Congestion and price competition in the cloud

Transcription

1 Congetion and rice cometition in the cloud Jonatha Anelmi, Danilo Ardagna, Adam Wierman 3 BCAM, Politecnico di Milano, 3 California Intitute of Technolog Abtract Thi aer rooe a model to tud the interaction of ricing and congetion in the cloud comuting marketlace. Secificall, we develo a three-tier market model that cature a marketlace with uer urchaing ervice from Softwarea-Service SaaS rovider, which in turn urchae comuting reource from either Provider-a-a-Service PaaS rovider or Infratructure-a-a-Service IaaS rovider. Within each level, we define and characterize cometitive equilibria. Further, we ue thee characterization to undertand the relative rofitabilit of SaaS and PaaS/IaaS and to undertand the imact of rice cometition on the uer exerienced erformance. Our reult highlight that both of thee deend fundamentall on the degree to which congetion reult from hared or dedicated reource in the cloud. We evaluate the inefficienc of uer erformance b tuding the rice of anarch and how that it grow with the non-linearit of the congetion function for cloud reource. I. INTRODUCTION The cloud comuting marketlace ha evolved into a highl comlex economic tem made u of a variet of ervice. Thee ervice are ticall claified into three categorie: i In Infratructure-a-a-Service IaaS, cloud rovider rent out the ue of hical or virtual erver, torage, network, etc. To delo alication uer mut intall and maintain oerating tem, oftware, etc. Examle include Amazon EC and Rackace Cloud. ii In Provider-a-a-Service PaaS, cloud rovider deliver a comuting latform on which uer can develo, delo and run their alication. Examle include Google A Engine and Microoft Azure. iii In Software-a-a-Service SaaS, cloud rovider deliver a ecific alication ervice for uer. There are a huge variet of SaaS olution thee da, uch a ervice, ERP, etc. Examle include ervice uch a Gmail and Google Doc. Naturall, each te of cloud ervice IaaS, PaaS, SaaS ue different ricing and contracting tructure, which ield a comlicated economic marketlace in the cloud. For examle, Amazon comuting ervice are billed on an hourl bai, while ome other Amazon ervice e.g., queue or datatore are billed according to the data tranfer in and out [5], [6]. Google ricing i alied on a er alication or uer er month bai and more comlex billing rule are alied if monthl quota are exceeded [5]. Further adding to the comlexit of the cloud marketlace i the fact that a articular SaaS i likel running on to of either a PaaS or IaaS. Thu, there i a multi-tier economic interaction between the PaaS or IaaS and the SaaS, and then between the SaaS and the uer. Thi multi-tier interaction wa illutrated roentl b the recent crah of IaaS rovider Amazon EC, which in turn brought down dozen of roent SaaS rovider [4], []. A a reult of the comlicated economic marketlace within the cloud, the erformance delivered b SaaS rovider to conumer deend on both the reource allocation deign of the ervice itelf a traditionall conidered and the trategic incentive reulting from the multi-tiered economic interaction. Imortantl, it i imoible to earate thee two comonent in thi context. For examle, uer are both rice-enitive and erformance-enitive when chooing a SaaS; however the bulk of the erformance comonent for a SaaS come from the back-end Iaa/PaaS. Further, the IaaS/PaaS doe not charge the conumer, it charge the SaaS. Additionall, there i cometition among SaaS rovider for conumer and among IaaS/PaaS rovider for SaaS rovider, which ield a cometitive marketlace that in turn detere the reource allocation of infratructure to uer, and thu the erformance exerienced b uer. Contribution of thi aer Thi aer aim to introduce and analze a tlized model caturing the multi-tiered interaction between uer and cloud rovider in a manner that exoe the interla of congetion, ricing, and erformance iue. To accomlih thi, we introduce a novel three-tier model for the cloud comuting marketlace. Thi model, illutrated in Figure, conider the trategic interaction between uer and SaaS rovider the firt and econd tier, in addition to the trategic interaction between SaaS rovider and either IaaS or PaaS rovider the econd and third tier. Of coure, within each tier there i alo cometition among uer, SaaS rovider, and IaaS or PaaS rovider, reectivel. To the bet of our knowledge, thi i the firt aer that jointl conider the interaction and the equilibria ariing form the laer of the full cloud tack i.e., uer, ervice, and infratructure/latform. The detail of the model are rovided in Section II and III but, briefl, the ke feature are: i uer trategicall detere which SaaS rovider to ue deending on a combination of erformance and rice; ii SaaS rovider comete b trategicall detering their rice and the IaaS/PaaS rovider the ue in order to maximize rofit, which deend on the number of uer the attract; iii IaaS/PaaS rovider comete b trategicall detering their rice to maximize their rofit; iv the erformance exerienced b the uer i affected b the congetion of the reource rocured at the IaaS/PaaS choen b the SaaS, and that thi congetion i a reult of the combination of congetion at dedicated reource, where congetion deend onl on traffic from the SaaS, and hared reource, where congetion deend on the total traffic to the IaaS/PaaS. The comlex nature of the cloud marketlace mean that the model introduced in thi aer i necearil comlicated too. To highlight thi, note that an analtic tud of the model entail characterizing equilibria within each of the three tier, in a context where deciion within one tier imact rofit and thu equilibria at ever other tier. Due to the comlexit of the model, in order to be able to rovide analtic reult, we need to conider a limiting regime. Motivated b the huge, and growing, number of SaaS

2 rovider and the comarativel maller number of IaaS/PaaS rovider, the limiting regime we conider i one where the number of uer and the number of SaaS rovider are both large ee Section III for a formal tatement. Under thi aumtion, we can attain an analtic characterization of the interacting market which ield intereting qualitative inight. More ecificall, with our anali we eek to rovide inight to the following fundamental quetion: i How rofitable are SaaS rovider a comared to PaaS/IaaS rovider? Doe either have market ower? ii How good i uer erformance? I the economic tructure uch that increaed cometition among cloud rovider ield efficient reource allocation? iii How doe the degree to which cloud reource are hared/dedicated imact i and ii? Our anali highlight a number of imortant qualitative inight with reect to thee quetion, and we dicu thee in detail in Section IV. For examle, our anali how that if congetion i doated b the congetion at hared reource, the cloud market doe not function well, i.e., rovider can be rofitable but ervice are unrofitable and thu have no reaon to articiate in the market. In contrat, if congetion i doated b congetion at dedicated reource, market function well, i.e., rovider and ervice are both rofitable and ervice receive the bulk of the rofit. However, in