A novel revenue optimization model to address the operation and maintenance cost of a data center
 Snehanshu Saha^{1}Email author,
 Jyotirmoy Sarkar^{2},
 Avantika Dwivedi^{1},
 Nandita Dwivedi^{1},
 Anand M. Narasimhamurthy^{3} and
 Ranjan Roy^{4}
https://doi.org/10.1186/s1367701500508
© Saha et al. 2015
Received: 15 June 2015
Accepted: 18 December 2015
Published: 19 January 2016
Abstract
Enterprises are enhancing investments in cloud services setting up data centers to meet growing demand. A typical investment is of the order of millions of dollars, infrastructure and recurring cost included. This paper proposes an algorithmic/analytical approach to address the issues of optimal utilization of the resources towards a feasible and profitable model. The economic sustainability of such a model is accomplished via CobbDouglas production function. The production model seeks to answer questions on maximal revenue given a set of budgetary constraints. The model suggests minimum investments needed to achieve target output.
Keywords
CobbDouglas production function Data center Concavity Returns to scale Profit function Cost functionMotivation and background
IT operations are integral to most business organizations around the world. The business communities need to rely on the information systems to run their organizational operations. Therefore, a company may incur loss due to disruptions and unavailability of information systems. It is necessary to have an IT infrastructure, which houses all the information systems to minimize all kind of disruptions and obstacles related to information systems. This reliable IT infrastructure is called data center. The cost to run a data center is generally associated with power, cooling, networking and storage equipment. A data center houses thousand of information and computing systems deployed in computer racks. A rack is an Electronic Industries Association enclosure, which is 2 meters high, 0.61 meters wide and 0.76 meters deep. A standard rack accommodates 4042 computing units and dense rack configuration servers (Blade rack) will accommodate 200 computing units. The heat dissipated by a standard rack is 10 KW and Blade rack will dissipate heat up to 30 KW. Hence, a data center containing 2000 racks may require 20 MW power [1]. Increasingly, PCbased computing and storage services are relocating to Internet services. While early Internet services were mostly informational, many recent Web applications offer services that previously resided in the client, including email, photo and video storage and office applications. The shift from PCbased computing services to serverside computing is driven primarily not only for the improvements in services, such as the ease of management (no configuration or backups needed) and ubiquity of access (a browser is all you need), but also by the advantages it offers to vendors. Now a days, Software as a service provides faster application development because it is easier for software vendors to make changes and improvements. Instead of updating millions of clients, vendors need to coordinate improvements and fixes inside their data centers and can restrict hardware deployment to a few welltested configurations. Moreover, data center economics allows many application services to run at a low cost per user. For example, servers may be shared with thousands of active users (and many more inactive ones), resulting in better utilization. Similarly, the computation itself may become cheaper in a shared service (e.g., an email attachment received by multiple users can be stored once rather than many times). Finally, servers and storage in a data center can be easier to manage than the desktop or laptop equivalent because they are under the control of a single, knowledgeable entity [2]. Though each data center is different based on the operations, facilities and the average cost per year to operate, a large data center costs between $10 million to $25 million. 42 % of costs are associated with hardware, software, uninterrupted power supplies, and networking. 58 % of the expenses is due to heating, air conditioning, property and sales tax. In a traditional data center, most of the cost is consumed by infrastructure for maintenance. A few surveys have estimated the maintenance cost up to 80 % of the total cost. As data centers have become important aspects in business organization, it is imperative to examine the costrevenue dynamics and design an effective way to optimize it. Cisco in its Global Cloud Index (GCI) Reports2012, forecasted data center traffic to shoot up to 554 exabytes (EB) per month by 2016, from 146 exabytes. There are a few techniques to optimize cost and profit. CobbDouglas is a widely used production model, but to the best of our knowledge, has never been used in the study of optimization issues arising in data centers.
Introduction & overview
The authors find it necessary to discuss different aspects of data centers and optimization, before proceeding to relevant scholarly work available in the public domain.
Data center key subsystems
 1.
Continuous power supply.
 2.
Air conditioning.
 3.
Network Connectivity.
Power is essential to provide uninterrupted services throughout the year. At large data centers, electricity is supplied either from a grid or from onsite generators. The electricity supplied to the computer racks is dissipated as heat. Therefore, a cooling system is required to mitigate the heat. Generally data centers have chillers to supply cold water, which is used for air conditioning. Network connectivity is necessary for data transmission within and outside of a data center. The power subsystem consists of a grid and backup generator. The network subsystems comprise of all the connectivity except rack switches, whereas cooling subsystem includes chiller and air conditioning system.
Traditional data center
Running a traditional data center is expensive in comparison to a cloud data center. Thousands of applications are running in traditional data centers along mixed hardware tool. Maintaining existing infrastructure consume the biggest chunk of the total cost and multiple management tools are required for operation and management. Lately, traditional data centers are being used by internet service providers for housing their own or third party servers. Traditionally data centers were either constructed for meeting the purpose of large organizations or Networkneutral data centers. These facilities establish interconnection of carriers and act as regional fiber hubs, providing services to local business in addition to hosting content servers.
Cloud data center

Constructed for serving different objectives.

Built to a different scale.

Created at a different time than the traditional data center.

Unlike traditional data center, these are responsible for executing and managing different workload.

