A Study of Revenue Cost Dynamics in Large Data Centers: A ...pesitsouthscibase.org/static_files/papers/study-revenue-cost.pdf · A Study of Revenue Cost Dynamics in Large Data Centers:

A Study of Revenue Cost Dynamics in Large Data Centers: AFactorial Design Approach

Gambhire Swati SampatraoSudeepa Roy DeyBidisha GoswamiSai Prasanna M. S.Snehanshu Saha

PESIT Bangalore South CampusDepartment of Computer Science and Engineering

Bangalore, Karnataka 560100, India

ABSTRACTRevenue optimization of large data centers is an open and challeng-ing problem. �e intricacy of the problem is due to the presenceof too many parameters posing as costs or investment. �is paperproposes a model to optimize the revenue in cloud data center andanalyzes the model, revenue and di�erent investment or cost com-mitments of organizations investing in data centers. �e modeluses the Cobb-Douglas production function to quantify the bound-aries and the most signi�cant factors to generate the revenue. �edynamics between revenue and cost is explored by designing anexperiment (DoE) which is an interpretation of revenue as func-tion of cost/investment as factors with di�erent levels/�uctuations.Optimal elasticity associated with these factors of the model formaximum revenue are computed and veri�ed . �e model responseis interpreted in light of the business scenario of data centers.

CCS CONCEPTS•Mathematics of computing→Convex optimization; •Appliedcomputing→ Forecasting;

GENERAL TERMSDesign, Modeling, Performance

KEYWORDSCobb-Douglas production function, Replication, Design of Exper-iment (DoE), Cloud Computing, Data Center, Infrastructure as aService (IaaS), Optimization.ACM Reference format:Gambhire Swati Sampatrao, Sudeepa RoyDey, BidishaGoswami, Sai PrasannaM. S., and Snehanshu Saha. 2017. A Study of Revenue Cost Dynamics in

Author’s addresses: G. S. Sampatrao, S. R. Dey, B. Goswami, Sai Prasanna M. S. andS. Saha, Department of Computer Science and Engineering, PESIT Bangalore SouthCampus, Bangalore, Karnataka, India - 560100.Communicating author: [email protected] to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor pro�t or commercial advantage and that copies bear this notice and the full citationon the �rst page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permi�ed. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior speci�c permission and/or afee. Request permissions from [email protected], Cambridge, UK© 2017 ACM. 978-1-4503-4774-7/17/03. . .$15.00DOI: h�p://dx.doi.org/10.1145/3018896.3056772

Large Data Centers: A Factorial Design Approach. In Proceedings of In-ternational Conference on Internet of �ings, Data and Cloud Computing,Cambridge, UK, March 22, 2017 (ICC2017), 14 pages.DOI: h�p://dx.doi.org/10.1145/3018896.3056772

1 INTRODUCTION�e data center has a very important role in cloud computing do-main. �e costs associated with the traditional data centers includemaintenance of mixed hardware pools to support thousand of ap-plications, multiple management tools for operations, continuouspower supply, facility of water for cooling the power system andnetwork connectivity, etc. �ese data centers are currently usedby Internet service providers for providing service such as infras-tructure and so�ware. Along with the existing pricing, a new setof challenges are due to the up-gradation , augmenting di�erentdimensions of the cost optimization problem.

Most of the applications in the industry are shi�ing towardscloud system, supported by di�erent cloud data centers. I & T Busi-ness industries assume the data center to function as a factory-likeutility that collects and processes information from an operationalstandpoint. �ey value the data that is available in real time tohelp them update and shape their decisions. �ese industries doexpect that, the data center needs to be fast enough to adapt to new,rapidly deployed, public facing and internal user applications forseamless service. �e technology standpoint demands the currentdata centers to support mobility, provisioning on demand, scala-bility, virtualization and the �exibility to respond to fast-changingoperational situations. Nevertheless, from an economic viewpoint,a few years of edgy �scal conditions have imposed tight budgets onIT organizations in both public and private sectors. �is compelsthem to rethink about remodeling and smart resource management.�e expectation in price and performance from clients needs tobe maximum while expectation (within the organization)in termsof cost has to have a minimum. �is is the classic revenue costparadox. Organizations expect maximum output for every dollarinvested in IT. �ey also face pressure to reduce power usage as acomponent of overall organizational strategies for reducing theircarbon footprint. Amazon Web Services(AWS) and other data cen-ter providers are constantly improving the technology and de�nethe cost of servers as the principle component in the revenue model.For example, AWS spends approximately 57% of their budget to-wards servers and constantly improvise in the procurement pa�ern

ICC2017, March 22, 2017, Cambridge, UK G. S. Sampatrao et al.

of three major types of servers, as pointed out by Greenberg et.al.[4] . �e challenge that a data center faces is the lack of accessof basic data critical towards planning and ensuring the optimuminvestment in power and cooling system. Ine�cient power usage,including the sub-optimal use of power infrastructure and over in-vestment in racks and power capacity, burdens the revenue outlayof organizations. Other problems include Power and Cooling excur-sions i.e. availability of power supply during pick business hoursand identifying the hot-spots to mitigate and optimize workloadplacement based on power and cooling availability. Since, energyconsumption of cloud data centers is a key concern for the owners,energy costs (fuel) continue to rise and CO2 emissions related tothis consumption have become relevant. �is was observed by Zhaoet. al. [9]. �erefore, saving money in the energy budget of a clouddata center, without sacri�cing Service Level Agreements (SLA) isan excellent incentive for cloud data center owners, and would atthe same time be a great success for environmental sustainability.�e ICT resources, servers, storage devices and network equipmentconsume maximum power.

