Learning Curves and Stochastic Models for Pricing and Provisioning Cloud Computing Services

Transcription

1 T Learning Curves and Stohasti Models for Priing and Provisioning Cloud Computing Servies Amit Gera, Cathy H. Xia Dept. of Integrated Systems Engineering Ohio State University, Columbus, OH 4310 {gera., he paradigm of loud omputing has started a new era of servie omputing. While there are many 111researh efforts on developing enabling tehnologies for loud omputing, few fouses on how to strategially set prie and apaity and what key omponents are leading to suess in this emerging market. In this paper, we present quantitative modeling and optimization approahes for assisting suh deisions in loud omputing servies. We first show that learning urve models an be helpful to apture the providers' ost redution with eonomy of sale. Suh models also help understand the potential market of loud servies and explain quantitatively why loud omputing is most attrative to small and medium businesses. We then present a stohasti model and a revenue management formulation to address the priing and resoure provisioning deisions for the loud servie providers. The approah enables the loud servie provider a quantitative framework to obtain management solutions and to learn and reat to the ritial parameters in the operation management proess by gaining useful business insights. Key words: loud omputing; learning urves; stohasti modeling; priing; provisioning History: Reeived Aug. 18, 010; Reeived in revised form Aug. 30, 010; Aepted Sep. 13, 010; Online first publiation Nov. 1, Introdution Fueled by the dramati growth in onneted devies, real-time data streams, and the adoption of servie oriented arhitetures and Web.0 appliations, loud omputing is rapidly gaining momentum as an alternative to traditional IT. Cloud omputing represents an emerging servie paradigm in whih large pools of systems are linked together to provide IT servies on behalf of lients (see, e.g. Hayes (008). The key features of loud omputing are on-demand availability and pay per use poliy. On-demand availability gives a ustomer the flexibility to inrease or derease apaity on the fly to meet its demand. It allows ompanies to trade fixed osts for variable usage based osts. Often the fixed osts of infrastruture are large and prohibitive to small to medium size businesses. Cloud omputing also promises to redue the human administration and development osts. In this emerging loud omputing servies marketplae, Amazon, Google, IBM, Mirosoft and Salesfore.om are leading the way with innovative servie offerings. Several priing and apaity planning strategies have emerged amongst loud omputing servie providers. For example, it has been reported that Salesfore.om uses 10 times fewer servers than Amazon to support the same number of lients (see tehrunh.om). While there are many researh efforts on developing enabling tehnologies (suh as virtualization, standardization, parallel omputing, and open soure software) for loud omputing, few fous on what leads to these various different priing and provisioning strategies, how to strategially set the proper priing and apaity, and what key omponents are leading to suess in this emerging yet highly ompetitive market? Aording to IDC's reent Cloud servies user survey (see IDC) of 44 IT exeutives/cios and their line-ofbusiness olleagues, the top two attributes users want from loud servies providers are ompetitive priing and offering performane-level assuranes. While the value delivered by loud servies are lear to the small and medium businesses, the loud provider may have trouble priing and operating profitably. To attrat business, the loud provider must offer a prie visibly lower than what the ustomer believes it is spending on its in-house management of IT. The provider must also provision suffiient apaity to satisfy the servie level agreement with the ustomers. Hene, ost redution and revenue generation are ritial to the suess of a loud servie provider. Consequently, strategi solutions for priing, provisioning, and delivering quality of servie are the determinant omponents of the business proess in order to win in the ompetitive emerging market. 99