Not constrained by limitations of traditional data centers.
The cost associated with cloud data centers are composed of three factors. Labor cost takes the smallest chunk of total operation cost, nearly 6 % of the total cost, whereas power and cooling cost and computing costs are 20 % and 48 % respectively. Other costs account for remaining 26 %. Cloud data centers add new cost, unlike traditional data centers [3].
Data center tiers

Data centers related to Corporate house.

Data centers responsible for computer infrastructure as a service (IaaS) and hosting Web application.

Data centers that provide services of TurnKey Solutions

Data centers, where Web 2.0 has been implemented.
Data center optimization

Lack of Access to Basic Data: Accurate data is required for achieving shortterm and longterm capacity planning to ensure organizations don’t overinvest or underinvest in power and cooling infrastructure. Specifically, IT industries need the ability to keep track of individual and aggregated server power usage and temperature data at any point of time and historical trends as well.

Inefficient Use of Power: Server power is typically over allocated and racks are under populated in worstcase. This may create a situation where power infrastructure is inefficiently used and contributes to overinvestment in racks and power capacity. IT solutions need to figure out a mechanism to control power consumption below theoretical peak values so that kilowatt capacity of each rack may be fully utilized.

Power and Cooling Excursions: The availability of services during businesscritical time is a top priority though power supply disruption or cooling tower failures can affect business operations massively. IT must be able to survive such failures to avoid downtime and deliver reliably on service level agreements (SLAs).

