Economic Models for Cloud Service Markets


 Coral Chrystal Price
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1 Economc Models for Cloud Servce Markets Ranjan Pal and Pan Hu 2 Unversty of Southern Calforna, USA, 2 Deutsch Telekom Laboratores, Berln, Germany, Abstract. Cloud computng s a paradgm that has the potental to transform and revolutonalze the next generaton IT ndustry by makng software avalable to endusers as a servce. A cloud, also commonly known as a cloud network, typcally comprses of hardware (network of servers) and a collecton of softwares that s made avalable to endusers n a payasyougo manner. Multple publc cloud provders (ex., Amazon) coexstng n a cloud computng market provde smlar servces (software as a servce) to ts clents, both n terms of the nature of an applcaton, as well as n qualty of servce (QoS) provson. The decson of whether a cloud hosts (or fnds t proftable to host) a servce n the longterm would depend jontly on the prce t sets, the QoS guarantees t provdes to ts customers, and the satsfacton of the advertsed guarantees. In ths paper, we devse and analyze three nterorganzatonal economc models relevant to cloud networks. We formulate our problems as non cooperatve prce and QoS games between multple cloud provders exstng n a cloud market. We prove that a unque pure strategy Nash equlbrum (NE) exsts n two of the three models. Our analyss paves the path for each cloud provder to ) know what prces and QoS level to set for endusers of a gven servce type, such that the provder could exst n the cloud market, and 2) practcally and dynamcally provson approprate capacty for satsfyng advertsed QoS guarantees. Keywords: cloud markets; competton; Nash equlbrum Introducton Cloud computng s a type of Internetbased computng, where shared resources, hardware, software, and nformaton are provded to endusers n an on demand fashon. It s a paradgm that has the potental to transform and revolutonalze the IT ndustry by makng software avalable to endusers as a servce []. A publc cloud typcally comprses of hardware (network of servers) and a collecton of softwares that s made avalable to the general publc n a payasyougo manner. Typcal examples of companes provdng publc clouds nclude Amazon, Google, Mcrosoft, EBay, and commercal banks. Publc cloud provders usually provde Software as a Servce (SaaS), Platform as a Servce (PaaS), and Infrastructure as a Servce (IaaS).The advantage of makng software avalable as
2 2 a servce s threefold [], ) the servce provders beneft from smplfed software nstallaton, mantenance, and centralzed versonng, 2) endusers can access the software n an anytme anywhere manner, can store data safely n the cloud nfrastructure, and do not have to thnk about provsonng any hardware resource due to the lluson of nfnte computng resources avalable on demand, and 3) endusers can pay for usng computng resources on a shortterm bass (ex., by the hour or by the day) and can release the resources on task completon. Smlar beneft types are also obtaned by makng both, platform as well as nfrastructure avalable as servce. Cloud economcs wll play a vtal role n shapng the cloud computng ndustry of the future. In a recent Mcrosoft whte paper ttled Economcs of the Cloud, t has been stated that the computng ndustry s movng towards the cloud drven by three mportant economes of scale: ) large data centers can deploy computatonal resources at sgnfcantly lower costs than smaller ones, 2) demand poolng mproves utlzaton of resources, and 3) multtenancy lowers applcaton mantenance labor costs for large publc clouds. The cloud also provdes an opportunty to IT professonals to focus more on technologcal nnovaton rather than thnkng of the budget of keepng the lghts on. The economcs of the cloud can be thought of havng two dmensons: ) ntraorganzaton economcs and 2) nterorganzaton economcs. Intraorganzaton economcs deals wth the economcs of nternal factors of an organzaton lke labor, power, hardware, securty, etc., whereas nterorganzaton economcs refers to the economcs of market competton factors between organzatons. Examples of some popular factors are prce, QoS, reputaton, and customer servce. In ths paper, we focus on nterorganzatonal economc ssues. Multple publc cloud provders (ex., Amazon, Google, Mcrosoft, etc.,) coexstng n a cloud computng market provde smlar servces (software as a servce, ex., Google Docs and Mcrosoft Offce Lve) to ts clents, both