In this paper, a revenue model with the cost function based onthe sample data is proposed. �is uses Cobb-Douglas productionfunction to generate a revenue and pro�t model and de�nes theresponse variable as production or pro�t, a variable that needs to beoptimized.�e response variable is the output of several cost factors.�e contributing factors are Server type and power and coolingcosts. �e proposed model heuristically identi�es the elasticityranges of these factors and uses a ��ing curve for empirical veri�-cation. However, the cornerstone of the proposed model is the in-teraction and dependency between the response variable, Revenueor Pro�t against the two di�erent types of cost as input/predictorvariables.

�e remainder of the paper is organized as follows. Section 2discusses the related work, highlighting and summarizing di�erentsolution approaches to the cost optimization problem in data center.In Section 3, Revenue Optimization in Data Center is discussed.�is section explains the Cobb-Douglas production function whichis the backbone of the proposed revenue model. Section 4 talksabout DoE that used to build and analyze the model . Section 5elucidates the critical factors of revenue maximization in the Cobb-Douglas model. In section 6, the impact of the identi�ed factorsin the proposed design is discussed. �e detailed experimentalobservation on IRS,CRS and DRS is provided in Section 7 . Section8 describes various experiments conducted for validation. Section9 is about predictive analysis to forecast the revenue from theobservation. Section 10 concludes our work.

2 RELATEDWORKCloud data center optimization is an open problem which has beendiscussed by many researchers. �e major cloud providers suchas Amazon, Google and Microso� spend millions for servers, sub-station power transformers and cooling equipments. Google [1]has reported $ 1.9 billion in spending on data centers in the yearof 2006 and $2.4 billion in 2007. $45 million in 2006 for data centerconstruction cost has been spent by Apple [2] while $606 millionon servers, storage and network gear and data centers was the ap-proximate cost incurred by Facebook [3]. Budget constraints force

the industries to explore di�erent strategies that ensures optimalrevenue. A variety of solutions have been proposed in the literatureaimed towards reducing the cost of the data centers. Ghamkhariand Mohsenian-Rad [5] highlight the trade-o� between minimizingdata center’s energy expenditure and maximizing their revenue foro�ered services. �e paper signi�cantly identi�es both the factorsi.e minimization of energy expense and maximization of revenuecost. �eir experimental design, however, could not present anyanalysis regarding contribution of factors to revenue generation.Chen et.al. [6] proposed a model that optimizes the revenue i.eexpected electricity payment minus the revenue from participatoryday-ahead data response. �e author proposes a stochastic opti-mization model identifying the constraints of other cost factorsassociated with data centers. �is may always not be applicable toreal cloud scenario where on-demand, fast response is a need andthe elasticity of cost factors has signi�cant contribution. Toosi et al.[7] have addressed the issue of revenue maximization by combiningthree separate pricing models in cloud infrastructure. Every cloudprovider has a limitation of its resources. �e authors propose aframework to maximize the revenue through an optimal alloca-tion which satisfy dynamic and stochastic need to customers byexploiting stochastic dynamic programming model. [8] argues thata �ne-grained dynamic resource allocation of VM in a data centerimproves be�er utilization of resources and indirectly maximize therevenue. �e authors have used trace driven simulation and shownan overall 30% revenue increment. Another possible solution in-volves migration and replacement of VM’s; Zhao et al. [9] proposedan on-line VM placement algorithm for enhancing revenue of thedata center. �e proposed framework has not discussed the powerconsumption of VM for communication and migration which actu-ally has a huge impact on price. Saha et.al. [10] have proposed anintegrated approach for revenue optimization. �e model is utilizedto maximize the revenue of service provider without violating thepre-de�ned QoS requirements, while minimizing cloud resourcecost. �e formulation uses the Cobb-Douglas production function[11], a well known production function widely used in econom-ics. Available scholarly document in the public domain emphasizethe need for a dedicated deployment model which meets the costdemand while maintaining pro�tability.

3 REVENUE OPTIMIZATION AND DATACENTERS

�e Cobb-Douglas production function is a particular form of theproduction function [11]. �e most a�ractive features of Cobb-Douglas are: Positively decreasing marginal product,Constant out-put elasticity, equal to β and α for L and K, Constant returns toscale equal to α+β .

Q(L,K) = ALβKα (1)�e above equation represents revenue as a function of two vari-ables or costs and could be scaled up to accommodate a �nitenumber of parameters related to investment/cost as evident fromequation (2). �e response variable is the outcome. e.g. Revenueoutput due to factors such as cost, man-hours and the levels of thosefactors. �e primary and secondary factors as well as replicationpa�erns need to be ascertained such that the impact of variationamong the entities is minimized. Interaction among the factors

A Study of Revenue Cost Dynamics in Large Data Centers: A Factorial Design Approach ICC2017, March 22, 2017, Cambridge, UK

need not be ignored. A full factorial design with the number ofexperiments equal to

k∑i=1

ni

would capture all interactions and explain variations due to tech-nological progress, the authors believe. �is will be illustrated inthe section titled Factor analysis and impact on proposed design.

A. Production MaximizationConsider an enterprise that has to choose its consumption bundle(S, I, P, N) where S, I, P and N are number of servers, investmentin infrastructure, cost of power and networking cost and coolingrespectively of a cloud data center. �e enterprise wants to maxi-mize its production, subject to the constraint that the total cost ofthe bundle does not exceed a particular amount. �e company hasto keep the budget constraints in mind and restrict total spendingwithin this amount.�e production maximization is achieved using Lagrangian Multi-plier. �e Cobb-Douglas function is:

f (S, I ,N , P) = kSα I βPγN δ (2)

Let m be the cost of the inputs that should not be exceeded.

w1S +w2I +w3P +w4N =m

w1: Unit cost of serversw2: Unit cost of infrastructurew3: Unit cost of powerw4: Unit cost of network

Optimization problem for production maximization is:

max f (S, I , P ,N ) subject to m

�e following values of S, I, P and N thus obtained are the valuesfor which the data center achieves maximum production undertotal investment/cost constraints.

