Storage Buy-or-Lease Decisions in Cloud Computing ... - IEEE Xplore

7TH EURO-NF CONFERENCE ON NEXT GENERATION INTERNET

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Storage Buy-or-Lease Decisions in Cloud Computing under Price Uncertainty Loretta Mastroeni and Maurizio Naldi, Senior Member, IEEE,

Abstract—Cloud computing allows to lease computational resources on demand on the network instead of buying them. Storage is a major computational resource that may be leased under the cloud computing approach. An economical model is provided for the analysis of the Buy-or-Lease decision for storage in cloud computing. The model introduces a probabilistic approach to evaluate future prices and the replacement needs for disk arrays (in contrast to the current literature which considers deterministic prices and fixed replacement rates). The median Differential Net Present Value is proposed as the decision variable. A simulation analysis is conducted that shows that the cloud computing approach appears more profitable for medium-sized companies with longer investment plans. The risks associated to wrong decisions is evaluated through the Value-atRisk tool. Index Terms—Cloud computing, Cost models, Virtualization, Storage.

I. I NTRODUCTION LOUD computing is an operational paradigm by which computing resources may be provided on demand over a computer network. A user of computing resources may therefore operate without owning the resource, but rather employing resources owned by others. By resorting to cloud computing, computational tasks may be carried out with an infrastructure that is partially owned and partially leased from a provider. Cloud computing is therefore an essential element of a virtualization strategy. The IT infrastructure is deployed in a provider’s data center as virtual machines, so that cloud computing represents an infrastructure-as-a-service (IaaS) paradigm. It was popularized in 2006 by Amazon’s Elastic Compute Cloud service, which offered virtual machines for 0.10$ an hour. Nowadays it represents an established means for the cloud user to reduce its capital costs. In fact, it introduces flexibility in the use of resources, by adding or removing capacity in the IT infrastructure to meet peak or fluctuating service demands, while paying only for the capacity actually used [1]. Clouds can be public, private, or hybrid. Private clouds allow users belonging to the same administrative domain to use computing resources scattered throughout that domain, so to achieve flexibility and efficiency. However, the most widespread employment of clouds is in the public version, where commercial cloud providers offer a publicly accessible

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M. Naldi is with the Department of Computer Science, University of Rome at Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy e-mail: [email protected]. Loretta Mastroeni with the Department of Economics, University of Rome Tre, Via Silvio D’Amico 77, 00145 Roma, Italy e-mail: [email protected].

remote interface to their proprietary infrastructure. In hybrid clouds the local infrastructure is supplemented with computing capacity leased from an external public cloud. Examples of management systems for private and hybrid clouds are described in [2]. When resorting to a public cloud, the cloud user has to pay a price for the leased resource. A marketplace has therefore been envisaged for the computing resources leased in cloud computing [3] [4]. Such prices are essential elements for the decisions taken by the cloud user. In fact, the prospective cloud user has to compare the leasing approach embedded in the cloud computing paradigm with the alternative decision to buy the computing resource and operate its own infrastructure. A natural comparison criterion is to consider the economics of the two alternative solutions, and solve the Buy-or-Lease dilemma. The economic evaluation of the leasing solution is essential for the development of the cloud paradigm. In fact, as hinted above, the main promise of that paradigm is to reduce the capital costs for the cloud user. One of the most relevant computational resources that may be leased in a cloud computing marketplace is storage. In this case we talk of data-as-a-Service (daaS) or data-Storage-as-aService (dSaaS) [1]. An example of such on-demand storage service is Amazon.coms S3 service, which lets users store digital objects on the online repositories at a price of 0.14$ per GB per month for the first TB (as of March 2011). In this case the Buy-or-Lease decision has to be taken by comparing the cost of buying the array of disks and running them versus paying a price for leasing by adopting the cloud approach. In this paper we deal with the comparison of the Buy versus the Lease approach on the basis of their respective economics. We aim at providing guidelines, based on the characteristics of the prospective cloud user and its memory requirements, to plan the use of cloud computing rather than the massive purchase of disks. An economical model to compare the two approaches has been proposed in [5], which takes into account both the capital expenses (CAPEX) and the operational expenses (OPEX), and provides a decision criterion based on the Net Present Value (NPV) of the two investment alternatives. However, in that model the future prices of disks are considered as deterministic quantities, as well as the future leasing prices and the disk replacement rates due to failures, while they are actually unknown and should be treated as random. In addition, that deterministic model does not consider the possibility that the decision turns out to be wrong, since no deviations from the deterministic prices are taken into account, and is therefore useless to evaluate the risk associated to that decision criterion. In this paper we revisit that model, by accounting for the