2 In the literature, priing and revenue management problems have reeived muh attention over the last deade in many areas, ranging from the airline yield management to hotel industry to retail stores (see Petruzzi and Dada (1999)). There exists a vast literature on revenue management, and interested readers an be referred to, e.g. Talluri and van Ryzin (004) for detailed reviews. In the ontext of priing and revenue management of utility omputing servies in the on-demand paradigm, there has been some reent work. Gurnani and Karlapalem (001) studies the eonomi viability of pay-per-use liensing of pakaged software over the internet by using a monopoly priing model. Paleologo (004) aounted for the impat of unertainty in the deision proess in the priing of on-demand omputing servies and proposed a novel methodology ``prie-at-risk.'' Dube et al. (007) studies the ompetitive equilibrium in the omputing servie market when ompetition is on both priing and quality of servie, and ustomers further fae the outsouring or inhouse hosting deisions. Many studies also address the queueing effets in the problem domain. Mendelson (1985) studied the effet of ongestion due to queueing effets and users' related osts on priing and apaity deisions at a omputing enter with a single proessor. Mieghem (000) studied the optimal pries and servie quality grades of a servie provider faing heterogeneous utlity-maximizing ustomers in queueing models under either full information or asymmetri information. Hampshire et al. (00) used queueing models to solve QoS provisioning and priing problems under Erlang loss queueing models. The above literature review is by no means omplete. However, the priing and resoure provisioning problem remains hallenging in the loud omputing paradigm in a number of ways. Often, a loud servie provider offers a menu of servie templates or onfigurations from whih ompanies hoose. The loud provides at least three distint resoures whose ombination makes the final produts on the menu. The templates are omposed of various amounts of three resoures: CPU, memory and storage. Clients may hoose to purhase any ombination of these templates. The loud provider gives ustomers the guarantee that their template will be available and unompromised by other ustomers' usage. The resoure provisioning is therefore of multiple dimensions (CPU, memory, bandwidth, storage). Different ustomers may have different requirements on eah of these dimensions, making the provisioning problem muh more diffiult. The Cloud flexibility paradigm allows ustomers to inrease and derease apaity on the fly adds further ompliations on the provisioning task. Lastly, as an emerging market, the elastiity of demand (to prie and quality of servie) in loud omputing is not well understood. Due to lak of historial data, demand will be initially diffiult to foreast. Also, it will take time for eonomies of sale to be realized. In this paper, we outline quantitative modeling and optimization approahes that might lead to viable solutions to these hallenges. In setion, we propose to use a learning urve model to apture the eonomy of sale in loud omputing servies. Suh models an also be used to study ustomers' loud adoption deisions and help understand quantitatively why loud omputing is most attrative to small and medium businesses. In setion 3, we introdue a stohasti model that aptures the dynamis in loud omputing servies. We then study the steady-state behavior of suh systems and present a revenue management formulation that helps to find the optimal priing and provisioning solutions. We further demonstrate the approah via numerial examples and provide management solutions that lead to useful business insights.. Learning Curve Model for Cloud Computing In loud omputing, a signifiant amount of expertise and learning is involved in operating large distributed data enters effiiently. In this setion, we present a learning urve model that aptures the prodution ost redution with eonomy of sale in loud omputing. We further use the model to study ustomers' loud adoption deisions and show quantitatively why loud omputing is most attrative to small and medium businesses..1 Learning Curves Learning urves, first proposed in Wright (1936), are used to determine the redution in osts per unit as experiene is gained while produing produts with mature and immature proess tehnologies (Blanett 00). It is a useful model to explain the eonomy of sale and has been suessfully applied to a wide variety of industries ranging from aerospae to auto and ship building to semiondutor manufaturing (Russell and Taylor 003). The learning urve model assumes that as the number of prodution units are doubled the marginal ost of prodution dereases by a fixed fator (or perentage). One minus this fator is alled learning fator (or learning oeffiient). Mathematially, it boils down to a widely aepted power law model, where the marginal ost of produing the x-th unit, denote by (x), is given by 100