both cae, our anali highlight another iue with the current market tructure: the interaction of ervice and rovider market erve to rotect inefficient rovider. That i, even if one rovider i extremel inefficient comared to other, the inefficient rovider till obtain ignificant rofit. Finall, another danger that our anali highlight i that the market tructure tudied here can ield ignificant erformance lo for uer, a comared with otimal reource allocation. Secificall, the rice cometition among ervice and rovider ield inefficient reource allocation. Relationhi to rior work There i a large literature that focue on trategic behavior and ricing in cloud tem and, more generall, in the internet. Thi area of network economic of network game i full of increaingl rich model incororating game theoretic tool into more traditional network model. The following urve rovide an overview of the modeling and equilibrium concet in ticall ued networking game, and additionall include an overview of their alication in telecommunication and wirele network [3], [7], [5]. In the context of cloud tem ecificall, an increaing variet of network game have been invetigated and three main area of attention in thi literature are reource allocation [8], [4], load balancing [], [7], [8], [], and ricing [], [9], [4], [7]. It i thi lat line of work that i mot related to the current aer. Within thi ricing literature, the mot related aer to our work are [], [7], [9], [4], [3], [7]; ee alo the reference therein. Each of thee aer focu on deriving the exitence and efficienc a meaured b the rice of anarch of ricing mechanim in the cloud. For examle, [9] conider a two-tier model caturing the interaction between SaaS and a ingle IaaS, and tudie the exitence and efficienc of equilibria allocation. Similarl, [], [7], [4] conider twotier model caturing the interaction between uer and SaaS or between SaaS and PaaS/IaaS, and tud the exitence and efficienc of equilibrium allocation. Thu, the quetion aked Uer λ x x x S Fig.. Service SaaS rice: α rice: α S rice: α S Model overview. P Provider PaaS/IaaS Price: β P Price: β P in thee and other aer are imilar to thoe in our work. However, the model conidered in thi aer i ignificantl more general than rior work tuding the cloud marketlace. Secificall, we cature the three-tier cometing dnamic between uer, SaaS, and IaaS/PaaS imultaneoul. Further, we model the ditinction between congetion from hared and dedicated reource. Neither of thee factor wa tudied in the reviou work; and both lead to novel qualitative inight about the cloud marketlace while imultaneoul reenting ignificant technical challenge to overcome. II. MODEL OVERVIEW AND NOTATION Our model rereent a three-tier cloud marketlace and focue on the interaction between PaaS or IaaS rovider termed rovider, SaaS rovider termed ervice, and final cutomer termed uer. We dicu each of the three tier in the following, tarting with rovider. The model i detailed, and o Figure rovide a high level diagram. Thi ection focue on the non-cometitive aect of the model and the next one introduce, and begin to characterize, the cometitive equilibria conidered at each level. A. Provider The rovider in our model rereent the PaaS and IaaS in the cloud. That i, the ell comuting infratructure to ervice SaaS in a manner uch a done b Amazon EC or Google A Engine. We model cometition among P rovider, where each rovider ell caacit to ervice at a rice of β er unit of caacit and time. For imlicit, we aume throughout that ervice are all intereted in one unit of caacit. Thi ricing etu i a imlified model of the ricing cheme that i currentl emloed b Amazon EC [5]. The total traffic erved b rovider i denoted b. Note that i deendent on both the fraction of ervice that chooe rovider, which we denote b g, and the number of uer that chooe thoe ervice. Since g i the fraction of ervice chooing rovider, g S i the number of ervice that ue rovider, where S denote the number of ervice. We ue thi articular formulation becaue we are intereted in letting the number of ervice grow large in our anali. The rofit for rovider, er unit of time, i thu: Provider-Profit = β g S. Note that we do not model the oerating cot of the rovider, but thi could be incororated eail into the above equation.

3 3 B. Service The ervice in our model la the role of SaaS. That i, the ell a ervice to uer, but bu the comuting infratructure ued to rovide thi ervice from a rovider. We model cometition among S interchangeable ervice, where each ervice i old for a rice α to uer. The uer traffic to ervice i denoted b x. Each ervice i run on exactl one of the P rovider, and the rovider choen b ervice i denoted b f. For imlicit, we aume each ervice i intereted in one unit of ervice caacit that rovider ell at rice β. The rofit for ervice, er unit of time, i thu: Service-Profit = α x β f. Note that we do not model the oerating cot of the ervice, but thi could be incororated eail into the above equation. C. Uer The uer in our model la the role of the cutomer of ervice SaaS. Since thee ervice have enormou uer bae, we conider a model with cometition among infinitelman uer, each of which control an infiniteimal amount of traffic, i.e., a non-atomic routing game. In total, the uer make u an inelatic arrival rate of λ. Uon arriving to the tem, each uer elect exactl one ervice. When making thi election there are two factor that come into la: the rice charged b the ervice α and the congetion-deendent cot l x, f, where x = x,..., x S i the amount of traffic going to each ervice and f i the ervice-to-rovider maing. Thi cot can be interreted a the mean dela or mean reliabilit of ervice. We ue the term congetion cot and latenc interchangeabl. It i imortant to highlight the deendence of the congetion cot on both x and f in the etting of thi aer. Secificall, the congetion cot at ervice could be affected b both the number of uer chooing and alo the number of uer chooing other ervice deloed on the ame rovider of, i.e., f. Thi i becaue rovider ma have ome dedicated reource for ervice uch a virtual machine, erver, or even a cluter of comuter; but other reource at the rovider are necearil hared uch a networking comonent and oibl torage. In thi aer, in order to obtain analtic reult, we limit ourelve to a articular form of interaction between hared and dedicated reource: l x, f = l f x + ˆl f f. So, l cature the congetion at dedicated reource congetion deend on onl the conidered ervice traffic and ˆl cature the congetion at hared reource congetion deend on the total traffic to the conidered rovider. Of coure, one could anale generalization of the form given in ; however thi form alread cature the qualitative interaction of thee te of workload. In fact, the relative magnitude of congetion in hared and dedicated reource how u roentl in the reult rovided in Section IV. Some additional technical aumtion widel ued in the literature, e.g., [], [3] we require are that l : R + R + and ˆl : R + R +, are continuoul-differentiable, increaing and convex; l = ˆl = for all ; and l x < for all x and ˆl < for all. Given the above, we model