S =mα

w1(1 + β + γ + δ ) (3)

I =mβ

w2(1 + α + γ + δ ) (4)

P =mγ

w3(1 + α + β + δ ) (5)

N =mδ

w4(1 + α + β + γ ) (6)

�e above results are proved in Appendix 1 [12].Since we are considering the equation with two factors only, the

equation(11) is re-framed. �e equation can be rewri�en as

f (S, P) = ASαPβ (7)

= ASαP (1 − α) (8)For A = 1, pro�t maximization is achieved when:

(1) ∂y∂S =

αS (α−1)K (1−α )k = αY

k(2) ∂y

∂K = (1−α )S (α )K (1−α )k= (1−α )Yk

B. Pro�t MaximizationConsider an enterprise that needs maximize its pro�t. �e Pro�tfunction is:

π = p f (S, I ,N , P) −w1S −w2I −w3P −w4N

Pro�t maximization is achieved when:

(1) p ∂f∂S = w1 (2) p ∂f

∂I = w2 (3) p ∂f∂P = w3 (4) p ∂f

∂N = w4

�e following values of S, I, P and N are obtained:

S =(pkα1−(β+γ+δ )ββγγ δδ

wβ+γ+δ−11 w

−β2 w

−γ3 w−δ4

) 11−(α+β+γ +δ ) (9)

I =(pkαα β1−(α+γ+δ )γγ δδ

w−α1 wα+γ+δ−12 w

−γ3 w−δ4

) 11−(α+β+γ +δ ) (10)

P =(pkαα ββγ 1−(α+β+δ )δδ

w−α1 w−β2 w

α+β+δ−13 w−δ4

) 11−(α+β+γ +δ ) (11)

N =(pkαα ββγγ δ1−(α+β+γ )

w−α1 w−β2 w

−γ3 w

α+β+γ−14

) 11−(α+β+γ +δ ) (12)

which is the equation for data center’s pro�t maximizing quan-tity of output, as a function of prices of output and inputs.y increases in price of its output and decreases in price of its inputsi� :

1 − (α + β + γ + δ ) > 0α + β + γ + δ < 1

�erefore, the enterprise will have pro�t maximization at the phaseof decreasing returns to scale. It is shown in [? ], that pro�t max-imization is scalable i.e. for an arbitrary n, number of input vari-ables(constant), the result stands as long as

∑ni=1 αi < 1; where αi

is the ith elasticity of the input variable xi . Constructing a mathe-matical model using Cobb-Douglas Production Function helps inachieving the following goals:

(1) To forecast the revenue with a given amount of investmentor input cost.

(2) Analysis of maximum production such that total cost doesnot exceed a particular amount.

(3) Analysis of maximum pro�t that can be achieved.(4) Analysis of minimum cost /input to obtain a certain output.

�e model empowers the IaaS entrepreneurs (while establishingan IT data center) estimate the probable output, revenue and pro�t.It is directly related to a given amount of budget and its optimiza-tion. �us, it deals with minimization of costs and maximization ofpro�ts too.�e assumption of those 4 factors (S,I,P,N) as the inputs relevant tothe output of an IaaS data center is consistent with the work-�ow


of such data centers. Again, α , β , γ and δ are assumed to be out-put elasticity of servers, infrastructure, power drawn and networkrespectively. A quick run down of the analytical work reveals theapplication of the method of Least Squares by meticulously follow-ing all the necessary mathematical operations such as making theProduction Function linear by taking log of both sides and apply-ing Lagrange Multiplier and computing maxima/minima by partialdi�erentiation (i.e., computing changes in output corresponding toin�nitesimal changes in each input by turn). In a nutshell, the ana-lytical calculation of Marginal Productivity of each of the 4 inputshas been performed. Based on the construction, the mathematicalmodel is capable of forecasting output, revenue and pro�t for anIaaS data center, albeit with a given amount of resource or budget.

3.1 ObservationsDoes this model anyway contradict the established laws of neo-classical economics anyway?

Neo-classical economics at abstract level, postulates AverageCost Curve (AC) to be a U-shaped curve whose downward partdepicts operation of increasing returns and upward the diminishingreturns. Actually, it is the same phenomenon described from twodi�erent perspectives; additional applications of one or two inputswhile others remaining constant, resulting in increase in outputbut at a diminishing rate or increase in marginal cost (MC) andconcomitantly average cost (AC).

Figure 1: Input vs Cost-1

Figure-1 shows that the cost is lowest or optimum where MCintersects AC at its bo�om and then goes upward.

Figure-2 shows that equilibrium (E) ormaximumpro�t is achievedat the point where Average Revenue Curve (a le� to right downwardcurve, also known as Demand Curve or Price Line as it shows grad-ual lowering of marginal and average revenue intersects (equals)AC and MC at its bo�om, i.e., where AR=AC=MC. Here, the regionon the le� of point E, where AR > AC depicts total pro�t. �erefore,E is the point of maximization of pro�t where AR = AC.�e data [12] of Table 6 displaying Data Center Comparison forDRS has been accumulated from the �gure 3.

Figure 2: Input vs Cost-2

Figure 3: Data Center Comparison Cost

Additional data �les uploaded to gitHub, an open repository,[12] documents detailed costs associated with di�erent data centerslocated in di�erent cities. Along with that, the maximum revenue,which is achievable using the Cobb-Douglas function, is shown.�e optimal values of the elasticity constants are also visible in twocolumns. Additional �les contain the proof of scalability of ourmodel.We have partitioned all the segments of Data Center costs into twoportions. Considering Labor, property sales tax, Electric powercost as infrastructure and combining Amortization, heating air-conditioning as recurring, we have recalculated the costs of allthe data centers. �e cost of running data center in New York ishighest as its annual labor cost, sales and power costs are higherthan any other cities. �e operating costs of data center in citiessuch as Rolla, Winston-Salem, and Bloomington are ranging within$11,000,000 to 12,500,000, are almost equal.