3 x ( ) log Kx α = (1) where K denotes the ost of the first unit, and α (0,1) is the learning fator. The total ost of produing x units, denoted by C(x), an then be obtained by integrating the above marginal ost funtion for 0 to x, whih gives 1+ log α Kx C( x) = () 1 + log α In loud omputing, ompanies like Amazon have developed extensive in-house knowledge whih they leverage to sell servies at a margin. It is estimated that Amazon urrently is running 30,000 EC instanes, and the ost per instane is urrently $.08/hr. Their original ost per EC instane was estimated to be $.0/hr. This represents an 80% learning fator. It has been reported [3] that for a typial data enter, as the total number of servers in house doubles, the marginal ost of deploying and maintaining eah server dereases 10-5%, thus the learning fators are typially within the range (0.75, 0.9).. Customer Cloud Adoption Deision Suppose the prodution ost funtion C s (x) of the servie provider is given by: 1+ log α s Kx s C ( x) = s (3) 1 + log α s where α s denotes the learning fator of the servie provider, and K s is the prodution ost of 1st unit inurred to the servie provider. The total ost on the learning urve divided by the number of prodution units, then gives the unit prie that the servie provider would harge to its ustomers. Assume ustomers an meet their omputing requirements in two ways: either in-house or outsoured via loud servie. Eah ustomer makes his/her deision of adopting loud servie or not based on his ost tradeoffs. The ost for the ustomer to meet x units of servie in-house is given by 1+ log α Kx C ( x) = (4) 1 + log α where α is the learning fator of the ustomer, K is the prodution ost of 1st unit by the ustomer. Hene, if the ost of doing in-house is less, then ustomer will not opt for loud omputing. So, hene, if a ustomer demands x units of servie, then he will only aept loud omputing if C ( x) > xπ (5) where π is the prie per unit time for a unit of servie harged by the (loud) servie provider. Solving above inequality gives: x K 1 log α < = (1 + log α ) π : η( π ) The learning urve model explains naturally why loud servie offering is mostly attrative to small and medium businesses. Figure 1 plots the total ost for two ompanies with different learning urve fators. Although both ompanies have the same ost of first unit as $1500, the total ost of prodution for red (α = 0.75) is muh lower than for blue (α = 0.8). Consequently, the red ompany an sell units at muh lower ost than blue ompany. Figure shows the ost omparison of the two servie options for a ompany, where the onave line orresponds to the in-house ost funtion based on the learning urve model while the straight line orresponds to the linear ost using loud servie. We see that it would be better off by adopting loud servie if its demand is smaller than η( π ), thus loud servie is generally attrative to small to medium businesses. In order to get under the servie provider's unit prie, one would have to be deploying tens of thousands servers in house. From the servie provider's point of view, η( π ) indiates the potential market share it an apture under a given unit prie π. Figures 3 and 4 show how η( π ) hanges as a funtion of π for different values of parameters α and K respetively. From the figures, we see that the demand is sensitive to both the learning fator and the prie. The harder it is for ustomers to learn, the more will ome to loud servies. On the other hand, inappropriate (6) 101