the uer a imizing their effective-cot when chooing a ervice : Uer-Effective-Cot = α + l x, f = α + l f x + ˆl f f. Thi form i motivated b conidering the congetion cot a being in currenc unit, and i a common modeling aumtion in the literature on congetion game; e.g., [], [3]. III. MODELING AND ANALYZING COMPETITION A crucial comonent of our model i caturing the trategic behavior of uer, ervice, and rovider. In thi ection, we define the equilibrium concet we tud within each level. Clearl, thee equilibrium concet are quite entangled. So, we tart b defining the uer equilibrium and build to the rovider equilibrium. Along the wa, we derive reult characterization and exitence reult for each of the equilibrium concet introduced. Throughout, roof are differed to the aendix for the ake of conciene. A. Cometition among uer Since each uer carrie an infiniteimall-mall amount of traffic, we model cometition among uer a a nonatomic routing game; e.g., []. In thi context, effective-cot imizing cometition ield a Wardro equilibrium, i.e., a traffic allocation that follow from Wardro rincile [6]: the effective cot of each uer i identical and imum at each ued ervice. Uing the notation of our model, a uer equilibrium i defined a follow. Definition. Conider a fixed ervice maing f and ervice rice α. A vector x UE = x UE α, f [, λ] S i a uer equilibrium if l f x UE = :x UE > :f = xue xue + ˆl f f UE + α { lf x UE = λ. = UE,, + ˆl f UE f + α }, : x UE > Thi i a well etablihed equilibria concet, e.g., [], and it i ea to obtain a trong characterization of the uer equilibrium uing a otential function-baed argument howing that condition are the KKT condition of a trictl-convex otimization roblem. Prooition. Conider a fixed ervice maing f and ervice rice α. There exit a unique uer equilibrium, which i given b the otimizer of x [ x ] l f zdz + α x + :f x = =, x = λ. ˆl zdz, Imortantl, becaue otimization 3 i trictl convex, the unique uer equilibrium i efficientl comutable []. 3

4 4 B. Cometition among ervice We model cometition among ervice a an oligoolitic ricing game between S rofit-maximizing ervice a in [], [7]. In articular, we aume that each ervice et a rice and elect a rovider in order to maximize it rofit, ielding a Nah equilibrium where no ervice ha a unilateral incentive to deviate. Service have two choice rice and rovider, and thu the equilibrium need to conider both. Becaue rice ma fluctuate at a fater time cale than the choice of a rovider, we focu on a two tage equilibrium and firt define a erviceequilibrium rice vector before then uing thi concet to define a ervice-equilibrium ditribution. Service-equilibrium rice vector: We define a erviceequilibrium rice vector a follow. Definition. Conider a fixed ervice maing f. Then, α SE = α SE f i a ervice-equilibrium rice vector if α SE argmax α, α x UE α, α SE, f,. 4 Etablihing the exitence of a ervice-equilibrium rice vector i harder than in the cae of uer equilibria. To highlight thi, note that in ome ecial cae, a ervice-equilibrium rice vector i equivalent to the notion of oligoolitic equilibrium in [], for which exitence i hown onl when the latenc function are linear. So, when characterizing ervice-equilibrium rice vector, we alo limit ourelve to linear latenc function. Within thi context, the following rooition how the exitence of a ervice-equilibrium rice vector even when ˆl, rovided that both l and ˆl are linear. It i likel that exitence can be guaranteed outide of thee retriction a well; however, a in [], [7], there are cae where an equilibrium doe not exit. Prooition. Conider l f x = ã f x and ˆl = â for all and. There exit a unique ervice-equilibrium rice vector. We can alo characterize the tructure of ervice-equilibrium rice vector in a more general context, rovided the exit. Prooition 3. Conider a fixed ervice maing f. If α SE i a ervice-equilibrium rice vector, then α SE atifie 5 for all, where = f and x = x UE α SE, f i a uer equilibrium. The rice tructure decribed b 5 i clearl cumberome. Given that the ervice-equilibrium ditribution and the rovider equilibrium defined later both build on thi characterization, it i imortant to find a imler rereentation if we hoe to be able to obtain analtic reult. The imortant obervation we ue to obtain uch a imlification i that, in ractice, the number of ervice tend to be ver large while the number of rovider i comarabl mall. Thu, a etting with S P i reaonable. Further, it haen that 5 imlifie dramaticall when we conider a large number of ervice. Thee obervation lead u to conider the following large-tem limit. Definition 3. The large-tem limit i defined via a caling arameter n N, with n, where in the nth tem: i The uer arrival rate i nλ. ii There are ns ervice and where ervice, S +, S +,..., n S + chooe the ame rovider, for all =..., S. iii The dedicated and hared reource caacit of each rovider cale roortionall with n, i.e., the latenc obe l x = l f x + ˆl f f n, for all. Thi limiting regime correond to a ituation where our tem i relicated n time. Thi i a claic aroach in economic [6] and in our cae ield the interretation that S i the number of normalized ervice, each of which behave like a non-atomic laer. In thi regime, the characterization of the rice vector equilibrium become manageable. Prooition 4. Conider a fixed ervice maing f. In the large-tem limit, there exit at mot one ervice-equilibrium rice vector α SE and α SE x UE l f x UE,, 6 where x UE = x UE α SE, f i a uer equilibrium. Not onl i the rice tructure in 6 more manageable than 5, it alo highlight ome imortant inight. Secificall, 6 tate that, in the large-tem limit, ervice make rofit onl becaue of the congetion on dedicated reource, congetion at hared reource i irrelevant. In other word, the negative externalitie exerted to uer due to the congetion on hared reource i not rofitable for ervice. Service-equilibrium ditribution: We have now defined and characterized the equilibrium with reect to one of the choice that ervice face, i.e., rice. What remain i to define and characterize the equilibrium for how ervice ditribute over rovider, i.e., the ervice-equilibrium ditribution. Clearl, the ervice-equilibrium ditribution deend on both the uer equilibrium and the ervice-equilibrium rice vector. The following lemma highlight that all the ervice maed to the ame rovider have identical ervice-equilibrium rice and incog traffic. Lemma. For an, uch that f = f, α SE f = α SE f. Furthermore, x UE α SE, f = zf UE for all where UE def z = z UE,..., zp UE i the unique olution of { l z :z UE UE + ˆl },g > g Sz UE + αse = l z UE + ˆl g Sz UE + α SE, : z UE, g > g Sz UE = λ z UE > if and onl if g >, 7 and α SE = α SE f i a ervice-equilibrium rice of an ervice that i maed on rovider, for all. The above lemma imlie that ervice are inditinguihable from the rovider tandoint and allow u to define a ervice-equilibrium ditribution in term of g, the roortion of ervice uing each rovider, intead of f, the exact maing of ervice to rovider. Thi i an imortant hift ince we are focued on the large-tem limit. In the following, recall that g = g f = { : f = } /S.