In �gure 4, X axis represents α ; Y axis presents β and Z axisdisplays the Revenue. �e graph demonstrates an example of con-cave graph. α and β are the output elasticity of infrastructure and


Figure 4: Revenue Function Graph of Data Center Compari-son Cost of DRS

recurring costs. �e recurring and infrastructure costs of data cen-ters located in Huntsville-Ala, Rolla-Mo, Bloomington-Ind, and SanFrancisco-Ca have been used to plot in the above graphs. We cansee the revenue is healthier where α , β are both higher than 0.7.�e max revenue is lying in the region where α , β are proximal to0.8 and 0.1 respectively. We choose these values! �is selection isveri�ed later by deploying Least Square like ��ing algorithms, asdiscussed in Section VI.

Figure 5: Revenue Function Graph of Data Center Compari-son Cost of CRS

�e graphs (Figure 5) portray the e�ects of Cobb-Douglas pro-duction function over cost incurred in di�erent data centers locatedin di�erent cities. As par pictorial representation, there is not muchdi�erencewith DRS though the revenues obtained in CRS are higherin comparison to DRS. �e observation is visible through the data

available in table. Similar to the other graphs, the X, Y, Z axesrepresent α , β and Revenue respectively.

Figure 6: Revenue Function Graph of Data Center Compari-son Cost of IRS

Figure 6 depicts the revenue under the constraint, Increasingreturn to scale (IRS) where the sum of the elasticities is more than1. Like the previous �gures, the elasticities are represented by theX, Y axes and Z represents the revenue, which has been calculatedusing Cobb-Douglas function.Additional �le [12] contains detailed information about data centercomparison costs for IRS,DRS and CRS, including revenue data,cost and optimal constraints. Please refer [11] for a quick tutorialon IRS, DRS and CRS.

Figure 7 is the graphical representation of Annual AmortizationCosts in data center for 1U server. All units are in $. We haveextracted fairly accurate data from the graph and represented intabular format (Table IX). Maximum revenue and optimal elasticityconstants are displayed in the same table. Additional �le [12] showsthe Optimal constants for DRS.

�e Revenue graph (Figure 8) displays the range of revenue in ac-cordance to the data of annual amortized cost of di�erent years.�eco-ordinate axes represent α , β and Revenue respectively. Servercost and Infrastructure cost are combined together as infrastructurecost, whereas Energy and Annual I & E are clubbed as recurringcost. α represents elasticity constant of infrastructure and β de-notes elasticity constant of recurring cost. �e recurring cost andinfrastructure cost of the years 1992, 1995, 2005, and 2010 have beenused to calculate revenue. �e revenue rises drastically in region ofα , β being greater than 0.5 in comparison to any other region. �epeak of the graphs indicate the maximum revenue located in theregion, where α , β are approximating 0.8 and 0.1 or vice-versa.

In the Figure 9, the co-ordinate axes represent α , β , and revenuerespectively. Slight di�erence is observed in the range of elasticities.Maximum revenue lies in the area, where (α is approximately 0.9and β is close to 0.1 or vice versa. �e revenue data, elasticities anddi�erent cost segments are displayed in tabular format[11].


Figure 7: Annual Amortization Costs in data center for 1Userver

Figure 8: Revenue Function Graph of Annual AmortizedCost

We observe that there is no major di�erence between revenuesduring the years 1992 to 1995. But the revenue becomes almost 3fold between the years 2000 and 2010. Server cost remains constantthroughout the years but signi�cant changes are noticed in othercost segments namely Energy cost, Infrastructure and Annual I &E.



In Figure 10, the maximum revenue is re�ected in the regionwhere (α and β nearby 1.8 and 0.1 respectively. In case of IRS, theoptimal revenue surges ahead of CRS and DRS. Revenue becomesalmost �ve times from the year 2000 to 2010. It displays two-foldjump from 2005 to 2010.

4 DESIGN OF EXPERIMENTS (DOE) ANDIMPACT ON THE PROPOSED MODEL

Factor analysis is an e�cient way to understand and analyze data.Factor analysis contains two types of variables, latent and mani-fest. A DoE paradigm identi�es the latent(unobserved) variableas a function of manifest(observed) variable. Some well knownmethods are principal axis factor, maximum likelihood, generalized


least squares, unweighted least square etc. �e advantage of us-ing factor analysis is to identify the similarity between manifestvariables and latent variable. �e number of factors correspond tothe variables. Every factor identi�es the impact of overall variationin the observed variables. �ese factors are sorted in the order ofvariation they contributed to overall result. �e factors which havesigni�cantly lesser contribution compared to the dominant onesmay be discarded without causing signi�cant change in output. �e�ltration process is rigorous but the outcome is insightful.

4.1 Factor Analysis for Proposed model�e proposed revenue model exploits factorial analysis to identifythe major factors among the inputs(Cost of Servers and Power). Ta-ble I, Table II and Table III have been created from the data available.Equation (13) describes our basic model with two factors each withtwo levels, de�ning all combinations for the output variable. Facto-rial design identi�es the percentage of contribution of each factor.�ese details can be used to understand and decide how the factorscan be controlled to generate be�er revenue. �e Cobb-Douglasproduction function provides insight for maximizing the revenue.�e paper[10] explains the coe�cient of the latent variables as acontributor of output function as evident from equation (2). In thegiven equation, (α , β , γ and δ are the parameters which are respon-sible for controlling the output function Y. However to generatethe output function y=f(S,I,N,P), the threshold level of minimumand maximum value needs to be bounded. �e contribution of α, β , γ and δ towards output function Y is not abundantly clear.�erefore, it is relevant to study the e�ects of such input variableson the revenue in terms of percentage contribution of each variable.An e�cient, discrete factorial design is implemented to study thee�ects and changes in all relevant parameters regarding revenue.Revenue is modeled depending on a constant(market force), a bunchof input variables which are quantitative or categorical in nature.�e road-map to design a proper set of experiments for simulationinvolves the following:

• Develop a model best suited for the data obtained.• Isolate measurement errors and gauge con�dence intervals

for model parameters.• Ascertain the adequacy of the model.