4 priing an definitely drive away demand from the loud servie provider. Similarly, the demand is sensitive to the first unit prodution ost K. The higher the ost of first unit, the more will ome to loud servie. Thus, ustomers with less expertise in omputing servies would tend to find loud servie attrative. Figure 1 Learning Curves under Different Learning Fators Figure Cost Comparison: In-house vs. Cloud Servie 3. oint Priing and Resoure Provisioning In this setion, we study the problem of priing and resoure provisioning from the servie provider s point of view. We first introdue a stohasti model that aptures the flexibility of allowing ustomers to upgrade and downgrade their servies dynamially. We then study the steady-state behavior of suh systems and present a revenue management formulation that helps to find the optimal priing and provisioning solutions. We further demonstrate the approah via numerial examples and provide management solutions that lead to useful business insights. 3.1 The Model Consider a single loud servie provider who provides lasses of servie templates. For simpliity, we onsider the ase where all lasses are defined as multiple units of some base instane. A lass servie template onsists of m units of base instane, with =1,...,. Assume the servie lasses are ordered suh that m 1 < m <... < m. For example, a base instane at Amazon EC is omprised of 1.7 GB of memory, 1 EC Compute Unit (1 virtual ore with 1 EC Compute Unit) and 160 GB of instane storage. 10

5 Assume ustomers desiring lass i servies arrive aording to a Poisson proess with the rate λ (with λ to be speified, and it depends on both the prie and the desired servie quality). Figure 3 η(π) for varying α s Figure 4 η(π) for varying K s To apture the flexibility of allowing ustomers to inrease or derease resoure requirements on the fly in loud omputing servies, we onsider the following probabilisti model. Assume that, after holding the servie template for an exponential amount of time with mean 1/μ, a ustomer upgrades/downgrades to servie template k with probability p k and terminate the servie with probability 1 p. The matrix P = [p k k, 1, k ] is transient so that every ustomer eventually leaves, thus I-P is invertible. The holding times for lass i templates are i.i.d. random variables, independent of the arrival proesses and other lasses. We assume the loud servie provider operates in a market with imperfet ompetition. This is suitable for loud omputing as it is a new market and the provider may hold a temporary monopoly, or the market may allow for produt differentiation. In the monopoly or new produt ase under imperfet ompetition, the provider an influene demand by varying its prie. We assume the demand rate for lass servie templates depends only on the prie π through a non-inreasing funtion λ (π ) where π denotes the prie harged for providing servie template per unit time. An alternative view of suh demand funtion is to assume that ustomers (of lass ) have i.i.d. reservation prie with distribution F, where the reservation prie represents the maximum prie that a ustomer is k = 1 103

6 willing to pay. Therefore, the expeted demand rate at prie π is a F (π ), where a is the potential demand rate of lass ustomers before the prie is announed. The above reservation funtion an be derived using the learning urve model we disussed earlier. Sine different ustomers tend to have different apaity requirements and learning apabilities, it requires extra efforts to understand the distribution of these variables. One an rely on marketing analysis, suh as panel studies, survey researh to obtain some sort of estimation on these parameters. In this paper, we simply assume that λ (π ) is given as a funtion of π and fous on the revenue management problem. Cloud servie providers are ompelled to deliver a quality of servie (QoS) defined by the underlying servie level agreement. Typial servie level agreements are termed suh that a ustomer should have aess to its desired servie with probability1 ε. We onsider an availability guarantee where the loud provider guarantees that a ustomer should have aess to its desired apaity with probability 1 ε, unompromised by other ustomers' usage at all times. The servie provider needs to hoose proper pries for eah servie lass, π =( π 1,..., π ). The servie provider also needs to provision suffiient apaity so as to meet the availability guarantee. 