Higher density endowed computing environments are more efficient but can lead to localized hot spots during periods of peak utilization. Organizations need to identify and mitigate hot spots and optimize workload placement based on power and cooling availability and efficiency [5].
In this paper, we propose a revenue model, which is best suited for the data centers in the current scenario. Mathematically, we have established the relevance of the model with the data centers. We find CobbDouglas model (CD) to be most suitable for optimizing the cost associated with the data centers. 3D graphs of cost data collected from various sources, have been generated for better understanding of the model. The remainder of the paper is organized as follows: Related work in the same field by other authors and Industry practices have been discussed in Section Introduction & overview. The analytical foundation of our proposed decision model has been elaborated in Section Related work. The decision model has been applied on data center’s real time data set and results have been analyzed in Section Analytical foundations of the decision model. We have compared our model with other mathematical models used in data center and explored the ways to overcome the drawbacks of CD function in conclusion Section Results and discussion. Appendix contains the mathematical proofs of optimal revenue and cost, computation of elasticities and relevant Matlab codes.
Related work
Industry practices
Enterprises make investments in millions of dollars on setting up data centers. The elite list includes many Fortune 500 companies Google, Apple, Facebook Inc., Amazon.com, and Microsoft, to name a few. Each of the four new Google data center projects unveiled in 2007, cost an estimated $600 million, which includes capital investment for construction, infrastructure, and servers for two data center buildings. In its earnings reports, Google reported $1.9 billion spent on data centers in 2006 and $2.4 billion in 2007 [11]. Apple operates a data center in Newark, California, which it acquired in 2006 for approximately $45 million at a significant discount to its construction cost. The data center occupies 108,000 square feet of total space. Apple intends to invest more than $1 billion over the next 10 years on its 183acre data center campus in Maiden, North Carolina [12]. Facebook has invested more than $1 billion in the infrastructure that powers its social network, which now serves more than 845 million users a month around the globe. The company spent $606 million on servers, storage, network gear and data centers in 2011, and spent another $500 million in 2012 [13]. In his research at Microsoft and Amazon Web Services, Hamilton has focused on cost models for operating hyperscale data centers. His presentation at the Amazon open house reviewed cost assumptions for an 8 megawatt data center, which could include 46,000 servers. The cost was estimated at $88 million (about $11 million per megawatt), inclusive of monthly operating costs for a facility, which is dominated by the cost for servers (57 %), followed by power and cooling (18 %) and electric power (13 %) [14].
A similarity between cloud and traditional data center is that both can be used for data storage. Cloud is an example of offpremises computing, whereas data centers are being used on premise storing system. Nowadays, data centers are effectively being utilized in cloud computing. Cloud services are now being provided through data centers, which house cloud services and cloudrelated resources. Cloud service providers also own data centers, which is located in different geographical location, for provisioning of uninterrupted services in case of outage and unpredictable situations. IaaS (Infrastructure as a service), which provides facilities like virtual machines, storage and load balancing maintains a large pool of resources in data centers. Data centers, which are largely being used for cloud computing are called cloud data center. Lately the demarcation of the terms has disappeared and all are referred as data centers. Existing data centers are often restructured with modern equipment so that it can take advantage of greater performance and energy efficient facilities of cloud computing. The entire process of modernization of data center is called data center transformation [15].
Cloud services are scalable, implying it will allocate resources based on your demands. We are considering the storage usage, which may range from terabytes to petabytes. Say, for example, if one organization need store 4000 GB (4 TB) of data in Amazon S3 (Simple Storage Service) then it would cost $118.50 per month for disk space (Considering Amazon charges $0.03 per GB for the first TB and $0.0295 per GB for the next 49 TB). Apart from storage charges, cloud service providers also charge the network usage as sometimes it requires to transfer data out of storage. Amazon S3 charges $0.090 per GB to transfer data up to 10 TB whereas google charges $0.11, adding $720 for AWS storage or it may incur $880 for google storage. API requests such as get, put, delete, copy, post etc. may also incur some cost. The charges may vary from $0.005 to $0.01 per 1000 or 10,000 requests based on the cloud service providers. There are escalating demands for data center space as services such as big data are migrating to cloud. Recent lease activities by the big data center players reveal the high demand for cloud services. Rackspace has leased 58,000 square feet at Digital Reality’s 69 acre data center park in Dallas. The rising demand is attracting investment from non traditional players into the data center space. Cousins Properties has transformed its 170,000 square feet of unused space out of a total of 1 million square foot American Cancer society center in Atlanta into a data center [16].
Since, energy consumption of cloud data centers is a key concern for owners owing to rising energy costs (fuel), CO2 emissions related to this consumption have become relevant [17]. Therefore, saving money in the energy budget of a cloud data center, without sacrificing Service Level Agreements (SLA) is an excellent incentive for cloud data center owners, and would at the same time be a great success for environmental sustainability. The ICT resources, servers, storage devices and network equipments consume maximum power. Processors [18] are the main contributors to the server’s power consumption whereas other components [19] like multiple level caches, RAM, I/O activities also contribute to the total power consumption of the server. The storage devices range from a single hard disk to SAN (Storage Area Network) devices, which consume a significant amount of power. The other significant contributors to power consumption are network equipments which includes routers and switch fabrics.
Academic work
James Hamilton [20] has shown that, quite significantly, power is not the largest cost, if the amortization cost of power, cooling infrastructure for 15 years and new server amortization cost over 3 years are taken into consideration. He concluded that, cooling amortization and server amortization monthly payments have been computed using 5 % per annum cost and server hardware costs are the largest. But power infrastructure cost will rise and server hardware cost may fall, resulting in the domination of power cost over all other data center expenses in not so distant future. Generally, a typical data center comprises 100 fully loaded racks with the current generation 1U servers needing $1.2 million for power and an additional $1.2 million for cooling infrastructure per annum. Moreover, $1.8 million annual cost is incurred due to maintenance, amortization of power and cooling equipment. Thus, power is the most significant cost of the data center while server hardware contributes to the biggest chunk of the total operating cost. These two cost factors are primarily considered for calculation of output elasticities using 3D graphs for three phases of returns in the enterprise lifecycle.
Returns are generally used to measure the corresponding change in output subsequent to change in physical dose of an input. Every enterprise has an initial increasing returns to scale, followed by constant returns and finally decreasing returns. The reasons for the occurrence of these phases are described later in the section titled “Analytical foundations of the decision model”.
CobbDouglas function has been widely used in economics and various sectors. Askhan Hassani has used this production function in construction management, in construction schedule crashing and project risk analysis related to duration of construction projects [21]. Moyazzem, Ajit and Tapati have used CobbDouglas function to decide the most suitable functional form of production process for major manufacturing sectors of a country. They have applied CobbDouglas model with additive error and multiplicative error term [22]. DeMin Wu [23], have shown the exact distribution of the indirect least squares estimator of the coefficients of the CobbDouglas production function within the context of a stochastic production model of MarschakAndrews type. Efstratios Rappos, Stephan and Rudlof have proposed integer programming optimization model of data center for determining the optimal allocation of data components among a network of Cloud data servers in such a way that it minimizes the total costs of additional storage, estimated data retrieval costs and network delay penalties [24]. Geooptimization technique considers the geographical location of the servers and customers while trying to optimize cost of cloud services [25]. Combinatorial optimization method has been developed to determine the best allocation process of virtual servers to target servers or virtual resource to actual resources [26]. Efforts have been made to reduce the cost of electricity in data center under multiple electricity market environment without compromising quality of services. The model proposed in Distributed Internet Data Centers in a MultiElectricityMarket Environment is an example of constrained mixed integer programming [27]. Budget constraints force organizations to explore strategies that yield optimal revenues.The proposed production model using CobbDouglas production function [28–30] is very relevant by paving an optimal way to attain the maximum revenue. In this paper, four major segments of the cost associated with data centers such as server, infrastructure, power, and network [31] are considered for optimization. Power is the fastest growing cost among all other costs. Several initiatives have been contemplated to curtail the cost associated with power. Dynamic smart cooling techniques, equipped with temperature aware cooling algorithm has been adopted to reduce the cost. Using scale processor and system power enables data centers save energy and reduce cost [32]. Here in this paper, an attempt is made to achieve the dual goal of profit maximization and cost minimization within certain constraints. It is proved mathematically that cost minimization can be achieved at increasing return to scale, whereas profit maximization can be attained at decreasing return to scale. The CobbDouglas production function, which has been used rigorously in this revenue model, is endowed with a flexible functional form and less restriction over output elasticity.
Next, we define key terminologies used in the scientific investigation & deployment of our model in the revenue optimization problem.
Key terminologies & techniques 
Mathematical optimization 
Optimization is a technique to select the best element from a set of available alternatives in the field of mathematics, computer science, economics or management science [33]. An optimization problem can be represented in various ways. 
Given: a function f:A→R from set A to the real numbers An element x _{0} in A such that f(x _{0})≤f(x) for all x in A (minimization). f(x _{0})≥f(x) for all x in A (maximization). The optimization technique is useful for modeling many real world problems In the above formulation, the domain A is called search space of function f and elements of A are called candidate solutions or feasible solutions. The function is suitably termed as cost function, revenue function or utility function based on the area of interest. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution. 
Computational optimization techniques 
To solve a particular problem, researchers may use an algorithm that will find the solution in finite steps. Iterative method will converge to the optimal solution and heuristic will give an approximate solution to a problem used in optimization scenario, is combinatorial algorithms. 
Iterative methods: 
Iterative methods are usually applied to solve problems of nonlinear programming. The iterative methods differ according to the use of Hessians, Gradients or function values. Evaluating Hessians and Gradient helps improve the rate of convergence of the functions, but such methods introduce computation complexity to each iteration. In some cases, the computational complexity may be very high. One major criterion for optimizers is the number function evaluations required as this often may require large computational efforts. The derivatives some time give detailed information for such optimizers but are hard to calculate. 
Methods that evaluate Hessian: 
∙ Newton’s method. 
Methods that evaluate Gradient: 
∙ QuasiNewton method. 
∙ Conjugate gradient method. 
∙ Interior point method. 
∙ Gradient descent. 
∙ Sub gradient method. 
∙ Ellipsoid method. 
∙ Reduced gradient method. 
Gradient descent method has been used in the proposed work to compute optimal costs. 
∙Increasing returns to scale: In the initial phase, the output may increase in a higher proportion [34]. This phase is called the phase of increasing returns. This change occurs as: 
1 Greater application of the variable factor ensures better utilization of the fixed factor. Actually, this enables the utilization of idle capacity (potential) of the fixed factor. 
2 It facilitates better division of the variable factor. 
3 It improves coordination between the factors. This paper establishes that cost minimization of an enterprise that invests on servers, infrastructure, network, power, etc. is achieved at this phase. The 3D plots obtained are neither concave nor convex. 
4 Constant returns to scale: An increase in one input may yield an increase in corresponding output in the same proportion. However, this phase rarely happens and even if it occurs, it would be for a very negligible period. Actually, it is only a passing phase between increasing and diminishing returns. 
∙Decreasing returns to scale: Ultimately, the phase of decreasing or diminishing returns will set in, whereby the deployment of an additional input will result into increase in output but at a diminishing rate or lower ratio [34]. 
This happens because: 
1 As more and more units of a variable factor are combined with the fixed factor, the latter gets overutilized. Hence, the rate of corresponding growth of output goes on diminishing. 
2 Factors of production are imperfect substitutes of each other. The divisibility of their units is not comparable. 
3 The coordination between factors get distorted so that marginal product of the variable factor declines. 
Our work proves that profit maximization of an enterprise is achieved in this phase [Fig. 7] 
The marginal product is the change in total output owing to a unit change in the input of a variable factor. It is also shown that marginal product increases for the initial phase, i.e. increasing returns to scale, subsequently stabilizes for constant returns and finally decreases for last phase, i.e., decreasing returns to scale. 
Analytical foundations of the decision model
We propose necessary results that will be used to model production, cost and profit of the cloud data center.
Theorem 1: production maximization
Consider an enterprise that has to choose its consumption bundle (S, I, P, N) where S, I, P and N are number of servers, investment in infrastructure, cost of power and networking cost respectively of a cloud data center. The enterprise wants to maximize its production, subjected to the constraint that the total cost of the bundle does not exceed a particular amount. The company has to keep the budget constraint in mind and keep total spending within this amount.
The above results are proved in Appendix 1.
A quick heuristic for CRS: Revenue optimization
If we consider again the case of constant return to scale, where all the elasticities of different cost components are equal. \(y=\prod _{i=1}^{n}x_{i}^{\alpha _{i}}\), where all α _{ i } are equal and \(\sum \alpha _{i}=1\). In such scenario, the response variable or output turns out to be the geometric mean of all inputs.
Theorem 2: cost minimization
Therefore, the enterprise will have cost minimization at the phase of increasing returns to scale.
Global Minima for cost minimization: a heuristic approach
The above partial derivatives are used in gradient descent method for cost minimization.
 1.
procedure GRADIENTDESCENT()
 2.
\(\frac {\partial c}{\partial \alpha }\leftarrow \left (w_{1}y_{\textit {tar}}^{\frac {1}{\alpha }} Y^{\frac {\beta }{\alpha }}ln\left (\frac {y_{\textit {tar}}}{Y^{\beta }}\right)\right)\frac {1}{\alpha ^{2}}\)
 3.
\(\frac {\partial c}{\partial \beta }\leftarrow \left (w_{1}y_{\textit {tar}}^{\frac {1}{\alpha }} Y^{\frac {\beta }{\alpha }}ln y_{\textit {tar}}ln\left (\frac {y_{\textit {tar}}}{Y^{\beta }}\right)\right)\frac {1}{\alpha ^{3}}\)
 4.
repeat
 5.
\(\alpha _{n+1}\leftarrow \alpha _{n}\delta \frac {\partial c}{\partial \alpha }\)
 6.
\(\beta _{n+1}\leftarrow \beta _{n}\delta \frac {\partial c}{\partial \beta }\)
 7.
α _{ n }←α _{ n+1}
 8.
β _{ n }←β _{ n+1}
 9.
until (α _{ n+1}>0)(β _{ n+1}>0)(α _{ n+1}+β _{ n+1}>1)
 10.
end procedure
Using the above algorithm, the optimal values of α,β and cost have been computed (cf. Results and discussion).
Theorem 3: profit maximization
Profit maximization is achieved when:
(1) \(p\frac {\partial f}{\partial S}=w_{1}\) (2) \(p\frac {\partial f}{\partial I}=w_{2}\) (3) \(p\frac {\partial f}{\partial P}=w_{3}\) (4) \(p\frac {\partial f}{\partial N}=w_{4}\)
Therefore, the enterprise will have profit maximization at the phase of decreasing returns to scale. It is later shown in Appendix 4, that profit maximization is scalable provided for an arbitrary, n, number of input variables (constant), the result stands as long as \(\sum _{i=1}^{n}\alpha _{i}<1\); where α _{ i } is the i ^{ t h } elasticity of the input variable x _{ i }.
To maximize l o g(y), if l o g(A) is negative, α needs to be as small as possible (since α>0) else α must be as large as possible. Same applied to β (Table 1).
Maximization of CD function for fixed A and B
A, B  α,β 