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randomness of both the prices and the failure rates. We provide therefore a modelling tool that allows the prospective cloud user to take a Buy-or-Lease decision in a probabilistic context, and evaluate the risks associated to that decision should it turn out to be wrong. We put forward the use of the median Differential Net Present Value as a decision variable in the Buy-or-Lease problem. We find that companies with longer time horizons (e.g., making a business plan for the next ten years) and medium size can get the larger benefits from turning to storage virtualization. Instead, the Buy option may be more suitable for large companies, especially over shorter time horizons. In addition, we introduce in this context the notion of Value-at-Risk to quantify the risk associated to the Buy-or-Lease decision, which results to be maximum in the region where the two approaches lead to very close economical performances. The paper is organized as follows. In Sections II and III we describe the model that incorporates the economical items associated to either the Buy or the Lease decision. We describe in detail the probabilistic modelling of future prices and of disk failures in Sections IV through VI. The analysis of the decision criterion is carried out through simulation in Section VIII for the scenario depicted in Section VII. II. T HE D IFFERENTIAL N ET P RESENT VALUE The decision to lease the storage devices from the cloud rather than buy it has to be taken on the basis of their respective economic values. An established tool to conduct such analysis is the Net Present Value [6]. In this section we review that tool and show how it can be applied to compare the two decisions in this context. In the following the entity using storage resources, be they leased from the cloud or bought, is generically referred to as the company. The owner of the data center that provides storage resources on demand is referred to as the lessor. Any investment project can be examined through the sequence of its cash flows, where positive values represent profits and negative values represent expenses. Since the project may extend over a number of years, we have to consider that the same amount of cash has a different value depending on the time when it takes place: the longer it is in the future the lower its value. Such principle represents the time value of money: future cash flows are reduced (discounted) to take into account their relatively lower value with respect to cash flows closer in time. The Net Present Value of a project is the algebraic sum of all the cash flows pertaining to that project, adequately discounted to account for their occurrence in time, i.e., the algebraic sum of the present values of all cash flows NPV =

N X t=0

Ft , (1 + k)t

(1)

where Ft is the cash flow at time t, k is the discount rate, and the project has a lifetime of N years (as customary, we assume that all the cash flows are lumped in one year intervals). According to the NPV criterion a project is better the larger its NPV. In our case the two projects to compare are the purchase and use of an array of storage devices versus the use

of leased storage devices of the same kind. In order to compute their respective NPVs we have to identify and evaluate the cash flows associated to them. In the following we describe the cash flows categories of interest in this context, adopting the classification employed in [5]. In the case of the Buy decision, the company incurs capital expenses to buy the disks and operational expenses to run the disk farm. Such expenses extend over the whole lifetime of the project, since new disks have to be bought over time to cope with the growing memory needs. At the same time those disks are employed to manage and sell services, and therefore contribute to generate profits. And, when the project is finished, the disks have however a residual (salvage) value, since they can be sold on the secondary market. If we indicate the sequence of profits by {Gt }, that of the capital expenses by {Ct }, the operational expenses by {Obuy,t } and the salvage value by S, the NPV under the Buy decision is NPVbuy =

N X Gt − Ct − Obuy,t t=0

(1 + k)t

+

S . (1 + k)N

(2)

Instead, if the company decides to lease storage, it does not incur any capital expenses, but has to pay a periodic price for the leasing contract (which we indicate by the sequence Lt ). It does incur operational expenses, though reduced with respect to the Buy case (though the leasing cost is itself a recurring cost that may be classified under the operational expenses, in this context we prefer to highlight it and account for it separately). And it can expect to get the same profits as in the Buy case. But it cannot make a profit by selling the devices on the secondary market at the end of the project. The NPV under the Lease decision is therefore NPVlease =

N X Gt − Lt − Olease,t t=0

(1 + k)t

.