3. Steady-State Behavior Different from many ovariane-based modeling approahes, the Partial Least Squares approah to Strutural Mathematially, one an view the ustomer flexibility model as a servie with phase type distribution. That is, ustomers (of all types) arrive aording to a Poisson proess with mean arrival rate λ=λ λ. Upon arrival, eah ustomer follows a finite state ontinuous time Markov hain (CTMC) on the state spae {0, 1,..., }, where states 1,..., are transient and state 0 is absorbing. State i represents a ustomer using servie template i, i=1,...,. One reahing the absorbing state 0, the ustomer leaves the system. The initial distribution is (0, α), where α=(α 1, λ 0 0 i..., α l ) and αi =. The CTMC has infinitesimal generator as, where S k =μ p k for,k=1,...,, and t= λ t S -S1. Here 1 represents the unity (olumn) vetor with all entries being 1. Let X denote the soourn time of a ustomer until absorption. Then X follows the so-alled phase-type distribution PH(α, S) (see e.g. Latouhe and Ramaswami (1999)). The mean soourn time E[X] is given by: EX 1 [ ] = α S 1. Now view the above soourn time as the servie time for the loud provider to serve eah ustomer until he leaves the system. If the servie provider has an infinite amount of resoure available, the number of ustomers in system at any time t is the same as the number of ustomers of a M/G/ queue where the servie time for eah ustomer is X with phase-distribution PH(α, S). Denote N the total number of ustomers in the system in steady state, and N the number of ustomers in state, =1,...,. Based on the theory on M/G/ (see e.g Gross and Harris (1985)), the total number of ustomers in system is Poisson distributed with mean ρ : = λe[ X]. Furthermore, N, the number of ustomers of type i in system 1 is also Poisson distributed with mean ρ, where ρ λ( αs = )( ), and ρ = ρ. Note that N = 1 1,..., N are independent Poisson random variables. Consequently, in steady-state, the probability that the system is in state (n 1,..., n ) is given by: n n ρ ρ ρ ρ π ( n,..., n ) = e e 1 = (7) n! n! = 1 = Resoure Provisioning The servie provider needs to provision suffiient in house apaity so as to meet the servie availability guarantee 1 ε. We next deide the total apaity B based on the steady-state behavior of the system. 104

7 In steady state, N, the total number of ustomers in system is Poisson distributed with parameter ρ. Label the N ustomers randomly and let Y i be the apaity requirement (in terms of number of base instanes) of the i-th ustomer, i=1,..., N. Sine a randomly seleted ustomer is of type with probability q = ρ / ρ, thus q = ρy = m, with probability q, = 1,...,. / ρ i Hene, the total apaity requirement to the system is given by Y. The goal of apaity planning is to i = 1 i hoose B large enough suh that N P{ Y > B} ε i (8) i= 1 N is a ompound Poisson (CP) random variable with mean ρe[ Y ]. Notie that Y For onveniene, we i = 1 i denote suh a ompound Poisson random variable by CP( ρ,y ). To deide the proper level of apaity B, we utilize a result derived in Madiman and Kontoyiannis (005), whih gives an upper bound on the tail distribution of a ompound Poisson random variable. We will state the theorem in Madiman and Kontoyiannis (005), first and then derive a losed form expression on how to selet B to ahieve availability guarantees 1 ε. N Theorem 1: Suppose U has a distribution CP(ν, W), where the base distribution (of W) has finite support, m = max( i Z : W i > 0) < + Let f : Z + be S-Lipshitz, i.e. Df( x) f ( x 1) f( x) S t P[ f ( U ) E[ f ( U )] t] exp 3ν m = + for all x. Then if t [ 0, (3 ν m ) / ] 3 (9) N Now onsider U = Y whih is a ompound Poisson random variable CP( ρ,y ). Note that the Y i = 1 i i 's has a finite support m=max{m : =1,..., }. Choose the idential funtion f(x)=x, whih is learly S-Lipshitz. We an then apply the above theorem to hoose a B that satisfies (8). It suffies to let B N = E [ Y ] + t = ρ q m t i= 1 i +, where t is hosen suh that = 1 t ln() ε = 3 3ρm We then have 3 1/ t = [ 3 ρm ln( ε )] (10) But (9) is only valid if t is in the range given in the theorem. Thus, we need to make sure 3 1/ 3ρm [ 3 ρmln( ε )] Solving the above inequality will give us, 4 ρm ln() ε (11) 3 Condition (11) an be easily satisfied in the loud omputing setting. For example, to ahieve servie availability 99.95% (thus ε = 0.05%) the right hand side of (11) will be in the order of 10. Sine ρ and m are typially large in loud servies, (e.g. the mean load ρ is typially in the order of hundreds or thousands, and m an be in the order of 10's or 100's), thus ondition (11) an be easily satisfied. 105