A<1,B<1  [ α _{ min },β _{ min }] 
A<1,B>1  [ α _{ min },β _{ max }] 
A>1,B<1  [ α _{ max },β _{ min }] 
A>1,B>1  [ α _{ max },β _{ max }] 
The RHS of (18) is similar in form to (16) and hence same rule can be applied i.e. If \(\prod _{i=1}^{N}u_{i}<1\) then α=α _{ min } else α=α _{ max }. The term involving β can be minimized similarly and independently. The only remaining step is to determine the permissible ranges. Let ε be the smallest value that α and β can take. Suppose in the above example, \(\prod _{i=1}^{N}u_{i}<1\) and \(\prod _{i=1}^{N}v_{i}>1\). We know that α should be minimized and β should be maximized. Since α+β<1, let α+β=1−δ, where δ is a small nonnegative number. We then have α _{ min }=ε and β _{ max }=1−δ−ε.
Results and discussion
As mentioned earlier, server and power/cooling costs form the biggest chunk of the total cost. These two inputs are considered for computing the values of the elasticities using 3D plots. However, the results obtained hold good for any number of inputs. It is also possible to aggregate the inputs into two broad categoriesoperational expenditure and capital expenditure and use these as the two inputs in the proposed cost model. Operational expenditures include the recurring costs like power/cooling, server management etc; whereas capital expenditure includes initial investment such as new server cost, infrastructure costs etc.
The data associated with data center costs from various sources have been accumulated and CobbDouglas function is applied on varying elasticities to find the optimal solution for revenue of data center, and finally revenue maximization is demonstrated graphically. All simulation results have been generated by a computer system using Matlab.
 1)Increasing Returns to Scale
 2)
Constant Returns to Scale
 3)
Decreasing Returns to Scale
Case 1: increasing returns to scale
Applying the constraints: α+β>1 α>0 β>0to the function: f=k x ^{ α } y ^{ β }
Using fmincon function of matlab [Appendix 5], the values of elasticities for which revenue is maximized for each year are obtained.
In Table 2, all units are in (123)B. The optimal revenue for all the years is obtained at α=1.8 and β=0.1 Using these results, 3Dsimulations are created and the corresponding graphs are obtained.
Simulation output for IRS
Year  New server  Power and cooling  Elasticity (α)  Elasticity (β)  Max. revenue 