5 5 l f x :f ˆl α SE = x l + l x f :f x + x 5 l l x f :f :f= x l ˆl + x :f= + l x + l f :f ˆl l x x f f :f= Definition 4. Conider fixed rovider rice β. Then g SE = g SE β i a ervice-equilibrium ditribution if it olve { α SE z UE β = max α SE z UE β }, : g SE > gse g SE =, :g SE > where z UE = z UE g SE atifie 7, and α SE z UE l z UE, i.e., it atifie 4. 8 = Note that thi definition correond to a variation of Wardro equilibrium [], [6], imilar to the concet ued for the uer equilibrium. Thi etting i reaonable ince ervice are inditinguihable and their number i large, o no articular ervice ha meaurable market ower. Secificall, in thi equilibrium the roortion of ervice on each rovider are uch that the rofit of all ervice are equal and maximal. We tre that Definition 8 i deendent on the caling in Definition 5. The term S that aear in the definition of z UE g SE hould be interreted a the normalized number of ervice. So, Sg SE can be an non-negative number. We now move to characterizing ervice-equilibria ditribution. Proving the exitence of a ervice-equilibrium ditribution i comlicated b the dicontinuitie in g that emerge in the equation characterizing z UE g, i.e., 7. The following rooition give a condition for which the ervice-equilibrium ditribution i unique, if it exit, and can be found via a trictl-convex otimization roblem. Prooition 5. Conider fixed rovider rice β. Let g β, β be the unique otimizer of the following trictl-convex otimization roblem g, l Sg g =, = λ. + ˆl zdz + Sβ g If g SE β > and gβ > for all, then g SE β = g β i the unique ervice-equilibrium ditribution. Under the aumtion that g SE β > for all, the otimizer of 9 coincide with the condition that define a ervice-equilibrium ditribution Definition 4, which i a remarkable and nontrivial roert of our model. It will be hown in Lemma that the aumtion g SE β > for all i true if β form a rovider equilibrium defined next. C. Cometition among rovider To cature cometition among rovider, we conider rice cometition among P rovider that are rofit maximizing. Thi ield the following Nah equilibrium formulation. 9 Definition 5. The vector β P E i a rovider equilibrium if β P E argmax β β g SE β, β P E S,. Imortantl, thi equilibrium embed the equilibrium at the ervice level, which in turn cature the equilibrium at the uer level. A uch, it i the mot informative, but alo the mot difficult to tud. Characterizing the exitence of rovider equilibrium i a ver difficult tak. Thi i highlighted b oberving the tructure of otimization, which i non-concave even when latenc function are linear. A a reult, we do not rovide analtic reult regarding exitence. However, it i oible to obtain reult characterizing rovider equilibria, a done in Prooition 3 for ervice-equilibrium rice in a ecial cae. In articular, the following rooition etablihe the tructure of the rovider equilibrium rice in the cae of two rovider and linear latenc function. The tructure rovided b thi reult i rovide a contructive numerical method to check exitence of a rovider equilibrium, a we dicu next, and ii i crucial for characterizing the uer erformance and ervice/rovider rofit, a we do in Section IV. Lemma. If β i a rovider equilibrium, then g SE β > for all. Prooition 6. Let P =, lf x = ã f x and ˆl = â, for all,. Then z β P E = g z ã g 4g ã g S, =, â + ã Sg where g = g SE β P E i a ervice-equilibrium ditribution and z = z UE g atifie 7. An imortant conequence of Prooition 6 i that it allow numerical invetigation of rovider equilibria. In articular, uon ubtituting into condition 8, one obtain a nonlinear tem comoed of four equation and four unknown, i.e., the and g. Subtituting the olution of thi tem back in, one obtain a gue for a rovider equilibrium. To check whether or not thi gue a β i a rovider equilibrium, one can check numericall, i.e., whether or not β i the bet-reone to β. Uing thi aroach we have numericall tudied random intance which â and ã choen uniforml at random from [.,]. In all cae, we found a unique olution to thi tem with the reulting rovider rice forg a bet-reone, which verifie that rovider equilibria ticall exit when latencie are linear. IV. PROFITABILITY AND EFFICIENCY To thi oint, we have introduced a model for the cloud comuting marketlace, and characterized the equilibria that

6 6 reult from cometition within thi etting. Our goal in thi ection i to ue thi characterization to undertand the imact of thee interacting market on all the artie involved, i.e., the uer, ervice, and rovider. In articular, our goal i to tud the three quetion outlined in the Introduction. Note that the anali reented in the following cruciall ue Prooition 5 and 6, which characterize the ervice and rovider equilibria reectivel. A. Profitabilit of ervice and rovider In order to tud the relative rofitabilit of ervice and rovider, we need to make ue of a detailed characterization of rovider equilibria, and o our focu i on the ecial cae of P = and linear latenc function, a thi retriction i necear to al Prooition 6. To al Prooition 6, we firt need to comute the uer equilibrium traffic allocation and the ervice equilibrium rice vector and ditribution. Thee are characterized b condition 8 where the β are given b 6. One can how that the reulting tem of equation can be exreed in term of the olution a olnomial of order four. Thu, finding the olution of thi tem i oible both analticall and numericall. However, the reulting exreion for g SE i cumberome and thu omitted. Intead, we focu on rereentative ecial cae and numeric examle in order to highlight the inight one can learn from the exreion. In articular, in the following we focu on the cae of mmetric and ammetric latencie. Smmetric latencie: To begin, we conider the ecial cae when latenc function are mmetric. In thi etting, we obtain imle, informative characterization of the rovider equilibrium and ervice equilibrium rice vector. Corollar. Let P =, l f x = ãx and ˆl = â, for all,. Then, g SE β P E =,, αse f = ã λ S, with f uch that gf = g SE β P E, z UE g SE β P E = λ S, λ S, and λ β P E 4âãS = S Sâ + 4ã Imortantl, the above reult ield after ome algebra the rofit of ervice and rovider, reectivel: 4ã 3Sâ λ Service-Profit = ã 3 Sâ + 4ã S Provider-Profit = λ âã 4 Sâ + 4ã To obtain inight from thee formula, let u focu on the imact of congetion at hared veru dedicated reource. The