(3)

The decision criterion in the Buy-or-Lease problem is then to opt for Buy if NPVbuy > NPVlease , and for Lease otherwise. Rather than computing the NPVs in the two cases and selecting the decision yielding the larger NPV, we can compute the difference between the two NPVs. Since the profits Gt are the same in the two cases, they cancel each other in the differential NPV, thus removing the need for their estimation. This differential NPV is therefore easier to compute. If we mark the present values of each group of cash flows by an overline, we can express the differential NPV as follows ∆NPV = NPVbuy − NPVlease = G − C − Obuy + S − (G − Olease − L)

(4)

= S + L + (Olease − Obuy ) − C The decision criterion is therefore to opt for the Buy decision if ∆NPV > 0, and for the Lease decision otherwise. III. P RESENT VALUE COMPONENTS In Section II we have set the general formula that provides the variable used in the decision criterion. However, we have to define exactly each of its components. In this section we provide such definitions.


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A. Capital Expenses Capital expenses are incurred just if the company decides to buy storage rather than lease it. In this case the company has to buy both the storage devices themselves (the disks) and the disk controller. Since the need for memory evolves over time, the disks are not bought upfront just in the first year of operations. Rather, their purchase is distributed over the years. We need therefore a model for the evolution of storage needs over the years. We also have to take into account that existing disks are subject to failures over their lifetime and have to be replaced. The overall number of disks bought during a year includes therefore both the disks bought to cope with the growing storage needs and those bought to replace the failed ones. We assume, as in [5], that the storage needs grow linearly over time. We indicate the overall memory needed at time t = 0, 1, . . . , N as Mt , expressed in GB. Though disk sizes tend to grow over the years, we assume that all the disks have the same capacity Ω, expressed again in GB. The number of working disks needed at time t is Dt = dMt /Ωe. As hinted at the beginning of this section, disks may fail and have to be replaced. Hence we have to consider the number Rt of disks to be replaced at time t. In [5] failures (and therefore replacements) are assumed to take place deterministically, at a fixed rate each year. This assumption is quite unrealistic, since failures occur randomly. In this paper we take therefore the more realistic assumption that Rt is a random variable. In Section VI we describe the model we employ for the number of failures taking place each year. Since the company has to buy new disks to cope with growing memory needs and to replace the existing ones, the overall number of disks to be purchased at time t is the sum At = Dt − Dt−1 + Rt . In addition to the disk, the company has to buy a disk controller (which we assume to be bought upfront), whose cost is Ccont . If we indicate the market unit price per GB of storage capacity at time t by Pt , the present value of the capital expenses is N X At ΩPt C = Ccont + . (5) (1 + k)t t=0 B. Operational Expenses In addition to CAPEX, the company incurs operational expenses to run the disks. These expenses are incurred, though to a different extent, both in the Buy and in the Lease cases. Namely, under the Buy case we consider both power costs and personnel costs. For power costs we consider a unit cost δ per kWh, and a power consumption Econt for the controller and Edisk for each disk (both expressed in kW). The personnel cost for year t is indicated as a multiple λ of the unit cost Ht . The present value of OPEX under the Buy decision is then Obuy =

N X 365 · 24 · δ(Econt + Edisk Dt ) + λHt t=0

(1 + k)t

lessor, but still incurs a portion of the personnel costs. We indicate by φ the multiple of the unit personnel cost incurred. The present value of OPEX under the Lease decision is then Olease =

N X t=0

φHt . (1 + k)t

(7)

When computing the differential NPV we can consider the difference of the two OPEX totals O = Olease − Obuy =

N X (φ − λ)Ht − 365 · 24 · δ(Econt + Edisk Dt ) t=0

(1 +

k)t

(6)

If the company leases the disks through the cloud, it then incurs no energy costs, since these pertain to the data center’s

(8)