1996  62  5  1.8000  0.1000  1977.88 
1997  65  5  1.8000  0.1000  2153.48 
1998  62  10  1.8000  0.1000  2119.84 
1999  60  10  1.8000  0.1000  1998.35 
2000  65  10  1.8000  0.1000  2308.04 
2001  55  15  1.8000  0.1000  1779.35 
2002  45  15  1.8000  0.1000  1239.91 
2003  47  15  1.8000  0.1000  1340.86 
2004  50  20  1.8000  0.1000  1542.58 
2005  52  20  1.8000  0.1000  1655.42 
2006  55  20  1.8000  0.1000  1831.28 
2007  56  30  1.8000  0.1000  1969.92 
2008  57  30  1.8000  0.1000  2033.69 
2009  58  30  1.8000  0.1000  2098.37 
2010  59  40  1.8000  0.1000  2227.09 
2011  60  40  1.8000  0.1000  2295.50 
2012  60  40  1.8000  0.1000  2295.50 
It is seen from the data set that if new server cost is increased by 1 unit from year 2007($56B) to 2008($57B), the revenue changes by 63 units and when it is increased by 1 unit from year 2008($57B) to 2009($58B), the revenue changes by 64.68 units. This proves that marginal product of input (new server cost) increases in increasing returns to scale.
Case 2: constant returns to scale
Simulation output for CRS
Year  New server  Power and cooling  Elasticity (α)  Elasticity (β)  Max. revenue 