n terms of the nature of an applcaton, as well as n qualty of servce (QoS) provson. The decson of whether a cloud hosts (or fnds t proftable to host) a servce n the longterm would (amongst other factors) depend jontly on the prce t sets, the QoS guarantees t provdes to ts customers 3, and the satsfacton of the advertsed guarantees. Settng hgh prces mght result n a drop n demand for a partcular servce, whereas settng low prces mght attract customers at the expense of lowerng cloud provder profts. Smlarly, advertsng and satsfyng hgh QoS levels would favor a cloud provder (CP) n attractng more customers. The prce and QoS levels set by the CPs thus drve the enduser demand, whch, apart from determnng the market power of a CP also plays a major role n CPs estmatng the mnmal resource capacty to meet ther advertsed guarantees. The estmaton problem s an mportant challenge n cloud computng wth 3 A cloud provder generally gets requests from a cloud customer, whch n turn accepts requests from Internet endusers. Thus, typcally, the clents/customers of a cloud provder are the cloud customers. However, for modelng purposes, endusers could also be treated as customers. (See Secton 2)
3 3 respect to resource provsonng because a successful estmaton would prevent CPs to provson for the peak, thereby reducng resource wastage. The competton n prces and QoS amongst the cloud provders entals the formaton of noncooperatve games amongst compettve CPs. Thus, we have a dstrbuted system of CPs (players n the game), where each CP wants to maxmze ts own profts and would tend towards playng a Nash equlbrum 4 (NE) strategy (.e., each CP would want to set the NE prces and QoS levels.), whereby the whole system of CPs would have no ncentve to devate from the Nash equlbrum pont,.e., the vector of NE strateges of each CP. However, for each CP to play a NE strategy, the latter should mathematcally exst. In ths paper, we address the mportant problem of Nash Equlbrum characterzaton of dfferent types of prce and QoS games relevant to cloud networks, ts propertes, practcal mplementablty (convergence ssues), and the senstvty analyss of NE prce/qos varatons by any CP on the prce and QoS levels of other CPs. Our problem s mportant from a resource provsonng perspectve as mentoned n the prevous paragraph, apart from t havng obvous strategc mportance on CPs n terms of sustenance n the cloud market. Related Work: In regard to market competton drven network prcng, there exsts research work n the doman of multple ISP nteracton and tered Internet servces [2][3], as well as n the area of resource allocaton and Internet congeston management [4][5][6]. However, the market competton n our work relates to optmal capacty plannng and resource provsonng n clouds. There s the semnal work by Songhurst and Kelly [7] on prcng schemes based on QoS requrements of users. Ther work address multservce scenaros and derve prcng schemes for each servce based on the QoS requrements for each, and n turn bandwdth reservatons. Ths work resembles ours to some extent n the sense that the prce and QoS determned can determne optmal bandwdth provsons. However, t does not account for market competton between multple provders and only focus on a sngle servce provder provdng multple servces,.e., the paper addresses an ntraorganzaton economcs problem. However, n ths paper, we assume sngleservce scenaros by multple servce provders. In a recent work [8], the authors propose a queueng drven gametheoretc model for prceqos competton amongst multple servce provders. The work analyzes a duopolstc market between two servce provders, where provders frst fx ther QoS guarantees and then compete for prces. Our work extends the latter cted work n the followng aspects: () we generalze our model to ncorporate n servce provders, (2) we address two addtonal game models whch are of practcal mportance,.e., prceqos smultaneous competton and prces fxed frst, followed by QoS guarantees competton, (3) we provde an effcent technque to compute multple equlbra n games, and (4) our models explctly characterze percentle performance of parameters, whch s specfc to cloud networks provsonng resources on a percentle bass. We also want to emphasze the fact that research on prce/qos competton amongst organzatons s not 4 A group of players s n Nash equlbrum f each one s makng the best decson(strategy) that he or she can, takng nto account the decsons of the others.