C. Salvage value Under the Buy decision the company owns its own disks. At the end of the time period considered for analysis, it is however left with a number of disks it would not have under the Lease decision. Such disks have a value on the secondary market. In order to perform a level comparison between the two decisions, we have to take into account the salvage value of the disks and include it among the positive contributions to the NPV under the Buy decision. As in [5] we assume that the sale is accomplished at the end of the analysis period, and the salvage value is a fixed percentage of the market price for a new disk of the same capacity, regardless of the actual age of the disk sold on the secondary market. If we indicate by γ < 1 that percentage (the salvage factor), the present value S of the disks sold at time N is γΩDN PN . (9) S= (1 + k)N IV. P RICE FLUCTUATIONS Under the Buy decision the company undertakes a sequence of purchases over time to cope with its growing memory needs. When deciding whether to buy or to lease (i.e., at time 0), the company knows the current market price of disks but cannot know in advance what the future price of disks will be. However, an extensive survey conducted on SATA disk prices every week over more than 5 years has shown that the price follows a decaying trend which is approximately exponential [5]: Pt ' P0 e−βt , (10) with P0 = 0.3$ and β = 0.438 In the deterministic approach taken in the Buy-or-Lease analysis conducted in [5], the price resulting from expr. (10) is assumed to derive the Buy-orLease decision. However, the prediction stated by expr. (10) is of course approximate, since real prices fluctuate around the values predicted by that exponential curve. Here we abandon the deterministic model, and assume instead that the future price is described by a random process with expected value defined by expr. (10): E[Pt ] = P0 e−βt .

.

.

(11)

Namely, we assume that the price is described by a Geometric Brownian Motion [7]: Pt = P0 eνt+σW (t) = P0 e(µ−σ

2

/2)t+σW (t)

,

(12)


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with W (t) being a Wiener process. The terms ν and σ 2 are respectively the drift and variance of the process. Since the expected value of such process is E[Pt ] = P0 eµt ,

(13)

we can easily match the characteristics of this process with our assumption (11) on the expected value of the price. Namely, the match is perfect by setting µ = −β. At any time t the price follows a lognormal distribution around its expected value decaying according to expr. (11). A sample picture of the resulting unit price behaviour is shown in Fig. 1. 0.3 0.25

Price [$]

0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

Time [years] Fig. 1.

Sample decay of storage unit prices

V. L EASING COST If the company decides to lease storage rather than buy it, a major item in the profit-and-loss computation is the leasing cost, i.e., the sum the company has to pay to the lessor, the data center’s owner that lets storage. Such sum covers all the operational costs strictly associated to the storage device itself during that period, e.g., the electrical power needed to run the devices. In this section we describe how we account for such cost in the NPV computation. In [5] the leasing cost was assumed to be deterministic and proportional to the overall memory leased. Moreover, the leasing unit cost was not assumed to be related to the market price of the storage device. We consider those to be two relevant shortcomings of that model and propose an alternative model for the leasing cost that removes both. The issue of pricing is a major one for the development of cloud computing and has been dealt with in the literature. In [3] the concept of a marketplace for computing resources has been introduced, where such resources are leased for even small time intervals. Since storage is a computing resource, we can extend that concept for the daaS case. However, in [3] an auction mechanism is envisaged to assign resources and set prices. Though auctions may suit the needs of the market players, and help the lessor extract as much value as possible from its resources, they may not be the best tool to consider when planning Buy-or-Lease decisions. In fact, the auction outcome may vary considerably depending on the number and

interests of the participants, which makes plans based on them quite unreliable. We opt instead for a different approach, where the price is set by the lessor. Relying on a price advertised by the lessor allows for much more reliable plans. We therefore expect the price for leasing storage to be based on a publicly available market price of that asset. In fact, in [8] the correct price of leasing an asset over a unit period of time (say from time t − 1 to time t) was set as the difference Lt−1 = Pt−1 − St−1,t ,

(14)

where St−1,t is the present value (at the beginning of the leasing period t − 1) of the residual value of the asset at the end of the leasing period t. The residual value of the asset at a future time is random, since it accounts for all the future variations of the asset’s value. We can assume that all those future variations are perfectly accounted for in the future price, so that the residual value can be written as 1−d Pt−1 − (Pt−1 − Pt ) = Pt−1 , (15) St−1,t = 1+k 1+k where we have introduced the one-period depreciation factor d = (Pt−1 − Pt )/Pt−1 , which is random since it depends on the future unknown asset’s price at time t. However, we know from the measurements reported in [5] (see Section IV), that the price fluctuates in time around an exponential decay curve: E[Pt ] = P0 e−βt .