8 Therefore, in order to provide servie availability 1 ε, it suffies to provision apaity B suh that 3 1/ B = ρ q m + [ 3 m ln( ε )] ρ (1) = 1 Observe from (1) that the planned apaity B onsists of two terms. The first term is the expeted value of resoure requirement in steady state whih is linear to the mean load ρ. The seond term ensures a safety margin in the order of ρ to absorb the demand variability where the oeffiient depends on the target availability ε. We believe suh square root staff result an be extended to the more general ase where the ustomer soourn times are generally distributed. 3.4 The Revenue Management Problem We are now ready to formulate the oint priing and resoure provisioning problem as a revenue management problem for the loud servie provider from a steady-state point of view. The profit earned by the servie provider is equal to the revenue minus the ost of providing the servie. Under a ommitted servie availability 1 ε, revenue is equal to π ( )(1 ). 1 ρ π ε = This is the summation of the prie harged per hour per ustomer for eah servie lass multiplied by the average number of ustomers in that lass thinned by a ε fration of demand loss. In order to ensure servie availability 1 ε, the provider will provision a total apaity B (in terms of number of base instane) given by (1). The ost of providing servies at apaity level B is given by the provider's learning urve C s (B) as defined in (3).We then have the following optimization problem: max ( )(1 ) C ( B) π, B πρ π ε s (13) Note that apaity B also depends on the pries π indiretly through ρ 's. 3.5 Numerial Results In this setion we demonstrate through a small example how optimal pries and resoure provision solutions an be obtained via the optimization framework as formulated in (13). Although the example may appear to be a bit simple, the solution provides interesting insights into the behavior of the optimal poliy. Consider a loud servie provider that offers three different types of templates: basi, medium and large. A basi servie template onsists of one unit of the base instane, a medium template onsists of 4 units of base instane, while a large template onsists of 8. That is: m 1 =, m = 4 and m 3 = 8. The total arrival rate is 100 ustomers/hr, where 90% of the arrivals are interested in basi templates, 8% are interested in medium size templates, and % are interested in large. Upon entering the system, eah ustomer holds its desired servie template for an exponential amount of time with mean 0 hours, and then upgrades and downgrades his servie probabilistially aording to transition matrix P, where P = For simpliity, we assume the provider has deided to ust set the base prie π for using one unit of the base instane per unit time. Hene, π 1 = π, π = 4π, and π 3 = 8π. Assume further that ustomers' hold reservation prie against the base prie, and the reservation prie is exponentially distributed with parameter γ. This will lead to a simple prie funtion λπ ( ) = λe βπ where β > 0 is prie sensitivity parameter. Suh exponential demand-to-prie elastiity funtion has been widely used in many previous studies, see, e.g. Gallego and van Ryzin (1994). In this example, we assume β = 0.8. We are varying π from $0.005 to $1.5 per instane per hour. For example, Amazon's prie for the small instane is $0.1 per instane per hour EC. We alulate the resulting λ( π ) and then alulate λ for eah servie i lass aording to Poisson splitting.. 106

9 The prodution ost of the servie provider for provisioning a total B units of apaity, C s (B), is given by the learning urve model (3). Using Amazon (EC) as the example, we assume the prodution ost of first unit is K s =/hr, and the target servie availability is set to 99.95%. We will show the profit under different learning oeffiient by varying the learning fator from 75% - 85%. Figure (5) plots the variation of profit vs. prie harged under different fators. Observe that with a learning fator 6 α = 75%, the optimal unit prie π is less than $0.10/hr, and the total (monthly) profit is around $ With a * 6 learning fator α = 80%, π $0.$ / hr, where the total profit dropped to $ When the learning fator is set to α = 85%, it is no longer profitable at all unit pries ranging from 0 to $1.4/hr. We see that the lower the learning fator (i.e. the faster the servie provider an ut down its prodution ost), the lower the optimal prie the provider an offer, and the more profitable it will be. Sine a lower prie will attrat more demands, whih yields a higher total profit. 4. Conlusion In summary, we have presented quantitative modeling and optimization approahes for managing loud omputing servies. We showed that learning urve models an be useful to model the providers' ost redution with eonomy of sale. Suh models also help understand ustomers' loud adoption deisions and explain quantitatively why loud omputing is most attrative to small and medium businesses. We further showed that stohasti models and revenue management formulation an help the priing and resoure provisioning deisions for the loud servie providers. The approah enables the loud servie provider a quantitative framework to obtain management solutions and to learn and reat to the ritial parameters in the operation management proess by gaining useful business insights. Figure 5 Profit vs. Prie under Different Learning Fators 107