imact of thee can be een b varing the contant â and ã. If â ã â ã then congetion i doated b congetion at the hared dedicated reource. A firt obervation i that the rovider rofit increae a â increae i.e., hared reource become more doant, and converge to S λ S ã when â. In contrat, the ervice rofit decreae a â increae and become negative for large enough â. Thu, rovider are rofitable when hared reource doate, but it i unrofitable for ervice to articiate in the cloud marketlace in thi etting. In contrat, if we let ã grow, i.e., dedicated reource doate, then both rovider and ervice rofit increae. Further, a ã ervice rofit alo grow unboundedl and rovider rofit converge to λâ. Thu, both ervice and rovider are rofitable if dedicated reource doate, but ervice extract a doant hare of the rofit. The fact that ervice receive a doant hare of the rofit i intereting given that cometition i much larger in the ervice market recall that S i the normalized number of ervice. Ammetric latencie: The cae of ammetric latencie i not a imle a the mmetric cae tudied revioul. Thu, we retrict ourelve to a articular form of ammetr that i articularl illutrative: â â, ã, ã. In thi cae, congetion i doated b the hared reource at rovider, and congetion at rovider i in balance between hared and dedicated reource in comarion to rovider. Thu, there i a doant rovider in thi cae which ha more efficient infratructure. The quetion we look at in the anali i the extent to which thi doant rovider can exloit thi increaed efficienc in the marketlace. The firt te of the anali i to characterize the rovider equilibrium rice in the ame etting a Prooition 6. Uing that â i large, we get g β P E S g 3 ã, =,. 5 To obtain a clean characterization of the rofit, we conider the cae when the ammetr become extreme, i.e., â and â = ã = ã =. Thi allow the following characterization of the rofit for the ervice and rovider. Corollar. Let P =, l f x = ãx and ˆl = â, for all,. Further, conider â and â = ã = ã =. Then, Provider-Profit 3 4 S λ S Provider-Profit 3S λ 6 S Service-Profit 3 λ 4 S A firt comment on Corollar i that it i in comlete alignment with 3 and 4 for the cae of mmetric rovider. Secificall, in the cae of mmetric rovider, ervice are unrofitable and rovider are rofitable when hared reource doate congetion. That i exactl what we ee in thi ammetric etting a well. However, Corollar alo highlight omething intereting about the cometition among rovider. In articular, the market tructure rotect the inefficient rovider b limiting the market ower of the doant rovider. In articular, deite the fact that the ga in efficienc i extreme, rovider can onl extract at mot four time the rofit of rovider, and rovider till extract ignificant rofit. The reaon for thi i that, even though the latenc function at rovider i ver tee, rovider till manage to obtain at leat /3 of the ervice ee the roof. Imortantl, becaue of the teene, thee are ervice with ver little uer traffic; however rovider can till obtain ignificant rofit from them. Imortantl, the inight rovided b Corollar hold more generall a well, a illutrated b Figure, which illutrate rovider and ervice rofit when â <. Though we have onl conidered one form of ammetr above, thi etting i quite rereentative. In articular, the other etting we have conidered all reinforce the imact of hared v. dedicated reource on rofitabilit highlighted b Corollar and the limited market ower attainable in the rovider market highlighted b Corollar. Thu, we omit them due to ace contraint.

7 7 Fig.. Profit 5 Provider Provider Service â Service and rovider rofit when rovider are ammetric. B. Efficienc of uer erformance The reviou anali focue on the imact of the marketlace on ervice and rovider, we now hift to the imact of the market tructure on the uer erformance. In articular, our goal i to undertand how rice cometition among ervice and rovider imact the erformance exerienced b uer. To erform thi tud, we contrat the erformance at equilibrium to the erformance achievable if the allocation of traffic to ervice and rovider wa erformed otimall. In articular, we tud the following ratio, which i often termed the rice of anarch [9]. To tate the definition we need to firt define the network latenc, our meaure for the aggregate uer-exerienced erformance: lz, g def = λ Sg z l z + ˆl Sg z 7 Thi form follow from uing Lemma and recalling that Sg z i the total traffic to rovider, which i normalized b the caling arameter n in the large tem limit. Note that Sg z = λ. Definition 6. We define the rice of anarch P oa a P oa def lz UE, g SE = u β lz, g 8 P E where g SE = g SE β P E and z UE = z UE g SE atif 8 when β = β P E, and z, g arg x,g lz, g g Sz = λ g =. 9 While man reult on the rice of anarch of non-atomic routing game are known, the multi-tier tructure of our model add ignificant comlexit in deriving uch reult. To highlight the challenge, note that the maing g and g SE β SE differ in general, which make anali difficult. However, the following traightforward lemma highlight that it i oible to bound the rice of anarch b focuing on cae with fixed ervice-to-rovider maing. Lemma 3. P oa P oa g SE β P E, where and def P oa g = lzue g, g lz, g g, g z g arg z lz, g g Sz = λ. Uing thi lemma, we can obtain lower bound on the rice of anarch b tuding the rice of anarch in the imler cae of fixed ervice-to-rovider maing. In thi cae, we can rove the following bound. Theorem. Let ɛ >. Conider P =, l = ã k, and ˆl = â k. Then, Ωk ɛ u P oa g SE β P E k +, where the u i taken over S, λ, ã, â, β P E. Thu, P oa Ωk ɛ. Theorem highlight that the rice of anarch grow quickl a the non-linearit of the latenc function grow. Thi behavior i not unexected, a a imilar reult hold for claical non-atomic routing game []. However, the conequence in thi etting i imortant: it highlight that the cloud marketlace can ield equilibria that have inefficient uer erformance. Note that Theorem rovide a negative reult; however we conjecture that oitive reult are alo oible. In articular, we conjecture that the rice of anarch remain mall in the cae of linear latenc function, imilarl to what i oberved in claical non-atomic routing game. In fact, when P =, our numerical tudie uort the claim that P oa 7/6. However, roving uch a reult i difficult. V. CONCLUDING REMARKS In thi aer we develo a three-tier market model for a