(16)

We can therefore approximate the depreciation factor as the deterministic value E[Pt ] = 1 − e−β , (17) d'1− E[Pt−1 ] so that the residual value can itself be approximated as e−β Pt−1 . (18) 1+k We can now revisit the general expression for the leasing price (14) and obtain an approximate expression for it, which has though the advantage of being computable since all the terms are known: e−β Lt−1 = 1 − Pt−1 . (19) 1+k St−1,t '

The expr. (19) tells us that the leasing price has to be proportional to the market price of the storage device. In that expression the proportionality coefficient includes the effects of depreciation (through β) and of the time value of money (through k). Actually, unlike the leasing contract considered in [8], we have to take into account that in a leasing contract for storage the lessor incurs a number of additional expenses, since the storage facility is actually run by the lessor. In order to account for those additional expenses as well, we assume that Lt−1 = ηPt−1 , (20) −β

e where the coefficient η > 1 − 1+k is the leasing surcharge. We can get some insight into the possible range of values for the leasing surcharge. We already know an estimate of the depreciation factor. In fact, in [5] the β factor was estimated as 0.438, so that the corresponding estimate of the depreciation


factor is d = 0.355. Since in [5] the unit leasing cost is set as 0.15$ per GB per month, the disk price at time 0 is assumed to be 0.3$, and the discount rate is set as k = 0.01, we can compute the value of the leasing surcharge implied in those assumptions. In fact, we have η = 0.15 · 12/0.3 = 6, while under depreciation and time value-of-money effects only we would have from expr. (19) 1 + k − e−β ' 0.361 (21) 1+k We note a large difference between the implied leasing surcharge obtained through typical market values and its lower bound obtained through eq. (21). The main reason for that is that the larger market-based value takes into account the transfer of OPEX from the company to the lessor, where OPEX represent the dominant portion of costs. η=

VI. D ISK FAILURES As hinted in Section III-A, disks are subject to failures. Failure data are typically reported through the Annual Replacement Rate (ARR), i.e., the percentage of disks replaced each year. Just a few extensive studies are reported in the literature on disk failures. We cite two studies, one of which also considered in [5], whose conclusions are based on the observation of more than one hundred thousand disks. Namely, in [9] the failure data are categorized by the disk age. The reported ARRs vary from a minimum of 1.7% for disks in their first year of operation to a maximum of 8.6% for 3-year old disks. In [10] a wider range is reported, with ARRs in the 0.5%-13.5% range (3% being the most frequent value). In [5] a deterministic replacement rate has been considered, namely 3%. Since this is quite unrealistic, we have considered a probabilistic replacement model instead. In our model all the disks have the same probability to fail during a year, and fail independently. The number of failures Ft at time t follows therefore a binomial distribution: Dt−1 x P[Ft = x] = p (1−p)Dt−1 −x x = 0, 1, . . . , Dt−1 , x (22) where p is the marginal failure probability of a disk during a year. Here we consider the ARR as a good estimate of such probability. VII. S IMULATION SCENARIO The model described in the previous sections includes a number of parameters, which must be set to describe the context of interest. In this paper we consider two cases, representing respectively a medium-sized company and a large company. We conduct an analysis of the Buy-or-Lease decision through simulation. In this section we describe the two cases and set the values for the parameters to be employed in the simulation. The two cases follow those considered in [5]. By medium-sized company we mean a company running from tens to hundreds of servers, while a large company is expected to run some thousands of servers. The main difference between the two lies in the size of memory requirements.

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Parameter Initial memory [TB] Yearly additional memory [TB] Disk capacity [TB] Disk power [kW] Controller cost [$] Controller power [kW] Energy cost [$/kWh] Salvage factor γ Personnel cost Ht [$] Personnel factor λ − φ Marginal failure probability Discount rate