For the sake of simplicity and convenience the factors S-I andP-N were grouped together as two factors. �e question of scalingdown impacting the model performance would be asked is not thelimitation of the model. Additional �les, [12] reveal a proof whichconsidersn number of factors for the samemodel and the conditionsfor optimality hold,n being arbitrary. Additionally, the conditionsobserved for two factors can be simply scaled to condition for nfactors. Since we consider the equation with two factors only,theequation can be rewri�en as:

f (S, P) = ASαPβ (13)

= ASαP (1 − α) (14)For A = 1, Pro�t maximization is achieved when:

(1) ∂y∂S =

αS (α−1)K (1−α )k = αY

kPro�t maximization is achieved when:

(2) ∂y∂K = (1−α )S

(α )K (1−α )k = (1−α )Yk

At this point, we note that both the factors,servers and power canbe controlled by alpha. �e rate of change in both the parameters inthe Cobb-Douglas equation can be determined. We have to choosethe alpha value in such a way that the pro�t maximization wouldnot drop below the threshold value.

�e 22 factorial design�e following are the factors, with 2 levels each:

(1) Cost of Power and Cooling, Factor 1 .(2) Type of Server categorized based on cost of deployment,

Factor 2.

Level Range (in Million Dollars)Low 5-15High 16-40

Table 1: Factor 1 Levels

Level Range (in Million Dollars)Type 1 45-55Type 2 56-65

Table 2: Factor 2 Levels

Let us de�ne two variables xA and xB as:

xA =

{−1, if Factor 1 is low1, if Factor 1 is high

xB =

{−1, if Factor 2 is of Type-11, if Factor 2 is of Type-2

�e Revenue y (in Million Dollars) can now be regressed on xA andxB using a nonlinear regression model of the form:

y = q0 + qAxA + qBxB + qABxAxB (15)

�e e�ect of the factors is measured by the proportion of totalvariation explained in the response.

Sum of squares total (SST):

SST = 22qA2 + 22qB2 + 22qAB2 (16)

Where:22q2A is the portion of SST that is explained by Factor 1.22q2B is the portion of SST that is explained by Factor 2.22q2AB is the portion of SST that is explained by the interactions ofFactor 1 and Factor 2.�us,

SST = SSA + SSB + SSAB (17)

Fraction of variation explained by A = SSA

SST(18)

Fraction of variation explained by B = SSB

SST(19)


Fraction of variation explained by AB = SSAB

SST(20)

Our choice of elasticity depends on the dynamics between thefactors and the benchmark constraints of optimization. CRS, forexample requires the sum of the elasticities to equal 1 and DoEreveals that factor1 contributes to the response variable to a lesserextent compared to factor 2. �erefore, in order that revenue growthmay be modeled in a balanced fashion, elasticity value for factor1 has been set to much higher compared to factor 2. �e samephenomena is observed in the cases of IRS and DRS and identicalheuristic has been applied to predetermine the choice of elasticities.�e authors intend to verify the heuristic choices through ��ingand regression in the la�er part of the manuscript.

Read Data Factor 1 Factor 2 Alpha BetaCRS 24.5 75.8 0.9 0.1DRS 2.44 97.36 0.8 0.1IRS 5.86 93.62 1.8 0.1

Table 3: Elasticity and percentage contribution of cost fac-tors

5 EXPERIMENTAL OBSERVATIONS5.1 Experiment 1 : IRS

Power and CoolingServer Low HighType-1 1509.63 1676.48Type-2 2062.39 2153.34Table 4: Experiment 1: IRS

Computation of E�ects:Substituting the four observations in the model, we obtain thefollowing equations:

1509.63 = q0 − qA − qB + qAB1676.48 = q0 + qA − qB − qAB2062.39 = q0 − qA + qB − qAB2153.34 = q0 + qA + qB + qAB

Solving the above equations for the four unknowns, the Regressionequation obtained is:

y = 1850.46 + 64.45xA + 257.4xB − 18.9xAxB (21)If we spend on a server for deployment and capacity building ,rev-enue is positively a�ected. Cloud business elasticity depends onresources and capacity and promise of elastic service provisioningis a function of hardware and so�ware capability .

Allocation of Variation:

SST = 22qA2 + 22qB2 + 22qAB2

= 2264.452 + 22257.42 + 22−18.92

= 283063

�e result is interpreted as follows:�e e�ect of Factor 1 on Revenue is 5.86%�e e�ect of Factor 2 on Revenue is 93.62%�e e�ect of interactions of Factors 1 and 2 on Revenue is 0.5%

5.2 Experiment 2: CRS

Power and CoolingServer Low HighType-1 43.5 47.5Type-2 50.82 55.38Table 5: Experiment 2: CRS

�e regression equation obtained is:y = 49.3 + 2.14xA + 3.8xB + 0.14xAxB (22)

�e result obtained a�er factor analysis is interpreted as follows:�e e�ect of Factor 1 on Revenue is 24.5%�e e�ect of Factor 2 on Revenue is 75.8%�e e�ect of interactions of Factors 1 and 2 on Revenue is 0.1%

5.3 Experiment 3: DRS

Power and CoolingServer Low HighType-1 29.47 31.99Type-2 33.6 36.87Table 6: Experiment 3: DRS

�e regression equation is:y = 41.9 + 1.77xA + 11.18xB + 0.5xAxB (23)

�e result obtained a�er factor analysis is interpreted as follows:�e e�ect of Factor 1 on Revenue is 2.44%�e e�ect of Factor 2 on Revenue is 97.36%�e e�ect of interactions of Factors 1 and 2 on Revenue is 0.19%

�e results suggest that there is no signi�cant interaction be-tween the factors. �us, all further analysis henceforth will be doneignoring the interaction factor.