4 4 new n the economcs doman. However, n ths paper we model networkng elements n prce/qos games va a queueng theoretc approach and analyze certan prce/qos games that are manly characterstc of Internet servce markets. Our proposed theory analyzes a few basc nterorganzatonal economc models through whch cloud servces could be prced under market competton. The evoluton of commercal publc cloud servce markets s stll n ts ncepton. However, wth the ganng popularty of cloud servces, we expect a bg surge n publc cloud servces competton n the years to come. The models proposed n ths paper take a substantal step n hghlghtng relevant models to the cloud networkng communty for them adopt so as to approprately prce current and future cloud servces. In practce, scenaros of prce and/or QoS competton between organzatons exst n the moble network servces and ISP markets. For example, AT&T and Verzon are competng on servce,.e., Verzon promses to provde better coverage to moble users than AT&T, thereby ncreasng ts propensty to attract more customers. Smlarly, prce competton between ISPs always exsted for provdng broadband servces at a certan gven bandwdth guarantee. Regardng our work, we also want to emphasze ) we do not make any clams about our models beng the only way to model nterorganzatonal cloud economcs 5 and 2) there s a dependency between ntraorganzatonal and nterorganzatonal economc factors, whch we do not account n ths paper due to modelng smplcty. However, through our work, we defntely provde readers wth a concrete modelng ntuton to go about addressng problems n cloud economcs. To the best of our knowledge, we are the frst to provde an analytcal model on nterorganzatonal cloud economcs. Our Contrbutons  We make the followng contrbutons n ths paper.. We formulate a separable enduser demand functon for each cloud provder w.r.t. to prce and QoS levels set by them and derve ther ndvdual utlty functons (proft functon). We then defne the varous prceqos games that we analyze n the paper. (See Secton 2.) 2. We develop a model where the QoS guarantees provded by publc CPs to endusers for a partcular applcaton type are prespecfed and fxed, and the cloud provders compete for prces. We formulate a noncooperatve prce game amongst the players (.e., the cloud provders) and prove that there exsts a unque Nash equlbrum of the game, and that the NE could be practcally computed (.e., t converges). (See Secton 3.) 3. We develop a noncooperatve gametheoretc model where publc cloud provders jontly compete for the prce and QoS levels related to a partcular applcaton type. We show the exstence and convergence of Nash equlbra (See Secton 4). As a specal case of ths model, we also analyze the case where prces charged to Internet endusers are prespecfed and fxed, and the cloud provders compete for QoS guarantees only. The models mentoned 5 We only model prce and QoS as parameters. One could choose other parameters (n addton to prce and QoS, whch are essental parameters) and a dfferent analyss mechansm than ours to arrve at a dfferent model.
5 5 n contrbutons 3 and 4 drve optmal capacty plannng and resource provsonng n clouds, apart from maxmzng CP profts. (See Secton 4.) 4. We conduct a senstvty analyss on varous parameters of our proposed models, and study the effect of changes n the parameters on the equlbrum prce and QoS levels of the CPs exstng n a cloud market. Through a senstvty analyss, we nfer the effect of prce and QoS changes of cloud provders on ther respectve profts, as well as the profts of competng CPs. (See Sectons 3 and 4.) 6 2 Problem Setup We consder a market of n competng cloud provders, where each provder servces applcaton types to endusers at a gven QoS guarantee. We assume that endusers are customers of cloud provders n an ndrect manner,.e., Internet endusers use onlne softwares developed by companes (cloud customers), that depend on cloud provders to servce ther customer requests. Each CP s n competton wth others n the market for servces provded on the same type of applcaton w.r.t functonalty and QoS guarantees. For example, Mcrosoft and Google mght both serve a word processng applcaton to endusers by provdng smlar QoS guarantees. Here, the word processng applcaton represents a partcular type. For a gven applcaton type, we assume that each end user sgns a contract wth a partcular CP for a gven tme perod 7, and wthn that perod t does not swtch to any other CP for gettng servce on the same applcaton type. Regardng contracts between a CP and ts endusers, we assume that a cloud customer forwards servce requests to a cloud provder on behalf of endusers, who sgn up wth a cloud customer (CC) for servce. The CP charges ts cloud customer, who s turn charges ts endusers. We approxmate ths twostep chargng scheme by modelng a vrtual onestep scheme, where a CP charges endusers drectly 8. In a gven tme perod, each CP postons tself n the market by selectng a prce p and a QoS level s related to a gven applcaton type. Throughout the paper, we assume that the CPs compete on a sngle gven type 9. We defne s as the dfference between a benchmark response tme upper bound, rt, and 6 We study Nash equlbrum convergence as ts proves the achevablty of an equlbrum pont n the market. We emphasze here that the exstence of Nash equlbrum does not mply achevablty as t may take the cloud market an eternty to reach equlbrum, even though there may exst one theoretcally. 7 In ths paper, the term tmeperod refers to the tme duraton of a contract between the CP and endusers. 8 We assume here that prces are negotated between the CP, CC, and endusers and there s a vrtual drect prce chargng connecton between the CP and ts endusers. We make ths approxmaton for modelng smplcty. 9 In realty, each CP may n general servce several applcaton types concurrently. We do not model ths case n our paper and leave t for future work. The case for sngle applcaton types gves nterestng results, whch would prove to be useful n analyzng the multple concurrent applcaton type scenaro.