Medium Company

Large Company

1 1

10 10 1 0.01

1500 0.5

2000 0.7 0.04 0.1 70000

0.5

1 0.03 0.05

TABLE I S IMULATION SCENARIO

We report in Table I the values of parameters for the two cases. For both cases we have considered price fluctuations as modelled by the Geometric Brownian Motion model described in Section IV, with µ = −0.438 and σ = 0.2. All the results reported in Section VIII have been obtained with 100000 simulation runs. VIII. D ECISION REGIONS AND VALUE - AT-R ISK The aim of the model we have described in the previous sections is to help the decision maker in the Buy-or-Lease decision. The criterion we employ is based on the differential NPV we have described in Section II. However, the employment of that criterion in a deterministic context, such as that adopted in [5], provided a clear-cut figure to decide whether to buy or to lease: a positive value of ∆NPV led to the Buy decision. Instead, when we account for uncertainties in future prices, we are not left with a single value for ∆NPV, but rather with a distribution of values. We have therefore to define a new decision variable in this probabilistic context. In addition, whatever the decision we take based on this new decision variable, it may happen that the decision proves wrong, simply because of the uncertainty associated to future prices. We have therefore to assess the non-zero risk incurred in that decision. In this section we define the variables employed to take the decision and to assess the risk associated to it, and report the results obtained by simulation for a number of cases. As described in Section IV the market price of storage in the future is essentially a random quantity. As a consequence, both the CAPEX and the Differential NPV are random quantities. Hence, our analysis according to the model described in Section II doesn’t provide us with a single number, but rather with a distribution of values. Two examples of such distributions, obtained by simulation, are reported in Fig. 2 and Fig. 3 respectively for the CAPEX and the Differential NPV. The curves reported in those figures pertain to a large company (as defined in Section VII) when analysed over a time horizon of 5 years (i.e., N = 5) and with a leasing price set with a leasing surcharge η = 25. We see that both distributions are unimodal and skewed to the right, as expected, since the CAPEX are a linear combination of lognormal random variables, and the Differential NPV includes CAPEX as a major item.

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Shorter time horizons are associated to larger values of the differential NPV. We have then the somewhat counterintuitive result that companies with a short time horizon should opt to buy storage facilities rather than lease them.

0.0003

0.0002 50000 0.0001

0 5000

6000

7000

8000

9000

10000

11000

12000

CAPEX [$] Fig. 2.

Empirical Probability Density Function of CAPEX

Median Differential NPV [$]

Probability Density Function


Time horizon 10 years 5 years

0 -50000 -100000 -150000 -200000 -250000 50

100

150

200

4x10-6

Fig. 4. Median Differential Net Present Value for a medium-sized company 3x10-6

2x10-6

300000

-6

200000

Time horizon 1x10

0 -200000

-100000

0

100000

200000

300000

400000

Differential NPV [$] Fig. 3.

Empirical Probability Density Function of Differential NPV

Differential NPV [$]

Probability Density Function

Leasing Surcharge

10 years 5 years

100000

0

-100000

-200000

Being faced with a distribution of values as the outcome of the NPV analysis, we are forced to employ an aggregate indicator for that distribution as a decision variable to take the Buy-or-Lease decision. A natural choice is the median value of ∆NPV, i.e., that value separating the higher half of the probability distribution from the lower half. The NPV criterion in a probabilistic context can therefore be restated as follows: the company decides to buy the storage devices if the median value of ∆NPV is positive. In the following we \ For any choice of values indicate such median value as ∆NPV. for the context parameters (i.e., those shown in Table ??) and of the time horizon, we can therefore compute the value \ and compare it against the zero value that serves of ∆NPV as a divide between the two decision regions. Alternatively, if we are faced with a decision involving a specific context parameter, we can consider the values of that parameter that lead to a positive (or negative) median ∆NPV and associate them to a Buy (Lease) decision. In Fig. 4 and Fig. 5 we see the median ∆NPV for the two archetypal companies and two time horizons, as the leasing surcharge (hence, the leasing price) grows. We see that the median ∆NPV is a linear function of the leasing surcharge.

20

25

30

35

40

45

Leasing Surcharge Fig. 5.