5.4 Randomizing the dataSince there was insu�cient data to conclude the e�ects of thefactors on revenue, we had to generate more data by discoveringthe distribution of the real data set and generating random datafollowing the same distribution. Our experiment has found that theoriginal data follows the Normal distribution (Figure 11 and 12).

�e tables VI,VII represent the random data that was gener-ated and corresponding revenue values calculated using the Cobb-Douglas model for IRS, CRS and DRS respectively.

�e Chi Square- Goodness of �t test was performed on the actualand generated data to con�rm the data trend.


Figure 11: �e Original and Generated Server Data that fol-lows Normal Distribution

Figure 12: �e Original and Generated Power and CoolingData that follows Normal Distribution

�e Null Hypotheses H0: A�er adding noise to the originaldata set, the data follows Gaussian Distribution. If H0=1, the nullhypothesis is rejected at 5% signi�cance level.

if H0=0, the null hypothesis is accepted at 5% signi�cance level.�e result obtained, H0=0 assures us that the data indeed follows

Gaussian Distribution with 95% con�dence level.

5.5 ReplicationsReplicates are multiple experimental runs with identical factorse�ings (levels). Replicates are subject to the same sources of vari-ability, independent of each other.In the experiment, two replications were conducted on the realdata and generated data(r=2), taking into consideration that it is a22 factorial design problem. �e results obtained are at par withthe results obtained from factorial analysis conducted for the orig-inal data. Replication, the repetition of an experiment on a largegroup of subjects, is required to improve the signi�cance of an

New Server Power and Cooling Revenue68 38 2860.8544 25 1253.1762 12 2158.8559 37 2209.8049 10 1387.8854 20 1771.7959 18 2056.1878 25 3512.0873 25 3117.2849 10 1387.8875 21 3216.1261 19 2195.1757 28 2019.7261 34 2326.7156 34 1994.7456 11 1781.8866 12 2566.97

Table 7: Revenue for IRS


Table 8: Revenue for CRS

experimental result. If a treatment is truly e�ective, the long-termaveraging e�ect of replication will re�ect its experimental worth.If it is not, then the few members of the experimental populationwho may have reacted to the treatment will be negated by the largenumbers of subjects who were una�ected by it. Replication reducesvariability in experimental results, increasing their signi�cance andthe con�dence level with which a researcher can draw conclusionsabout an experimental factor [13]. Since this was a 22 Factorialproblem, 2 replications had to be performed. Table IX documentsthe results obtained a�er replications were performed for IRS, CRSand DRS respectively.



Table 9: Revenue for DRS

IRS CRS DRSNew Server 81.9 66.21 62.19

Power and cooling 12.48 31.43 35.72Interaction 3.05 .35 0.000013

Error 2.55 1.99 2.02Table 10: Percentage variation for IRS, CRS and DRS

It is observed that the contribution of two factors towards thetotal variation of the response variable is consistent between thereal data and the simulated random data.

5.6 Con�dence intervals for e�ects�e e�ects computed from a sample are random variables andwould be di�erent if another set of experiments is conducted. �econ�dence intervals for the e�ects can be computed if the varianceof the sample estimates are known.

If we assume that errors are normally distributed with zero meanand variance σ 2e , then it follows from the model that the yi ’s arealso normally distributed with the same variance σe .

�e variance of errors can be estimated from the SSE as follows:

s2e =SSE

22(r − 1)�e quantity on the right side of this equation is called the Mean

Square of Errors (MSE). �e denominator is 22 (r − 1), which is thenumber of independent terms in the SSE.�is is because the r error terms corresponding to the r replicationsof an experiment should add up to zero. �us, only r − 1 of theseterms can be independently chosen.

�us, the SSE has 22 (r − 1) degrees of freedom. �e estimatedvariance is Sq0 = SqA = SqAB =

Se√22r

�e con�dence interval for the e�ects are :

qi ∓ t[1−α/2;22(r−1)]sqi

�e result obtained for the range is as follows:(61.21, 62.15)(-10.58, -9.64)(15.78, 16.72)(-13.36, -12.42)against the actual values 61.68, -10.11, 16.25, -12.89. None of thecon�dence intervals included 0 fortifying the goodness of the ex-periment.

5.7 Principal Representative Feature(PRF)�e PRF primarily identi�es the contributors in the system whichhas maximum variance and tries to identify a pa�ern in a givendata set which is unique.

�e �rst principal component accounts for as much of the vari-ability in the data as possible, and each succeeding componentaccounts for as much of the remaining variability as possible.

�ough primarily used for dimensionality reduction, the PRFhas been exploited here to �gure out the contributions of eachfactor towards the variation of the response variable. �e authorsdon’t intend to ignore one of the two input parameters and that’snot how the method should be construed. Since data trends haveevidence of normal behavior, PRF was used as an alternative tofactor analysis. If Shapiro Wills test for normalcy revealed non-normal behavior, ICA could have been used to understand howeach factor contributes to the response variable, ”y”.

�e PRF conducted on the generated data gave the followingresults: variation explained by �rst factor (New server) is 66% 2ndfactor (P&C) explains 34.08% of the variation.

5.8 Non-parametric EstimationAparametric statistical test is one that makes assumptions about theparameters (de�ning properties) of the population distribution(s)from which one’s data are drawn, while a non-parametric test isone that makes no such assumptions.

�e tests involve estimation of the key parameters of that distri-bution ( the mean or di�erence in means) from the sample data. �ecost of fewer assumptions is that non-parametric tests are generallyless powerful than their parametric counterparts.