1996  62  5  0.9000  0.1000  48.2003 
1997  65  5  0.9000  0.1000  50.2943 
1998  62  10  0.9000  0.1000  51.6598 
1999  60  10  0.9000  0.1000  50.1575 
2000  65  10  0.9000  0.1000  53.9041 
2001  55  15  0.9000  0.1000  48.2987 
2002  45  15  0.9000  0.1000  40.3181 
2003  47  15  0.9000  0.1000  41.9273 
2004  50  20  0.9000  0.1000  45.6222 
2005  52  20  0.9000  0.1000  47.2613 
2006  55  20  0.9000  0.1000  49.7084 
2007  56  30  0.9000  0.1000  52.6116 
2008  57  30  0.9000  0.1000  53.4564 
2009  58  30  0.9000  0.1000  54.2997 
2010  59  40  0.9000  0.1000  56.7509 
2011  60  40  0.9000  0.1000  57.6159 
2012  60  40  0.9000  0.1000  57.6159 
In Table 3, all units are in (123)B. The optimal revenue for all years are obtained at α=0.9 and β=0.1 Using these results, 3Dsimulations are created and the graphs of Constant Return to scale are obtained.
It is evident from the data set that if new server cost is increased by 1 unit from year 2007($56B) to 2008($57B), the revenue changes by 0.84 units. And when it is increased by 1 unit from year 2008($57B) to 2009($58B), the revenue changes by 0.84 units. This proves that marginal product of input (new server cost) is constant in constant returns to scale.
Case 3: decreasing returns to scale
Applying the constraints: α+β<1 α>0 β>0to the function: f=k x ^{ α } y ^{ β }and using fmincon function of matlab [Appendix 5], the values of elacticities are obtained for which revenue is maximized for each year.
Simulation output for DRS
Year  New server  Power and cooling  Elasticity (α)  Elasticity (β)  Max. revenue 

1996  62  5  0.8000  0.1000  31.9014 
1997  65  5  0.8000  0.1000  33.1305 
1998  62  10  0.8000  0.1000  34.1911 
1999  60  10  0.8000  0.1000  33.3059 
2000  65  10  0.8000  0.1000  35.5084 
2001  55  15  0.8000  0.1000  32.3519 
2002  45  15  0.8000  0.1000  27.5536 
2003  47  15  0.8000  0.1000  28.5291 
2004  50  20  0.8000  0.1000  30.8517 
2005  52  20  0.8000  0.1000  31.8351 
2006  55  20  0.8000  0.1000  33.2961 
2007  56  30  0.8000  0.1000  35.1773 
2008  57  30  0.8000  0.1000  35.6789 
2009  58  30  0.8000  0.1000  36.1788 
2010  59  40  0.8000  0.1000  37.7474 
2011  60  40  0.8000  0.1000  38.2584 
2012  60  40  0.8000  0.1000  38.2584 
It can be seen from the data set that if new server cost is increased by 1 unit from year 2007($56B) to 2008($57B), the revenue changes by 0.50 units. And when it is increased by 1 unit from year 2008($57B) to 2009($58B), the revenue changes by 0.49 units. This proves that marginal product of input(new server cost) decreases in decreasing returns to scale.
Gradient descent output for costminimization
Year  New server  Power and cooling  Elasticity (α)  Elasticity (β)  Min. Cost 

1996  62  5  1.3732  0.0091  56.2716 
1997  65  5  1.3729  0.0102  58.0163 
1998  62  10  1.3732  0.0091  56.2716 
1999  60  10  1.3747  0.0031  55.3465 
2000  65  10  1.3729  0.0102  58.0163 
2001  55  15  1.3724  0.0123  51.9474 
2002  45  15  1.3712  0.0170  45.7922 
2003  47  15  1.3745  0.0039  47.5236 
2004  50  20  1.3742  0.0049  49.2799 
2005  52  20  1.3726  0.0116  50.1856 
2006  55  20  1.3724  0.0123  51.9474 
2007  56  30  1.3737  0.0069  52.7832 
2008  57  30  1.3750  0.0017  53.6839 
2009  58  30  1.3722  0.0131  53.7031 
2010  59  40  1.3735  0.0080  54.5258 
2011  60  40  1.3747  0.0031  55.3465 
2012  60  40  1.3747  0.0031  55.3465 
In the gradient descent calculation, we assumed the target revenue as $120B and unit cost of new server installation as 0.6.
Conclusion
In this paper, the proposed production model using CobbDouglas production function is used to quantify boundaries of the inputs (cost of servers, networking, infrastructure and power), for which the maximum revenue subject to the constraint that the total cost does not exceed a particular amount, is attained. These values for the inputs are mentioned in Eqs. 2 through 5, and can be used to obtain the maximum revenue by substituting in the production model. Similarly, the production model is also used to obtain the minimum total cost subject to a certain amount of production (output) that has to be achieved. This value of total minimum cost can be computed using Eq. 9. Further, the inputs contributing to the maximum profit are deduced, as evident from Eqs. 11 through 14. Computation of revenue function using the production model, a subsequent operation, becomes straight forward enough.
Hence, the total cost required to achieve the maximum revenue, the minimum cost required to achieve a predefined revenue as well as the total cost required to achieve the maximum profit are successfully calculated. These strategies can be used for deploying existing resources optimally so that a responsible and beneficial balance can be achieved over the longer term. Economic sustainability implies utilizing the assorted assets of the company efficiently to deliver functioning profitability over a sustained period.
Finally, this paper shows simulations for a given data set, and it is seen that for every phase (IRS, CRS or DRS), there exists a common optimal value of elasticities (α and β) for annual data, which maximizes the revenue. It is also observed from the 3D graphs that the production function is concave for constant and decreasing returns to scale this signifies the enterprise will definitely reach a point where its profit will be maximum for a particular investment. Whereas, it is neither concave or convex for the increasing return to scale, which signifies that the enterprise will reach a point where it is able to minimize the investments made and achieve a target output. Therefore, it can be seen from these 3D graphs and optimal output elasticities that the production model proposed agrees with the current practices in industry.
It has also been established that the behavior of the marginal product of an input depends on the phase of the enterprise. It will rise during increasing returns to scale phase, stay stable for constant returns phase and fall for the last phase of decreasing returns to scale.
An enterprise investing in the data center may be willing to suffer a decrease in production by 10 % or less, at the cost of a major decrement in total investment made on the inputs. This compromise with the yearly production can increase the overall profit of the enterprise drastically. This result needs confirmation by analyzing the different values of inputs for a deviation of revenue ranging from 0–10 %. For example, consider the investments made in the year 2009, (Table 1). The maximum revenue is $36.1788B. If the enterprise decides to decrease investment on servers to $55B and on power & cooling to $28B, the revenue decreases to $34.4354B. Therefore, the output decreases by $1.7434B (that is 4.8 %), but at the same time, the cost of input decreases by a total of $2B. Thus, overall profit increases. Tables 2 and 3 fortify our observation further. Table 2 shows that an increase in the cost of server while keeping power and cooling costs did not impact the revenue. In fact, the revenue increased. Table 3 demonstrates the fact that a decrease in server cost coupled with a twofold increase in power plus cooling cost did not affect revenue in a negative way. Gradient descent ensures that cost is achieved to be minimal by manipulating the elasticities and the increase in revenue is observed accordingly.
In the context of the our proposed model, nature of technological progress is not very important as we have not instrumented this in our model. Curvature violation is a major issue, in case of flexible functional form. We expect the global curvature conditions to be consistent with economic theory when estimations of cost, profit, revenue are required from a functional form. Along with that, the task of maintaining the flexibility of functional form is also necessary. Translog function is a generalized form of CobbDouglas function, a flexible functional form providing second order approximation. Both the CobbDouglas and Translog functions are linear in parameters and can be estimated using least squares methods. It is possible to impose restrictions on the parameters (homogeneity conditions). CD functions are simplistic, assumes all firms have same production elasticities and that subsitution elasticities equal 1. Translog function, which is a generalization of CobbDouglas, suffers from curvature violation. But we have not implemented Translog function in our model. Multiple test cases run on several different values don’t indicate any curvature violation (cf. Fig. numbers 1–5).