Median Differential Net Present Value for a large company

In both Fig. 4 and Fig. 5 we have employed the leasing surcharge as a driving parameter. Actually this is the only parameter that is set by the lessor, and is completely outside the influence of the cloud user. It is therefore a convenient parameter that the cloud user can adopt to take its decision. Since the market price at the time of the Buy-or-Lease decision is known, we can formulate our decision criterion in terms of the leasing surcharge η through expr. (20). We can compute the \ if the actual leasing surcharge η0 that leads to a zero ∆NPV: \ > 0 and the company leasing surcharge is η > η0 , then ∆NPV should opt for a Buy decision (the leasing price is too high). The value η0 is therefore the minimum leasing surcharge that leads the company to a Buy decision (or, alternatively, the maximum leasing surcharge tolerable for the Lease decision). It is anyway the limit between two decision regions: the higher that limit leasing surcharge, the more profitable the Lease decision. In Fig. 6 we report the value of such limit


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leasing surcharge for both company sizes as the time horizon varies. We first note that η0 grows slightly superlinearly as the time horizon grows confirming that the longer is the business lifetime, the more convenient is the leasing solution. In addition, we see that the limit leasing surcharge is quite higher for a medium sized company than for a large one (roughly fourfold): as the size of memory requirements grows, the virtualization option appears less profitable. It is however to be noted that the depreciation and time-of-value effects alone would lead to a leasing surcharge η = 0.386 (when the discount rate is k = 0.05), quite smaller than the range observed in Fig. 6.

Limit Leasing Surcharge

160

Company size Medium Large

140 120 100 80 60 40 20 3

4

5

6

7

8

9

10

alternative ways: ( l : P[∆NPV < −l] ≤ 1 − α VaRα = l : P[∆NPV > l] ≤ 1 − α

\ >0 if ∆NPV \ < 0. if ∆NPV (23) The VaR provides us with the level of exposure, but its interpretation depends on the value we adopt for the confidence level α. If α is very close to 1, the resulting VaR can be taken as a proxy for the maximum loss the company faces when taking the wrong decision. We report in Fig. 7 and Fig. 8 the observed VaR when the time horizon is 10 years, respectively for a mediumsized company and a large one, as a function of the leasing surcharge. In both cases we have considered two quite large values for the confidence level, namely α = 0.95, 0.99. We see that the risk is very small when the leasing surcharge is either very low or very high. In fact, in those extreme situations, the leasing price is such to make one of the two decisions extremely more profitable than the other. In between, we have instead a range of values with a mixed behaviour. We see that the risk grows as we approach the limit leasing surcharge η0 (see Fig. 6 for comparison). In fact, when η = η0 the median ∆NPV is zero, and the two decisions are practically indifferent. In other terms, the company has exactly a 50% probability of taking the wrong decision, i.e., of incurring larger costs.

Operating time [years] 250000

Minimum Leasing Surcharge for Buy-over-Lease decision

Though we use the median ∆NPV as the decision criterion, we must not forget the essential uncertainty on future prices that makes ∆NPV a random quantity. Therefore, the decisions taken on the basis of the median value of ∆NPV (or any other parameter of its distribution) may prove to be wrong. \ > 0 (which makes the company For example, even if ∆NPV decide to buy the storage devices), the future prices may lead to larger costs than what the company would incur by leasing. Taking decisions in such an uncertain context exposes the company to risks. In the case of the Buy decision, the risk is represented by actually having a negative ∆NPV, while in the Lease decision the risk is represented by the opposite situation, i.e., having a positive ∆NPV, because of the definition (4) of ∆NPV. Since in that definition the profits cancel out, having a ∆NPV of sign opposite to the expected one means that the company is facing larger costs than expected. We can then evaluate the risk by computing the amount of such larger costs. Since the differential ∆NPV is represented by a probability distribution (see, e.g., Fig. 3), we have again to resort to a parameter of that distribution. A common approach in risk analysis is to consider the Value-at-Risk (VaR) as a risk measure. Namely, if we define that probability as the confidence level, the VaR at the confidence level α ∈ (0, 1) is the smallest number such that the probability that the loss exceeds l is not larger than 1 − α [11]. Since we can err in either of two ways (by taking the wrong Buy or Lease decision) we have to define the VaR in the following two

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IX. C ONCLUSION We have proposed a probabilistic model for the Buy-orLease decision for storage in a cloud computing environment. Such model takes into account the random nature of both the evolution of prices over time and the occurrence of disk failures (which call for additional replacement expenses). The model represents therefore an advancement over current models for the same purpose, which are purely deterministic. We have adopted the median Differential Net Present Value as a decision variable in the Buy-or-Lease problem. We have employed that model in a simulation analysis to examine the profitability of either decision. We have found