10 Referenes Amazon Elasti Compute Cloud, IDC Enterprise Panel, IT Cloud Servies User Survey, pt.3: What Users Want From Cloud Servies Providers, August 008, available at Dube, P., Z. Liu, L. Wynter, and C. Xia. Competitive equilibrium in eommere: Priing and outsouring.computers & Operations Researh, 34: , 007 Gallego, G. and G. van Ryzin. Optimal dynami priing of inventories with stohasti demand over finite horizons. Management Siene, 40(8): , Aug Gross, D. and C. Harris. Fundamentals of Queueing Theory. ohn Wiley & Sons In.: New York, 1985 Gurnani, H. and K. Karlapalem. Optimal priing strategies for internet-based software dissemination. ournal of the Operational Researh Soiety, 5(1):64 70, 001 Hampshire, R.C., W. A. Massey, D. Mitra, and Q. Wang. Provisioning of bandwidth sharing and exhange. In Teleommuniations Network Design and Eonomis and Management: Seleted Proeedings of the 6 th INFORMS Teleommuniations Conferenes (Boa Raton, FL, pages Kluwer Aademi Publishers, Boston/Dordreht/London, 00 Hayes, R. Cloud omputing. Communiations of the ACM, 51:9 11, 008 Latouhe, G. and V. Ramaswami. Introdution to Matrix Analyti Methods in Stohasti Modelling. ASA SIAM, 1st edition,

11 Madiman, M. and I. Kontoyiannis. Conentration and relative entropy for ompound poisson distributions. In Proeedings of the 005 IEEE International Symposium on Information Theory (Adelaide, Australia), September 005 Mendelson, H. Priing omputer servies: Queueing effets. Communiations of the ACM, 8:31 31, 1985 Mieghem,.V. Prie and servie disrimination in queuing systems: Inentive ompatibility of g sheduling. Management Siene, 46: , 000 Paleologo, G. A methodology for priing utility omputing servies. IBM Systems ournal, 43(1):0 31, 004 Petruzzi, N. and M. Dada. Priing and the newsvendor model: a review with extensions. Operations Researh, 47: , 1999 Talluri, K. and G. van Ryzin. The Theory and Pratie of Revenue Management. Kluwer Aademi Publishers, 004 Tehrunh.om. The Effiient Cloud: All Of Salesfore Runs On Only 1,000 Servers, Mar 3, 009, available at Wright, T. Fators affeting the ost of airplanes. ournal of Aeronautial Siene, 4(4):1 18, 1936 Amit Gera is a M.S. student at The Ohio State University in Industrial & Systems Engineering Department with speialization in Operations Researh. Prior to this he got his B.Teh from Indian Institute of Tehnology, Madras. He has 4 years of experiene in stohasti modeling and system performane measures. He is urrently working on priing and apaity planning for Cloud Computing servie providers. His researh interests inlude stohasti modeling, revenue management and deision analysis. He is an ative member of student hapter INFORMS at The Ohio State University. Cathy H. Xia is an assoiate professor in the Department of Integrated Systems Engineering at The Ohio State University, with an adunt appointment from the Department of Computer Siene and Engineering. Dr. Xia speializes in performane modeling and analysis, stohasti proesses, and distributed management of omputing networks and servie systems. She reeived her M.S. in Statistis, and her Ph.D. in Operations Researh, both from Stanford University. Before oining OSU, she has worked as a researh sientist at IBM T.. Watson Researh Center for many years and has reeived numerous IBM researh and pratie awards. Dr. Xia has served NSF panels and a number of onferene ommittees. She is the program hair of INFORMS Midwest 011, luster hair on loud omputing for INFORMS Annual Conferene 010, and tutorial hair for ACM Sigmetris 008. Dr. Xia is the assoiate editor of the International ournal of Information Systems in the Servie Setor. Dr. Xia has obtained over 10 patents granted. She has published 1 edited book, 1 book hapter, and over 50 papers in peer-reviewed ournals and onferenes. 109