Translog function is very commonly used.

It is a generalization of the CobbDouglas function.

It is a flexible functional form providing a second order approximation.

CobbDouglas and Translog functions are linear in parameters and can be estimated using least squares methods.
l n q _{ i }=b _{0}+b _{1} l n x _{1i }+b _{2} l n x _{2i }+v _{ i }+u _{ i }Translog: l n q _{ i }=b _{0}+b _{1} l n x _{1i }+b _{2} l n x _{2i }+0.5b _{11}(l n x _{1i })^{2}+0.5b _{22}(l n x _{2i })^{2}+b _{12} l n x _{1i } l n x _{2i }+v _{ i }+u _{ i }

It is more difficult to interpret.

Translog function requires many parameters for estimation.

It can suffer from curvature violation.
The curvature violation is not evident for CobbDouglas in the case of CRS (Constant return to scale) and DRS (Decreasing return to scale). It may arise when we consider IRS (Increasing return to scale). This IRS curvature violation issue can be tackled using Monte Carlo simulation. In theory, it’s a technical possibility but the authors have not encountered in this study. In fact, Translog production functions requires estimation of many parameters: K +3+K(K +1)/2 and may suffer from curvature violations. On the other hand, CobbDouglas, having linear parameters (in logs) estimated or simulated at par with this assumption, and empirically speaking, we do not see too many curvature violations (on higher orders) for such production/cost functions. If there are, it sometimes arises due to added local (meaning industryspecific, product specific, as opposed to global meaning universalized) restrictions, which we haven’t instrumented in our structure and need not be concerned about. In cases of stochastic frontier analysis estimates with Translog (and therefore quadratic forms) the curvature violation is much more feasible than a Cobb Douglas production function. Considering all these points, CobbDouglas is a more suitable option in comparison to Translog as a foundation of our mathematical model.
This production model obtained from CobbDouglas function might also prove to be helpful in forecasting the revenue of a cloud data center. An enterprise planning to setup a data center may want to know the approximate revenue for a particular capital expenditure on the inputs. The future revenue estimates may be determined by using this production function [Appendix 7].
Chandrakant et al in their paper Cost Model for Planning, Development and Operation of a Data Center have considered four key cost components (space, power, cooling, operation) in total cost calculation. Individual cost component has been discussed in details based on their dependency on different parameters such as amortization cost, maintenance cost and the influence on the total cost. These apart, other cost factors (Licensing cost, Personnel cost, Operation cost) have also been incorporated in the cost model. In contrast, our proposed model is not dependent on the number of cost components. The CobbDouglas model can be expanded as it can accommodate any number of cost factors. Our model not only highlighted cost optimization but also had shown how to achieve profit maximization, revenue maximization. Using real world data set we have established the efficiency of our suggested mathematical model. Jim Gao, in his Machine Learning Applications for Data Center Optimization paper, rigorously used machine learning application to model data center performance and energy efficiency. Various challenges related to data center have been discussed and neural network has been implemented to build the mathematical framework. Energy efficiency is the prime objective of the optimization model and the performance of the model is limited by the quality and quantity of data inputs like any other machine learning applications. In contrast, our model is independent on the training set and methods of training the machine.
where K ^{′}= ICT capital K ^{′′}= nonICT capital. K denotes capital and L denotes labor. According to the assumption, sectoral Solow progress B in the sector using ICT capital is associated with K’/L (hybrid sectoral capital intensity). The parameter B(K’/L) implies the ICT sector, characterized by capitalsaving technological progress which can be found in various fields such as computer chip (Moore’s law which says that the power of computing chip will double within 3 or more recently 2 years) and fiber optical cables. The information and communication technology (ICT) and other few factors are the key reasons behind the technological progress.
Additional remarks & future work
Technological progress
AWS and other data center providers are constantly improving the technology and define the cost of servers as the principle component in the revenue model. For example, AWS [31] spends approximately 57 % of their budget towards servers and constantly improvise in the procurement pattern of three major types of servers. The model used by the authors need to look at such situation i.e include three different input variables for the costs of three different types of servers. In essence, a whole body of future work may include incorporating a lot of input parameters and subject the Cobb Douglas model to multiple levels of the same parameter, namely power cost. This leads to a possibility of interesting analytical exercise, coupled with a full factorial design of an experiment.
Experimental design