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Value-at-Risk for a large company (10 years)

out that the decision to resort to cloud computing appears more profitable for medium-sized companies rather than for large ones, and for longer investment plans rather than short ones. In particular we can identify the maximum leasing price such that the leasing option is still profitable over the Buy decision. We also show that the change from a deterministic to a probabilistic approach allows us to recognize the presence of risks associated to either decision. We have evaluated such risk through the well known Value-at-Risk risk measure. Such risk reaches its maximum in the region where the two decisions lead to nearly equal profits. R EFERENCES [1] B. Furht and A. Escalante, Eds., Handbook of Cloud Computing. Springer, 2010. [2] B. Sotomayor, R. S. Montero, I. M. Llorente, and I. T. Foster, “Virtual infrastructure management in private and hybrid clouds,” IEEE Internet Computing, vol. 13, no. 5, pp. 14–22, 2009. [3] J. Altmann, C. Courcoubetis, G. D. Stamoulis, M. Dramitinos, T. Rayna, M. Risch, and C. Bannink, “Gridecon: A market place for computing resources,” in Grid Economics and Business Models, 5th International Workshop, GECON 2008, Las Palmas de Gran Canaria, Spain, ser. Lecture Notes in Computer Science, J. Altmann, D. Neumann, and T. Fahringer, Eds., vol. 5206. Springer, 26 August 2008, pp. 185–196.

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[4] M. M. Hassan, B. Song, and E. Huh, “A market-oriented dynamic collaborative cloud services platform,” Annales des T´el´ecommunications, vol. 65, no. 11-12, pp. 669–688, 2010. [5] E. Walker, W. Brisken, and J. Romney, “To lease or not to lease from storage clouds,” IEEE Computer, vol. 43, no. 4, pp. 44–50, 2010. [6] D. G. Newman, T. G. Eschenbach, and J. P. Lavelle, Engineering Economic Analysis , 9th ed. Oxford University Press, 2004. [7] N. H. Chan and H. Y. Wong, Simulation techniques in financial risk management. John Wiley, 2006. [8] M. H. Miller and C. W. Upton, “Leasing, buying, and the cost of capital services,” The Journal of Finance, vol. 31, no. 3, pp. 761–786, June 1976. [9] E. Pinheiro, W.-D. Weber, and L. A. Barroso, “Failure trends in a large disk drive population,” in 5th USENIX Conference on File and Storage Technologies, FAST 2007. USENIX, February 13-16, 2007, pp. 17–28. [10] B. Schroeder and G. A. Gibson, “Disk failures in the real world: What does an mttf of 1, 000, 000 hours mean to you?” in 5th USENIX Conference on File and Storage Technologies, FAST 2007. USENIX, February 13-16, 2007, pp. 1–16. [11] C. Alexander, Value-at-Risk Models. Wiley and Sons, 2009.

Maurizio Naldi graduated cum laude in 1988 in Electronic Engineering at the University of Palermo and then received his Ph.D. in Telecommunications Engineering from the University of Rome at Tor Vergata. After graduation he pursued an industrial career, first at Selenia (now Alenia) as a radar designer (1989-1991), and then in the Network Planning Departments of Italcable (1991-1994), Telecom Italia (1995-1998), and Wind Telecomunicazioni (19982000) where he was appointed Head, Traffic Forecasting & Network Cost Evaluation Group. Since 2000 he is with the University of Rome at Tor Vergata, where he is now Aggregate Professor. His current research interests lie in the areas of Network Economics and Computational Advertising. He is a Senior Member of IEEE, and a member of MAA.

Loretta Mastroeni graduated cum laude in Mathematics at the University of Rome and then she attended the Scuola di Alta Matematica F. Severi in Rome. After graduation she started her academic career, first as a Tenured Researcher at the University of Rome Tor Vergata, then, since 1998, as an Associate Professor at the University of L’Aquila. Since 2002 she is Associate Professor at the Department of Economics at the University of Rome Roma Tre, where she holds the courses Mathematical Economics, Mathematical Finance. Her main research interests focus on the pricing of financial derivatives (especially financial and real options), duality theory in economics, optimal control techniques applied to biological systems. She has acted as a referee for a number of peer-reviewed journals (Mathematical Finance, Journal of Economic Dynamics and Control, and others). She is the author of several publications and she is currently engaged in a number of research projects.