Apart from the conclusion obtained above, we perform the non-parametric estimation which does not rely on assumptions that thedata are drawn from a given probability distribution.

�e �gures 13, 14 and 15 represent the results obtained from thenon- parametric estimation for the data A1: New Server and A2:Power & Cooling, corresponding to IRS, CRS and DRS respectivelyand also visualizes the interaction between the factors.

�e �gures 16, 17 and 18 represent the results obtained from thenon- parametric estimation for the data New Server and Power &Cooling, corresponding to IRS, CRS and DRS respectively on thegenerated data set and also visualizes the interaction between thefactors.

�e above �gures suggest no interaction between the factors,which is in agreement with the results obtained in the previoussections.


Figure 13: Non-parametric Estimation for IRS-Original Data

Figure 14: Non-parametric Estimation for CRS-OriginalData

Figure 15: Non-parametric Estimation for DRS-OriginalData

6 EXPERIMENTSLet the assumed parametric form be y = K + α log(S) + β log(P).Consider a set of data points.

Figure 16: Non-parametric Estimation for IRS-GeneratedData

Figure 17: Non-parametric Estimation for CRS-GeneratedData

Figure 18: Non-parametric Estimation for DRS-GeneratedData

lny1 = K ′ + αS ′1 + βP ′1...

......

...

lnyN = K ′ + αS ′N + βP ′N


whereS ′i = loд(S

′i )

P ′i = loд(P′i )

If N >3, (23) is an over-determined system. One possibility is aleast squares solution. Additionally if there are constraints on thevariables (the parameters to be solved for), this can be posed as aconstrained optimization problem. �ese two cases are discussedbelow.

(1) No constraints : An ordinary least squares solution. (24) isin the form y = Ax where,

x =[K ′ α β

]Ty =

y1.

.

yN

(24)

and

A =

1 S ′1 P ′1...

1 S ′N P ′N

(25)

�e least squares solution for x is the solution thatminimizes

(y −Ax)T (y −Ax)It is well known that the least squares solution to (24) isthe solution to the system

ATy = ATAx

i.e.x = (ATA)−1ATy

In Matlab the least squares solution to the overdeter-mined system y = Ax can be obtained by x = A \ y.�e following is the result obtained for the elasticity valuesa�er performing the least square ��ing:

IRS CRS DRSα 1.799998 0.900000 0.799998β 0.100001 0.100000 .099999Table 11: Least square test results

(2) Constraints on parameters : �is results in a constrainedoptimization problem. �e objective function to be mini-mized (maximized) is still the same namely

(y −Ax)T (y −Ax)�is is a quadratic form in x. If the constraints are linear

in x, then the resulting constrained optimization problemis a�adratic Program (QP). A standard form of a QP is :

min xTHx + f T x (26)s.t.

Cx ≤ b Inequality ConstraintCeqx = beq Equality Constraint

Suppose the constraints are that α and β are >0 and α +β ≥ 1. �e quadratic program can be wri�en as (neglectingthe constant term yT y ).

min xT (ATA)x − 2yTAx (27)s.t.

α > 0β > 0

α + β ≤ 0In standard form as given in (29), the objective function

can be wri�en as :

xTHx + f T x (28)where

H = ATA and f = −2ATy�e inequality constraints can be speci�ed as :

C =

0 −1 00 0 −10 1 1

and

b =

001

In Matlab, quadratic program can be solved using the

function quadprog.�e below results were obtained on conducting �a-

dratic Programming.

IRS CRS DRSK 3.1106 0 0α 1.0050 0.9000 0.8000β 0.1424 0.1000 0.1000

Table 12: �adratic Programming results

7 PREDICTION AND FORECASTINGLinear regression is an approach for modeling the relationship be-tween a dependent variable y and one ormore explanatory variablesdenoted by x. When one explanatory variable is used, the model iscalled simple linear regression. When more than one explanatoryvariable are used to evaluate the dependent variable, the model iscalled multiple linear regression model. Applying multiple linearequation model to predict a response variable y as a function of 2predictor variables x1,x2 takes the following form:

y = b0 + b1x1 + b2x2 + e (29)Here, b0,b1,b2 are 3 �xed parameters and e is the error term.Given a sample,(x11,x21,y1),�(x1n ,x2n ,yn ) of n observations the model consist offollowing n equations


y1 = b0 + b1x11 + b2x21 + e (30)y2 = b0 + b1x12 + b2x22 + e (31)y3 = b0 + b1x13 + b2x23 + e (32)yn = b0 + b1x1n + b2x2n + e (33)

So, we have

©«y1...

yn

ª®®¬ =©«

1 x11 · · · xk1...

.... . .

...

1 x1n · · · xkn

ª®®¬©«

b1...

bk

ª®®¬ +©«

e1...

en

ª®®¬where k = 1...17 (34)

Or in matrix notation: y = Xb + e

Where:• b : A column vector with 17 elements areb0,b1, ...,b16

• y: A column vector of n observed values of y = y1, ...,yn

• X : An n row by 17 columnmatrix whose (i, j+1)th elementXi, j+1 is 1 if j is 0 else xi j

Parameter estimation:b = (XTX )−1(XTy) (35)

Allocation of variation:

SSY =n∑i=1

y2i (36)

SS0 = ny2 (37)SST = SSY − SS0 (38)

SSE = yTy − bTXTy (39)SSR = SST − SSE (40)

whereSSY=sum of squares of YSST=total sum of squaresSS0=sum of squares of ySSE=sum of squared errrorsSSR= sum of squares given by regressionCoe�cient of determination:

R2 =SSR

SST=

SST − SSESST

(41)

Coe�cient of multiple correlation

R =

√SSR

SST(42)

�e interaction term is ignored in this case, since the experimentsdescribed earlier in the paper have clearly indicated that there isno signi�cant interaction between the predictor variables. Hence,intercept=0.