6 6 the actual response tme rt,.e., s = rt rt. For example, f for a partcular applcaton type, every CP would respond to an enduser request wthn 0 seconds, rt = 0. The response tme rt may be defned, ether n terms of the expected steady state response tme,.e., rt = E(RT ), or n terms of φ percentle performance, rt (φ), where 0 < φ <. Thus, n terms of φpercentle performance 0, P (RT < rt (φ)) = φ. We model each CP as an M/M/ queueng system, where enduser requests arrve as a Posson process wth mean rate λ, and gets servced at a rate µ. We adopt an M/M/ queueng system because of three reasons: ) queueng theory has been tradtonally used n request arrval and servce problems, 2) for our problem, assumng an M/M/ queueng system ensures tractable analyses procedures that entals dervng nce closed form expressons and helps understand system nsghts n a noncomplex manner, wthout sacrfcng a great deal n capturng the real dynamcs of the actual arrvaldeparture process, and 3) The Markovan nature of the servce process helps us generalze expected steady state analyss and percentle analyss together. Accordng to the theory of M/M/ queues, we have the followng standard results [7]. and rt = µ λ, () rt (φ) = ln( φ ) µ (φ) λ, (2) µ = λ + rt, (3) µ (φ) =λ + ln( φ ) rt (φ) Equatons 2 and 4 follow from the fact that for M/M/ queues, P (RT < rt (φ)) = φ = e (µ λrt(φ)). Wthout loss of generalty, n subsequent sectons of ths paper, we conduct our analyss on expected steady state parameters. As mentoned prevously, due to the Markovan nature of the servce process, the case for percentles s exactly smlar to the case for expected steady state analyss, the only dfference n analyss beng due to the constant, ln( φ ). Thus, all our proposed equlbrum related results hold true for percentle analyss as well. Each cloud provder ncurs a fxed cost c per user request served and a fxed cost ρ per unt of servce capacty provsoned. c arses due to the factor λ n Equaton 3 and ρ arses due to the factor rt n the same equaton. In ths sense, our QoSdependent prcng models are queuengdrven. A cloud provder charges pr to servce each enduser request, where pr ɛ [pr mn, pr max ]. It s evdent that 0 As an example, n cloud networks we often assocate provsonng power accordng to the 95th percentle use. Lkewse, we could also provson servce capacty by accountng for percentle response tme guarantees. (4)
7 7 each CP selects a prce that results n t accrung a nonnegatve gross proft margn. The gross proft margn for CP s gven as pr c ρ, where c + ρ s the margnal cost per unt of enduser demand. Thus, the prce lower bound, pr mn, for each CP s determned by the followng equaton. pr mn = c + ρ, =,..., n (5) We defne the demand of any CP, λ, as a functon of the vectors pr = (pr,..., pr n ) and s =(s,..., s n ). Mathematcally, we express the demand functon as λ = λ (pr, s) =x (s ) y pr j α j (s j )+ j β j pr j, (6) where x (s ) s an ncreasng, concave, and thrce dfferentable functon n s satsfyng the property of nonncreasng margnal returns to scale,.e., equalszed reductons n response tme results n progressvely smaller ncreases n enduser demand. The functons α j are assumed to be nondecreasng and dfferentable. A typcal example of a functon fttng x (s ) and α j (s j ) s a logarthmc functon. We model Equaton 6 as a separable functon of prce and QoS vectors, for ensurng tractable analyses as well as for extractng the ndependent effects of prce and QoS changes on the overall enduser demand. Intutvely, Equaton 6 states that QoS mprovements by a CP result n an ncrease n ts enduser demand, whereas QoS mprovements by other compettor CPs result n a decrease n ts demand. Smlarly, a prce ncrease by a CP results n a decrease n ts enduser demand, whereas prce ncreases by other competng CPs result n an ncrease n ts demand. Wthout loss of practcal generalty, we also assume ) a unform ncrease n prces by all n CPs cannot result n an ncrease n any CP s demand volume, and 2) a prce ncrease by a gven CP cannot result n an ncrease n the market s aggregate enduser demand. Mathematcally, we represent these two facts by the followng two relatonshps. y > j β j,=,..., n (7) and y > j β j,=,..., n (8) The long run average proft for CP n a gven tme perod, assumng that response tmes are expressed n terms of expected values, s a functon of the prce and QoS levels of CPs, and s gven as P (pr, s) =λ (pr c ρ ) ρ rt s, (9) The proft functon for each CP acts as ts utlty/payoff functon when t s nvolved n prce and QoS games wth other competng CPs. We assume n ths paper that the proft functon for each CP s known to other CPs, but none of