cloud marketlace. Our focu i on a etting including uer urchaing ervice from SaaS rovider, which in turn urchae comuting reource from either PaaS or IaaS. Within each level we define and characterize cometitive equilibria. Further, we ue thee characterization to undertand the rofitabilit of SaaS and PaaS/IaaS, and to undertand the imact of rice cometition on uer exerienced erformance. The reult in thi aer rereent a tarting oint for the anali of the novel model reented here, and we hoe that the model i of interet in it own right for future reearch. In articular, it cature a rich interaction between ricing and congetion acro multile market; and there are man oen quetion that remain. For examle, our reult require for technical reaon a variet of imlifing aumtion. It would be quite intereting to relax thee, e.g., tud P > and characterize equilibrium exitence and uniquene for nonlinear latenc function. Additionall, our rice of anarch anali focued on lower bound, but uer bound are alo imortant but eegl difficult to obtain. Finall, a articularl imortant future direction i to tud the contrat between different ricing cheme ued b ervice and rovider uing the model in thi aer. REFERENCES [] D. Acemoglu and A. Ozdaglar. Cometition and efficienc in congeted market. Math. Oer. Re., 3: 3, 7. [] E. Altman, U. Aeta, and B. Prabhu. Load balancing in roceor haring tem. In Proc. of ValueTool, age, 8. [3] E. Altman, T. Boulogne, R. El-Azouzi, T. Jiménez, and L. Wnter. A urve on networking game in telecommunication. Comut. Oer. Re., 33:86 3, 6. [4] Amazon EC Outage Reveal Challenge of Cloud Comuting. htt: // cloudtime.org/ / 7/ 3/ amazon-outage-rik-comuting/. [5] Amazon EC Pricing. htt:// aw.amazon.com/ ec/ ricing/. [6] Amazon Simle Queue Service Amazon SQS. htt://aw.amazon.com/ q/.

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[5] Google A Engine Pricing. htt:// cloud.google.com/ ricing/. [6] O. D. Hart. Monoolitic cometition in a large econom with differentiated commoditie. Review of Economic Studie, 46: 3, 979. [7] M. Haviv. The Aumann-Shael ricing mechanim for allocating congetion cot. Oeration Reearch Letter, 95: 5,. [8] Y.-J. Hong, J. Xue, and M. Thottethodi. Dnamic erver roviioning to imize cot in an iaa cloud. In Proc. of ACM SIGMETRICS, age 47 48,. [9] E. Koutouia and C. Paadimitriou. Wort-cae equilibria. In STACS, volume 563 of Proc. of STOC, age 44 43, Januar 999. [] NetworkWorld. Amazon outage one ear later: Are we afer? htt: // new/ / 47-amazon-outage html. [] N. Nian, T. Roughgarden, E. Tardo, and V. V. Vazirani. Algorithmic Game Theor. Cambridge Univerit Pre, New York, NY, USA, 7. [] T. Roughgarden and E. Tardo. How bad i elfih routing? J. ACM, 49:36 59, Mar.. [3] Y. Song, M. Zafer, and K.-W. Lee. Otimal bidding in ot intance market. In Proc. of IEEE INFOCOM, age 9 98,. [4] F. Teng and F. Magoule. A new game theoretical reource allocation algorithm for cloud comuting. In Advance in Grid and Pervaive Comuting, age 3 33,. [5] A. van den Nouweland, P. Borm, W. van Goltein Brouwer, R. Bruinderink, and S. Tij. A game theoretic aroach to roblem in telecommunication. Manage. Sci., 4:94 33, 996. [6] J. G. Wardro. Some theoretical aect of road traffic reearch. Proceeding of the Intitute of Civil Engineer, :35 378, 95. [7] B. Yolken and N. Bambo. Game baed caacit allocation for utilit comuting environment. In Proc. of ValueTool, age 8, 8. APPENDIX A. Proof of Prooition With reect to multilier L,, L and L 3,, the KKT condition of 3 read l f x + α L,f L L 3, =, ˆl + L, =, lu feaibilit contraint and comlementarit lackne. Subtituting the econd equation into the former, we obtain condition. Since 3 i a trictl convex otimization roblem, there exit a unique imizer; and thu a unique uer equilibrium. B. Proof of Lemma For contradiction, aume that α SE < α SE. Then, x UE α SE, f > x UE α SE, f b the definition of uer equilibrium. Given the tructure of α SE f 5, which i roven in Prooition 5, we have that α SE f i trictl increaing in x. Thi follow becaue the fraction that multilie x in 5 doe not change when x varie and x + x i ket contant. Conequentl, α SE > α SE, which i a contradiction. Further, the ame argument give that α SE > α SE doe not hold, and o α SE = α SE. But, if α SE = α SE, then 5 enure that x = x. Since and are general, thi hold true in general, and x = f Sg f. Subtituting x = f Sg f = z in Definition, we get condition 7. Exitence and uniquene of a vector z UE that olve 7 follow eail b uing the otential function method imilarl to the roof of Prooition. C. Proof of Prooition Conider Formula 5. If the latencie are linear, the righthand term can be rewritten a x ã + c, where = f and c i ome contant. Let α be uch that α i given b Formula 5 where the x atifie the following condition { lf x + ˆl f f + x ã f + c } :x > = l f x + ˆl f f + x ã f + c, : x > :f x = =,, x = λ. Since l = ˆl = for all, reviou condition read l f x + ˆl f f + x ã f + c = l f x + ˆl f f + x ã f + c, > :f = x =,, x = λ. 3 and form a linear tem with S + P unknown and S + P indeendent equation, thu there exit a unique olution a x and it i uch that x > for all. We claim that α, which i now well-defined becaue α = x ã f + c for all, i a ervice-equilibrium rice vector. To rove thi, we ue Definition and how that no ervice ha the incentive in changing it rice. Thi amount to check that for all : α argmax α argmax α,l 3, α x α x UE α, α, f lx + ˆl f + α L 3, = lx + ˆl f + α L 3,, :f x = =,, x = λ L 3,, L 3, x =, 4 which follow b uing that x UE α, α, f i the otimizer of the trictl convex otimization roblem 3 and ubtituting it KKT condition in the contraint of the otimization in 4. One can check that the oint α, L 3, =, atifie the KKT condition of the otimization roblem 4, for all. Note that if L 3, =,, then 4 become a trictl concave otimization roblem, which mean that there exit a unique otimizer. Since 4 i non-concave or nonconvex, thi allow u to a that α, L 3, =, i a local maximum. However, we now rove that α, L 3, =, i actuall a global maximum. B contradiction, aume that L 3, > for ome. Then, x = b comlementarit lackne, but thi imlie that the rofit of rovider i zero, in contrat with reviou cae which wa oitive. Thi rove exitence. Now, the fact that L 3, =, in an tationar oint of the Lagrangian of 4 and that 4 i trictl concave roblem when L 3, =, rove uniquene.

