’t assume any particular service access technology. Table 2 shows the assumed tiered monthly pricing structure. For illustrative purposes, we don’t consider data uploading or downloading costs. Single-user computers Single-user computers make up the vast majority of all shipped disk storage capacity. 2 For our study, we assume the same user owns and operates the single-user computer. This user’s storage requirement grows at a moderate rate of 100 Gbytes per year. Therefore, regardless of where data is stored, we assume the level of effort in managing the system/data is approximately equal in both cases because of the low storage volume involved—that is, = 0. We assume the user must purchase a new disk controller (C = $1,000), specified to consume 0.5 kW of power. Also, as storage is required, the user will purchase 2.5 2.0 1.5 1.0 0.5 y = 1.2984e – 0.0012 X Disk price Exponential (disk price) 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 Start date: 20 April 2003 Days Figure A. Weekly SATA disk price data collected from Pricewatch.com from 20 April 2003 to 19 August 2008. range [0, 1]. In equation form, this gives us the salvage value S = γ * Ω * ⎡V T ⎤ Ω ∗ K ∗ e –0.438 ∗ T , simplifying into ΔNPV as shown by Equation 1 in the main text. References 1. O.A. Lamont and R.H. Thaler, “Anomalies: The Law of One Price in Financial Markets,” J. Economic Perspective, vol. 17, no. 4, 2003, pp. 191-202. 2. B. Schroeder and G. Gibson, “Disk Failures in the Real World: What Does an MTTF of 1,000,000 Hours Mean to You?” Proc. 5th Usenix Conf. File and Storage Technologies (FAST 07), Usenix Assoc., 2007, pp. 1-16. COROLLARY FOR ESTIMATING THE UPPER-BOUND PRESENT VALUE We derive the following corollary from the economic Law of One Price, which states, “In an efficient market, identical goods will have only one price.” Corollary 1. The upper bound of the present values for the purchase and lease price equilibrium in an efficient market is derived from the risk-free interest rate, I . F Informal proof: In an efficient market, an arbitrager has no opportunity to make a risk-free profit. Assuming an efficient market in which financing at the risk-free rate is possible for a subset of agents, the proof is by contradiction. Thus, if the present value of the lease price is higher than that derived from the risk-free rate, I , F an arbitrager can purchase disks from monies borrowed at the riskfree rate, lease at this higher price, and pocket the risk-free profit. Also, if the present value of the purchase price is higher than that derived from the risk-free rate, I , an arbitrager can purchase a disk F from monies borrowed at the risk-free rate, sell the disk at some future instance at this higher price, and pocket the risk-free profit. APRIL 2010 A Computer Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page M S BE aG F 47
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