8 8 the CPs know the values of the parameters that other competng CPs adopt as ther strategy. Problem Statement: Gven the proft functon for each CP (publc nformaton), how would each advertse ts prce and QoS values (wthout negotatng wth other CPs) to endusers so as to maxmze ts own proft. In other words, n a compettve game of profts played by CPs, s there a stuaton where each CP s happy wth ts (prce, QoS) advertsed par and does not beneft by a postve or negatve devaton n the values of the advertsed par. In ths paper, we study games nvolvng prce and QoS as the prmary parameters,.e., we characterze and analyze the exstence, unqueness, and convergence of Nash equlbra. Our prmary goal s to compute the optmal prce and QoS levels offered by CPs to ts endusers under market competton. Our analyss paves the path for each cloud provder to ) know what prce and QoS levels to set for ts clents (endusers) for a gven applcaton type, such that t could exst n the cloud market, and 2) practcally and dynamcally provson approprate capacty for satsfyng advertsed QoS guarantees, by takng advantage of the property of vrtualzaton n cloud networks. The property of vrtualzaton entals each CP to allocate optmal resources dynamcally n a fast manner to servce enduser requests. Usng our prcng framework, n each tme perod, cloud provders set the approprate prce and QoS levels after competng n a game; the resultng prces drve enduser demand; the CPs then allocate optmal resources to servce demand. Remark. We decded to not analyze a compettve market,.e., where CPs are prce/qos takng and a Walrasan equlbrum results when demand equals supply, because a compettve market analyss s manly applcable when the resources traded by an organzaton are neglgble wth respect to the total resource n the system [9][0]. In a cloud market ths s defntely not the case as there are a few cloud provders and so the resource traded by one s not neglgble wth respect to the total resources traded n the system. Therefore we analyze olgopolstc markets where CPs are prce/qos antcpatng. We consder the followng types of prceqos game models n our work.. CP QoS guarantees are prespecfed; CPs compete wth each other for prces, gven QoS guarantees. (Game ) 2. CPs compete for prce and QoS smultaneously. (Game 2) 3. CP prce levels are prespecfed; CPs compete for QoS levels. (Game 3). Game 3 s a specal case of Game 2 and n Secton 4, we wll show that t s a Game 2 dervatve. Lst of Notatons: For reader smplcty, we provde a table of most used notatons related to the analyss of games n ths paper.
9 9 Symbol U = P pr pr pr c λ ρ rt rt φ s s s x () α j() Meanng Utlty functon of CP Prce charged by CP per enduser Prce vector of CPs Nash equlbrum prce vector Cost ncurred by CP to servce each user Arrval rate of endusers to CP cost/unt of capacty provsonng by CP response tme upper bound guarantee response tme guarantee by CP percentle parameter QoS level guarantee provded by CP to ts users QoS vector of CPs Nash equlbrum QoS vector ncreasng, concave, and a thrce dfferentable functon nondecreasng and dfferentable functon Table. Lst of Symbols and Ther Meanng 3 Game  Prce Game In ths secton we analyze the game n whch the QoS guarantees of CPs are exogenously specfed and the CPs compete for prces. Game Descrpton Players: Indvdual cloud provders; Game Type: Noncooperatve,.e., no nteracton between CPs; Strategy Space: Choosng a prce n range [pr mn, pr max ]; Player Goal: To maxmze ts ndvdual utlty U = P Our frst goal s to show that ths game has a unque prce Nash equlbrum, pr (an nstance of vector pr), whch satsfes the followng frst order condton P pr = y (pr c ρ )+λ,, (0) whch n matrx notaton can be represented as M pr = x(s)+z, () where M s an n n matrx wth M =2y, M j = β j, j, and where z = y (c + ρ ). We have the followng theorem and corollary regardng equlbrum results for our game. The readers are referred to the Appendx for the proofs. Theorem : Gven that the QoS guarantees of CPs are exogenously specfed, the prce competton game has a unque Nash equlbrum, pr, whch satsfes Equaton. The Nash equlbrum user demand, λ, for each CP evaluates to y (pr c ρ ), and the Nash equlbrum profts, P, for each CP s gven by y (pr c ρ ) 2 ρ. rt s Corollary : a) pr and λ are ncreasng and decreasng respectvely n each of