Develop a model best suited for the data obtained.

Isolate measurement errors and gauge confidence intervals for model parameters.

Check the adequacy of the model.
Response variable is the outcome, e.g. revenue output due to factors such as cost, manhours and the levels of those factors. The primary and secondary factors as well as replication patterns need to be ascertained such that the impact of variation among the entities is minimized. Interaction among the factors need not be ignored. A full factorial design with the number of experiments equal to \(\sum _{i=1}^{k}n_{i}\) would capture all interactions and explain variations due to technological progress, the authors believe. Here, n denotes the number of factors and k stands for different levels each factor may have [37].
Appendix 1
Appendix 2
Appendix 3
Appendix 4
w _{ i }: Unit cost of inputs
Therefore decreasing returns to scale, is validated
Appendix 5
Matlab Code for Increasing return to scale:
A = [ 11;−1−1;−10;0−1];b = [ 1.9;−1.1;−0.1;−0.1];x0 = [ 0.4;0.1]; [ x,fval]=fmincon(@cobbfun,x0,A,b);function f = cobbfun(x)%CobbDouglas function with k = 1%f is a representation of CobbDouglas function.% x(1), x(2) is representing elasticity constant of new server spending and power/cooling cost respectively. f=−62^{ x(1)}∗5^{ x(2)};end
Matlab Code for Constant return to scale:
A = [ −10;0−1];b = [−0.1;−0.1];Aeq = [ 11];beq = [ 1]x0 =[ 0.4;0.1]; [x,fval]=fmincon(@cobbfun,x0,A,b,Aeq,beq);function f = cobbfun(x)%CobbDouglas function with k =1% f is a representation of CobbDouglas function.%x(1), x(2) is representing elasticity constant of new server spending and power/cooling cost respectively. f=−62^{ x(1)}∗5^{ x(2)};end
Matlab decreasing returns to scale:
A = [ 11;−10;0−1];b = [ 0.9;−0.1;−0.1];x0 = [ 0.4;0.1]; [ x,fval]=fmincon(@cobbfun,x0,A,b);function f = cobbfun(x)%CobbDouglas function with k = 1%f is a representation of CobbDouglas function.%x(1), x(2) is representing elasticity constant of new server spending and power/cooling cost respectively. f=−62^{ x(1)}∗5^{ x(2)};end
Appendix 6
Both conditions for concave function are satisfied by decreasing and constant returns to scale. Therefore, the graph obtained for decreasing and constant returns to scale is concave, while for increasing returns the graph is neither concave nor convex.
Appendix 7
Forecasting the revenue of the data center
These 5 equations are used for determining the values of α,β,γ,δ a n d k. Substituting these values of α,β,γ,δ a n d k in Eq. (64), the equation of curve of best fit for the economic data of the established data centers is obtained.
Now, if an enterprise planning to setup a data center can predict its approximate revenue for a particular set of investments on the 4 inputs.
Appendix 8
3D Plot Code for Decreasing returns to scale:
syms xm ym;dy = 0.001;dx = 0.001; f=7000000.^{ x } m.∗4700000.^{ y } m; [ xm,ym]=meshgrid(.1:dx:.9,.1:dy:.9); f(xm+ym>0.9)=NaN;surf(xm, ym, f, ‘EdgeColor’, ‘none’)
3D Plot Code for Constant returns to scale:
syms xm ym;dx = 0.001;dy = 0.001; f=1600.^{ x } m.∗270.^{ y } m; [ xm,ym]=meshgrid(.1:dx:.9,.1:dy:.9); f(xm+ym>1)=NaN;surf(xm, ym, f, ‘EdgeColor’, ‘none’)
3D Plot Code for Increasing returns to scale:
syms xm ym;dx = 0.001;dy = 0.001; f=65.^{ x } m.∗5.^{ y } m; [ xm,ym]=meshgrid(.1:dx:1.9,.1:dy:1.9); f(xm+ym<1.1)=NaN; f(xm+ym>1.9)=NaN;surf(xm, ym, f, ‘EdgeColor’, ‘none’)
Notes
Declarations
Acknowledgements
The authors wish to thank Professor B.P. Vani, Institute for Social and Economic Change, for carefully reading the manuscript and offering insightful suggestions. The authors are grateful to Dr. Saibal Kar, Centre for Studies in Social Sciences, Calcutta and IZA, Bonn for reading the paper and giving useful feedback.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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