9 D. Proof of Prooition 3 We begin with a technical lemma. Lemma 4. Let f be given and α = α SE f be a erviceequilibrium rice vector. Then, α x UE α, f >,. Proof: To begin, aume that α =. It follow that there exit ɛ > uch that each ervice ha the incentive in etting rice α = ɛ. To ee thi, note that, b the aumtion l f = ˆl f =, it follow that x UE, f >,. Take α = ɛ = l x UE, f >. In thi cae, l x UE, f < l x UE ɛe, f, where e i the unit vector in direction, becaue of the monotonicit of the latencie. In other word, ɛ < l x UE ɛe, f, which imlie x UE ɛe, f > and then ɛx UE ɛe, f >. Now, to finih the roof, we how that if there exit uch that α x UE α, f >, then α x UE α, f >,. To rove thi art, we roceed a in Lemma 4. of []. Let K def = α +l x UE α, f, which i oitive b aumtion. Aume α x UE α, f = for ome and conider the rice α = K ɛ for ome mall ɛ >. Uing that l f = ˆl f =, necearil x UE αf, f >, which violate the hothei that the rofit of i zero. Uing the above lemma, we can now rove Prooition 3. Let f be given and α = α SE f. Uing that α x UE α, f > for all b Lemma 4, ervice-equilibrium rice of ervice are the otimizer of the following maximization max α,x α x l x + ˆl + α = l f x + ˆl f f + α, :f x = =, S = x = λ 5 where = f. Imoing to zero the artial derivative of the Lagrangian of reviou otimization roblem, with reect to multilier L,, L,, L 3 R we obtain x = L, α = L, l x L, L 3 L, lf x L,f L 3 = :f L,ˆl + L, = :f = L,ˆl + L, =. 6 Subtituting L, from the third equation, and exliciting L, reectivel L, from the fourth fifth equation, after ome algebra we get 5. E. Proof of Prooition 4 B Lemma, we have and α SE = α SE 7 in 5, we get x = x S+ = x S+ = x n S+,. 7 S+ = αse S+ = αse c n S+,. Subtituting α = x l x + x, {,..., S} 8 c where c def = S n l f x =:f n ˆl S + 9 l x f =:f c def = S n l f x =: f n : f = l x l x n ˆl S l x f =:f + n S l x =:f= +n S l ˆl l x f x f =:f= +n S l x f = 3 and, with a light abue of notation, we have ued = ns n =:f x = = S =:f x =. A n, c and c, and we conclude that 6 hold. Now, utting 6 in, we get the condition { lf x + ˆl } f f + x l x f = :x > l f x + ˆl f f + x l f x, {,..., S} : x > :f = x =, S = x = λ x,, 3 which coincide with the KKT condition of otimization roblem x S = x lf x + P = S =:f = x =, S = x = λ. ˆl zdz 9 3 Problem 3 i trictl convex becaue the l and the ˆl are increaing and convex b aumtion, and therefore there exit a unique otimizer []. F. Proof of Prooition 5 Conider the otimization roblem g,z z Sg l z + g =, g Sz = λ. zsg ˆl z dz + Sβ g 33 Imoing the artial derivative of the Lagrangian of 33 to zero and with reect to multilier L, L, L 3,, L 4, for all, after ome algebra we get Sβ Sz l z L 3, = L, l z + ˆl z Sg + z l z L 4, = L,. 34 Now, uing that the L 3, and the L 4, are non-negative and that the comlementarit lackne condition enure L 3, g = and L 4, =, condition 34 can be written a {Sβ Sz l } :g > z = Sβ Sz l z, : g > 35 { l z + ˆl z Sg + z l z } :z > = l z + ˆl z Sg + z l z, : z >.