10 0 the parameters {c,ρ,=, 2,..., n}, and b) pr s j l j (M ) l x lj (s j). = y λ s j =(M ) j x j (s j) Corollary mples that ) under a larger value for CP s degree of postve externalty δ, t s wllng to make a bolder prce adjustment to an ncrease n any of ts cost parameters, thereby mantanng a larger porton of ts orgnal proft margn. The reason s that competng CPs respond wth larger prce themselves, and 2) there exsts a crtcal value 0 s 0 j rt such that as CP j ncreases ts QoS level, pr and λ are ncreasng on the nterval [0,s 0 j ), and decreasng n the nterval [s 0 j, rt). Senstvty Analyss: We know the followng relatonshp P s j =2y (pr c ρ ) pr s j (2) From t we can nfer that CP s proft ncreases as a result of QoS level mprovement by a competng CP j f and only f the QoS level mprovement results n an ncrease n CP s prce. Ths happens when P ncreases on the nterval [0,s 0 j ] and decreases on the remanng nterval (s0 j, rt]. In regard to proft varaton trends, on ts own QoS level mprovement, a domnant trend for a CP s not observed. However, we make two observatons based on the holdng of the followng relatonshp P s j =2y (pr c ρ ) pr s j ρ (rt s ) 2 (3) If a CP ncreases ts QoS level from 0 to a postve value and and ths results n ts prce decrease, s equlbrum profts become a decreasng functon of ts QoS level at all tmes. Thus, n such a case s better off provdng mnmal QoS level to ts customers. However, when CP s QoS level ncreases from 0 to a postve value resultng n an ncrease n ts prce charged to customers, there exsts a QoS level s b such that the equlbrum profts alternates arbtrarly between ncreasng and decreasng n the nterval [0,s b ), and decreases when s s b. Convergence of Nash Equlbra: Snce the prce game n queston has a unque and optmal Nash equlbra, t can be easly found by solvng the system of frst P order condtons, pr = 0 for all. Remark. It s true that the exstence of NE n convex games s not surprsng n vew of the general theory, but what s more mportant s whether a realstc modelng of our problem at hand results n a convex game. Once we can establsh that our model results n a convex game, we have a straghtforward result of the exstence of NE from game theory lterature. Ths s exactly what we do n the paper,.e., to show that our model s realstc and ndeed leads to a convex game thus leadng further to the exstence of NE. 4 Game 2  PrceQoS Game In ths secton, we analyze the game n whch the CPs compete for both, prce as well as QoS levels. In the process of analyzng Game 2, we also derve Game
11 3, as a specal case of Game 2, and state results pertanng to Game 3. Game Descrpton Players: Indvdual cloud provders; Game Type: Noncooperatve,.e., no nteracton between CPs; Strategy Space: prce n range [pr mn s ; Player Goal: To maxmze ts ndvdual utlty U = P We have the followng theorem regardng equlbrum results. 4yρ Theorem 2: Let rt 3, pr max ] and QoS level (x ) 2, where y = mn y,ρ = mn ρ, x = max x (0). There exsts a Nash equlbrum (pr,s ), whch satsfes the followng system of equatons: P pr = y (pr c ρ )+λ =0,, (4) and satsfes the condton that ether s (pr ) s the unque root of x (s )(pr ρ c ρ )= f pr (rt s ) 2 c + ρ ( + rt 2 x (0)) or s (pr )=0otherwse. Conversely, any soluton of these two equatons s a Nash equlbrum. Senstvty Analyss: We know that s (pr ) depends on x (s ) and pr. Thus, from the mplct functon theorem [] we nfer that the QoS level of CP ncreases wth the ncrease n ts Nash equlbrum prce. We have the followng relatonshp for pr >c + ρ ( + rt 2 x (0)), s (pr )= x (s ) x (s > 0, (5) )(pr c ρ ) ρ (rt s ) 2 whereas s (pr ) = 0 for pr < c + ρ ( + We also notce that for rt 2 x (0)). pr >c + ρ ( + rt 2 x (0)), s ncreases concavely wth pr. The value of s (p ) obtaned from the soluton of the equaton x (s )(pr c ρ )= f ρ (rt s ) 2 pr c + ρ ( + can be fed nto Equaton 5 to compute the prce rt 2 x (0)), vector. The system of equatons that result after substtuton s nonlnear n vector pr and could have multple solutons,.e., multple Nash equlbra. Inferences from Senstvty Analyss: Games, 2, and 3 gves us nonntutve nsghts to the prceqos changes by ndvdual CPs. We observe that the obvous ntutons of equlbrum prce decrease of competng CPs wth ncreasng QoS levels and vceversa do not hold under all stuatons and senstvty analyss provde the condtons under whch the counterresult holds. Thus, the ntrcate nature of noncooperatve strategy selecton by ndvdual CPs and the nterdependences of ndvdual strateges on the cloud market make cloud economcs problems nterestng. Convergence of Nash Equlbra: Snce multple Nash equlbra mght exst for the prce vectors for the smultaneous prceqos game, the tatonnement