�e ratio of Training data : Test data is 90-10 as the data availableis less. However, the ratio could be changed to 80-20, 70-30 withthe increasing size of the data available.

�e elasticity values obtained exactly match with the valuesobtained earlier and holds good for the test data set as well. �eR squared test conducted for validation yields 0.99. �is indicatesexcellent �t.

IRS CRS DRSSSY 16984.7190 269.9263 217.7689SSO 999.1011 269.5206 217.4285SST 15985.6179 .4056 0.3403SSR 15984.61683 0.4056 0.3374

R squared .9999 .9999 .9913Alpha 1.8 0.9 0.8Beta 0.08 or 0.1 0.099 or 0.1 0.08 or 0.1

Table 13: Multiple Linear Regression Results

8 CONCLUSIONWith the increase in utility computing , the focus has now shi�edon cost e�ective data centers. Data centers are the backbone toany cloud environment that caters to demand for uninterruptedservice with budgetary constraints. AWS and other data centerproviders are constantly improving the technology and de�ne thecost of servers as the principle component in the revenuemodel. Forexample, AWS spends approximately 57% of their budget towardsservers and constantly improvise in the procurement pa�ern ofthreemajor types of servers. Here in this paper, we have shown howto achieve pro�t maximization and cost minimization within certainconstraints. We have mathematically proved that cost minimizationcan be achieved at the phase of increasing return to scale, whereaspro�t maximization can be a�ained at the phase of decreasingreturn to scale. �e Cobb Douglas model which is a special case ofCES model is used by the authors as revenue model which looks atsuch situation i.e include two di�erent input variables for the costsof two di�erent types of servers.

�e factors, number of servers (S) and investment in infrastruc-ture (I) were combined to cost of deploying new server. �e othertwo factors, cost of power (P) and networking cost (N) were com-bined to cost of power and cooling. Our work has established thatthe proposed model agrees with optimal output elasticity with real-time data set. As server hardware is the biggest factor of totaloperating cost of data center and power is the most signi�cant costamong other cost segments of data center, we have taken these twocost segments prominently in our output elasticity calculation. �eanalytic exercise, coupled with a full factorial design of an exper-iment quanti�es the contribution of each of the factors towardsthe revenue generated. �e take away factor for a commercialdata center from this paper is that the new server procurementand deployment cost plays a major role in the cost revenue dynam-ics. Also, that the response variable is a function of linear predictors.

A weakness of the model is the inability to predict the techno-logical progress as a variable. Another weakness that the authorswould like to point out is the curvature violation of the Convexfunctional form when the number of features or input parametersgrow. Since the prediction of the constant technological progress


cannot be precisely modeled, the experiments performed takingthe randomly generated data proves that the model used by theauthors is valid to encompass the developments due to technologi-cal progress. Also, one can’t guarantee the optimal values of theelasticity empirically.

�e paper is potentially a good working tool for the entrepreneurempowering them with e�cient/optimal resource allocation forall the inputs. �ere could be di�erent combinations of resourceallocation, even for a speci�c quantum of output. �e concavity ofthe Production Possibility Curve ensures that. �e proposed modelhas shown to be amenable to higher scale. �us, any signi�cantincrease in the budget, consequently scale of investment on inputs,does no way invalidate the conclusion. Again, the fundamentallaw of variable proportions and its ultimate, the law of diminishingreturns, has been found to be operative.

REFERENCES[1] Google, How much do Google data centers cost? , [WWW document ]. Re-

trieved Sepetember 10, 2016 from h�p://www.datacenterknowledge.com/google-data-center-faq-part-2/.

[2] Apple, How Big is Apple’s North Carolina Data Center? [WWW document ],Retrieved Sepetember 13, 2016 from h�p://www.datacenterknowledge.com/the-apple-data-center-faq/.

[3] Facebook, How much Does Facebook Spend on its Data Cen-ters? [ WWW document ], Retrieved Sepetember 19, 2016 fromh�p://www.datacenterknowledge.com/the-facebook-data-center-faq-page-three/,as accessed on 26 February 2016

[4] Albert Greenberg, James Hamilton, David A.Maltz and Praveen Patel.2009. �e Cost of a Cloud: Research Problems in Data Center Networks, ComputerCommunication Review, 39, 1, 68-73, 2009

[5] Mahdi Ghamkhari and Hamed MohsenianRad, Energy and Performance Manage-ment of Green data centers. 2013. A Pro�t Maximization Approach,IEEE transac-tions on Smart Grid, 4, 2, 1017 - 1025, 2013

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[8] Yuan Zhang, Xiaoming Fu, K K and Ramakrishnan. 2014. Fine-Grained Multi-Resource Scheduling in Cloud Data Centers, 2014 IEEE 20th International Work-shop on Local & Metropolitan Area Networks (LANMAN), DOI: 10.1109/LAN-MAN.2014.7028622

[9] Liaping Zhao, Liangfu Lu, Zhou Jin and Ce Yu. 2015. Online Virtua Machine Place-ment for Increasing Cloud Providers Revenue IEEE Transactions on ServicesComputing, 1, 1, PrePrints, doi:10.1109/TSC.2015.2447550

[10] Snehanshu Saha, Jyotirmoy Sarkar, Anand MN , Avantika Dwivedi, NanditaDwivedi, Ranjan Roy and Shirisha Rao. 2016. A Novel Revenue OptimizationModel to address the operation and maintenance cost of a Data Center. Journalof Cloud Computing, Advances, Systems and Applications. 5, 1, 1-23, 2016; DOI10.1186/s13677-015-0050-8

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[12] Table 1-Dataset, h�ps://github.com/swatigambhire/demorepo, as accessed on 26February 2016

[13] Raj Jain. 1991. �e Art of Computer System Performance Analysis. 1-720, Wiley;ISBN: 978-0-471-50336-1

Received 2017; accepted 2017

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