10 Furthermore, in an otimal olution of 33, we have z > for all. In fact, aug that z = for ome, we have L 4, = L <, b 34, becaue there alwa exit uch that z > for which 34 and L 4, z = enure that L = l z + ˆl z Sg +z l z, which i trictl oitive becaue the latencie are increaing. Therefore, ubtituting in 35 that z > for all and ince in an otimal olution g > for all b hothei, we get condition β z l z = β z l z, l z + ˆl z Sg + z l z = l z + ˆl z Sg + z l z,. 36 Condition 36, together with the contraint of 33, coincide with the condition 8 that define a ervice-equilibrium ditribution if g SE >,. Therefore, ervice-equilibrium ditribution g SE = g SE β P E, rovided that the exit, are exactl the otimizer of 33. With the change of variable z = Sg, 33 i equivalent to 9. Uing that the l and the ˆl are convex, one can check that the objective funtion of 9 i trictl convex the Heian i trictl oitive emi-definite. Since 9 i defined on a non-emt and convex domain and that the g are oitive in an otimal olution, 9 i a trictl-convex otimization roblem; ee []. G. Proof of Lemma Let β be a rovider equilibrium. Then, β. In fact, if β = then g SE = where = argmax ã thi i evident from 8. But doe not make an rofit becaue β = and thu ha the incentive to deviate. If β > for at leat rovider, then again the tructure of 8 enure that all the other rovider make trictl oitive rofit all of them have the incentive in increaing of ɛ their rice, i.e., β g SE β > for all, which imlie gse β >. H. Proof of Prooition 6 Let rice vector β = β, β be a rovider equilibrium. Uing that in a rovider equilibrium g β > for all b Lemma, the value of β that maximize the rofit of rovider are the otimizer of the following maximization roblem a PROB max β g, β g S ã Sg β = ã Sg β, g + g =, ã Sg + â = ã Sg + â, + = λ 37 Without lo of generalit we tud PROB. The Lagrangian of PROB, with reect to multilier L i R, i =,..., 5, read L def = β g S+ L ã Sg β ã Sg + β + L g + g + L 3 ˆl + ã Sg ˆl ã Sg + L 4 + λ L 5 β, 38 where L 5. Imoing the artial derivative with reect to β, g, g,, of L to zero, we obtain that the following condition are necearil atified in a tationar oint of L: g S = L L 5 β S = L ã S g 3 + L L 3 ã Sg L ã S g 3 + L + L 3 ã Sg =, L S g L S g ã + L 3 â + ã Sg + L 4 = ã L 3 â + ã Sg + L 4 =. 39 Now, aume that the otimizing β i greater than zero. Then, L 5 = b comlementarit lackne condition. Uing g S = L and ubtituting L reectivel L 4 from the econd fifth equation in the firt fourth, we obtain β S = g S S g ã 3 L 3 Sg ã g S S g ã + c â + ã+c Sg = L 3 4 After ome algebra, thi ield. Note that thi i not enough to claim that the otimizing β atifie becaue we aumed β oitive. To comlete the roof we need to argue that β i oitive. Lemma 5. The right-hand term of i non-negative. Proof: To rove the tatement, we how that ã + ã S g 3 ã S g 3 Sg + â + ã Sg + â 4 S ã + ã S g S g. Conidering â = â =, after ome algebra the left-hand ide of reviou inequalit become /g /g. I. Proof of Corollar Uing the linearit of the latenc function and ubtituting in the definition of ervice-equilibrium ditribution, in a rovider equilibrium we have the following condition ã ã = Sg Sg ã g g ã S g 3 Sg S â + ã 4 Sg ã + â = ã + â, 43 Sg Sg g + g =, + = λ with g,,. B ubtitution, one can check that g = /, = λ/,, olve the above equation. To rove the tatement, we need to how that that i the unique olution. For contradiction, aume that >. Then, b 43, g < g and the left-hand ide of equation 4 i oitive. Since ã S g 3 ã Sg S / ã â + Sg i non-negative b Lemma 5, the right-hand ide of equation 43 become nonoitive, i.e., a contradiction. Therefore, g = /, = λ/,, i the unique olution and ubtituting in we get.

11 J. Proof of Corollar Subtituting 5 in 8, after ome algebra we get ã z g = ã z g g g ã z + â Sg z = ã z + â Sg z g + g = Sg z + Sg z = λ, 44a 44b which mean that in a olution UE, g SE of 44 we have 3 gse, Therefore, even though the latenc at rovider one i ver large, cometition doe not revent it in hoting at leat /3 of the ervice, though the reulting traffic Sg z UE will be mall becaue 44a can be rewritten a uing 44b Sg z UE = ã Sg ã Sg + â + â + ã Sg + â λ, 46 which aroache zero when â. Aume that â = a and â = ã = ã =. The olution of the non-linear tem that i obtained b ubtituting and 6 in 8 can be exreed in term of a olnomial of order four. In articular, uing Male, one obtain that in a olution of uch non-linear tem, g SE = z where z i a root, in [, ], of the fourth-order olnomial 3a S λ + Oaz 4 a S λ + Oaz 3 6a S λ + Oz + 36 S λ + 9 S λ 8 8 S λ S λ + Oa = Oa z Dividing b a, one get that the olution of thi olnomial converge to the olution of 3z z 3 = a a. Here, all four olution are aarentl feaible, but the one of interet i z = /3 becaue of 45. Uing g SE = /3 7 λ 4 S and that z UE ee 46 we get β P E = g SE and ubtituting in the definition of ervice and rovider rofit, we get the exreion in the tatement. K. Proof of Theorem Let g = g SE be a ervice equilibrium ditribution defined with reect to generic rovider rice. The following inequalitie rove P oa g k + with reect to an model intance: lz UE g, g = Sg z UE :g > k + :g > k + :g > k + :g > = k + lz, g. ã z UE k + â Sg z UE k Sgz Sg z UE ã z UE k UE + â z k dz Sgz Sg zã z k + â z k dz Sg ã z k+ + â Sg z k+ 47a In 47a, we ue that z UE g imize :g Sg > z l z + Sg z ˆl zdz. Thi follow becaue the condition 7 that define z UE g when α SE = z l z recall that g i a ervice-equilibrium ditribution, i.e., the ervice-equilibrium rice in the large-tem limit, are the KKT condition of the trictl-convex otimization roblem z :g Sg > z l z + Sg z ˆl zdz :g Sg > z = λ. 48 Moving to the lower bound, we rove that Ωk ɛ P oa g. To do thi, we conider the exhibit a u S,λ,ã,â,=, bad examle. Our examle i the following: ã = â = δ >, â = k ã =. Uing the KKT condition of 48, which are the defining condition of z UE g, and, we get Sg z UE = λ Sg = λ δ ã S k g k +kã λ S k g k +kã S k g k + â /k + â /k + +kã S k g k +k /k g S + +k/k g S ã S k g k + â /k + â /k + ã S k g k /k δ + â + â /k Finall, ubtituting into the definition of P oa g we obtain P oa g k δ +k /k k+ Sg + + +k/k k S k g k + +k/k Sg Sg k+ Sg + k S k g k + Sg = Sg++k+/k Sg + Sg+k Sg + λ + Sg g S + g S k+ k+ k+ Sg + Sg ++k /k Sg + Sg ++k /k k+, Sg k Sg + Sg, + 49 where the lat inequalit ue that + k +/k k. Now, b mmetr, one ma have choen the cae ã = â = δ >, â = k ã = and obtained again 49 written with reect to g intead of g. Therefore, lim u k ã,â,=, P oa g k max k Sg Sg+ Sg + Now, let ɛ >. Since there exit S uch that max ɛ, the roof i finihed.. 5 Sg Sg + >.