12 2 scheme [9][2] can be used to prove convergence. Ths scheme s an teratve procedure that numercally verfes whether multple prce equlbra exst, and unqueness s guaranteed f and only f the procedure converges to the same lmt when ntal values are set at pr mn or pr max. Once the equlbrum prce vectors are determned, the equlbrum servce levels are easly computed. If multple equlbra exst the cloud provders select the prce equlbra that s componentwse the largest. Regardng the case when CP prce vector s gven, we have the followng corollary from the result of Theorem 2, whch leads us to equlbrum results of Game 3, a specal case of Game 2. Corollary 2. Gven any CP prce vector, pr f, the Nash equlbrum s(pr f ) s the domnant soluton n the QoS level game between CPs,.e., a CP s equlbrum QoS level s ndependent of any of ts compettors cost or demand characterstcs and prces. When s (pr f ) > 0, the equlbrum QoS level s ncreasng and concave n pr f, wth s (prf )= x (s). x (s)(prf c ρ) 2ρ (rt s ) 3 We observe that Game 3 beng a specal case of Game 2 entals a unque Nash equlbrum, whereas Game 2 entals multple Nash equlbra. 5 Concluson and Future Work In ths paper, we developed nterorganzatnal economc models for prcng cloud network servces when several cloud provders coexst n a market, servcng a sngle applcaton type. We devsed and analyzed three prceqos gametheoretc models relevant to cloud networks. We proved that a unque pure strategy Nash equlbrum (NE) exsts n two of our three QoSdrven prcng models. In addton, we also showed that the NE s converge;.e., there s a practcally mplementable algorthm for each model that computes the NE/s for the correspondng model. Thus, even f no unque Nash equlbrum exsts n some of the models, we are guaranteed to fnd the largest equlbra (preferred by the CPs) through our algorthm. Regardng convergence to Nash equlbra, t s true that t could take a long tme for convergence of Nash equlbra (computng NE s PPAD Complete [8]), however n 95% of the cases n practcal economc markets, NE s acheved n a decent amount of tme. Our prceqos models can drve optmal resource provsonng n cloud networks. The NE prce and QoS levels for each cloud provder drves optmal enduser demand n a gven tme perod w.r.t. maxmzng ndvdual CP profts under competton. Servcng enduser demands requres provsonng capacty. As a part of future work, we plan to extend our work to develop queueng optmzaton models to compute optmal provsoned resources n cloud networks. Once the optmal values are computed, the power of vrtualzaton n cloud networks makes t possble to execute dynamc resource provsonng n a fast and effcent manner n multple tme perods. Thus, our prcng models are specfcally suted to cloud networks. As a part of future work, we also plan to extend
13 3 our analyss to the case where cloud provders are n smultaneous competton wth other CPs on multple applcaton types. 6 Appendx Proof of Theorem. Proof: For a gven servce level vector s, each CP reserves a capacty of rt =. Consder the game G wth proft/utlty functons for each CP rt s represented as P =(x (s ) y p j α j (s j )+ j (β j p j )(pr c ρ ) W, (6) where 2 P pr pr j ρ W = rt s Snce = β j, the functon P s supermodular. The strategy set of each CP les nsde a closed nterval and s bounded,.e., the strategy set s [pr mn, pr max ], whch s a compact set. Thus, the prcng game between CPs s a supermodular game and possesses a Nash equlbrum [3]. Snce y > j β j, =,..., n (by Equaton 7), 2 P > 2 P pr 2 j pr pr j and thus the Nash equlbrum s unque. Rewrtng Equaton and usng Equaton 6, we get λ = y (pr c ρ ). Substtutng λ n Equaton 9, we get P = y (pr c ρ ) 2 ρ Q.E.D. rt s Proof of Corollary. Proof: Snce the nverse of matrx M,.e., M exsts and s greater than or equal 0[4], from pr = M (x(s)+z) (Equaton ), we have pr s ncreasng n {c,ρ =, 2,..., n}. Agan, from Lemma 2 n [4], we have δ y (M ) 0.5 δ <, where δ s the degree of postve externalty 2 faced by CP from other CP (prce, QoS) parameters, and t ncreases wth the β coeffcents. Ths leads us to pr c = pr ρ = y (M ) = δ > 0. Therefore, we show n another dfferent way that pr s ncreasng n {c,ρ,=, 2,..., n}. Snce M exsts and s greater than or equal to 0, we agan have λ c = λ ρ = y ( pr ρ ) = y ( pr ) = y (δ ) < 0, from whch we conclude that λ s decreasng n c {c,ρ,=, 2,..., n}. Part b) of the corollary drectly follows from the fact that the nverse of matrx M,.e., M exsts, s greater than or equal 0, and every entry of M s ncreasng n β j coeffcents. Q.E.D. Proof of Theorem 2. A functon f : R n R s supermodular f t has the followng ncreasng dfference property,.e., f(m,m ) f(m 2,m ), ncreases n m for all m >m 2 n (pr, pr j). The readers are referred to [6] for more detals on supermodularty. 2 A postve externalty s an external beneft on a user not drectly nvolved n a transacton. In our case, a transacton refers to a CP settng ts prce and QoS parameters.
14 4 Proof: To prove our theorem, we just need to show that the proft functon P s jontly concave n (pr,s ). Then by the NashDebreu theorem [5], we could nfer the exstence of a Nash equlbra. We know the followng results for all CP P pr = y (pr c ρ )+λ (7) and P θ = x (s )(pr c ρ ) ρ (rt s ) 2 (8) Thus, 2 P = 2y pr 2 < 0, 2 P = x s 2 (s )(pr c ρ ) 2ρ < 0, (rt s ) 3 x (s ). We determne the determnant of the Hessan as 2y (x ρ ) ρ f the followng condton holds: 2 P s pr = (s )(pr c 0 (the suffcent condton for P (rt s ) 2 to be jontly concave n (pr,s )), 4y ρ pr 2 (x (s )) 2 4y 3 ρ rt mn s (x (s )) 2 = 3 4y ρ, (9) (0))2 where the last equalty follows from the fact that x > 0 and x s decreasng. Now snce pr = pr (s ), by Theorem t s n the closed and bounded nterval [pr mn, pr max ] and must therefore satsfy Equaton 5. Agan from Equaton 7, we have P s as s tends to rt, whch leads us to the concluson that s (pr ) s the unque root of x (s )(pr c ρ )= ρ f pr (rt s ) 2 c + ρ ( + rt 2 x (0)) or s (pr ) = 0 otherwse. Q.E.D. Proof of Corollary 2. Proof: Substtutng pr max = pr mn = pr f nto Theorem 2 leads us to the fact that s(pr f ) s a Nash equlbrum of the QoS level competton game amongst CPs and that t s also a unque and a domnant soluton, snce s(pr f ) s a functon of pr, c, and ρ only. (Followng from the fact that s (pr ) s the unque root of x (s ρ )(pr c ρ )= f pr (rt s ) 2 c + ρ ( + or rt 2 x (0)) s (pr ) = 0 otherwse.) Q.E.D. (x References. M. Armbrust, A. Fox, R. Grffth, A. D. Joseph, R. H. Katz, A. Konwnsk, G. Lee, D. A. Patterson, A. Rabkn, I. Stoca, and M. Zahara. Above the clouds: A Berkeley Vew Of Cloud Computng. Techncal Report, EECS, U. C. Berkeley, S. C. M. Lee and J. C. S. Lu. On The Interacton and Competton Among Internet Servce Provders. IEEE Journal on Selected Areas n Communcatons, 26, S. Shakkota and R. Srkant. Economcs Of Network Prcng Wth Multple ISPs. IEEE/ACM Transactons on Networkng, 4, P. Hande, M. Chang, R. Calderbank, and S. Rangan. Network Prcng and Rate Allocaton Wth Contentprovder Partcpaton. In IEEE INFOCOM, L. Jang, S. Parekh, and J. Walrand. Tmedependent Network Prcng and Bandwdth Tradng. IEEE BoD, 2008.
15 6. J. K. MackeMason and H. R. Varan. Prcng Congestble Network Resources. IEEE Journal on Selected Areas n Communcatons, 3, D. Songhurst and F. Kelly. Chargng Schemes For Multservce Networks. 5th Internatonal Teletra?c Congress, P. Dube, R. Jan, and C. Touat. An Analyss of Prcng Competton For Queued Servces Wth Multple Provders. ITA Workshop, H. R. Varan. Mcroeconomc Analyss. Norton, M. E. Wetzsten. Mcroeconomc Theory: Concepts and Connectons. South Western, W.Rudn. Prncples of Mathematcal Analyss. Mc.Graw Hll, K.Arrow. Handbook of Mathematcal Economcs. North Holland, X. Vves. Nash Equlbrum and Strategc Complementartes. J. Mathematcal Economcs, 9, F. Bernsten and A. Federgruen. Compartve Statcs, Strategc Complements, and Substtutes n Olgopoles. Journal of Mathematcal Economcs, 40, D.Fudenberg and J.Trole. Game Theory. MIT Press, D. M. Topks. Supermodularty and Complementarty. Prnceton Unversty 7. D. Bertsekas and R. Gallager. Data Networks. Prentce Hall Inc., C. Daskalaks, P. W. Goldberg, and C. H. Papadmtrou. The Complexty of Computng A Nash Equlbrum. SIAM Journal of Computng. 39(),
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